Simply false

“A solver's job is to return the most accurate answer the arithmetic allows, so an inner solve should be run as tightly as the format permits.”

At an iterate 2.50·10⁻³ from a root that is known exactly by construction, inner tolerances of 10⁻⁴ and 10⁻¹⁴ both land 1.200·10⁻⁵ from it — the same four digits — at 426 and 1,129 conjugate gradient iterations. The refusal is fed the claim that eleven decades of inner accuracy move the resulting point, and required to fail.

Tested in The problem that arrives again · the sequence of solves series · refused by a factor of nine, claimed on a problem with nothing to oversolve

“A symmetric matrix that is difficult to factorise is difficult because of its numbers, so a better pivot rule or a higher precision will get through it.”

The factorisation is run at κ(H) = 1, where H is the identity and every number in the matrix is of order one, and the claim that it completes is fed the result. It fails at row n + 1, exactly as it does at κ(H) = 10¹⁰.

Tested in The zero that is not a missing entry · the Saddle-point systems series · refused by a saddle-point matrix that Cholesky completes on

“An n × n eigenvalue problem has n eigenvalues, and its eigenvectors can be taken as a basis for the space the matrix acts on.”

The claim is fed a quadratic eigenvalue problem of size 6 and asked for its 6 eigenvalues. The computed spectrum has 12, and the assertion that it has 6 fails. A second refusal takes 7 of the eigenvectors in 6 dimensions and asks whether they are independent; the smallest singular value of the 6 × 7 matrix of them is 10⁻¹⁵⁷ of the largest.

Tested in A matrix that depends on its own eigenvalue · the polynomial eigenvalue series · refused by a quadratic eigenvalue problem with as many eigenvalues as it has rows

“The McMillan degree is defined as the rank of the Hankel matrix of Markov parameters, so computing it means forming that matrix and taking its numerical rank.”

A model with a state dimension of 24 and a transfer function with exactly six poles is handed to both routes. The Hankel matrix of its Markov parameters returns rank two, because CAᵏB grows like ‖A‖₂ᵏ and its entries span 1.9·10³⁷; the Loewner matrix of samples of H, which never sees A, returns six with four orders of gap.

Tested in A model that is a rational function · the transfer function series · refused by the Markov-parameter rank of a model whose degree is known

“A deterministic program, given the same input twice, returns the same output twice.”

The same 1,024 numbers presented in 400 different orders, summed by an ordinary eight-piece reduction, return 72 distinct binary64 values. The refusal is fed the assertion that the count is one and required to reject it.

Tested in The same program, twice · the reduction order series · refused by the claim that an ordinary reduction is order-independent

“Removing the rounding removes the cost, so an exact elimination is a slow version of the same computation.”

On a twelve by twelve matrix with one-digit entries all three routes return the determinant 2,269,932,161,262 exactly. The widest number formed is 40 bits by fraction-free elimination and 1,422,165 bits by unreduced rational elimination. An assertion that the widest intermediates agree within a factor of a hundred is fed that matrix and must reject it.

Tested in An answer with no error in it · the exact cost series · refused by the claim that the three exact routes are the same computation at different speeds

“A matrix with no zero entries has to be stored entry by entry.”

The 96 × 96 block between the intervals [0, 1] and [2, 3], for the kernel 1/r, has five singular values above 10⁻⁸ of the largest — so 960 numbers describe 9,216. The refusal is fed the same claim about a block of independent draws on the same points, where 96 of 96 survive the same cut, and required to fail.

Tested in A block nobody can call sparse · the Off-diagonal rank series · refused by a block of independent draws is low rank

“A randomised method with a probabilistic bound gives you the answer; the probability is a formality.”

The error moves by a factor of 1.60 across seeds at the same oversampling. The bound itself was not violated once in eighty seeds, which is reported rather than tuned — it is loose by 5.43× against the median, and a bound never violated at this sample size is not thereby deterministic. What the seed changes is the answer, not the guarantee.

Tested in A bound that holds with probability · the randomised series · refused by the claim that a randomised method returns one answer

“The eigenvalues of a Kronecker sum are the products of the eigenvalues of its terms.”

The whole spectrum of the two-dimensional Laplacian is computed from the assembled matrix and compared against the set of pairwise products of the one-dimensional spectrum. The two disagree at the first eigenvalue and everywhere after it, and the assertion that they match is fed that comparison and required to fail.

Tested in An index that is a pair · the kronecker series · refused by a sum is not a product

“Conjugate gradients is a direct method: run it n steps and it is exact.”

After exactly n = 30 steps on a κ = 10⁶ matrix the residual is 5.1·10⁻⁴, and the residual basis that was supposed to be orthogonal has ‖RᵀR − I‖ = 5.29. Finite termination is a theorem about exact arithmetic and this is the case that puts it under load: the well-conditioned model problem cannot test it, because convergence arrives long before step n and nothing is left to fail.

Tested in The rate the condition number predicts · the krylov series · refused by a claim that CG terminates in n steps

“The number of zero eigenvalues of a Laplacian is a property of the graph.”

A twenty-vertex path is required to have two zero eigenvalues, which it does not: it is connected, its second eigenvalue is 0.0246, and the assertion is fed that requirement and must reject it.

Tested in A matrix with no numbers in it · the graph laplacian series · refused by the claim that a connected graph has more than one zero eigenvalue

“The elimination order affects how long the factorisation takes, not how much memory it needs.”

On the same 12×12 grid: natural order 1,739 entries, RCM 1,354, nested dissection 1,413, minimum degree 1,026. The matrix itself has 408. Every one of those factorisations is equally accurate and one of them is a factor of 1.7 larger than another, counted symbolically from the elimination graph with no arithmetic at all and again by factorising — the two counts agree exactly, as integers.

Tested in The order decides the memory · the ordering series · refused by the claim that elimination order does not affect fill

“Two block eliminations of the same nonsingular system are two orderings of the same arithmetic, so they lose the same accuracy.”

Both methods are run on one system with κ(H) = 100, and the claim that their forward errors against the exact rational answer agree within two orders is fed the pair. It fails: they are 3.0·10⁻⁸ and 8.8·10⁻¹³.

Tested in Two ways to remove a constraint · the Saddle-point systems series · refused by the range-space method at κ(A) = 10⁵

“An a-priori error bound is a worst case, so a reduced model will normally do considerably better than 2Σσ and the bound is a safety margin rather than a prediction.”

‖H − Hᵣ‖∞ is measured against twice the sum of the σₖ beyond r, at every order from one to seven on a twenty-state model. The ratio is 1.0000 at every order, to four decimals. The bound is attained rather than approached, because removing one state costs exactly twice the Hankel singular value it removed and the sum is that result applied repeatedly.

Tested in The bound that is known in advance · the balanced truncation series · refused by the ratio of the bound to the measured error at seven truncation orders

“The components a computation can resolve are the components worth keeping.”

They are not the same set, and the gap is measurable. The Picard crossing — where |uₖᵀb| stops falling and levels at the noise floor — lands at 32, 32, 40 and 45 across noise levels of 1%, 0.1%, 0.01% and 0.001%. The truncation that actually minimises the error is 21, 28, 32 and 38. Every one of the last few resolvable components carries more error than signal.

Tested in Where the answer stops being in the data · the regularisation series · refused by the Picard condition claimed for a right-hand side that is noise

“The division by the previous pivot in a fraction-free elimination usually comes out exact and needs a fallback for when it does not.”

Every division performed by the elimination on five families at every size from three to twelve is checked for a zero remainder before the quotient is taken, and each intermediate is compared against the minor Sylvester's identity names, computed by cofactor expansion of the original matrix. An assertion that some step leaves a remainder is fed those matrices and must reject them.

Tested in Every intermediate is a minor · the Fraction-free series · refused by the claim that a fraction-free step can leave a remainder

“The normalised Laplacian is the combinatorial one divided by a degree, so their eigenvalues are proportional.”

On a sixty-vertex two-block graph the two second eigenvalues are 2.554 and 0.1538 — a ratio of 16.6 — while the degrees run from 10 to 22. The assertion that the ratio is one of the degrees is fed that graph and must reject it.

Tested in Two Laplacians of one graph · the graph laplacian series · refused by the claim that the normalised Laplacian is the combinatorial one divided by a degree

“The rank of a matrix is a property of the matrix.”

The same 128 × 128 kernel block needs 2 columns at 10⁻² and 9 at 10⁻¹⁴ — a factor of four and a half, from one matrix, with nothing about it changed between the two readings. The refusal is fed the assertion that the two counts are equal and required to fail.

Tested in A rank that is a number of digits · the Off-diagonal rank series · refused by the rank a block needs does not depend on the accuracy asked for

“A saving that comes from structure is a saving that costs nothing to state, so a small case shows it as well as a large one.”

The two-route check on the reshaped Kronecker product is asked to demonstrate an order-of-magnitude saving at d = 1 and n = 2, where the assembled and reshaped counts are four and four. The assertion that the reshaped route costs an order of magnitude less is fed that case and required to fail.

Tested in A solve that is d decompositions · the kronecker series · refused by one factor of size two saves nothing

“The right forcing term is a constant, and the work is to tune it for the problem.”

The cheapest constant costs 980 inner iterations and ends with a forward error of 2.43·10⁻⁸; the dearest costs 9,358 and ends at 2.79·10⁻¹⁰. The adaptive rule costs 1,009 and asks for tolerances from 0.90 at the first step to 4.2·10⁻³ at the last, so no constant reproduces it. The refusal is fed the claim that some constant matches it on both numbers at once, and required to fail.

Tested in A tolerance that reads its own residual · the inexact newton series · refused by the adaptive rule, claimed to be some constant

“A finer discretisation costs more to store, in proportion.”

At 32, 64, 128 and 256 points a side, an admissible kernel block needs 5, 5, 5 and 5 columns at 10⁻⁸ — so its storage grows like n while the block grows like n². The refusal is fed the assertion that the last count exceeds the first and required to fail.

Tested in The size the rank does not notice · the Off-diagonal rank series · refused by the rank of an admissible block grows with the block

“A block-diagonal preconditioner built from a saddle-point system's own blocks gives three eigenvalues, so any reasonable approximation to the Schur complement gives three too.”

The same preconditioner is built with S replaced by A diag(H)⁻¹Aᵀ, and the claim that the preconditioned spectrum has three distinct values is fed the result. It fails: the eigenvalues spread by 2.66 from the closed form and the iteration count goes from 3 to 11.

Tested in Three eigenvalues, and two are the golden ratio · the block preconditioning series · refused by a diagonal approximation to AH⁻¹Aᵀ

“The Hankel singular values are defined as √λ(PQ), so any accurate eigensolver applied to PQ computes them accurately, and the choice of route is a matter of convenience.”

Both routes are run on the same model. They agree for the first six values and then the product route stops descending, flattening at 9.7·10⁻¹⁰ against a prediction of σ₁√u = 2.3·10⁻⁹, while the square-root route continues to 10⁻¹⁶. Every value below that floor is a term in the error bound.

Tested in The product nobody had to form · the balanced truncation series · refused by the tail of the Hankel singular values from the product route

“Dividing a summation across more workers makes it less accurate, because there are more places for rounding to enter.”

The mean error over eight vectors falls from 4.8·10⁻¹⁴ with one accumulator to 8.1·10⁻¹⁵ with sixty-four, because each accumulator's walk is shorter. The refusal is fed the assertion that one accumulator is the more accurate and must reject it.

Tested in Where the disagreement comes from · the reduction order series · refused by the claim that a parallel reduction is the less accurate one

“Both graph Laplacians annihilate their known null vector exactly.”

On a thirty-vertex star the normalised Laplacian applied to the vector of square-rooted degrees returns 3.8·10⁻¹⁵ rather than zero. The assertion that it returns exactly zero is fed that graph and must reject it.

Tested in The vertex nobody solves for · the graph laplacian series · refused by the claim that both Laplacians have an exact null vector

“A better exact solver would keep the answer's parts as short as the matrix's entries.”

Every entry of the exact solution is multiplied back by det A and compared, as an integer, with the determinant of the matrix with that column replaced by the right-hand side. The two agree at every size from three to thirteen, so the entry IS that ratio and its denominator divides det A. An assertion that some denominator is longer than det A is fed those systems and must reject them.

Tested in The answer is longer than the question · the exact cost series · refused by the claim that the answer's length is a property of the algorithm

“A set of tensors of rank at most r contains the nearest point of it to any tensor, as the rank-k matrices do.”

The classification is fed a 2 × 2 × 2 tensor that is a sum of two rank-one terms with disjoint supports — an ordinary rank-two tensor whose distance to the rank-two set is zero and is attained — and asked to report rank three. It refuses, which is what separates the phenomenon from the arithmetic that surrounds it.

Tested in A nearest point that is not there · the tensor rank series · refused by a rank-two tensor is not the example

“A sketch finds the structure in the data.”

On a matrix with a flat spectrum the randomised rank-10 error is 1.0 — and so is the optimal one. Neither method achieved anything, and the failure belongs to the matrix rather than to the sketch. The figure draws which of three cases a matrix is in first, because every other claim in the field is downstream of the spectral decay.

Tested in Randomisation does not create structure · the randomised series · refused by the claim that sketching finds structure that is not there

“A random tensor of a given shape has a rank, in the way a random matrix has full rank almost surely.”

Four thousand 2 × 2 × 2 tensors with independent standard normal entries are classified exactly, by the sign of the hyperdeterminant, and the claim that the two ranks occur equally often is fed the measured share. It fails: the share is π/4, which is neither one nor a half.

Tested in A rank that is not a property of the tensor · the tensor rank series · refused by the split is not even

“The QR algorithm converges.”

Without a shift it does not. On [[0, 1], [1, 0]] the subdiagonal entry moves by 1.09·10⁻¹⁴ in two hundred iterations, which is to say it does not move at all — the two eigenvalues have the same modulus and the ratio governing convergence is one. Wilkinson's shift settles the same matrix in a single step. The shift is not an acceleration; it is what makes it an algorithm.

Tested in The algorithm the libraries actually run · the The QR algorithm series · refused by convergence claimed for the unshifted algorithm

“Any Krylov method applied to an ill-posed problem acts as a spectral filter, so its step count is a regularisation parameter in the same sense a truncation index is.”

Fitted against its own space — every power of σ, not only the even ones — GMRES's measured weights miss any degree-8 polynomial by 36% on a shifted blur. The refusal is fed the claim that the misfit is below 10⁻⁶ and required to fail.

Tested in The basis decides what a filter is · the GMRES series · refused by a spectral filter claimed for a method whose space is built from A

“A matrix built from a kernel and two clusters of points has low-rank off-diagonal blocks.”

Two parallel segments of length 8 a distance 16 apart, so q = ½ exactly as in the compressible case, with the kernel cos(40r)/r: 53 columns at 10⁻⁸ against 6 for 1/r on the same points. The refusal is fed the assertion that the oscillatory block needs at most twelve and required to fail.

Tested in The kernel with nothing to compress · the Off-diagonal rank series · refused by an oscillatory block is low rank

“A preconditioner for a constrained system has to approximate the whole matrix, so a badly conditioned constraint makes it a badly conditioned preconditioner.”

The nontrivial part of the preconditioned spectrum is computed at κ(A) = 1, 10², 10⁴ and 10⁶, and the claim that it moves is fed the four lists. It fails: the six values agree to 10⁻⁵ relative across all four.

Tested in A preconditioner that need not know the constraint · the block preconditioning series · refused by a constraint preconditioner at κ(A) = 10⁶

“Two linearisations of the same quadratic are interchangeable — they have the same eigenvalues, so a comparison of the spectra they compute will show whether the choice matters.”

Six routes are run on a well-scaled problem and their computed spectra compared against the closed form. The claim that the choice shows up in the answers is fed the six and fails: they agree to within a factor of seventeen at the fourteenth digit, which is noise. On a badly scaled problem the same six differ by a factor of forty at the fifth.

Tested in Six routes to one spectrum · the linearisation backward error series · refused by six routes told apart by their spectra at sensible units

“A two-sided projection and a one-sided one build reduced models from the same subspace, so the second solve per interpolation point is an optimisation rather than a requirement.”

Both projections are built at the same four points. The values agree with the original to 10⁻¹⁶ in each case. The two-sided model's derivative agrees to 2·10⁻⁸ and the one-sided model's is out by 4.6·10⁻⁵ — a factor of 2.3·10³ — so the second solve buys the Hermite half of the conditions and the first buys none of it.

Tested in Exact at the points that were named · the moment matching series · refused by the derivative of a one-sided reduced model at its own interpolation points

“A parallel reduction returns a different answer on every problem, so the effect can be shown on any vector.”

On 4,096 positive numbers the seven partitionings agree to a relative 1.6·10⁻¹⁵. The refusal is fed the assertion that positive data disagrees with itself and must reject it — which is why every quick demonstration of this effect is a demonstration of nothing.

Tested in The vector that hides it · the summation series · refused by the claim that any vector shows the spread

“A randomised method with a bound that holds with high probability gives an answer you can quote.”

The same matrix, the same right-hand side and the same rank of 8 return relative errors of 0.244, 0.376, 0.419 and 0.449 across four seeds. The refusal is fed two runs at the same rank with different seeds and required to reject the claim that they agree to six digits.

Tested in An answer that changes with the seed · the randomised series · refused by a randomised method treated as a function of its input

“Hadamard's bound is a loose worst case, so a modular routine budgeting against it is buying a large safety margin.”

On a Sylvester Hadamard matrix of order eight the bound is 25 bits and the determinant occupies 25: the bound is attained. An assertion that the bound exceeds the determinant's length by more than eight bits is fed that matrix and must reject it.

Tested in How many primes the answer needs · the modular lift series · refused by the claim that the bound is loose by more than a byte on every matrix

“A factorisation kept too long makes the answer gradually less accurate, so the cost of keeping it is a loss of accuracy that can be traded against work.”

At a drift of 0.01 a member, periods of 1 to 15 all reach the tolerance and cost 16.42 down to 9.27 MFlop; a period of 20 does not converge at all. Nothing between the optimum at five and the failure at twenty returns a worse answer — every one of them returns the same answer for more work, until one of them returns none. The refusal is fed the claim that the cost of staleness is accuracy, and required to fail.

Tested in A factorisation kept past its date · the reuse series · refused by a shelf life, claimed for a sequence with no drift

“To orthogonalise a matrix, take the Q from its QR factorisation. That is what QR is for.”

On an 8×8 matrix with κ = 1.5, the polar factor is 0.586 from A in the Frobenius norm and the sign-fixed QR factor is 0.658. The refusal is fed the claim that QR's Q is at least as near, and required to fail.

Tested in The nearest orthogonal matrix · the polar decomposition series · refused by QR's Q offered as the nearest orthogonal matrix

“A well-spread spectrum means a Krylov method will converge quickly.”

The cyclic shift has every eigenvalue on the unit circle — as well spread as a spectrum can be — and GMRES makes no progress on it for n − 1 steps. The residual is exactly 1 at every step until the last, checked to 10⁻⁹ at every size, and then the method solves the system exactly. A spectrum and a convergence rate are not the same fact about a non-normal matrix.

Tested in The spectrum that predicts nothing · the GMRES series · refused by a convergence claim made from the spectrum alone

“The sweep cut of the Fiedler vector is the best cut in the graph.”

On a thirteen-vertex random graph the sweep returns a conductance of 0.4545 and enumeration over all 8,190 subsets finds 0.3077. The assertion that the two agree is fed that graph and must reject it.

Tested in The vector that has to be rounded · the spectral partition series · refused by the claim that the relaxation and the combinatorial optimum agree

“A condition number of 10¹³ means the answer has at most three correct digits, because the forward error is bounded by the condition number times the unit roundoff.”

The system is solved and the answer compared against an exact rational solution of the same stored matrix. The claim that the error is near κ·u is fed the pair, and fails: κ is 3.04·10¹³ and the relative error is 9.4·10⁻¹⁶.

Tested in A condition number sent to infinity · the Interior-point conditioning series · refused by the augmented interior-point system at μ = 10⁻¹²

“Compensated summation makes a sum reproducible, because it removes the rounding that the order was changing.”

Kahan's loop over 400 permutations of one vector returns 119 distinct binary64 values — three times fewer than a plain loop's 303, and 119 more than one. The refusal is fed the assertion that a compensated sum is order-independent and must reject it.

Tested in The sum that cannot be wrong · the reproducible summation series · refused by the claim that accuracy buys reproducibility

“A nonzero determinant has a nonzero residue at every prime, so a modular elimination reporting zero has found a singular matrix.”

A matrix built as a unimodular conjugate of diag(1, …, 1, 30030) has determinant 30030 = 2·3·5·7·11·13. Its residue is exactly zero at five of the first twenty-five odd primes, and the matrix is nonsingular. The assertion is fed that matrix and must reject it.

Tested in A prime that divides the answer · the modular lift series · refused by the claim that a nonzero determinant has a nonzero residue at every prime

“Cheeger's inequality holds for the combinatorial Laplacian's second eigenvalue.”

On the complete graph of twenty vertices the combinatorial λ₂ is 20, so the lower half claims a conductance of at least 10, and every cut of a complete graph has conductance about a half. The assertion is fed that graph and must reject it.

Tested in A bound with a square root in it · the spectral partition series · refused by the claim that Cheeger's inequality holds for the unnormalised eigenvalue

“A quasi-optimal truncation is optimal in practice, so the factor in the bound is a formality.”

The measured error of a rank-six truncation is compared against the lower bound on the best possible error for the same multilinear rank, and the assertion that the two are equal to six digits is fed the comparison. It fails at a ratio of 1.708 against a permitted √3 — and the reason it fails is that the lower bound is weak, which a second measurement establishes.

Tested in A decomposition made only of SVDs · the multilinear rank series · refused by the lower bound is not attained

“Regularisation removes the noise and hands back the signal.”

Write the Tikhonov answer as V F Vᵀ x plus the regularised inverse applied to the noise. If regularisation handed back the signal, V F Vᵀ would be the identity. On the site's 64-point blur at the best λ for 0.1% noise its middle row puts 0.42 on the diagonal and spreads the rest over a kernel 2.82 points wide that dips to −0.075, and that kernel accounts for 0.1023 of a total relative error of 0.1051 — the noise for 0.0259.

Tested in A second blur, narrower than the first · the regularisation series · refused by a regularised answer claimed to be the truth plus filtered noise

“A preconditioner should be rebuilt once the matrix has drifted by more than a few per cent of its norm.”

On an 80×80 matrix at κ = 10⁴, a drift of 10⁻² of the Frobenius norm costs 38 preconditioned iterations placed on the small eigenvalues and 9 placed on the large ones, with ‖E‖/‖A₀‖ equal to fourteen digits in both cases. The refusal is fed the claim at a drift of 10⁻⁶, where both cost the same because neither costs anything, and required to fail.

Tested in Where the drift lands · the reuse series · refused by the end of the spectrum mattering, claimed below the drift where it does

“Two clusters of points that are next to each other interact weakly enough to compress.”

For [0, 1] and [1, 2] the separation ratio q = (½ + ½)/1 is exactly 1, so the expansion the compression rests on does not converge, and the block's rank climbs 9, 11, 12, 13 as the sampling is refined. The refusal is fed the assertion that q < 1 for that pair and required to fail.

Tested in Which pairs are allowed to be small · the admissibility series · refused by touching intervals are admissible

“A random sketch is a fixed piece of machinery, so it can be built once and applied wherever it is needed.”

With the sketch kept, ‖AΩ‖/‖A‖ is 2.62 at the first round and 9.0·10⁻¹⁵ at the second — the unit roundoff. With it redrawn it stays above 3, and after five rounds the errors are 0.284 and 0.113. The refusal is fed the same claim on twenty independent matrices, where one sketch and twenty give mean errors of 0.36184 and 0.35789, and required to fail.

Tested in The sketch that is spent · the sketching series · refused by the safety of reuse, claimed where the input depends on the draw

“A rule that watches how many iterations the last solve took is enough to decide when to rebuild a preconditioner.”

The growth rule rebuilds four times whatever a rebuild costs. At a setup worth 50 iterations that is 508 iteration-equivalents against the best fixed period's 542, and at 100 it is 708 against 727 — it wins. At a setup worth 5 it is 328 against 170, and at 200 it is 1,108 against 1,008 — it loses. The refusal is fed the claim that it wins everywhere, and required to fail.

Tested in What a rebuild is worth · the reuse series · refused by the growth rule, claimed to win at every setup cost

“The width of a pre-rounded summation is a free parameter, so its accuracy can be raised to whatever is wanted.”

The accumulator must hold a sum of n multiples of the spacing without rounding, so the width plus log₂n must stay under 53 bits. A caller asking for 48 bits on 4,096 terms is refused rather than given an answer that is neither accurate nor reproducible — which is the cost the policy's price does not show.

Tested in What determinism costs · the reproducible summation series · refused by a width that would overflow the accumulator

“A graph determines its spectral partition.”

Three seven-vertex cliques joined in a triangle have λ₂ = λ₃ to 1.7·10⁻¹⁵. Eight vectors from that eigenplane sweep to three different partitions. The assertion that they all give one partition is fed that graph and must reject it.

Tested in A partition decided in the last digit · the spectral partition series · refused by the claim that a repeated λ₂ still names one partition

“A residue determines the rational it came from, so reconstruction works at any modulus and gets better as the modulus grows.”

355/113 is reduced modulo primes just above 2³ through 2⁴⁴ and reconstructed from each residue. At every modulus below 2¹⁸ the reconstruction returns a different fraction or refuses; at and above it, 355/113 exactly, at every modulus tried. The assertion that the smallest modulus recovers it is fed that case and must reject it.

Tested in A fraction recovered from one remainder · the modular lift series · refused by the claim that a residue determines its rational at any modulus

“Generalised cross-validation needs nothing but the data.”

At equal noise size on the site's 64-point deconvolution, over 48 draws: GCV more than doubles the best available error on 3 draws when the noise is white and on 14 when neighbouring samples are correlated at ρ = 0.9, choosing λ at a median of 0.54 times the best. Whitening the problem by the noise covariance takes it back to 3. The rule needs nothing but the data only if the noise is white.

Tested in Noise that spares the answer and fools the rules · the regularisation series · refused by a parameter rule claimed indifferent to the noise's correlation

“The higher-order SVD is a tensor SVD, so its core is diagonal in the way a matrix's middle factor is.”

The core of a rank-four truncation of a smooth tensor is measured for the fraction of its energy off the superdiagonal, and the assertion that the fraction is below 10⁻⁸ is fed the result. It fails, and on a tensor with no structure the superdiagonal carries 0.4 per cent of the total.

Tested in The orthogonality that cannot be diagonal · the multilinear rank series · refused by the core is not diagonal

“A factorisation that exists under every ordering is as stable under every ordering, since the theorem does not distinguish between them.”

Five hundred random symmetric permutations are factorised and their residuals recorded. The claim that the best and worst are within a factor of 1.5 is fed the pair, and fails: they are 1.6·10⁻¹⁶ and 1.0·10⁻⁷.

Tested in The regularisation that legalises every order · the Quasi-definite series · refused by the worst of five hundred orderings

“A symmetric matrix can be factorised by taking its diagonal entries in some order — search harder if the first one is small.”

[[0, 1], [1, 0]] has eigenvalues ±1 and no nonzero diagonal entry. The refusal is fed the claim that diagonal pivoting finds a pivot somewhere in it, and required to fail.

Tested in When symmetry is not enough · the cholesky series · refused by a symmetric factorisation restricted to 1×1 pivots

“Any set of n − m independent directions annihilated by A is as good a basis for the null space as any other, since they all describe the same subspace and give the same answer.”

Three bases for one null space are built and their reduced Hessians formed. The claim that the naive one's condition number is within a factor of a million of the orthonormal one's is fed the pair, and fails: 3.80·10¹⁶ against 25.6.

Tested in The basis nobody chose on purpose · the null space basis series · refused by the first m columns taken as basic

“Whether a matrix can be compressed is a property of the matrix.”

The same 256 × 256 kernel matrix under one symmetric permutation: κ = 24.3948 and a Frobenius norm of 6.13996414·10³ in both numberings, and the hierarchical representation goes from 41 per cent of n² to 100 per cent of it. The refusal is fed the assertion that the shuffled one still stores under half and required to fail.

Tested in The same matrix, numbered twice · the admissibility series · refused by a shuffled numbering is still compressible

“The order of the rows of a least-squares problem does not affect its answer, since a permutation of the rows is a permutation of the residual and the norm is the same.”

One weighted problem is solved by Householder QR twice, differing only in whether the constraint rows are above or below the data rows. At τ = 10¹⁴ the two answers are 4.8·10⁻¹⁵ and 2.1·10⁻³ from the exact one — a factor of 4.3·10¹¹.

Tested in A constraint is a weight at infinity · the constrained least squares series · refused by the weighted rows placed last

“A more accurate computation is a more reproducible one, because both come from having less rounding error.”

Pre-rounded summation is bitwise identical over 400 permutations and 2.3·10⁻⁷ wrong; the ordinary reduction it replaces is 2.7·10⁻¹¹ wrong and returns 72 answers. The refusal is fed the assertion that the reproducible answer is the exact one and must reject it.

Tested in Accuracy and agreement are different properties · the reproducible summation series · refused by the claim that a reproducible sum is an exact one

“PageRank is the stationary distribution of the link graph's random walk.”

The routine is asked for α = 1 — the walk with no teleportation, which is what that claim describes — and refuses, because that chain has no unique stationary distribution on any graph with more than one closed class.

Tested in A ranking that is an eigenvector · the random walk series · refused by a teleportation parameter of one, which is the chain itself

“Rank is a property of an array of numbers, so an exact computation returns the rank.”

One six by six integer matrix, built as a unimodular conjugate of diag(1, 1, 1, 1, 2, 6), has rank six over the rationals, five modulo three and four modulo two, computed by exact elimination with no threshold anywhere. The assertion that every field returns the same rank is fed that matrix and must reject it.

Tested in The rank depends on the ring · the exact rank series · refused by the claim that rank is a property of the entries alone

“Orthogonalising is orthogonalising. Reordering the columns of a matrix before the factorisation cannot change the orthogonal matrix that comes back.”

On an 8×8 matrix with κ = 20, reversing the columns moves the sign-fixed Householder Q by 3.096 in the Frobenius norm, on matrices whose own norm is 2.828. The assertion that a QR's Q survives a column permutation is fed that pair and required to fail.

Tested in A test with no answer in it · the polar decomposition series · refused by a QR orthogonalisation claimed to be independent of the column order

“A few dozen noise draws are enough to say how badly a parameter-choice rule can do.”

Over the sixteen draws the earlier comparison used at 0.1% noise, cross-validation's worst choice cost 1.12 times the oracle's error. Over a thousand draws at the same noise it costs more than ten times the oracle on 57 of them and 144,000 times on the worst. A failure rate of 5.7% leaves sixteen draws clean with probability 0.39, so the clean sixteen were the likeliest single outcome and no evidence at all.

Tested in One draw in twenty · the Parameter choice series · refused by sixteen draws claimed to bound the worst case of generalised cross-validation

“A Hessian with negative eigenvalues cannot belong to a constrained minimum, so an indefinite H rules the minimum out before anything else is computed.”

The saddle-point matrix of a problem whose Hessian has four negative eigenvalues is factorised and the claim that it has no minimum is fed the count. It fails: ten positive pivots and four negative, which is the reduced Hessian's count plus (4, 4, 0), and the reduced Hessian's eigenvalues run from 0.5 to 5.

Tested in A minimum the Hessian cannot see · the Saddle-point systems series · refused by a constrained minimum ruled out because the Hessian is indefinite

“A backward-stable algorithm gives the exact answer to a nearby problem, so the answer is as good as the data deserves.”

On a 40×40 Kac–Murdock–Szegő system at ρ = 0.999, the smallest perturbation of any kind that makes the computed solution exact is 2.0·10⁻¹⁷ relative and is rank one, constant along none of its diagonals. The smallest one that is itself a symmetric Toeplitz matrix is 5.6·10⁻¹², and the smallest one that is a member of the family the problem was actually posed in does not exist. The refusal is fed a Toeplitz matrix that is not positive definite, where the recursion divides by a residual variance that has gone negative.

Tested in A nearby problem of the wrong kind · the structured backward error series · refused by a Levinson recursion run on a Toeplitz matrix that is not positive definite

“A matrix you can only apply is a matrix you cannot factorise.”

At 512 unknowns the construction performs 256 products with an operator it never forms and returns a representation at 4.03·10⁻⁷, against 5.49·10⁻⁸ for the decomposition of every block — a factor of 7.3 and never better. The refusal is fed the assertion that a sample beats the decomposition of the same rank and required to fail.

Tested in Built from products alone · the sketching series · refused by a sample beats the decomposition

“A function of a sum of variables is a product of functions of them, so anything that separates at one cut separates at all of them into one term.”

The train ranks of the sine of a sum of d variables are computed at every cut and the assertion that all of them are one is fed the result. They are all two — the addition formula gives two terms at every cut and not one — and the assertion fails at the first.

Tested in The format that does not notice the dimension · the tensor train series · refused by a sum inside a sine is not separable

“An expression with one value has one cost, so the order in which its products are taken is a detail of the implementation.”

The classic six-matrix chain is priced under every evaluation order by a dynamic program over subsets. The cheapest is 15,125 multiply-adds and the dearest is 512,793,750, and the assertion that the two are within one per cent of each other is fed the pair and fails by a factor of 33,900.

Tested in The order the products are taken in · the contraction series · refused by the order is the cost

“A set of shifts spread evenly across the spectrum covers it, so it is a reasonable default when nothing is known about where the eigenvalues cluster.”

Eight equally spaced shifts on a spectrum spanning three decades give max|r| = 0.85, against 0.052 for eight geometric ones — sixteen times worse, which the iteration pays squared. The measured ADI errors are 5.9·10⁻¹ and 2.0·10⁻³, a factor of 290 for the same eight solves.

Tested in Where to put the poles of a rational function · the gramian decay series · refused by the rational factor of eight equally spaced shifts

“A computed eigenpair with a residual at the rounding level has been computed accurately, whatever preprocessing the solver applied to the problem first.”

An eigenvalue of A − λI + γ√(λ+c)I is computed by approximating the square root and linearising. Its residual against the approximated problem is 8·10⁻¹⁵. Its distance from the true eigenvalue is 5.9·10⁻⁴, and the residual against the original problem — one further evaluation of √ — is 3.0·10⁻⁵.

Tested in The problem the solver was actually given · the approximation before linearisation series · refused by the residual of an eigenpair against the problem the solver was handed

“Two IEEE-754 conforming implementations of the same expression return the same answer, because the standard specifies the result of every operation.”

ad − bc with entries near 2²⁷ returns 0 from the contracted form and 1 from the fused one; both are conforming, since the standard specifies each operation and not which operations a compiler emits. A residual computed from the same expression inherits the same form and cannot report on it, which is the refusal.

Tested in One multiply the compiler removed · the fma contraction series · refused by the claim that a residual can see which form was compiled

“The PageRank power iteration contracts by α at every step, whatever the graph.”

On a connected thirty-six-vertex grid the measured contraction at α = 0.5 is 0.404, not 0.5, because the Google matrix's second eigenvalue is α·λ₂(P) and λ₂(P) is below one. The assertion that the rate is α is fed that graph and must reject it.

Tested in The rate is the second eigenvalue · the random walk series · refused by the claim that the power iteration contracts by α on any graph

“Leverage is a diagnostic that reads the same on any fit: flag the observations above the usual rule of thumb, and the shape of the design does not come into it.”

The leverages sum to the number of columns, so the average is p/m and nothing else. At sixty observations of three columns the average is 0.0500 and a row at h = 0.5 is ten times it; at thirty-two observations of eight columns the average is 0.2500 and the identical row sits exactly on the rule of thumb. The refusal is fed a design of twenty observations for eight columns — where the average leverage is already 0.4, every row is influential, and the axis says nothing — and required to fail.

Tested in Influence is decided before the data · the leverage series · refused by a hat-matrix figure on a design with too few observations

“The nearest orthogonal matrix to a noisy estimate of a rotation is a rotation.”

Twenty points of thickness 10⁻² are rotated and measured with noise 0.1. In the figure's 300 trials, 102 polar factors of the cross-covariance have determinant −1. The assertion that the polar factor is always a rotation is run over two hundred such alignments on a separate seed and required to fail.

Tested in A rotation that comes back mirrored · the polar decomposition series · refused by a polar factor of a cross-covariance claimed to be a rotation

“A derivative penalty is what makes the L-curve's corner a reliable choice of regularisation parameter.”

On the step signal at 0.1% noise a first-difference penalty takes the corner's median cost from 1.53 to 1.003. On two smooth bumps it takes it from 29 to 4.45, and on four spikes it makes it worse, from 1.004 to 1.04. In all fifteen pairings of five signals and three penalties the corner sits where noise is between 12% and 21% of the plotted norm, so what decides its cost is where the oracle's share sits — a property of the signal and the penalty together, and the refusal is fed the claim that every pairing lands within 10%.

Tested in The corner reads the norm it is drawn in · the Parameter choice series · refused by the L-curve corner claimed to land on the oracle whatever the penalty and the signal

“A preconditioner changes how quickly an iteration converges, not what it converges to, so a better preconditioner can only make an iterative solve cheaper.”

Conjugate gradients on a 64-point deconvolution at 1% noise, preconditioned by AᵀA + αI, reaches a best error of 0.1426 at α = 0.1, 0.1412 at 10⁻³, 1.353 at 10⁻⁶ and 719 at 10⁻¹², where the unpreconditioned run reaches 0.1426. The refusal is fed the claim that the preconditioned best error matches the plain one at every shift and required to fail.

Tested in A preconditioner that arrives past the answer · the iterative regularisation series · refused by a preconditioner claimed to change the speed of a stopped iteration and not its answer

“A preconditioned matrix whose only eigenvalue is one is effectively the identity, so a Krylov method converges in a single step.”

With the block upper-triangular preconditioner and the exact Schur complement every eigenvalue of P⁻¹K is one, and GMRES takes two steps on every one of eighteen systems whose κ(H)κ(A) is at most 10⁶: the first leaves a relative residual between 10⁻⁵ and 0.32, the second reaches rounding. The refusal is fed the claim that the run converges in one step and required to fail.

Tested in One eigenvalue and two steps · the block preconditioning series · refused by a single eigenvalue offered as a single step

“A long plateau at a small residual is slow convergence, so the remedy is more iterations.”

Twenty thousand sweeps are run against the border-rank tensor and the assertion that the largest rank-one term stays within half again of its value at sweep ten is fed the trace. It fails: the term grows by a factor of 2.16 while the error falls by a factor of 4.8, and neither has finished.

Tested in An iteration that walks out of the set · the alternating least squares series · refused by a swamp diverges

“A sketch is a cheap substitute for a decomposition, so the random matrix is a detail of the implementation.”

The number of random numbers a dense Gaussian sketch of every mode needs is counted against the tensor's own entry count, and the assertion that the sketch is smaller is fed six indices at eight points a side. It fails: 1,769,472 random numbers against 262,144 entries.

Tested in Sketching what is never unfolded · the sketching series · refused by a dense sketch outgrows its tensor

“A preconditioner that takes fewer iterations is a better preconditioner.”

At κ = 20.9 the tightest preconditioner on the sweep takes 2 steps against the cheapest one's 13, and costs 1.16 million multiplications against 0.48 million. The refusal is fed the assertion that the tighter one costs less and required to fail.

Tested in The accuracy worth paying for · the hierarchical solve series · refused by fewer iterations is less work

“A dense Schur complement has to be stored as a dense matrix.”

On a 15 × 15 grid the Schur complement on the middle column is 100 per cent nonzero, so the sparsity field's result stands; and its cross block needs 5 columns of 7 at 10⁻⁸. The refusal is fed the assertion that the complement is under half nonzero and required to fail.

Tested in The fill that is not independent · the fill series · refused by the fill is sparse

“A small determinant means a matrix is close to singular, and a determinant of one means it is not.”

A tenth of the identity at n = 60 has a determinant of 10⁻⁶⁰ and a condition number of exactly 1; the refusal is fed the claim that such a matrix is nearly singular and required to fail.

Tested in The number that decides nothing · the determinant series · refused by a small determinant read as evidence that a matrix is nearly singular

“A diagonalisation of an integer matrix by unimodular operations gives its invariant factors, so any elimination reaching a diagonal has computed them.”

The elimination reaches a diagonal whose entries do not divide one another, and the divisibility chain has to be imposed afterwards by replacing consecutive entries with their gcd and lcm. Each diagonal is checked entry by entry against the gcds of the minors, and an intermediate diagonal that has not been repaired fails that comparison.

Tested in What a determinant does not determine · the normal forms series · refused by the claim that a unimodular diagonalisation is a Smith normal form

“A matrix that changes in only p of its entries between two solves is a small update, so a factorisation of the first is a good preconditioner for the second.”

The factorisation at μ is carried to σμ and refined twelve times, and the claim that the residual comes down is fed the result. It fails at σ = 0.5 already; at σ = 0.1 the first reused step leaves a relative residual of 3.8·10⁹.

Tested in What survives one step of the barrier · the reuse series · refused by one barrier step at σ = 0.1

“A symmetric matrix has well-conditioned eigenvectors.”

It has well-conditioned eigen*values*, and the two are different quantities in the same matrix. At a perturbation of 10⁻⁶ the eigenvalue shift is 3.867·10⁻⁷ at every gap; the eigenvector's angle runs from 3.06·10⁻⁶ to half a radian as the gap closes. Fed to the refusal at a gap of 10⁻⁸, the claim reports an angle of a radian and a half.

Tested in The gap decides the eigenvector · the eigen conditioning series · refused by the claim that symmetry protects the eigenvectors as well as the eigenvalues

“A tolerance is an implementation detail, chosen for convenience and unrelated to the accuracy of the answer.”

Three tolerances from three fields, swept: each moves the backward error it introduces by at least three decades and the work not done by up to 90 per cent, with fitted slopes of 0.039, 0.051 and 0.225 of the work a decade — within a factor of six of each other in fields sharing no arithmetic. The refusal is fed a drop tolerance of one, which discards the entire strict lower triangle and leaves a factor that preconditions nothing, reported as an ordinary point on the curve.

Tested in The zero you are allowed to write · the deliberate zero series · refused by a drop tolerance that leaves a preconditioner with nothing in it

“A sketch that preserves a least-squares problem's objective to within (1 + ε) preserves everything about the problem to within (1 + ε), constraints included.”

The same constraint is imposed exactly outside the sketch and then written as a weight and sketched with the objective. Feasibility is 1.7·10⁻¹⁶ in the first case and 1.7·10⁻⁸ in the second — eight orders, on a problem where the objective is within a quarter of its optimum either way.

Tested in The half of a problem a sketch may touch · the sketching series · refused by a constraint written as a weight of 10⁸ and sketched

“Respecting a structure costs stability: the smallest perturbation that keeps a problem's structure is far larger than the smallest one that does not, so a structured method is trading accuracy for tidiness.”

An explicit symmetric perturbation is constructed for each computed eigenpair of a palindromic quadratic, checked to be symmetric and to remove the residual, and its norm compared with the unstructured backward error. The claim that the structured one is an order of magnitude larger fails: the worst ratio over twelve pairs is 1.41.

Tested in A perturbation that keeps the symmetry · the structured backward error series · refused by a structured backward error an order of magnitude above the unstructured one

“Clustering the poles of a rational approximant towards the singularity is always the right choice, so a code should cluster as tightly as it can.”

Clustering wins by factors of 2.2 to 9.0 on a target set that stands off from the branch point. On one that reaches to within a hundredth of it, the same clustering rule wins nothing — ratios of 0.72 to 0.92 — because poles nearer the cut than any sample point produce basis functions the fit cannot tell apart.

Tested in An error committed before the arithmetic · the approximation before linearisation series · refused by a clustered set of poles on a target set that touches the branch point

“A power iteration on a stochastic matrix converges.”

A directed nine-cycle at α = 0.999999, started from a concentrated teleport vector, is still 1.8·10⁻⁶ from converged after four hundred steps. The assertion that it has converged is fed that chain and must reject it.

Tested in A chain with no stationary vector · the random walk series · refused by the claim that a power iteration converges on any chain

“Orthogonalising each block with Householder, which cannot lose orthogonality, makes block Gram–Schmidt as stable as Householder.”

On a 64×16 matrix in blocks of four whose blocks are each exactly orthonormal, built at κ = 10⁸, block classical Gram–Schmidt with Householder inside returns a basis 5.6·10⁻⁴ from orthogonal on the seed the check uses and 4.2·10⁻³ at the median of three. The assertion that such a basis is orthogonal to 10⁻¹² is fed that matrix and required to fail.

Tested in A stable block is not a stable basis · the Gram–Schmidt series · refused by a block Gram–Schmidt claimed stable because each block is orthogonalised by Householder

“Refining the discretisation of a problem makes the computed answer more accurate.”

A Gaussian deconvolution of fixed physical width, with noise of 0.1% per sample and data computed to quadrature accuracy, solved with no regularisation on grids of 12 to 48 points and scored against the continuous signal: the error falls from 0.266 at 12 points to 0.130 at 26, is 0.267 at 34, 5.04 at 40 and 761 at 48, over sixteen noise draws. Without noise the same grids keep improving to 0.100 at 40 points; the refinement that helps the discretisation is the refinement that lets the noise in.

Tested in The grid was the first filter · the regularisation series · refused by a finer grid claimed to give a better unregularised answer

“The number of iterations is the regularisation parameter of an iterative method, so a step count means the same thing from one iteration to another.”

Landweber and CGLS reach best errors of 0.1414 and 0.1426 on the same problem at 1% noise, at steps 1,778 and 20. The assertion that two iterations reaching the same answer take comparable numbers of steps is required to fail, and it does, by a factor of 89.

Tested in A step that is not a unit of work · the iterative regularisation series · refused by a step count compared across two iterations as a unit of work

“An interior-point method finds out which constraints are active by converging, so the active set is known at the end of the run and not before.”

A crossover is attempted from every iterate of a fifteen-iteration run and the claim that only the last two certify is fed the result. It fails: the first certified iterate is the first, where the method's own iterate has a relative error of 0.22 and the certified point one of 4·10⁻¹⁴.

Tested in The active set before the digits · the Interior-point conditioning series · refused by an active set that waits for the last iteration

“A reduced basis is orthogonal, so its orthogonality defect goes to one.”

The defect is the product of the basis vectors' lengths divided by the lattice determinant, and Hadamard's inequality makes it at least one for every basis of every lattice, with equality only when the vectors are mutually orthogonal. An assertion that a reduced basis achieves a defect below 0.999 is fed the reduced basis and must reject it.

Tested in A basis that describes its lattice badly · the lattice reduction series · refused by the claim that a reduced basis has an orthogonality defect below one

“A CP decomposition is unique, so its factors can always be read as components.”

Six fits are run from six starting points on a 2 × 2 × 2 rank-three tensor, whose k-ranks sum to six against a required eight. All six reach the tensor to the rounding level and the worst agreement between two of them about the factors is 0.021. The assertion that every pair agrees to 0.99 is fed that number and fails.

Tested in A factorisation that is unique for once · the uniqueness series · refused by uniqueness needs the condition

“A symbolic phase can only bound the fill, because the numeric phase may move a pivot and every moved pivot adds entries the analysis did not allocate.”

Three orderings are analysed and then factorised, and the claim that the counts differ is fed the pairs. It fails: 113 against 113, 63 against 63 and 63 against 63.

Tested in An ordering that does not wait for the numbers · the sparse pivoting series · refused by a quasi-definite matrix under three orderings

“Truncating after every operation makes a long chain of them dangerous.”

Thirty-two successive truncations of a running rank-4 object leave it within 1.034 of the best rank-4 approximation of the exact sum, in both of the two regimes measured, where a bound linear in the number of truncations would say 32. The refusal is fed the assertion that the excess exceeds two and required to fail.

Tested in The rounding that was not the problem · the recompression series · refused by a chain of truncations costs a factor

“A degree read from noisy samples is still a rank decision with a cliff in it.”

With relative noise of 10⁻¹⁰ on the samples, the Loewner matrix's singular values fall to the noise level rather than to the rounding level and the gap at the cut is 4.4 instead of 2·10⁸. The assertion that the gap survives is fed that data and must reject it.

Tested in A model with no matrices behind it · the Data-driven realisation series · refused by the claim that a degree read from noisy samples is still a decision with a cliff

“An exact solve returns an accurate answer, since it commits no rounding.”

An integer Hilbert system of order eight is perturbed by an exact relative 10⁻¹⁴ and solved over the rationals with no rounding anywhere. The answer differs from the unperturbed one by a relative 10⁻⁵. The assertion that an exact solve returns an answer accurate to the rounding level is fed that system and must reject it.

Tested in An exact answer to a measured problem · the exact cost series · refused by the claim that an exact solve of measured data returns an accurate answer

“A truncation returns the rank it was asked for.”

A rank-three object is truncated to rank eight and the assertion that the result has rank eight is fed the answer. It has rank three, because a truncation can only discard, and the assertion is required to fail.

Tested in A knob calibrated in residuals · the recompression series · refused by a rank-3 object truncated to rank 8

“The fill left on a separator is compressible however its unknowns are numbered.”

The Schur complement on a 23-unknown separator is permuted symmetrically and the same cross block is asserted to need six columns of eleven at 10⁻⁸, as it does in the separator's own order. The assertion is fed the renumbered block and must fail.

Tested in The cliff behind the count · the fill series · refused by a shuffled numbering leaves a low-rank fill

“A compression ratio is a measure of success, so a format that hands back a small fraction of the numbers has kept the object.”

The rank-(2,2,2) higher-order SVD of an 8×8×8 array of independent normal entries is 56 numbers against 512 — a compression by 9.14 — and its relative error is 0.9581. The assertion that such a truncation is exact to 10⁻¹⁰ is fed that decomposition and must reject it.

Tested in A compression of 10¹⁴ that still does not fit · the multilinear rank series · refused by a truncation is not a decomposition

“A least-squares solution computed with Gram–Schmidt can only be as accurate as the orthogonality of its Q allows.”

On a consistent 40×8 problem at κ = 10⁸, modified Gram–Schmidt's Q is off orthogonal by 8.3·10⁻¹⁰, which times κ is 0.083. Orthogonalising [A b] and reading Qᵀb out of the last column of R gives a forward error of 2.7·10⁻¹⁰ against an exact rational solve. The assertion that the error must be at least a tenth of κ‖QᵀQ − I‖ is fed that problem and required to fail.

Tested in The right-hand side as one more column · the Gram–Schmidt series · refused by least squares by modified Gram–Schmidt claimed to be only as accurate as its Q is orthogonal

“An unbiased estimate of the noise level serves the discrepancy principle as well as the noise level itself.”

The root mean square of the last eight coefficients uₖᵀb has a median of 0.98 of the true noise norm over 400 draws — unbiased to two figures — and handed to the discrepancy principle it doubles the error on 35 draws at 0.1% noise and 73 at 10%, the worst by a factor of 33,000. Thirty-two coefficients with the same median double it on none at 0.1%. What meets the cliff is the estimate's lower tail, and its median says nothing about that.

Tested in Thirty-two coefficients instead of a noise level · the Parameter choice series · refused by an unbiased noise estimate from eight coefficients claimed to keep the discrepancy principle off its cliff

“Regularisation methods that reach the same best error on a problem are interchangeable, so choosing between a penalty and a truncation is a matter of convenience.”

On answers of the form (AᵀA)^ν w the fitted slope of best error against noise is 0.705 for Tikhonov at ν = 2 and 0.704 at ν = 4, against 0.826 and 0.941 for truncation, and at ν = 4 and 10⁻⁶ noise Tikhonov's best error is 38 times truncation's. The refusal is fed the claim that Tikhonov's rate keeps pace with truncation's on a smooth answer and is required to fail.

Tested in The method that cannot use a smooth answer · the iterative regularisation series · refused by Tikhonov's error claimed to keep pace with truncation on a smooth answer

“A factorisation that is not tuned to the machine's cache cannot compete with one that is, so the block size must always be chosen for the hardware.”

At n = 96 over eight fast memories the recursive elimination moves 1.215, 1.277, 1.218, 1.161, 1.198, 1.097, 0.965 and 0.937 times the words of the best block from a scan of 1 to 24, without reading M. It does not beat every tuned block — it loses by 10% to 28% at six memories — and the refusal is fed the claim that it does, and required to fail.

Tested in The recursion that was never told the memory · the blocking series · refused by a recursion claimed to move fewer words than every tuned block

“A rational approximant beats a polynomial one on a function with a branch point, so a code facing an algebraic singularity should use rational approximation whatever region it is working on.”

At equal matrix size the ratio of the polynomial's error to the rational's runs 0.45, 1.4, 4.4, 1.5, 0.04 across five sizes on a target set that stops at the smallest eigenvalue — no consistent winner, the ordering flipping with the rounding. On a set reaching towards the cut the same ratios are 1.2, 2.9, 8.0, 24, 73.

Tested in Two approximants and one matrix size · the approximant choice series · refused by the two approximants at equal linearisation size on a target set clear of the cut

“The Collatz-Wielandt ratios bracket the Perron root for any nonnegative vector.”

The bracket is asked for on a vector with a zero entry, where the ratio (Ax)i/xi is not defined and the theorem does not apply. It refuses rather than returning a division by zero.

Tested in An eigenvector that must not change sign · the perron frobenius series · refused by a bracket asked for on a vector with a zero in it

“The computation is the weakest link in a computed ranking.”

On an eighty-page link graph, removing a single arc reorders eleven of the fourteen top comparisons while perturbing the teleport vector at the rounding level reorders none. The assertion that the ranking is at least as robust to its data as to its arithmetic is fed that graph and must reject it.

Tested in A ranking whose order is not determined · the perron frobenius series · refused by the claim that the computation is the weak part of a computed ranking

“The transforms in this solve are all applied to the same stored array, so which index each one runs along is bookkeeping rather than arithmetic.”

A 3 × 3 × 2 tensor is unfolded along its first mode and folded back along its second. The result has exactly the right shape and the wrong contents: ‖wrong − T‖ is 44.09 against ‖T‖ = 336.44. The assertion that folding along the wrong mode returns the tensor is fed that case and required to reject it.

Tested in Five indices are cheaper than two · the kronecker series · refused by the mode is not a formality

“The accuracy of a formatted factorisation is set by how many truncations it performs.”

The same depth sweep run at a second matrix size, where a tree of two levels performs no truncation and a tree of five performs a hundred. The assertion that the deep tree's ratio to the representation's error exceeds twice the shallow tree's is fed both rows and required to fail.

Tested in The count that is not the budget · the recompression series · refused by a hundred truncations cost twice what none do

“Hutch++ improves on Hutchinson, so it is the estimator to reach for.”

At a decay of 0.6 the two fitted exponents are −0.280 and −7.154, and at 96 products with A the errors are 5.71·10⁻² and 1.71·10⁻⁸. On a spectrum with no decay at all they are −0.615 and −0.495, and the deflating method is behind at every budget drawn, 6.21·10⁻² against 3.07·10⁻². The assertion that deflating first helps on any matrix is fed that second matrix and must reject it.

Tested in A rate that belongs to the matrix · the trace estimation series · refused by a deflation offered as an improvement on a matrix with nothing to deflate

“The cost of an iterative solve is a property of the matrix, the right-hand side and the tolerance.”

The same solve at thirteen partition counts takes between 674 and 690 iterations. On a matrix with κ = 4 all partitionings take the same 21 steps, which is the refusal: the assertion that any problem shows a spread of iteration counts is fed the well-conditioned case and must reject it.

Tested in A stopping test is a race · the stopping test series · refused by the claim that every problem is machine-dependent

“Restarting a Krylov method just means starting again — the information in the discarded space is lost and the method begins from scratch.”

The new starting vector is Π(A − θⱼI) applied to the old one, so the component along each eigenvector is multiplied by that polynomial evaluated at its eigenvalue: measured against predicted to 10⁻⁹ at every eigenvalue. The refusal is fed a restart with no shifts, which moves the starting vector not at all, and requires the claim that it is a restart to fail.

Tested in Restarting is a filter · the lanczos series · refused by a restart that discards nothing claimed to be a restart

“A preconditioner that narrows the range of good stopping steps makes a residual-based stopping rule less reliable, so faster convergence on an ill-posed problem is paid for in accuracy at the stop.”

Over twenty draws of the noise at 1% on the 64-point deconvolution, the discrepancy principle's median stop on conjugate gradients preconditioned by the cosine approximation truncated at τ = 0.032 is nearer that run's best than its median stop on the plain run is to the plain run's best — 1.03 against 1.07 over forty draws — although the preconditioned run has a quarter as many steps within 10% of its best. The refusal is fed the claim that the preconditioned stop is the worse of the two, and required to fail.

Tested in A stopping rule that follows the run it is given · the iterative regularisation series · refused by a preconditioner's narrower window claimed to cost the discrepancy principle accuracy

“A more accurate discretisation of an integral equation gives a more accurate computed answer, so discretisation error and noise can be dealt with separately.”

The blur is discretised on grids of 16 to 28 points by sampling the kernel and by integrating it against cubic splines, with the same nodal unknowns and the same data. With no noise the spline discretisation is the more accurate on every grid. At 1% noise per sample, over sixteen draws, its best unregularised grid reaches 0.1527 against the sampled grid's 0.1472. The refusal is fed the claim that the spline discretisation's best is the better, and required to fail.

Tested in A better discretisation is a weaker filter · the regularisation series · refused by a better discretisation claimed to give a better unregularised answer at every noise level

“Nested dissection's better asymptotics make it the faster ordering at the sizes a person draws, even if minimum degree gives less fill.”

On the 24×24 grid Laplacian, the total arithmetic of the Cholesky factorisation, counted as the sum of squared column counts, is 158,003 under nested dissection and 103,481 under minimum degree. The refusal is fed the claim that nested dissection's is the smaller, and required to fail.

Tested in An ordering that buys processors, not time · the ordering series · refused by nested dissection claimed to do less total work than minimum degree at every size drawn

“A matrix that attains the worst-case growth bound is a robust worst case, so a small perturbation of it grows almost as much.”

Wilkinson's 40 × 40 matrix grows by 2³⁹ = 5.5·10¹¹ under partial pivoting. Over ten draws of Gaussian noise with standard deviation 10⁻¹⁴ added to every entry, the median growth is 2. The refusal is fed the claim that the perturbed matrix's median growth is still above a thousand, and required to fail.

Tested in A worst case is as fragile as its margin · the growth series · refused by the growth bound's attaining matrix claimed to keep its growth under a perturbation of rounding size

“A recursive factorisation has no tuning parameter that matters for data movement: the width at which it switches to a simple kernel is chosen for function-call overhead and cannot change how many words it moves.”

On a 96-by-96 matrix with 144 words of fast memory the recursion moves 126,742 words with a base case of one column and 323,422 with a base case of twelve, performing the same 585,200 operations and choosing the same pivots in both. The claim that a base case of twelve moves within ten per cent of a base case of one is fed that matrix and required to fail.

Tested in The block size a recursion still has · the blocking series · refused by a base case claimed to cost a recursion nothing at any size

“Coefficients spread over many orders of magnitude make a flow problem harder whichever formulation is solved, and scaling to a unit diagonal removes the spread and leaves both formulations with the grid's own conditioning.”

On an 8 × 8 grid with resistances spread over six decades, both systems are scaled to a unit diagonal. The node equations' condition number is 4.6·10⁴ and the loop equations' on the least-resistance tree is 5.44. The claim that the node equations come within a factor of ten of the loops is fed that network and required to fail.

Tested in Spread resistances make the loops easy · the null space basis series · refused by the node equations claimed to be as easy as the loops once both are scaled to a unit diagonal

“Once a saddle-point matrix has been regularised so that every ordering factorises, the signs of its pivots still answer the second-order question about the problem that was posed.”

A saddle-point system with ten unknowns and four constraints whose reduced Hessian has one eigenvalue of −1 has nine positive and five negative eigenvalues. Regularised with H + 2I, its factorisation reports ten positive pivots — the count of a minimum. The refusal is fed the claim that the regularised count equals the true one, and required to fail.

Tested in A shift that certifies a saddle · the Quasi-definite series · refused by a regularised count of pivot signs claimed to certify the unregularised minimum

“A leverage computed to full precision gives one minus the leverage to full precision, so every diagnostic that divides by 1 − h is as accurate as the leverage it starts from.”

A row of a forty-by-six design is set to leverage 1 − 10⁻¹². Its leverage from a Cholesky factor of AᵀA is accurate to about one unit of roundoff, and one minus it is wrong by 1.11·10⁻⁴ of itself. The claim that one minus the computed leverage keeps ten significant digits there is fed that row and required to fail.

Tested in One minus a leverage is a subtraction · the leverage series · refused by one minus a computed leverage claimed to keep its digits near one

“Once the subtraction is avoided by reading 1 − h from the complementary block of a QR factor, it is exact to rounding on any design.”

A cubic fit to thirty points on [−1, 1] and one at x = 100 has 1 − h = 4.2·10⁻¹³ for the far point and κ(A) = 6.91·10⁵. The complement's row norm is wrong by 7.15·10⁻¹¹ of itself, against κ(A)·u = 7.67·10⁻¹¹. The claim that it is wrong by less than 10⁻¹⁴ is fed that design and required to fail.

Tested in One minus a leverage is a subtraction · the leverage series · refused by one minus a leverage from the complementary block claimed to keep every digit on any design

“An integer relation among measured numbers is found more reliably the more digits of them the search is given, so a caller should scale by as large a power of ten as the arithmetic allows.”

A relation with coefficients up to 30 among four numbers held as doubles is recovered in 8 of 8 trials when the numbers are scaled by 10¹², 4 of 8 at 10¹⁸ and 1 of 8 at 10⁴⁰. The same relation among the same numbers held as exact rationals is recovered 8 of 8 at every scaling from 10⁹ to 10⁴⁰. The assertion that recovery does not fall as the precision asked for rises is fed the double sweep and must reject.

Tested in A relation among digits that were not there · the lattice reduction series · refused by more digits is more evidence

“A warm start is the best available starting point for the next member of a drifting sequence, since the previous answer is the closest thing to it that has been computed.”

Over twenty members at a drift of 0.02, starting from the previous answer costs 160 inner steps and starting from the line through the last two costs 12 — the same accuracy, 1.5·10⁻⁹ against 1.5·10⁻⁹, and three vector operations more. The assertion that no start beats the previous answer is fed the pair and must reject.

Tested in A warm start is degree zero · the sequence of solves series · refused by the previous answer is the best start

“Run a Krylov eigensolver long enough and it finds every eigenvalue of a symmetric matrix.”

On a 24×24 matrix with an exactly doubled eigenvalue at 10, a single-vector run returns one copy at 12, 16, 20 and 24 steps — the last of those on a Krylov space of dimension 23. A block of two returns two. The refusal is fed a twenty-step run and required to reject the claim that it found the eigenvalue twice.

Tested in An eigenvalue one vector cannot see · the invariant subspace series · refused by the claim that running a single-vector method longer finds a repeated eigenvalue

“Between two low-precision formats, the one with more mantissa bits is the more accurate.”

It depends on where the data is, and the block formats are the case that makes it obvious. Across sixteen octaves of range inside a block the 2-norm of the error moves by 1.76× while the median entry's error moves by 62×, ending at exactly 1.00, with 383 of 640 entries deleted outright. A format's competence is a claim about the numbers it is given.

Tested in One exponent for thirty-two numbers · the block formats series · refused by the claim that the format with more bits always wins

“Effective resistance is defined between any two vertices of any graph.”

The routine is handed a graph in two pieces and asked for its resistance matrix. It refuses, because the grounded Laplacian is singular there and the resistance between components is infinite rather than large.

Tested in A distance computed by a solve · the effective resistance series · refused by a disconnected graph, where the resistance between components is infinite

“Oversampling is a refinement, so a method that skips it is the same method with a little more variance.”

With no extra columns at all the sketched decomposition's median error is 2.2338 times the exact one of the same rank. The assertion that the ratio is within a thousandth of one — that no oversampling is as good as some — is fed that row and required to fail.

Tested in What a single draw cannot report · the sketching series · refused by oversampling is not free to skip

“Memory is cheap, so writing down the coefficient matrix of a matrix equation and eliminating it is a reasonable way to solve one.”

At n = 20 the coefficient matrix costs 4.27·10⁷ operations to eliminate against the 4.8·10⁵ Bartels and Stewart's algorithm needs. The assertion that the two are within a factor of ten is fed the comparison and must reject it.

Tested in The elimination the matrix does not need · the matrix equation series · refused by the Kronecker form offered as a method rather than as a definition

“The Zolotarev factor is a worst case taken over an interval, so a real run lands orders below it and it cannot be used as an estimate of the error.”

The relative error of a low-rank ADI factor is measured against max|r|² for the same shifts on a twenty-state model, and the assertion is fed the claim that the first is a thousandth of the second. It must reject it, and does: across forty-four measurements on five model sizes the ratio runs from 0.7025 to 0.7915.

Tested in Bracketing an error nobody can measure · the gramian decay series · refused by an ADI error claimed orders below its own rational factor

“A contour method's probe block is a cost parameter, so a caller who wants more eigenvalues from a region pays for them with more probes and there is no cheaper route.”

With two probes and a contour holding twelve eigenvalues, one moment returns rank two — the ceiling. Two, three and four moments return four, six and eight, exactly K times the probe count, and the number of solves round the contour is 512 at every K because a higher moment is one more multiplication per quadrature point.

Tested in A ceiling with a knob on it · the nonlinear eigenvalue series · refused by the rank of a block Hankel of K moments against a probe block of two

“An ill-conditioned decomposition is a decomposition of an ill-conditioned object, so the tensor's own conditioning is the thing to look at.”

Along a sequence whose members all have nearly the same norm and the same rank, the condition number of the step a decomposition takes runs from 9.57 to 8,193. The assertion that it stays within half of its starting value is fed the two ends and fails by a factor of 856.

Tested in A tensor that cannot be decomposed · the conditioning series · refused by the tensor is not what is ill-conditioned

“A rank-revealing format compresses whatever it is given, so its storage is never worse than the array it replaces.”

A 6 × 6 × 6 array of independent normal entries is decomposed to 10⁻⁸ and the assertion that the train stores less than a quarter of the entries is fed the result. The train stores 288 numbers against 216 entries, so the assertion fails by a factor of more than five — and the same 288 is what a smooth reciprocal tensor of the same shape costs at the same tolerance.

Tested in The digit that costs more than the tensor · the tensor train series · refused by noise has no train ranks

“A compression tolerance is the error of the matrix it produces.”

On the unshifted kernel matrix this field's storage measurements are made on, at 128 unknowns and compressed at 10⁻⁸, the assembled representation's error in the Frobenius norm is 2.75·10⁻¹⁰ — thirty-six times smaller than the number that was typed. The assertion that the assembled error is the tolerance, to within a factor of a half, is fed that number and required to reject it.

Tested in The knob that moved two things · the hierarchical solve series · refused by the assembled error is the tolerance

“If two runs disagree, converge further — the disagreement is what is left of an incompletely converged answer.”

Across four tolerances the ratio between the best and worst of seven runs is 1.34, 1.48, 1.71 and 1.17 while the accuracy improves by a factor of 1.5 million. The refusal is fed the assertion that the tightest tolerance makes the runs agree and must reject it.

Tested in The tolerance that buys no agreement · the stopping test series · refused by the claim that converging further removes the disagreement

“A preconditioner that is cheap because it approximates the operator behaves like the exact preconditioner it approximates, only less sharply, so the construction that made the exact one invertible serves the cheap one too.”

Conjugate gradients on the 64-point deconvolution at 1% noise, preconditioned by the cosine-diagonal approximation of the blur shifted by α, reach best errors of 0.7486 at α = 10⁻³, 3.362 at 10⁻⁴ and 6.352 at 10⁻⁶, against 0.1426 unpreconditioned — where the exact shifted preconditioner reaches 0.1412, 0.1611 and 1.353. The refusal is fed the claim that one of those three shifts comes within 5% of the plain run's floor, and required to fail.

Tested in What a cheap preconditioner has to leave alone · the iterative regularisation series · refused by a fast-transform preconditioner made invertible by a shift, claimed to reach the plain run's floor

“A coarse grid with light regularisation is an alternative to a fine grid with heavier regularisation, since the grid and λ are two ways of setting the same amount of smoothing.”

At 0.1% noise per sample, over sixteen draws, the 30-point grid with its best λ reaches a continuous error of 0.1384 and the 96-point grid with its best λ reaches 0.1178. The refusal is fed the claim that the coarser grid is at least as good, and required to fail.

Tested in Where the grid hands over to λ · the regularisation series · refused by a coarse grid with light regularisation claimed to match a fine grid with the right λ

“A greedy ordering such as minimum degree leaves a measurable gap to the least possible fill even on a small, regular graph.”

On the 4×5 grid Laplacian the least number of factor entries over all 20! elimination orders, found by dynamic programming over the 2²⁰ sets of eliminated vertices and checked by counting the factor of the order it returns, is 76, and minimum degree with its ties broken by index reaches 76. The refusal is fed the claim that minimum degree's factor is the larger, and required to fail.

Tested in The least fill there is · the ordering series · refused by a heuristic ordering claimed to fall short of the least fill on a grid small enough to search

“The conditioning of a null-space method belongs to its basis: choose the basis with the best-conditioned Z, and the reduced problem is as well conditioned as that basis allows.”

On an 8 × 8 grid of resistances drawn over four decades, the breadth-first tree from the centre has the best-conditioned Z of six trees, κ(Z) = 5.82, and loop equations at κ(ZᵀHZ) = 683; the least-resistance tree has κ(Z) = 11.2 and loop equations at 143. The claim that the breadth-first tree's loop equations come within a factor of 1.5 of the least tree's is fed that network and required to fail.

Tested in The tree the resistances choose · the null space basis series · refused by a spanning tree chosen without the resistances claimed to condition the loop equations as well as the least-resistance tree

“A fit in which no single observation's deletion changes much has no influential observations: checking every point on its own checks them all.”

Thirty ordinary observations on [−1, 1] and a pair at x = 6, both raised by 3 above the line. Deleted one at a time, each twin's deleted residual is 0.54 and 0.69; deleted together they are 2.97 and 3.05. The claim that a twin's own deleted residual shows at least half of the shift it shares is fed that pair and required to fail.

Tested in Two observations that hide each other · the leverage series · refused by a pair of observations claimed to show its joint influence in single-deletion diagnostics

“Large growth under partial pivoting requires a matrix constructed to produce it; a well-conditioned matrix from a standard method keeps the growth small.”

The multiple-shooting matrix of y′ = My with M = [[−1/6, 1], [1, −1/6]], forty steps of 0.3 and the boundary condition y(0) + y(12) given, has 82 unknowns and a condition number of 8.3. Its growth factor under partial pivoting is 1.1·10⁴. The refusal is fed the claim that the growth is below 100, and required to fail.

Tested in The growth a boundary-value problem supplies · the growth series · refused by a well-conditioned matrix from a standard method claimed to keep partial pivoting's growth small

“The primal and dual perturbations of a quasi-definite regularisation are one knob set to one size, and they cost the answer about the same.”

On a saddle-point system with ten unknowns, four constraints and condition numbers of 100 in both blocks, the regularised solve's error is 1,451 times γ with only the zero block perturbed and 19.1 times δ with only H perturbed, at 10⁻⁸. The refusal is fed the claim that the two errors are within a factor of two, and required to fail.

Tested in The perturbation that does the work · the Quasi-definite series · refused by the two blocks of a quasi-definite regularisation claimed to cost the answer the same

“A contraction that has to fit in a given amount of memory should be evaluated in the order that holds the smallest intermediate, since that is the order most likely to fit.”

On twenty networks whose cheapest order is not already their leanest, minimising the largest intermediate costs a median of 1.549 times the cheapest order's arithmetic and requiring the same peak costs 1.146, with the two 5.167 times apart on one network. The assertion that the two questions have the same answer is fed the pair and must reject.

Tested in A ceiling is not a target · the contraction series · refused by minimising a peak that only had to be bounded

“The nearest lattice point to a target is found by writing the target in the basis and rounding each coordinate to the nearest integer.”

On a lattice whose basis is skewed by 40, rounding the coordinates lands a mean of 26.16 times further from the target than the nearest lattice point does, is the nearest point on none of 24 targets, and at worst lands 40.02 times too far. The identical arithmetic on the reduced basis is exact at every one. The assertion that the rounded point is the nearest is fed all 24 and must reject.

Tested in Rounding a coordinate in the wrong basis · the lattice reduction series · refused by rounding coordinates finds the nearest point

“The cost of solving a batch of related problems is decided by how many there are and how far apart they are, so a code has nothing to choose.”

The same sixteen problems, differing only in the order they are solved in, cost between 1.00 and 2.80 times each other under one rebuild rule, and a nearest-neighbour reordering matches the sorted cost to every digit on all eight shufflings at every drift. The assertion that the order does not change the arithmetic is fed the pair and must reject.

Tested in The order a batch arrives in · the sequence of solves series · refused by the order is not a decision

“The numerical rank of a matrix at a stated threshold is a property of the matrix and the threshold.”

One matrix returns ranks of 10, 11 and 12 across seven partitionings of the same inner products. On a matrix whose spectrum falls slowly enough that every singular value clears the threshold, all seven return 14 — which is the refusal: the assertion that a well-separated spectrum has an ambiguous rank is fed that case and must reject it.

Tested in A rank that depends on the thread count · the rank series · refused by the claim that any rank test is machine-dependent

“A random sample of a graph's edges, reweighted, approximates the graph.”

A barbell's edges are kept uniformly at random with probability a quarter and reweighted by four. The bridge is one edge among hundreds, it is missed, and the sample is disconnected — a quadratic form wrong by everything rather than by a factor. The assertion that the sample stays connected is fed that graph and must reject it.

Tested in A graph with a tenth of the edges · the effective resistance series · refused by the claim that any sample of a quarter of the edges is a sparsifier

“How a hierarchical representation's storage grows with the problem is a property of the format and the accuracy it is run at.”

At 128 unknowns and ε = 10⁻⁸ the same partition and the same truncation applied to a matrix of independent normal draws store 27,136 numbers against the matrix's own 16,384 — a ratio of 1.656, where the kernel matrix on the same points stores 10,112. The assertion that the ratio falls below 0.25 is fed the matrix of draws and required to fail.

Tested in The offset that moved the slope · the storage growth series · refused by a matrix of independent draws is compressible

“The structured backward error is the honest measurement, so it is the one to compare two solvers by.”

Charging one residual to less data can only make the quotient larger, and at every one of the forty-eight points measured here the structured backward error is the larger — 1.83·10⁻¹² against 1.93·10⁻¹⁷ for Levinson at n = 8, ρ = 0.999. Being larger does not make it a ranking: Levinson's is 103 times elimination's at n = 10, ρ = 0.98 and one sixteenth of it at n = 12, ρ = 0.8, and across forty-eight points it is the larger at twenty-eight of them. The refusal is fed the claim that restricting the perturbation can lower the backward error, computed on a quadratic eigenvalue problem where the same relation holds, and required to fail.

Tested in The number that cannot rank them · the structured backward error series · refused by the claim that a restricted backward error can be the smaller one

“A dozen Markov parameters is plenty to form a Hankel matrix and take its rank, because how many of them are taken is a matter of convenience rather than of information.”

A twenty-four-state model whose sensor sits ten grid points from its actuator is asked for a run of three parameters. All three are exactly zero, because information travels one grid point per application of the state matrix, and the sequence is refused rather than drawn as three zeros with a growth rate inferred from nothing.

Tested in The definition asks for more of what defeats it · the transfer function series · refused by a run that ends before the information arrives

“The componentwise condition number is a refinement of the normwise one — usually a bit smaller, occasionally worth computing, and never a different answer.”

Both numbers are computed for one matrix and the claim that they agree within six orders is fed the pair. It fails: κ₂ is 3.04·10¹³ and the componentwise number is 13.25, a ratio of 2.3·10¹².

Tested in Two condition numbers of one matrix · the scaling series · refused by the augmented interior-point matrix at μ = 10⁻¹²

“The cheapest way to evaluate a contraction is a property of the expression, so the order can be worked out once and used whenever that expression is evaluated.”

The order an exhaustive search chooses for the inner product of two tensor trains at rank 4 costs 8.011 times the optimum at rank 256, and the order it chooses at rank 256 costs 301.3 times the optimum at rank 2. The assertion that a plan compiled at one rank is within one per cent of optimal at every rank is fed both and must reject.

Tested in The plan that was right at rank four · the contraction series · refused by one order, optimal at every rank

“A lattice reduction's steps are unimodular operations on integers, so the lattice determinant is preserved exactly and is a sufficient check that the reduction was performed correctly.”

On the basis [[1, 0], [2⁷⁰ + 12345, 1]] with the size-reduction coefficient rounded through a double, the returned basis has an orthogonality defect of 12,345 where the exact reduction returns 1, and it fails the algorithm's own Lovász condition. The lattice determinant is 1 before and 1 after. The assertion that an unchanged determinant implies a reduced basis is fed that pair and must reject.

Tested in The knob and the rounding · the lattice reduction series · refused by the determinant certifies the reduction

“Asking a CP fit for more terms than the tensor has is harmless: the extra terms come back with negligible weights and can be dropped, the way a truncated SVD's extra singular values are negligible.”

A rank-three tensor fitted with four terms returns a smallest weight 4.7 times below the largest — not a rounding — and two terms whose cosine is 0.99927, against 0.515 at the exact rank. The assertion that the extra term's weight falls to the rounding level is fed six starting points and must reject on every one.

Tested in One term too many · the alternating least squares series · refused by the extra term is negligible

“A constrained solve can be checked by its feasibility: an answer that satisfies its constraints to the rounding level is an answer the routine computed properly.”

With a third constraint equal to the first plus 10⁻¹² of an independent direction, the null-space route returns an answer 1.16·10⁻⁴ from the exact one and the saddle-point route one 8.22·10⁻³ from it, while both satisfy every constraint to 10⁻¹⁵. The assertion that a feasibility below 10⁻¹² implies a correct answer is fed the pair and must reject.

Tested in Feasible and wrong · the constrained least squares series · refused by feasibility is a check

“The rule that chooses the Tikhonov parameter most accurately is the rule to read a preconditioner's truncation level off, since the two parameters differ only by a known constant.”

Over twenty-four draws of the noise at 1% on the 64-point deconvolution, generalised cross-validation's λ has a median error 1.034 times the oracle λ's and the discrepancy principle's is 1.041. Read as cutoffs at τ = λ/2, the discrepancy principle's worst draw reaches 1.12 times the unpreconditioned floor and GCV's reaches 198. The refusal is fed the claim that the better λ rule gives the better cutoff, and required to fail.

Tested in The rule that is wrong in the right direction · the iterative regularisation series · refused by a rule's accuracy as a λ rule, read as its accuracy as a cutoff rule

“The L-curve's corner is expensive because it locates the corner badly, so a rule that aims directly at the noise share the corner detects would be a repair for it.”

Across fifteen pairings of five signals with three penalties at 0.1% noise, the corner reads a noise share between 0.101 and 0.210. A rule aimed at a fixed share of 0.2239 — the best target for four spikes — costs 37.8 times the oracle on the two-bump signal, where the corner itself costs 25.1. The refusal is fed the claim that aiming at the corner's share is at least as cheap as the corner, and required to fail.

Tested in A rule that has to be told how good its answer will be · the Parameter choice series · refused by the corner's cost read as an error of detection rather than of target

“A change of units in the constraints hurts an interior-point method's active-set identification because it makes the step matrix ill-conditioned, so equilibrating the rows is the repair and there is nothing else to fix.”

Over six programmes with rows rescaled by up to 10³ either way, leaving the matrix untouched and only starting the iteration at the magnitudes the rows imply takes the mean iteration count from 74.3 to 18.8 — the same 18.8 it has at every spread including none — while the share of the run that certifies goes from 10% to 43% against equilibration's 64%. The refusal is fed the claim that the starting point leaves the iteration count where it was, and required to fail.

Tested in Two repairs for one symptom · the Interior-point conditioning series · refused by one cause offered for a symptom two repairs each partly cure

“Partial pivoting's value is concentrated in the first step, where a small leading entry would otherwise produce a huge multiplier, and the later searches are a formality.”

Over 60 standard normal 8×8 matrices, replacing the pivot search at step 1 with taking the first available row multiplies the median growth factor by 1.624, and doing the same at step 2 multiplies it by 1.545, at step 3 by 1.419 and at step 4 by 1.333. The refusal is fed the claim that some one step's removal costs more than three times, and required to fail.

Tested in Which of the choices is doing the work · the elimination series · refused by the stability of elimination attributed to a single load-bearing decision

“A Householder reflection is orthogonal by construction, so rounding anything about it perturbs the reflection without stopping it being one.”

With Q = I − βvvᵀ, a relative perturbation of 10⁻⁴ in β gives ‖QᵀQ − I‖ = 6.3·10⁻⁴ on 8×8 matrices at condition number 10⁸, where the same perturbation of every component of v gives 1.8·10⁻¹⁵. The refusal is fed the claim that a perturbed scalar leaves the factorisation orthogonal to 10⁻¹³, and required to fail.

Tested in One number that has to be right · the householder series · refused by the reflection's orthogonality claimed to survive an error in its scalar

“Because the operation count grows like the square of the fill, an ordering with slightly more fill but a more even distribution of column heights can do less arithmetic, so minimising fill and minimising work are different problems.”

Over forty random sparse graphs of sixteen vertices, with both minima found by a search over all 2¹⁶ subsets, one order attains both on thirty-nine. On the fortieth the least-fill order does 1.0099 times the least arithmetic, and the least-work order's fill equals the least fill. The refusal is fed the claim that the least-fill order's arithmetic exceeds the minimum by more than five per cent on some graph, and required to fail.

Tested in Two minima that are one minimum · the ordering series · refused by two orderings claimed to attain the two minima separately

“Along a converging Newton sequence the steps change slowly, so starting the inner solve from the previous step saves inner iterations.”

On the 200-unknown drifting problem at a forcing term of 0.1, starting the inner conjugate gradients from the previous accepted step costs 1,583 total inner iterations against 980 from zero — 61 per cent more — because the residual of the guess is 15 to 209 times the residual of zero. The refusal is fed the claim that the unscaled guess is no worse than no guess, and required to fail.

Tested in A guess worth two per cent · the inexact newton series · refused by the previous Newton step taken as a guess at the next one

“Hutch++'s advantage over Hutchinson comes from the deflation, so its published budget split can be treated as a constant of the method.”

At a budget of 48 products on a 60×60 matrix whose eigenvalues decay by 0.99 per step, the published third gives a median relative error of 1.8·10⁻² over 24 draws and spending nothing on the sketch gives 5.8·10⁻³ — so the split is worse than no deflation at all, by a factor of 3.05. The refusal is fed the claim that the third is within 1.2 times the best split at every decay measured, and required to fail.

Tested in The split nobody is in a position to choose · the trace estimation series · refused by a published budget split treated as a constant of the method

“A block method's width is a tuning parameter — run a narrow block for longer and it finds the same eigenvalues.”

A block of two on a triple eigenvalue returns two copies at every step count up to a basis of half the problem's dimension. The refusal is fed the claim that it returns three after thirty steps and required to fail.

Tested in How wide the block should be · the invariant subspace series · refused by a block narrower than the multiplicity claimed to converge eventually

“The accuracy of a dot product is a function of its length and its data.”

At 60 terms the mean relative error is one number with a cutoff of 64 and a different one with a cutoff of 128, on identical data — so the accuracy at a length is a property of which kernel that length selected. The refusal is fed the assertion that the two agree and must reject it.

Tested in The length that changes the kernel · the algorithm selection series · refused by the claim that a dot product's accuracy is a property of its length

“Which order the vertices are eliminated in does not change the work.”

On a thirty-six-vertex grid, minimum degree fills 71 edges and maximum degree fills 293. The assertion that eliminating the highest degree first fills less is fed that graph and must reject it.

Tested in Eliminating a vertex is a graph operation · the graph elimination series · refused by the claim that the ordering does not matter which way round

“Where the support points of a rational approximant are placed does not matter.”

On a target set reaching to within two per cent of a branch point, an evenly supported approximant of degree eight has error 9.5·10⁻⁶ and an adaptively supported one has 1.5·10⁻⁷. The assertion that the even one does as well is fed that problem and must reject it.

Tested in The points the algorithm chose · the adaptive interpolation series · refused by the claim that where the support points go does not matter

“A beam search improves as it widens: keeping more candidates than a greedy rule keeps can only bring the answer closer to the one an exhaustive search would find.”

On 24 of 60 random seven-tensor networks, some wider beam under the smallest-result rule returns a strictly dearer order than a narrower one — by up to 3.49 times in a single step of the width. The assertion that a network's excess falls monotonically in the width is fed all sixty and must reject.

Tested in The search that got worse as it widened · the contraction series · refused by a wider beam is a better beam

“The accuracy attainable in a constrained least-squares problem is governed by the condition number of A, so κ(A) is what a caller should compute before deciding whether to trust the answer.”

Two problems with κ(A) = 10¹² agreeing to twelve figures, six unknowns and two constraints each, return 4.73·10⁻¹⁶ and 3.03·10⁻⁴ from the same routine. The assertion that a shared κ(A) implies a shared attainable accuracy is fed the pair and must reject.

Tested in The condition number that does not know · the constrained least squares series · refused by κ(A) predicts the answer

“A preconditioner that reaches the answer in a quarter of the steps is doing the same thing the unpreconditioned run does, faster, so a cutoff below the one that works is simply too fast rather than wrong.”

At 1% noise on the 64-point deconvolution, over twelve draws, the unpreconditioned run's best iterate carries 23.2 of effective dimension. At τ = 10⁻¹ the preconditioned best carries 23.7 and 88% of the run's iterates land inside the unpreconditioned run's own range; at τ = 10⁻³ the best carries 32.1 and 5% do. A run that were merely fast would land on that range at a coarser spacing; this one leaves it. The refusal is fed the claim that the share landing on the range does not fall with the cutoff, and required to fail.

Tested in The parameter neither knob is · the iterative regularisation series · refused by a cutoff past the edge described as a run that is merely too fast

“Penalising two norms at once gives a regularisation with two parameters, so the pair should be swept and the extra freedom is worth the extra search.”

Over ten draws at 0.1% noise on each of five signals, the best (λ₁, λ₂) pair beats the better of the two single penalties by a median of between 1.0000 and 1.0314 times, and by at most 1.171 on any draw. On 30 to 70 per cent of draws the best pair has one of its two parameters at exactly zero. The refusal is fed the claim that mixing is worth at least a fifth as much as choosing, and required to fail.

Tested in A second penalty is not a second parameter · the Parameter choice series · refused by a two-penalty regularisation treated as a genuine two-parameter family

“Wilkinson's matrix shows that some matrices genuinely require exponential growth under any elimination, which is why the 2^(n−1) bound cannot be improved.”

On Wilkinson's matrix at n = 7 the greedy order gives a growth factor of 64, and the ordering 2,3,4,5,6,7,1 gives 2.0000 with every multiplier equal to one — an elimination that is legal by partial pivoting's own standard and that partial pivoting does not choose. Every one of the 5,040 orderings is scored to establish that the minimum is 2. The refusal is fed the claim that no ordering avoids the growth, and required to fail.

Tested in The order the greedy rule cannot choose · the elimination series · refused by Wilkinson's growth attributed to the matrix rather than to the order the greedy rule takes

“The compact WY form is algebraically identical to a product of reflections, so its orthogonality is structural in the same way and rests on nothing computed.”

With every entry of T perturbed by a relative 10⁻⁶, a block of sixteen reflectors on a 24×16 matrix at condition number 10⁸ gives ‖QᵀQ − I‖ = 9.8·10⁻⁶, against 3.9·10⁻¹⁵ with T as computed. The refusal is fed the claim that the perturbed blocked form stays orthogonal to 10⁻¹², and required to fail.

Tested in A triangle where the scalar was · the householder series · refused by the compact WY form's orthogonality claimed to survive an error in its triangle

“The hybrid ordering interpolates between minimum degree and nested dissection, so a shallow dissection depth gives most of the parallelism for a small fraction of the extra work.”

On a 24×24 grid, minimum degree does 103,481 operations with a critical path of 41,072 and full nested dissection does 149,517 with 34,572. One level of dissection followed by minimum degree does 127,956 operations with a critical path of 53,508 — more work than either end and a longer critical path than either. The refusal is fed the claim that one level is no worse than minimum degree on both counts, and required to fail.

Tested in The depth that is worse than both ends · the ordering series · refused by the hybrid claimed to interpolate monotonically between its two ends

“Because the estimator's standard error is computable from its own samples, a rule that stops when the relative standard error reaches a target delivers that relative accuracy.”

At a target relative standard error of 1% on a 60×60 matrix with eigenvalues decaying by 0.9, the median relative error over forty draws is 5.3·10⁻³ and the largest is 2.9·10⁻², with 80 per cent of draws inside the target. The refusal is fed the claim that the worst draw is inside the target, and required to fail.

Tested in A rule that reads only its own probes · the trace estimation series · refused by an estimated standard error read as a bound on the error

“A floor on the last step's forcing term declines only accuracy the outer loop cannot use, so it saves inner work and costs nothing.”

At an outer tolerance of 10⁻⁸ on the relative residual, the run with the floor reaches a forward error of 3.14·10⁻⁶ in 822 inner iterations and the run without it reaches 1.19·10⁻⁹ in 1,109 — 2,600 times more accurate for 35 per cent more work, with both runs' outer residuals below the tolerance. The refusal is fed the claim that the floor never costs a factor of three in the answer, and required to fail.

Tested in One line that buys a quarter of the run · the inexact newton series · refused by unused accuracy identified with accuracy nobody wanted

“The bound on a fraction-free elimination's intermediates is Hadamard's bound on the matrix, so every step's entries are held to the same length as the determinant.”

On a 10×10 integer matrix with entries between −6 and 6, the longest intermediate after the first step is 3 bits and Hadamard's bound on the whole matrix is 36. The bound that applies after step k is the one on (k+1)×(k+1) minors, which reads 5, 9, 12, 16, 19, 23, 26, 30, 33 and 36 — and the growth reads 3, 6, 8, 11, 14, 17, 20, 22, 24 and 27. The refusal is fed the claim that a single bound describes every step, and required to fail.

Tested in A bound on every intermediate at once · the Fraction-free series · refused by one bound for every step of an elimination whose steps produce minors of different sizes

“Tightening the tolerance of a hierarchical representation makes it keep more blocks dense, which is why its storage grows faster.”

At n = 512 on the smooth kernel, the strong admissibility partition holds 250 blocks — 156 low-rank and 94 dense — at every tolerance from 10⁻² to 10⁻¹², identically, because the partition is decided by the cluster tree's geometry before any singular value is looked at. The refusal is fed the claim that the dense count rises with the tolerance, and required to fail.

Tested in The partition that does not move · the storage growth series · refused by a partition claimed to respond to a tolerance it is decided before

“An optimality certificate obtained from an early iterate trades accuracy for iterations, so a method that wants the tightest answer should run its own test to the tightest tolerance it can.”

Over six programmes of twenty unknowns and forty constraints, the tightest μ test stops at iterate 15.0 with a relative error of 1.60·10⁻¹², and the crossover from the iterate at 1.5 returns a point whose relative error is 1.78·10⁻¹⁴. The refusal is fed the claim that the tightest tolerance on μ reaches the certificate's accuracy, and required to fail.

Tested in A test with no tolerance in it · the Interior-point conditioning series · refused by a certificate from an early iterate treated as a trade against accuracy

“Solving a matrix through a nearby circulant and a low-rank correction is as accurate as solving the matrix, provided the matrix itself is well conditioned.”

tridiag(−1, 2 + 10⁻⁸, −1) at n = 64 has κ = 1,712 and elimination solves it to 1.1·10⁻¹⁴. Solved through its periodic circulant by the rank-two Woodbury correction it is wrong by 1.2·10⁻⁴. The refusal is fed the claim that the correction reaches 10⁻¹⁰ there, and required to fail.

Tested in The circulant the problem did not contain · the circulant series · refused by the correction's accuracy claimed from T's conditioning

“Energy-minimising interpolation is a better interpolation than the classical algebraic formula, because it optimises what the classical one only approximates.”

On the isotropic operator the classical interpolation's energy is 347.00 and the minimum over its own sparsity pattern is 347.00. The refusal is fed the claim that minimisation improves on it by even 1% and required to fail.

Tested in The formula that was already optimal · the algebraic multigrid series · refused by an improvement claimed where the classical formula is already the minimiser

“A regression test with a tolerance above the machine's variation catches any real defect.”

A defect of one part in 10¹⁵ in one matrix entry produces answers overlapping the six the machine produces on its own, so no threshold separates them without failing correct builds. The refusal is fed the assertion that the smallest defect is visible and must reject it.

Tested in What a regression test can ask for · the Regression tolerance series · refused by the claim that a regression test can catch any defect

“Restricting which coefficients a perturbation may move costs the same factor at every eigenvalue.”

On an eight-mass overdamped chain the K-only backward error is 1.06 times the unrestricted one at the smallest eigenvalue and 45.7 times at the largest. The assertion that the ratio is one number for the whole polynomial is fed that spectrum and must reject it.

Tested in A perturbation that moves every coefficient · the linearisation backward error series · refused by the claim that the restriction costs the same factor at every eigenvalue

“The total stretch is the condition number of the tree-preconditioned system.”

A star tree of the complete graph on twelve vertices has total stretch far above its preconditioned condition number. The assertion that the two are equal is fed that pair and must reject it.

Tested in A preconditioner that is a tree · the graph elimination series · refused by the claim that total stretch is the condition number rather than a bound

“A run whose error keeps falling while its terms keep growing is at a boundary, so the elasticity crossing a threshold identifies a swamp.”

Six fits to genuinely rank-three tensors whose factors have cosine 0.9 all cross an elasticity of 0.25 over a twenty-sweep window, and all of them converge — one to 4.02·10⁻¹² in 3,388 sweeps. The assertion that a crossing identifies a run with nothing to converge to is fed those six and must reject.

Tested in The test that is a deadline · the alternating least squares series · refused by the threshold identifies the boundary

“Solving the optimality conditions of a constrained least-squares problem is the exact route, so any error in it comes from the elimination and a stable solver removes it.”

At κ(A) = 10¹¹ the optimality conditions solved in floating point return a relative error of 4.64·10⁻⁴, and the same conditions solved in exact rationals after the cross-product has been formed in floating point return 3.45·10⁻⁴. The assertion that an exact solve of the assembled system recovers the answer is fed the pair and must reject.

Tested in The reference was a method · the constrained least squares series · refused by the loss is in the solve

“Since every intermediate of a fraction-free elimination is a minor and a different pivot order produces different minors, choosing the pivot for size reduces the largest intermediate the run has to hold.”

Over twenty random 10×10 integer matrices with entries in ±6, the natural order makes 0 row exchanges, the largest-pivot rule 7 and the smallest-nonzero rule 7, and all three reach a peak of 29 bits — which is the length of the determinant, identical under all three because the determinant is. The refusal is fed the claim that some rule reaches a smaller peak, and required to fail.

Tested in Three orders and one last entry · the Fraction-free series · refused by a pivot rule claimed to reduce the peak of an exact elimination

“The leaf size of a hierarchical matrix's cluster tree is a performance parameter that trades against the tolerance, so a tighter tolerance wants a smaller leaf to keep the blocks compressible.”

On the smooth kernel at n = 512, the leaf minimising storage is 4 at a tolerance of 10⁻², 8 at 10⁻⁶ and 16 at 10⁻¹² — it rises rather than falls, because a higher rank makes a low-rank block dearer and leaves a dense one unchanged. The refusal is fed the claim that the best leaf at twelve digits is no larger than at two, and required to fail.

Tested in Two knobs on one number · the storage growth series · refused by a leaf size claimed to want to fall as the tolerance tightens

“A low-rank correction routed through a nearly singular base matrix loses at least the base matrix's condition number times the unit roundoff.”

At n = 64, tridiag(−1, 2cos(10π/64), −1) solved through the wrap whose twist is 10⁻⁶ of a turn from landing on both zeros of its symbol: κ of the wrap is 4.1·10⁷, κ·u is 4.5·10⁻⁹, and the correction's forward error is under 10⁻¹⁰. The refusal is fed the claim that the error is at least κ·u, and required to fail.

Tested in Two near-zeros cost less than one · the circulant series · refused by a nearly singular base claimed always to cost the correction its condition number

“An inertia-correction loop never stops on a saddle: it raises the shift until the count is right, and a right count is a minimum.”

At κ(A) = 10⁸, on 10 × 4 problems whose reduced Hessian has one eigenvalue at −μ for μ between 5.6·10⁻⁴ and 5.6·10⁻³, the loop stops below μ on draws at every curvature. The refusal is fed the claim that no trial stops below the curvature, and required to fail.

Tested in The shift that stops at the first right count · the Saddle-point systems series · refused by an inertia-correction shift claimed never to stop on a saddle

“Ranking a beam on an admissible estimate of the work still to be done removes the anomaly in which a wider beam returns a worse answer.”

Over 60 seven-tensor networks with the beam ranked on cost so far plus the largest group still to be paired — a lower bound on the remaining cost — some wider beam still returns a dearer order than a narrower one on 12. The refusal is fed the claim that none does, and required to fail.

Tested in A beam ranked on what remains · the contraction series · refused by an admissible estimate claimed to remove the width anomaly

“A worst case that rests on a pivot margin loses its growth as the margin divided by the noise, whatever the noise perturbs.”

The multiple-shooting matrix over an interval of 12 at a step of 0.3 has pivot margin 5.6·10⁻³ and growth 1.1·10⁴. With Gaussian noise of 10⁻⁴ added only to its nonzero entries, the median growth over ten draws is still 1.1·10⁴, where margin over noise is 56. The refusal is fed the claim that growth times noise over margin is below two there, and required to fail.

Tested in Noise the growth amplifies · the growth series · refused by the margin-over-noise law claimed for noise that keeps the zeros

“Weighting precise points by their precision makes an alignment behave like as many points as their information is worth.”

Five points with a tenth of the noise, weighted by 1/σ², carry the information of 515 equal points. At an effective noise ratio of 0.45 they mirror on 7.9 per cent of alignments; five hundred and fifteen equal points mirror on under two. The refusal is fed the claim that the weighted rate is within three points of the information count's, and required to fail.

Tested in Five precise points are five points · the polar decomposition series · refused by weighted points claimed to mirror like their information count

“An adaptive range finder with a probabilistic certificate stops within a few columns of the optimal rank.”

On an 80 × 80 matrix whose singular values fall by a factor of 0.8 a step, the best rank-11 approximation has spectral error below 0.1, and the certified rule stops at a median rank of 30 over five seeds. The refusal is fed the claim that it stops within five columns of eleven, and required to fail.

Tested in The rank a certificate charges · the randomised series · refused by an adaptive rank claimed to be near the optimal rank

“The probe norm is conservative enough that it can be trusted at more than its face value.”

Scaling the probes by 0.3 rather than 7.98 — reading each probe as three times smaller than it is — stops the rule at a median rank of 14 and leaves the basis with a true error above the tolerance on nine of twelve draws. The refusal is fed the claim that no draw misses, and required to fail.

Tested in The rank a certificate charges · the randomised series · refused by a probe trusted beyond its face value claimed to stay safe

“A tight enough solve tolerance makes the parabola through the last three answers a better start than the line through the last two, since the stored answers' error is then too small to matter.”

On the sequence whose roots move along a straight line, at a relative residual tolerance of 10⁻¹⁴, twenty members cost 17 inner steps from the line and 23 from the parabola. The refusal is fed the claim that the parabola needs fewer, and required to fail.

Tested in A straight path has nothing for a parabola to fit · the sequence of solves series · refused by a tolerance claimed to make degree two win on a straight path

“The line through the last two answers is always the better start, because the parabola amplifies the stored error twice as much.”

On a path bent by 0.01·sin 3t, at a tolerance of 10⁻¹², the line costs 107 inner steps and the parabola 73. The refusal is fed the claim that the line needs no more, and required to fail.

Tested in A straight path has nothing for a parabola to fit · the sequence of solves series · refused by the line claimed always to beat the parabola

“A reduced model that is guaranteed stable inherits the stability margin of the system it reduces.”

At Péclet number 10 the system's rightmost eigenvalue is −3.49 and the edge of its numerical range is −0.986; one-sided models are certified only to the second. The refusal is fed the claim that the edge is no further right than the spectrum, and required to fail.

Tested in Half the conditions and a certificate · the reduced stability series · refused by the certificate's margin claimed to be the spectrum's

“A more accurate discretisation of an integral equation keeps its advantage once the regularisation parameter is chosen well, so the discretisation and the regulariser are separate decisions.”

At 1% noise per sample on 96 points, with each discretisation's best λ, the sampled kernel reaches 0.1375 and the spline discretisation 0.1374. The refusal is fed the claim that the sampled error is more than 2% above the spline one, and required to fail.

Tested in The grid on which the discretisation stops mattering · the regularisation series · refused by a better discretisation claimed to keep its lead once λ is chosen

“Choosing the pivot column as well as the row changes the constants of threshold pivoting and not the argument: a looser threshold still buys fill with growth.”

On the 8×8 conflict grid, row-and-column threshold pivoting has growth 2.54 at τ = 0.001 and 1.23 at τ = 1. The refusal is fed the claim that the loose threshold's growth exceeds the strict one's a hundredfold, as it does for the row-only rule, and required to fail.

Tested in The column that was never fixed · the sparse pivoting series · refused by a column choice claimed to change the constants and not the argument

“Freedom to choose the column makes pivots safer, so it lowers the growth factor whatever rule breaks ties between equally cheap candidates.”

Over twenty random sparse 40 × 40 matrices at τ = 0.1, the median growth is 15.1 for the row-only rule and 24.8 for the row-and-column rule when both take the earliest of equally cheap candidates. The refusal is fed the claim that the second is no larger, and required to fail.

Tested in The column that was never fixed · the sparse pivoting series · refused by column freedom claimed to lower growth on its own

“A cutoff rule for a fast-transform preconditioner, once calibrated on one blur, carries to another: the discrepancy principle's λ halved keeps the unpreconditioned floor whatever the operator.”

Over twenty-four draws at 1% noise on a Gaussian blur 1.5 grid points wide, the cutoff at half the discrepancy principle's λ divides 36 directions against an answer that carries 32.6, and its worst draw reaches 1.181 times the unpreconditioned floor. The cutoff at λ itself divides 32 and its worst draw is 1.058; on the collection's own blur and on one 4 points wide it is 1.063 and 1.034. The refusal is fed the claim that the halved cutoff's worst draw on the narrow blur stays within 10% of the floor, and required to fail.

Tested in A count that marks the edge and not the pace · the iterative regularisation series · refused by a cutoff rule calibrated on one blur carried unchanged to a sharper one

“A preconditioned run leaves the unpreconditioned run's path when its stride passes a length the construction cannot exceed, so the count of preconditioned directions, which sets the stride, can be read as a speed limit.”

Over twelve draws at 1% noise, the first cutoff at which fewer than half a run's iterates land on the unpreconditioned path has a median stride of 5.71 effective dimensions a step on a blur 1.5 points wide, 3.44 on the collection's 2.5-point blur and 2.41 on a 4-point one. The refusal is fed the claim that the three strides agree to within 30%, and required to fail.

Tested in A count that marks the edge and not the pace · the iterative regularisation series · refused by the edge read as a stride the construction cannot exceed

“A subdivision search finds every root eventually — if a box cannot be decided, cut it smaller.”

A root sitting exactly on the first midpoint cut is on the boundary of every box the search ever asks about, and the Krawczyk verdict needs a root strictly inside. The refusal is fed the claim that the midpoint search separates the pair and required to fail.

Tested in Where the box is cut · the interval series · refused by a subdivision rule claimed to find a root that lies on its own boundary

“An answer known to be a whole number can be recovered by rounding a backward-stable computation.”

The spanning-tree count of the complete graph on twenty vertices is 2.62·10²¹. The elimination that computes it has a relative error of 4.9·10⁻¹⁵, and the nearest two representable numbers there are 3.4·10⁷ apart. The assertion that rounding recovers the count is fed that graph and must reject it.

Tested in A count that comes out of a determinant · the graph elimination series · refused by the claim that an answer known to be a whole number can be recovered by rounding

“Hyperbolicity is a structural class like symmetry, so a damping that makes a chain overdamped makes a longer chain of the same springs and masses overdamped too — the property is established once for the family and inherited at every size.”

A chain of eight masses is given β = 5.7, which certifies the same chain at seven masses and is 0.9898 of the critical damping 5.758770483 at eight. The certificate is searched for and there is none. The assertion that damping close to critical is close enough is fed that chain and must reject it.

Tested in A class a longer chain takes away · the hyperbolic quadratic series · refused by a certificate for a chain just under the critical damping

“An inertia count sees a constraint for as long as the constraint has numerical rank: the count and a rank test drop a dependent constraint at the same place.”

On 10 × 4 problems at a constrained minimum with curvature −1 along the fourth constraint's weak direction, the LDLᵀ count is first wrong at σ = 1.4·10⁻⁹ in the median, where a rank test at 10·u·‖A‖ keeps the constraint down to 1.1·10⁻¹⁵. The refusal is fed the claim that the count's first failure is below 10⁻¹⁴, and required to fail.

Tested in A constraint the count stops seeing · the Saddle-point systems series · refused by an inertia count claimed to drop a constraint where a rank test does

“A beam that is wide where its commitments are most damaging — at the first pairings — and narrow where they are nearly forced is at least as good as keeping one candidate throughout.”

At nine tensors, over 40 networks, widths 16, 8, 4, 2 and then 1 give a median cost of 1.263 times the exhaustive order's, against 1.062 for a width of one. The refusal is fed the claim that the wide-early median is no worse, and required to fail.

Tested in Widen the beam where the ranking is right · the contraction series · refused by a wide-early width schedule claimed never to lose to width one

“The smallest pivot margin a factorisation meets, divided by the growth at that step, says how much noise it takes to remove the growth.”

On the shooting matrix over an interval of 12 at a step of 0.3, the smallest margin over growth anywhere in the factorisation is 8.2·10⁻⁹, at the second-to-last step, and dense noise of 10⁻⁶ halves the growth — a hundred and twenty times the prediction. The refusal is fed the claim that the unrestricted minimum predicts the noise within a factor of three, and required to fail.

Tested in A margin the factorisation records · the growth series · refused by the unrestricted minimum margin claimed as the predictor

“An alignment with several thin directions is mirrored as often as that many independent coin flips, each at the one-direction rate, would make it.”

At a noise ratio z = 0.45 and twenty points, one thin direction's k × k determinant is negative on 3.0 per cent of draws, which compounded over two independent directions gives 5.8. Two thin directions together give 9.9. The refusal is fed the claim that the two agree within two points, and required to fail.

Tested in A mirror decided in the thin directions · the polar decomposition series · refused by several thin directions claimed to mirror as independent flips

“Any sketch with the same number of columns finds the same range as a Gaussian one, so the cheapest sketch is the right one.”

On a 64 × 64 matrix whose ten leading right singular vectors are ten particular columns, a width-20 sparse sketch with one nonzero a row leaves a median error of 3.33 times σ₁₁ over twenty seeds, against 0.41 for a Gaussian sketch of the same width. The refusal is fed the claim that the sparse median is within half again of the Gaussian's, and required to fail.

Tested in A sketch that finds the columns it can see · the randomised series · refused by a one-nonzero sketch claimed as good as a Gaussian whatever the matrix

“Choosing each member's start by evaluating the residual at both candidates pays for itself, because it almost always picks the start that needs fewer steps.”

Over thirty runs at a drift of 0.02 the residual rule needs 1,214 inner steps and makes 540 extra residual evaluations, a quarter of a step each on a dense chord iteration: 1,349 in all, against 1,246 for always taking the parabola. The refusal is fed the claim that the rule's total is no larger, and required to fail.

Tested in The degree the history chooses · the sequence of solves series · refused by a residual test claimed to pay for itself on a dense chord step

“A one-sided projection of a stable system is stable in whatever coordinates the state is written.”

The convection–diffusion system at Péclet number 10, written in coordinates rescaled by a diagonal matrix with condition number 1.3·10⁴, has the same eigenvalues and transfer function; one-sided models of order 1 are unstable at eleven placements of nineteen. The refusal is fed the claim that none is, and required to fail.

Tested in A certificate written in coordinates · the reduced stability series · refused by a Galerkin model claimed stable in any coordinates

“A sweep of the pivot search's width can be read for safety as well as for fill, and it says the narrow search is the safer one: over a sample of random sparse matrices the one-column search's worst growth factor is several times better than the full search's.”

One column gives back a fifth of the fill benefit at every threshold measured, which is the economy priced. The growth does not behave that way: the ratio of the narrow search's worst growth to the full search's is 0.17, 2.19, 0.75, 0.80 and 0.50 over five independent draws of the same experiment. The refusal is fed the claim that the narrow search has the better worst case on every draw, and required to fail.

Tested in How few columns the search needs · the sparse pivoting series · refused by a worst case claimed to order the two searches consistently

“A sparse QR factorisation is computed without its orthogonal factor, because storing it would destroy the sparsity that made the factorisation affordable — so the complementary-block route to one minus a leverage is unavailable to a sparse code, and the subtraction is the only route left.”

One row of the orthogonal factor is the Householder vectors applied in order to a unit vector, and a factorisation that cannot apply the orthogonal factor cannot solve a least-squares problem, so the vectors are kept. Rebuilt that way, the complement agrees with the one computed from a stored factor to within 10⁻¹⁸ at every leverage from 1 − 10⁻² to 1 − 10⁻¹⁵, in 900 operations. The refusal is fed the claim that the two differ, and required to fail.

Tested in The factor a sparse code keeps anyway · the leverage series · refused by the complement claimed to need a stored orthogonal factor

“The leaf size that minimises the operations a hierarchical matrix-vector product does is a different leaf from the one that minimises stored numbers, so a code has a choice to make between memory and speed.”

At n = 512 and eight digits the product does 160,128, 139,904, 135,936, 155,136 and 216,064 operations at leaves of 4, 8, 16, 32 and 64, against stored totals of exactly half each. Both are least at 16. The refusal is fed the claim that the two counts prefer different leaves, and required to fail.

Tested in A second objective that is the first one doubled · the storage growth series · refused by an operation count claimed to prefer a different leaf from the storage

“A fast-transform preconditioner made invertible by a shift has no edge: its runs sit off the unpreconditioned path at every shift, so there is no shift at which it can be used.”

On the 64-point deconvolution with a Gaussian blur 2.5 points wide at 1% noise, over eight draws, a shift whose square root is ten to the minus a quarter puts 97% of the run's iterates on the unpreconditioned path and reaches 1.001 times the unpreconditioned floor, in 16.5 steps against 19. The refusal is fed the claim that fewer than half of that run's iterates land on the path, and required to fail.

Tested in The shift had an edge, and the approximation moved it · the iterative regularisation series · refused by the shifted fast-transform construction described as having no edge

“A reading that represents the answer better gives a better regularised answer: add a breakpoint where the signal has a corner and the error falls.”

On a 96-point grid a piecewise-linear reading with a doubled node at each edge of the step represents the signal to 0.0007 of its size, against 0.069 with no breakpoint. With each reading's own best λ over sixteen draws at 1% noise it is recovered to 0.1473, against 0.1376 with no breakpoint. The refusal is fed the claim that the doubled-knot reading's error is below the smooth reading's, and required to fail.

Tested in A corner the penalty can afford · the regularisation series · refused by a breakpoint reading judged by how well it can represent the answer

“Once a breakpoint helps, its position can be taken from the data: scan it and keep the position whose fit has the least residual.”

On 48 points at 1% noise, a scan of the box's position by the residual of a fit at λ = 10^−1.5 puts it on the step on 2 draws of 16, and the median error with the box where the scan put it is 0.1346 — against 0.0207 with the box placed by hand and 0.1427 with no box at all. The refusal is fed the claim that the scanned box's error is within a factor of two of the hand-placed one's, and required to fail.

Tested in A corner the penalty can afford · the regularisation series · refused by a breakpoint placed by the residual at 1% noise claimed to recover the step

“A relation search can certify what it returns by the gap between its shortest and next-shortest vector, whatever the number of quantities: a large gap means a real relation.”

Among six numbers of sixty random digits with a planted relation of coefficients up to 30, rounded to doubles, 35 runs of 128 over eight precisions return an exact relation and one of the 35 has a gap of a digit or more; among three numbers 81 of 96 do. The gap a relation has room for falls as the double's sixteen digits are shared among more numbers. The refusal is fed the claim that at least half the found relations among six numbers clear a one-digit gap, and required to fail.

Tested in The room a relation has to stand out · the lattice reduction series · refused by a relation's gap read as a certificate at any number of numbers

“A capacitance matrix with condition number κ(S) multiplies the corrected solve's error by about κ(S), so the correction is only as good as its capacitance matrix is conditioned.”

For a pentadiagonal band corrected at rank four, with the wrap 10⁻⁷ of a turn from landing a single sample on a zero of its symbol, the 4 × 4 capacitance matrix has κ = 1.7·10⁸ and the pivoted correction's error is 29 times the cancellation times u. The refusal is fed the claim that the error reaches a hundredth of κ(S) times the cancellation times u, and required to fail.

Tested in The correction lost to its own two-by-two solve · the circulant series · refused by the capacitance matrix's condition number read as the correction's amplification

“A tensor's rank can be read off a rank sweep by the collinearity of the fitted terms: the first rank at which two terms become nearly parallel is one past the answer.”

Over six planted rank-three tensors at four noise levels, the last rank before the best fit's largest cosine between two terms passes 0.95 is three on 12 sweeps of 24. One tensor's own three terms have a cosine of 0.960 at its true rank, and two tensors' four-term fits stay at 0.872 and 0.883. The refusal is fed the claim that the collinearity rule names rank three on at least 20 of the 24, and required to fail.

Tested in The rank a sweep can vouch for · the alternating least squares series · refused by the collinearity of a fit's terms read as a rank rule across tensors

“Graphs with the same adjacency spectrum have the same Laplacian spectrum.”

C4 with an isolated vertex and the five-vertex star share the adjacency spectrum {2, 0, 0, 0, -2} and have different Laplacian spectra. The assertion that the two spectra agree is fed that pair and must reject it.

Tested in The spectrum is not the graph · the graph invariant series · refused by the claim that one cospectrality implies the other

“The error of an accumulation grows in proportion to the number of operations, because each one adds a rounding.”

A left-to-right sum against the number of terms fits a slope of 0.486; three thousand alternating rotations against the number of steps fit 0.554; a conjugate gradient residual recurrence against the largest iterate fits 0.507. Every bound written for them is linear. The refusal is fed the claim that the slope is 1, and required to fail on all three.

Tested in Three walks and one bound · the summation series · refused by the bound's slope, claimed as the drift's

“The admissibility constant is a third knob on a hierarchical matrix's storage, trading against the tolerance and the leaf size, so there is a setting of it that stores least.”

Swept from 0.2 to 1.4 at n = 512 and eight digits, the storage falls at every step — 225, 201, 170, 133, 133 and 120 numbers per unknown — and the loosest setting is the cheapest. What rises instead is the error against the dense matrix, from 3.28·10⁻¹⁰ to 1.13·10⁻⁹, at a tolerance that never moved. The refusal is fed the claim that some interior setting stores less than the loosest, and required to fail.

Tested in A geometry setting that is a second accuracy · the storage growth series · refused by a separation constant claimed to trade rather than to loosen

“A symmetric factorisation has strictly less freedom than an unsymmetric one — no column to choose independently of the row — so the conflict between choosing a pivot for sparsity and choosing it for stability must be at least as sharp, and a sparsity-driven ordering must pay for itself in growth.”

On a 28 × 28 saddle-point matrix the sparsest-available ordering holds 70 entries against the natural order's 113, with a growth of 1.28 against 1.83. At constraint densities of 2, 3, 4 and 6 nonzeros a row the entry counts are 74, 72, 74 and 86 against 111, 113, 118 and 142, and the growth is never worse by more than a fifth. The refusal is fed the claim that the sparser ordering pays in growth as it does without symmetry, and required to fail.

Tested in The freedom a symmetric factorisation does not have · the sparse pivoting series · refused by a sparse ordering claimed to cost stability on the symmetric family

“Two regularised solutions with the same effective dimension are the same answer: the dimension says how much of the data each admitted, and that is all that distinguishes them.”

On the 64-point deconvolution with a blur 2.5 points wide at 1% noise, over twelve draws, Tikhonov's solution matched to the effective dimension of the best conjugate-gradients iterate carries 1.38 times the iteration's noise and 0.951 times its bias, and on six problems the noise ratio runs from 1.22 to 1.42 while the whole error agrees to within five per cent. The refusal is fed the claim that the noise ratio at the answer's dimension is within 10% of one, and required to fail.

Tested in One arc, and what each filter pays to be on it · the iterative regularisation series · refused by two filters of equal effective dimension read as the same answer

“Because conjugate gradients and Tikhonov stop at the same effective dimension, Tikhonov's curve of error against dimension is the iteration's curve at every dimension.”

On the blur 1.5 points wide at 1% noise, at half the answer's effective dimension, Tikhonov's error is 1.37 times the iteration's at the same dimension over twelve draws, and at 0.3 of it the ratio across six problems runs from 1.03 to 1.68. The refusal is fed the claim that the ratio at half the answer's dimension is within 10% of one, and required to fail.

Tested in One arc, and what each filter pays to be on it · the iterative regularisation series · refused by the arc read as one curve at every effective dimension

“Generalised cross-validation is the best of the standard λ rules on a discretised deconvolution: its choice is within a few per cent of the oracle's, whatever the grid.”

Over sixteen draws at 0.1% noise per sample on a 48-point grid, GCV's median error is 1.003 times the per-draw oracle's and its worst is 2,944 times. On every grid of 28 points or fewer, at three noise levels, no draw is twice the oracle; on grids of 30 points and more 33 draws of 288 are. The refusal is fed the claim that GCV's worst draw on the 48-point grid is within three times the oracle, and required to fail.

Tested in The data count their dimensions, not the step's · the regularisation series · refused by generalised cross-validation judged by its median on a fine grid

“A grid can be chosen from the data: refine until the regularised solution stops using the dimensions the grid offers, and the grid is fine enough.”

At 0.01% noise the effective dimension of the solution at the discrepancy principle's λ is 28.8 on 34 points and between 28.9 and 29.6 on every grid from 40 to 96, and 34 is the first grid on which it uses no more than 85% of the grid. The best error any λ allows there is 0.1274, against 0.1121 on 40 points — 13.7 per cent worse. The refusal is fed the claim that the grid so chosen is within five per cent of the 40-point grid, and required to fail.

Tested in The data count their dimensions, not the step's · the regularisation series · refused by a grid chosen by the data's effective dimension claimed to be enough

“If a relation search returns the same vector when run at two precisions, the vector is a relation among the numbers.”

Among six numbers of sixty random digits with a planted relation of coefficients up to 300, rounded to doubles, the searches at N and N/10 digits return the same vector on 23 runs of 127 that do not return a relation — every one at fifteen digits or fewer, where an approximate relation is genuinely the lattice's shortest vector and survives losing a digit. The refusal is fed the claim that no accident agrees across the two precisions, and required to fail.

Tested in Two precisions guard the other edge · the lattice reduction series · refused by two precisions agreeing read as proof of a relation at any number of digits

“Searches at N and at N/10 digits of the same doubles share most of their rounding, so a relation among the rounding would come back at both.”

Over three to six numbers and coefficients up to 3, 30 and 300, 561 runs scaled to eighteen digits or more returned something other than a relation, and on none of them did the search a digit further in return the same vector. The refusal is fed the claim that at least one of them agrees, and required to fail.

Tested in Two precisions guard the other edge · the lattice reduction series · refused by the two scalings claimed to share enough rounding to agree spuriously

“Fitting a noisy tensor with one term too many, the extra term finds real structure in the noise, so the fitted terms are less collinear than in a noise-free overfit and the collinearity is a weaker diagnostic.”

Over six planted rank-three tensors, the best four-term fit's largest cosine between two terms has a median of 0.975 without noise and 0.992, 0.977 and 0.997 at 0.1%, 1% and 3% relative noise; the range at every level runs from about 0.87 to 0.999. The refusal is fed the claim that the median at 3% noise is at least 0.05 below the noise-free median, and required to fail.

Tested in A fit that has an answer and cannot stop · the alternating least squares series · refused by noise read as making one term too many less degenerate

“A stopping rule that reads the standard error of its own probes is worst calibrated at its loosest targets, where it stops on a standard deviation measured from the warm-up's few probes; the tighter target is the more reliable promise.”

Over 400 draws on a 60×60 matrix whose eigenvalues decay by 0.9, with a warm-up of eight and no margin, a 10% target is met on 75.8% of draws and a 3% target on 63.7%; at decays of 0.8 and 0.97 the loose target is met on 66.0% and 99.5% against 66.0% and 71.3%. The refusal is fed the claim that the loose target's coverage is more than five points below the tight target's at decay 0.9, and required to fail.

Tested in The miss a normal table already priced · the trace estimation series · refused by the loose target read as the worst-calibrated

“A sliding window that solves at every step can carry its corrected coefficients forward as the start of the next step's correction, because consecutive windows share all but one row and the previous answer is therefore close to the next.”

A 24-row window on six nearly parallel integer columns, κ(A) = 3.2·10⁶, response noise as large as the rows, a thousand steps. The window's exact answer changes by a median 0.79 of itself per step. Corrected once a step, a fresh seminormal start reaches a median error of 2.2·10⁻⁸ and the carried start 3.7·10⁻⁵. The refusal is fed the claim that the carried start ends no worse than the fresh one, and required to fail.

Tested in The answer the last window left · the sequence stability series · refused by the previous window's answer read as the better start

“The coordinates that break a Galerkin model's stability certificate are an artificial rescaling; a discretised operator stored in the nodal values a code actually uses carries the dissipativity of the problem it discretises, and its one-sided reduced models are safe.”

A centred-difference convection–diffusion operator at Péclet 100 on thirty interior nodes, cells graded 1,000 to 1 towards the outflow boundary, reduced by one-sided rational Krylov projection at orders 1, 2, 3 and 6 over nineteen shift placements from 0.1 to 100. In nodal coordinates seven models come back with a pole in the right half plane; projected in the inner product weighted by cell size, none do. The refusal is fed the claim that no nodal model is unstable, and required to fail.

Tested in The inner product the mesh already computed · the reduced stability series · refused by nodal coordinates on a graded mesh read as carrying the problem's dissipativity

“A block Gram–Schmidt whose Q is orthogonal to working precision is a stable least-squares solver, and one whose Q has lost orthogonality is not; orthogonality is the measure of a block QR's fitness.”

On 64×16 matrices with the ill-conditioning inside blocks of four, κ = 10⁶, two passes of block classical Gram–Schmidt with Cholesky QR inside return Q orthogonal to 1.2·10⁻¹⁵ and a least-squares solution 1.3·10⁵ times less accurate than Householder's. One pass of block modified Gram–Schmidt with b appended as a block, at κ = 10¹² with orthogonality lost to 6.3·10⁻⁵, is within 1.1 times Householder's. The refusal is fed the claim that the two-pass Cholesky variant solves the κ = 10⁶ problem within ten times Householder, and required to fail.

Tested in What the appended block inherits · the Gram–Schmidt series · refused by an orthogonal Q read as a stable least-squares solver

“A large growth factor that rests on exact ties between candidate pivots is an accident of the tie-break, removed by a perturbation the size of rounding.”

Wilkinson's matrix of order 24 grows by 2²³ = 8.4·10⁶ under partial pivoting and under threshold pivoting at every threshold from 0.99 to 0.1. Under partial pivoting, noise of 10⁻¹⁶ on its stored entries halves the median growth over eleven draws; at τ = 0.5 it takes noise of 0.18. The refusal is fed the claim that noise at rounding halves the growth at τ = 0.5, and required to fail.

Tested in A threshold that holds the growth still · the growth series · refused by tie-based growth read as removable at rounding under a threshold

“A recursive factorisation's base case trades data movement for call overhead: the cache wants the narrowest leaves and the processor the widest, so any base case wider than one column is bought with words.”

An 80-column matrix, which halves to leaves of exactly 10, factorised recursively across 144 words of fast memory: with a base case of 10 it moves 0.80 times the words of the same recursion split to single columns, with 356 calls against 335,606, and the same operations and pivots. The refusal is fed the claim that the base case of 10 moves at least the pure recursion's words, and required to fail.

Tested in The leaf that sits on the edge · the blocking series · refused by the narrowest leaf read as what the cache wants

“Once the capacitance system is solved stably, a banded matrix's corrected solve through its circulant wrap loses only what the cancellation in its last line costs: the intermediate solve is larger than the answer, and that ratio times u is the error.”

The squared Laplacian shifted by s = 10⁻⁸, n = 64, through its wrap twisted by half a step with the 4 × 4 capacitance system solved by pivoted elimination: the cancellation is 1.0·10⁴, so the cancellation times u is 1.1·10⁻¹²; the median forward error over five right-hand sides is 8.5·10⁻¹¹, 74 times that, and 0.27 times κ(wrap)·u = 3.1·10⁻¹⁰. The refusal is fed the claim that the error is within three times the cancellation times u, and required to fail.

Tested in A zero no twist can step around · the circulant series · refused by the cancellation read as the whole loss on a high-order zero

“A QR of the constraint matrix performed only when the inertia count has gone wrong is enough to make an inertia-correction loop's verdict trustworthy.”

Ten unknowns and four constraints of condition number 10⁸, a reduced Hessian with one eigenvalue at −μ for nineteen values of μ from about 10⁻⁸ to 10, eight draws each. The ordinary loop certifies 63 of the 152 saddles as minima. Consulting the null space whenever the count is wrong still certifies 57, the deepest at μ = 0.056; consulting it on every count certifies 5, none deeper than 1.8·10⁻⁸. The refusal is fed the claim that consulting on wrong counts leaves at most five, and required to fail.

Tested in A loop that asks the null space why · the Saddle-point systems series · refused by arbitration on wrong counts read as catching false certificates

“Smoothed aggregation is the standard fix for anisotropic operators, so it fixes the rotated one.”

On the same 31×31 grid at ε = 10⁻³ it converges at 0.193 with the anisotropy along an axis and at 0.789 with the same anisotropy at 45°. The reason is in the stencil, before any solver runs: the rotated operator's axis couplings are 0.5005 and its coupling along the anisotropy is 0.2498. The refusal is fed the claim that it converges there and requires it to fail.

Tested in Aggregating what the matrix calls strong · the anisotropy series · refused by the standard answer to anisotropy claimed to work at 45°

“A residual is enough to certify a computed answer, whatever machine produced it.”

Two reductions of one vector both have residuals under 10⁻⁶ relative and are different numbers. The refusal is fed the assertion that two answers with small residuals are the same answer, and must reject it.

Tested in Two machines, one certificate · the Regression tolerance series · refused by the claim that a small residual certifies agreement between runs

“An n × n quadratic eigenvalue problem has 2n eigenvalues, whatever its leading coefficient is.”

The claim is fed a chain of five masses with two of them removed and asked to agree that det Q has degree 10. The characteristic polynomial is interpolated exactly, in BigInt rationals, at eleven integer nodes; its degree is 8, and the assertion that it is 10 fails. Across every chain length from three to eight and every number of missing masses, twenty-seven settings in all, the degree is 2n − k and never 2n.

Tested in One mass removed, and one eigenvalue gone · the polynomial eigenvalue series · refused by degree 2n from a singular leading coefficient

“The estimator is right on four matrices in five, and the matrix it is wrong about is an elaborate construction nobody will meet.”

Over four hundred seeded 8×8 matrices the estimate is exact on 83.0 per cent and the worst underestimate in the sample returns 37.7 per cent of the truth; the constructed matrix returns 7.74 per cent, five times below anything the sample reached, and its entries take five distinct values at n = 16 — 1, ±12.92, 0.9 and 1.7. The refusal is fed the claim that the matrix which defeats the estimator has many distinct entries, and required to fail.

Tested in The tail a sample never reaches · the Condition-estimation series · refused by a counterexample dismissed as a contrived matrix

“A graph Laplacian's eigenvalues are real, because a Laplacian is symmetric.”

The Laplacian of a directed cycle on nine vertices has eigenvalues 1 − exp(2πik/9), computed by the real Schur form and matching the closed form to 1.2·10⁻¹⁵. Its largest imaginary part is 0.985. The assertion that the spectrum is real is fed that graph and must reject it.

Tested in A Laplacian that is not symmetric · the directed laplacian series · refused by the claim that a graph Laplacian's eigenvalues are real

“Forming a residual in exact arithmetic gives the exact residual.”

The same design at 4²⁴: the double solution's residual for the heavy row, with the subtraction and every product carried out in rationals, is still wrong by 1.6·10⁻². The exact answer rounded to doubles misfits the heavy row by more than its residual. The refusal is fed the claim that the exactly formed residual is right to 10⁻¹⁰, and required to fail.

Tested in The residual the solution cannot hold · the leverage series · refused by an exactly formed residual of a rounded solution read as exact

“Generalised cross-validation's degenerate minimum can be removed by refusing λ whose residual falls below a moderate fraction of its value at the real minimum.”

On the 64-point grid at 0.01% noise per sample, over sixteen draws, GCV restricted to λ whose residual is at least half the residual at its rightmost local minimum still misses the oracle by 327 times on its worst draw. Over all 528 draws a guard at one half leaves 17 draws more than twice the oracle and two more than a hundred times; only the guard at one — the rightmost minimum itself — leaves none above ten. The refusal is fed the claim that the guard at one half keeps the 64-point grid's worst draw within ten times the oracle, and required to fail.

Tested in The minimum on the right · the regularisation series · refused by a half-strength residual guard read as removing GCV's trap

“With its degenerate minimum removed, generalised cross-validation is as safe as the discrepancy principle.”

At 1% noise per sample the rightmost minimum still misses four draws by more than twice the oracle — 2.34, 5.01, 7.31 and 3.13 times on the 30-, 34-, 48- and 96-point grids — where the discrepancy principle's worst on any grid is 1.16. The refusal is fed the claim that guarded GCV stays within a fifth of the oracle at 1%, and required to fail.

Tested in The minimum on the right · the regularisation series · refused by guarded GCV read as matching the discrepancy principle's worst draw

“A saddle whose negative curvature is a hundredth of the regularisation can be told from a minimum by refinement only after hundreds of steps, long enough for its error to grow visibly.”

A saddle-point system with ten unknowns and four constraints, reduced curvature −0.01, regularised by δ = 7. Its refinement error grows by 1.0014 a step and takes 485 steps to double, but a least-squares fit of the logarithm of the residual over the last ten steps is positive from step 62 until the run ends at step 800, and on the minimum at the same δ it is negative from step 10. The refusal is fed the claim that the saddle's verdict needs more than 300 steps, and required to fail.

Tested in The residual turns before the error doubles · the Quasi-definite series · refused by a shallow saddle claimed to need its error's doubling time to be seen

“A beam over contraction orders needs width only where its ranking is a near-tie; keeping the candidates within a few per cent of the best is enough to reach the exhaustive order.”

Over sixty seven-tensor networks, a beam keeping every candidate within 25% of the best finds the exhaustive order on 31, one more than a width of one, and walking the exhaustive tree shows its own pairing is the ranking's first choice on 315 of 360 levels and more than a factor of two below it on the networks width one misses. The refusal is fed the claim that the 25% beam finds the exhaustive order on at least forty networks, and required to fail.

Tested in A near-tie is a factor of four · the contraction series · refused by near-ties claimed to be where a contraction beam needs its width

“At nine tensors no contraction beam priced between 142 and 1,380 pairings beats the wide-late schedule's median.”

Over forty nine-tensor networks the wide-late schedule prices 142 pairings for a median of 1.0556 times the exhaustive order. A beam keeping everything within a factor of three of the best prices 426 for a median of 1.0074, with a ninetieth percentile of 1.458 against wide late's 2.498. The refusal is fed the claim that no beam below 1,380 pairings has a lower median than wide late, and required to fail.

Tested in A near-tie is a factor of four · the contraction series · refused by wide late claimed unbeaten in the median below constant width sixteen

“A pivot rule that chooses the entry minimising the largest entry of the resulting trailing submatrix sees, at the first step of Wilkinson's matrix, the ordering that avoids its exponential growth.”

On Wilkinson's matrix at n = 8 with its column ties broken by 10⁻¹², every choice of first pivot leaves a largest active entry of 2 to eleven decimal places, the greedy row exactly 2 and the others 2 + 10⁻¹², so the look-ahead rule takes the greedy row and the growth is 128 = 2⁷. The refusal is fed the claim that the one-step rule keeps the growth at 4 or below, and required to fail.

Tested in One step ahead is one step short · the elimination series · refused by a one-step look-ahead claimed to find Wilkinson's good ordering

“A blocked Householder factorisation pays the compact-WY form's extra departure once per block, so its orthogonality degrades with the number of blocks.”

On 96 × 64 matrices at condition number 10⁸, eight blocks of eight whose own factors depart by a median of 3·10⁻¹⁵ to 5·10⁻¹⁵ end at ‖QᵀQ − I‖ = 1.42·10⁻¹⁴, where the sum of the eight is 3.29·10⁻¹⁴ and their root-sum-square 1.18·10⁻¹⁴; unblocked, the same factorisation ends at 2.91·10⁻¹⁴. The refusal is fed the claim that the accumulated departure reaches at least 0.8 of the sum, and required to fail.

Tested in Eight blocks and sixty-four reflections · the householder series · refused by a blocked factorisation's departures claimed to add block by block

“A compact-WY block containing a nearly dependent column has a badly conditioned triangle T, and that costs the block its orthogonality.”

With the fourth column of a 96 × 64 matrix set to the third plus 10⁻¹⁴ times a random vector, the first block's T has an entry of 8.6·10²⁵ and the block's own ‖QᵀQ − I‖ is 3.97·10⁻¹⁵, against 4.17·10⁻¹⁵ when the column is 10⁻² away. The refusal is fed the claim that the block's departure exceeds 10⁻¹⁰, and required to fail.

Tested in Eight blocks and sixty-four reflections · the householder series · refused by a badly scaled compact-WY triangle claimed to cost orthogonality

“The constant in the forcing rule's last-step floor sets a trade between inner work and forward accuracy, so a larger constant only buys more saving at the price of accuracy.”

On a 200-unknown problem at condition number 10⁵ and an outer tolerance of 10⁻⁸, the floor with c = 1 converges in 10 outer steps and 782 inner iterations; with c = 3 it runs to the 79-step cap without meeting the tolerance. The refusal is fed the claim that the c = 3 run converges, and required to fail.

Tested in A floor with a cliff at one · the inexact newton series · refused by a last-step floor above one claimed to cost only accuracy

“Coherence, the largest leverage of any column on the leading singular subspace, is what decides whether a one-nonzero sketch fails, so two matrices of equal coherence get the same sketch error.”

On a 64 × 64 matrix whose ten leading right singular vectors are ten coordinate vectors, turned towards a random basis by t = 0 and by t = 0.01, the coherence is 6.400 both times and the one-nonzero sketch's median error over thirty draws is 3.47 and 1.23 times σ₁₁. The refusal is fed the claim that the two medians agree to within half a unit, and required to fail.

Tested in The leverage that did not move · the randomised series · refused by the largest column leverage read as what breaks a one-nonzero sketch

“Reserving a bucket for each of the rank's heaviest columns removes the collisions that break a one-nonzero sketch on a coherent matrix, and so brings it to the Gaussian sketch's accuracy.”

On the exactly coherent matrix, a width-20 one-nonzero sketch with ten buckets reserved for the ten heaviest columns has a median error of 1.16 times σ₁₁ over thirty draws against the Gaussian sketch's 0.41. The refusal is fed the claim that the reserved sketch comes within a quarter of the Gaussian's median, and required to fail.

Tested in The leverage that did not move · the randomised series · refused by reserving the rank's worth of heavy columns claimed to cure a coherent one-nonzero sketch

“Upwinding adds artificial diffusion, and diffusion is dissipative, so an upwinded convection–diffusion operator is dissipative in nodal coordinates on any mesh and its Galerkin reduced models are certified stable.”

An upwind convection–diffusion operator at Péclet 100 on thirty interior nodes, cells graded 1,000 to 1 towards the outflow. The right edge of its numerical range in nodal coordinates is 6.78, and five of the 76 one-sided reduced models at orders 1, 2, 3 and 6 have a pole in the right half plane. The refusal is fed the claim that the nodal edge is negative, and required to fail.

Tested in The smaller cell downstream · the reduced stability series · refused by upwinding read as restoring the nodal certificate on a graded mesh

“A separator found from the graph alone is no better than the middle row of a grid and usually worse, so the depth-one hybrid's critical path grows when the separator has to be found rather than known.”

On the 24×24 grid Laplacian, one level of dissection with the separator taken as the middle row gives a critical path of 53,508; with the separator found as the middle level of a breadth-first level structure from a pseudo-peripheral vertex it gives 44,116. The refusal is fed the claim that the found separator's path is longer, and required to fail.

Tested in The halves were the price · the ordering series · refused by a found separator claimed to lengthen the depth-one critical path

“The depth-one penalty is the separator's dense block added to every path of the elimination tree.”

On the 24×24 grid the middle-row separator's own work is 4,900, while the depth-one critical path exceeds minimum degree's by 12,436. The refusal is fed the claim that the separator's work is more than half the penalty, and required to fail.

Tested in The halves were the price · the ordering series · refused by the separator's dense block read as the depth-one penalty

“A banded matrix whose symbol has a zero of fourth order cannot be solved through its circulant wrap as accurately as by elimination, because no twist keeps the wrap well conditioned.”

On the squared Laplacian at n = 64 with a shift of 10⁻⁸ and a half-step twist, the corrected solve is wrong by 8.5·10⁻¹¹ against elimination's 2.5·10⁻¹², and after one step of refinement against the band by 1.8·10⁻¹². The refusal is fed the claim that the refined route stays ten times behind elimination, and required to fail.

Tested in One step past the zero · the circulant series · refused by the corrected route claimed unable to reach elimination on a fourth-order zero

“An error that has moved by less than a per cent of itself over ten sweeps is within a tenth of where the fit will end, so a test in the rank decision's own tolerance can stop any fit safely.”

Six starts of an alternating fit at rank three on a noise-free rank-three tensor whose factor columns lean together at cosine 0.9. The window test at δ = 10⁻² stops five of them at errors near 10⁻² while the runs go on to reach 10⁻¹⁰ to 10⁻¹³. The refusal is fed the claim that every start stops within a tenth of the error it reaches, and required to fail.

Tested in A still error is not a settled one · the alternating least squares series · refused by a still error read as a settled one

“Symmetrising a directed graph gives the undirected Laplacian of the graph with its arrows removed.”

On a balanced digraph the two matrices agree entrywise to 10⁻¹⁴ at every size from eight to forty-eight, because the stationary distribution is then the degree distribution. On the two-block family with one arc back they differ by 0.163. The assertion that the difference is at the rounding level is fed that family and must reject it.

Tested in A conductance the arcs do not measure · the directed laplacian series · refused by the claim that symmetrising a digraph always gives the undirected Laplacian

“Cholesky QR has one boundary: the computable condition κ²u ≪ 1 separates the condition numbers where it works from the condition numbers where it does not.”

The figure that draws the erratic outcomes is asked for κ = 10², 10³, 10⁴ and 10⁵ and must reject the range, because below the licence nothing refuses and nothing comes back wholly non-orthogonal — both of the assertions that make it a figure about a cliff fail there. Above the licence there is no matching edge to be the other side of: on 256×8 matrices the first refusal is at κ = 2.2·10⁸ and a factorisation still comes back at 10¹³.

Tested in The licence is not the boundary · the algorithm selection series · refused by a figure about the cliff drawn where the cliff is not

“Rigour costs only width. A method that cannot prove a tight bound returns a wide one, so a rigorous statement is always available at some price in pessimism.”

A 14×14 Hilbert system verified at 16 significand bits returns no bound at all — the verification reports that it declined and carries no numeric bound to read. The assertion that a number is available there is fed exactly that case and is required to reject it.

Tested in Nine steps of pessimism · the interval series · refused by a bound read from a verification that refused

“A matrix built to fool the condition estimator fools condition estimation: any walk over the vertices of the 1-norm ball can be made to stop one column short by the same construction.”

The system whose inverse has a column of ones, an alternating column of 1-norm n·t and filler columns is estimated at 16.4, 7.7 and 5.1 per cent of its true κ₁ at n = 8, 16 and 24 by LAPACK's walk. The block estimator with two vectors, one of them random, returns the exact κ₁ on every one of twenty seeds at every size. The refusal is fed the claim that its worst seed at n = 16 is below half the truth, and required to fail.

Tested in Two columns see what one walk cannot · the Condition-estimation series · refused by the constructed matrix claimed to fool a block estimator

“The best leaf size of a hierarchical matrix is a function of the mean rank of its compressed blocks, so a table of best leaves collapses onto one curve when plotted against the mean rank at a fixed leaf.”

On 512 points, cos(120r)/r at 10⁻⁴ has a mean rank of 4.56 at a leaf of 16 and stores least at a leaf of 8; 1/r at 10⁻⁸ has 4.67 at a leaf of 16 and stores least at 16. Read at the leaf a halving would create — 4.08 and 4.67 against a threshold of 4 — the break-even rule separates them. The refusal is fed the claim that two cells within a fifth of a rank of each other at a leaf of 16 always want the same leaf, and required to fail.

Tested in A quarter of the leaf · the storage growth series · refused by the mean rank at a fixed leaf read as deciding the best leaf

“A least-squares line through five stored answers amplifies their error less than the line through two, so it is the better starting point for the next member of a sequence.”

Thirty twenty-member runs of a drifting nonlinear sequence of 120 unknowns, on paths bent by c·sin 3t for c from 0 to 10⁻¹ and solve tolerances from 10⁻⁶ to 10⁻¹⁴. Started from the least-squares line through the last five answers they take 1,772 inner steps in total; from the line through the last two, 1,669. The refusal is fed the claim that the fitted line takes fewer, and required to fail.

Tested in A fit wins where the steps were few · the sequence of solves series · refused by a least-squares fit claimed to beat the interpolant because it amplifies stored error less

“The sharpness p at which a roll-off between Tikhonov and truncation best matches conjugate gradients says how steep the iteration's polynomial effectively is.”

On six deconvolutions at twelve draws each, the iteration's local sharpness — half the slope of the log-odds of its factor against log σ, which is p at every height for every member of the family — is 1.01 to 1.09 where its factors are between 2% and 10%, and 2.0 to 4.9 where they are between 65% and 80%. The single p fitted to its factors drifts from 2.29 at the first step to between 1.71 and 1.88 at the answer. The refusal is fed the claim that the local sharpness varies by less than half across heights, on the blur 2.5 points wide at 1% noise at a third of the answer, and required to fail.

Tested in A tail from Tikhonov and a corner from truncation · the iterative regularisation series · refused by the iteration read as a Tikhonov with a sharper roll-off

“The augmented Lagrangian preconditioner's least work sits where the smallest generalised eigenvalue of the augmented Schur pair reaches about 0.06, so a rule can set γ to reach that value.”

On twenty-four systems with ‖H‖ = ‖A‖ = 1, κ(A) from 10 to 10⁴ and κ(H) from 10 to 10⁴, the γ of least outer-times-inner work on a quarter-decade grid runs from 0.032 to 3.2·10⁵ and the smallest generalised eigenvalue there from 0.0020 to 0.059 — a factor of thirty. A target of 0.06 costs the worst system 1.35 times its least work; the best target, 0.03, costs 1.29. The refusal is fed the claim that the optimum's ν lies within a factor of three on every system, and required to fail.

Tested in An augmentation read in the smallest eigenvalue · the block preconditioning series · refused by the least-work augmentation read as a fixed smallest generalised eigenvalue

“A recursive elimination given a base case of exactly the square root of M, less 2, puts its leaves at or below the edge, where the traffic is at or below the pure recursion's, on any size of matrix.”

On 100 columns at 144 words of fast memory, halving to a base case of 10 produces leaves of 6 and 7 and moves 1.054 times the pure recursion's words; over every size from 96 to 127 it moves more than the pure recursion on sixteen, up to 1.073. Cutting at multiples of 10 instead moves 0.828 times on 100 columns and between 0.787 and 0.856 on every size. The refusal is fed the claim that halving to the edge on 100 columns moves under 0.9 times the pure recursion's words, and required to fail.

Tested in Leaves cut to the edge on purpose · the blocking series · refused by a halving recursion's base case read as putting its leaves on the edge

“A stopping rule whose spread is estimated from a separate pilot can take its margin from the normal table, since the samples it averages are independent of the pilot.”

On a 60 × 60 matrix with eigenvalues 0.9 to the power k, over 400 draws, a two-stage rule that estimates the spread from four pilot probes and averages the number of fresh probes the normal margin of one standard error asks for lands inside a 3% target on 58.5% of draws, against the table's 68.3%; with Student's margin on three degrees of freedom, 1.197 instead of 1, it lands inside on 68.0%. The refusal is fed the claim that the normal margin on four probes covers within three points of the table, and required to fail.

Tested in A spread measured on probes it does not average · the trace estimation series · refused by a normal margin read as calibrating a rule whose spread comes from a small pilot

“With more samples than unknowns, generalised cross-validation's degenerate minimum near λ = 0 is gone by construction, because neither the residual nor m − t reaches zero.”

Over 240 draws at four samples per unknown — five grids of 24 to 96 unknowns, three noise levels, sixteen draws each — four draws still have their deepest GCV minimum at an interior λ left of the rightmost, and on 96 unknowns at 0.01% noise one of them misses the oracle by 877 times. The minimum at the floor of the scale, which the square systems produce on 21 draws, occurs on none. The refusal is fed the claim that four samples per unknown leave no dip and no hundredfold miss, and required to fail.

Tested in More samples take the floor and leave the dip · the regularisation series · refused by more samples than unknowns read as removing GCV's degenerate minimum

“An approximate relation that is the shortest vector of a relation lattice at D digits stays the shortest when digits are dropped, since a combination that cancels to D digits also cancels to fewer.”

Pooled over three to six numbers, coefficients up to 3, 30 and 300 and eight precisions from 6 to 30 digits, 62 of 781 runs that returned something other than an exact relation returned the same vector a digit further in. Two digits further in, 17 did, and three digits further in, 5. The refusal is fed the claim that the accidents that survive one dropped digit mostly survive two — at least four in five — and required to fail.

Tested in The digits between the two searches · the lattice reduction series · refused by an approximate relation read as staying shortest when digits are dropped

“Appending the right-hand side as one more block buys a least-squares solve by block modified Gram–Schmidt its full advantage over Qᵀb whatever the right-hand side is.”

On 64 × 16 problems with the ill-conditioning between blocks of four at κ = 10⁸, the appended block's error is 1.3·10⁷ times smaller than Qᵀb's when b is in the range of A. With a component outside the range of relative size 10⁻², it is 54 times smaller; at 1, Qᵀb's is the smaller. The refusal is fed the claim that the advantage stays above a million at a relative residual of one per cent, and required to fail.

Tested in The residual the appended block cannot remove · the Gram–Schmidt series · refused by the appended block's advantage read as independent of the residual

“A factorisation that records large growth on a step can switch to a stricter pivot search for the steps that follow, and so pay for the stricter search only where growth is being made.”

On the shooting matrix over an interval of 12 at a step of 0.3, partial pivoting grows by 1.1·10⁴, and no elimination step multiplies the active submatrix's largest entry by more than 1.284 — the growing mode's rate over one interval, e^(5h/6). A trigger that searches as rook pivoting does after any step rising by more than 1.25 holds the growth at 2.30; at 1.3 it never fires and the growth is 1.1·10⁴. The refusal is fed the claim that a trigger at 1.3 catches the growth, and required to fail.

Tested in A trigger finer than the growth · the growth series · refused by a growth trigger read as seeing the shooting matrix's growth in one step

“A right-hand side with no component along the growing direction hides the saddle from iterative refinement for ever.”

In double precision every correction is rounded and the rounding puts a component along every direction at every step. A right-hand side with the growing direction removed exactly gets the verdict at step 2,723 on the saddle of curvature −0.01, 1,102 at −0.1 and 181 at −1, within a fifth of a right-hand side with sixteen decades removed. The refusal is fed the claim that the verdict never comes, and required to fail.

Tested in The right-hand side that hides the saddle · the Quasi-definite series · refused by a right-hand side read as hiding a saddle from refinement for ever

“A pivot rule that looks further ahead is never worse than one that looks less far: charging the next step's cost as well as this one's can only improve on reading the pivot entry.”

On twenty-four random 8 × 8 integer matrices, the rule that charges this step and the cheapest next step has a median of 1.025 times the least any row order reaches, against the smallest pivot's 1.199 — and on one matrix it costs 1.871 times the least, where the smallest pivot costs 1.165 and its worst over all twenty-four is 1.609. The refusal is fed the claim that the two-step rule is never worse than the smallest pivot's worst, and required to fail.

Tested in What reading the next pivot buys · the Fraction-free series · refused by a longer look-ahead read as never worse

“A contour method's ceiling is the number of moments times the width of the probe block, so a caller who needs more eigenvalues out of a region buys them by widening the block, and a wider block is never worse than a narrow one.”

The ceiling is K·min(n, ℓ), and the min is not decoration. On a delay problem of size four with twelve eigenvalues inside a circle of radius six, one moment against a probe block of four, six, eight and twelve returns rank four every time, at 2,048, 3,072, 4,096 and 6,144 complex solves. The block of twelve with one moment returns four values and not one of them is within 0.75 of any eigenvalue inside the circle. The figure is fed a probe block of twelve on a problem of size four and refuses it.

Tested in Where a contour's budget should go · the nonlinear eigenvalue series · refused by a probe block the contour cannot afford

“On a nearly collinear window, the rounding of a recursive-least-squares update of the previous answer is too large for the updated answer to be a better start than a fresh solve.”

On a 24-row window of nearly parallel integer columns at κ(A) = 3.2·10⁶ with noise as large as the rows, the previous answer moved by the rank-two update the entering and leaving rows imply, then corrected once, has a median largest relative coefficient error of 3.3·10⁻⁹; a fresh seminormal solve corrected once has 2.2·10⁻⁸. The refusal is fed the claim that the updated start ends no better than the fresh one, and required to fail.

Tested in The step the two rows owe · the sequence stability series · refused by the rank-two update's rounding read as spoiling the predicted start

“Values small enough to underflow are too small for gradual underflow to change the answer; flushing them to zero costs only digits nobody uses.”

On a 48-state birth–death chain whose barrier's top is 2⁻¹⁶ of the near well, half precision with gradual underflow returns the far well's probability of 0.2454 with a relative error of 2.2·10⁻³, and flush to zero returns exactly 0. The refusal is fed the claim that the two agree to within a tenth of the answer, and required to fail.

Tested in The well on the far side of the band · the subnormals series · refused by values below the smallest normal read as too small for gradual underflow to matter to the answer

“Balancing makes a nonsymmetric matrix that a diagonal similarity would symmetrise as accurate to eigensolve as the symmetric matrix itself.”

The Kac matrix of order 80 is diagonally similar to a symmetric tridiagonal matrix whose eigenvalues come out within 1.5·10⁻¹² of the integers. Balanced, its eigenvalues come out within 3.4·10⁻⁶, because balancing leaves each off-diagonal pair unequal by up to nineteen times and the worst eigenvalue condition number at 5·10⁸. The refusal is fed the claim that the balanced route is within a hundred units of roundoff of the norm, and required to fail.

Tested in Balanced is not symmetric · the Exact ground truth series · refused by balancing read as symmetrising a matrix that a diagonal similarity makes symmetric

“The critical group refines the spanning-tree count the Laplacian spectrum determines, so it tells apart the graphs the Laplacian spectrum cannot.”

Over all 1,022 pairs of Laplacian-cospectral connected graphs on eight vertices, the critical group separates 435; none of the 361 pairs whose two groups are both cyclic can be separated, since a cyclic group is fixed by its order and the order is the shared tree count. The refusal is fed the claim that the group separates every pair, and required to fail.

Tested in A finer invariant that hears less · the graph invariant series · refused by the critical group read as separating every pair the Laplacian spectrum cannot

“A rank that changes with the field is a curiosity of specially constructed matrices; a random integer matrix has the same rank modulo a small prime as over the rationals.”

Of 3,000 random 16 × 16 0/1 matrices, the 96.8% that are invertible over the rationals are singular modulo two 70% of the time and modulo three 43% of the time. The refusal is fed the claim that under 5% of them lose rank modulo three, and required to fail.

Tested in The field decides it, usually · the exact rank series · refused by a rank that changes with the field read as a constructed curiosity

“Penalising more features of the solution at once — its size, its slope and its curvature — gives a better regularised solution than the best single penalty, once each parameter is tuned.”

Over forty draws of five signals, the best of a cube of fourteen values per parameter for the norm, the first difference and the second difference beats the best pair of penalties by a median of 1.0000 times on every signal and by at most 0.66%. The refusal is fed the claim that the third penalty buys at least five per cent on the median draw, and required to fail.

Tested in A third penalty on a flat floor · the Parameter choice series · refused by a third penalty read as buying what a second did not

“One step of iterative refinement of a fast circulant-wrap solve reaches elimination's accuracy whenever the wrap's condition number times the unit roundoff is well below one.”

The Toeplitz band of the fifth power of the shifted Laplacian's symbol at 16 points has a wrap whose condition number times u is 1.3·10⁻⁶, and its corrected solve needs three refinement steps to come within twice elimination's error; the third power at 128 points, at 3.2·10⁻⁵, needs one. The refusal is fed the claim that one step suffices at the fifth power's 1.3·10⁻⁶, and required to fail.

Tested in Where one step stops being enough · the circulant series · refused by one refinement step read as sufficing whenever the wrap's conditioning times u is small

“The best circle on which to count the eigenvalues in a gap is the one halfway between the eigenvalue inside and the eigenvalue outside, where the contour is as far from both as it can be.”

On the four-by-four delay problem the gap between the real eigenvalues 0.822 and 1.587 has its midpoint at 1.204 and its geometric mean at 1.142. Ten digits of the count cost 84 quadrature points at the midpoint and 27 at the geometric mean; on the next two gaps, 101 against 41 and 164 against 78. The claim that the midpoint is never dearer than the geometric mean is fed all eight gaps and fails on every one bounded by two real eigenvalues.

Tested in The circle between two eigenvalues · the nonlinear cost series · refused by the best circle halfway between two eigenvalues

“The ramp — the iteration's first-step filter used as a fixed method — matches conjugate gradients early in the run and loses the match near the answer, where the iteration's corner has softened.”

On six deconvolutions at twelve draws each, the ramp placed at the iterate's clipped sum of factors is within 0.044 of the iteration's error from 0.3 to 0.5 of the answer and within 0.016 from 0.7 to 1.0, closer near the answer than the second power of the earlier family (0.018), Tikhonov (0.031) or truncation (0.089). The refusal is fed the claim that the ramp is further from the iteration near the answer than twice the family's best, and required to fail.

Tested in The first step's filter, kept for the whole run · the iterative regularisation series · refused by the ramp read as losing the iteration near the answer

“How closely the two Hankel singular values of a dropped pair agree says whether the pair still reads the reduction error: the wider the split, the worse the reading.”

On six oscillators at damping 0.1, order four's first dropped pair has split by 0.43 and its two values added are within 3.6 per cent of the error; order two's has split by 0.15 and is 15.9 per cent off. The reading's departure is bounded instead by the other dropped modes' response under the error's peak, on all thirty cases measured to damping 0.15. The refusal is fed the claim that a wider split always means a worse reading, and required to fail.

Tested in Where a ringing mode stops being a pair · the balanced truncation series · refused by a discarded pair's split read as saying when its reading fails

“Balanced truncation's guarantee of a stable reduced model means its poles stay about as far from the imaginary axis as the full model's.”

At damping 0.1 every pole of the six-mode system is at least 0.1 from the axis. Its balanced truncation to six states, whose cut falls between two Hankel singular values agreeing to 0.7 per cent, has a pole 1.2 times ten to the minus four from it, and the reduction's largest error sits in a spike at zero frequency 7.13 high, twice what the dropped pair reads. The refusal is fed the claim that the reduced model's slowest pole is within a factor of ten of the full model's, and required to fail.

Tested in Where a ringing mode stops being a pair · the balanced truncation series · refused by a balanced truncation's guaranteed stability read as a stability margin

“The short leaf an aligned recursion cannot fill costs less at the top of the recursion, as a thin strip of a large product, than at the bottom as a narrow panel.”

On every size from 96 to 127 columns at 144 and 256 words, cutting the leftover off first moves more words than cutting it off last on every size that has a leftover — 29 of 29 at 256 words — and the same words on the sizes that do not. At 256 words the one-column sizes 99, 113 and 127 go from 0.947, 0.898 and 0.902 of the pure recursion's words to 0.978, 0.923 and 0.924. The refusal is fed the claim that the leftover-first cut is cheaper on the one-column sizes, and required to fail.

Tested in The short leaf belongs at the bottom · the blocking series · refused by the short leaf read as cheaper at the top of the recursion

“A base case near the cache's edge is near-optimal either side of it: a column too wide or too narrow costs about the same small amount.”

At 144 words the aligned cut's median over thirty-two sizes is 0.881 of the pure recursion's words one column under the edge, 0.823 at it and 2.478 one column over; at 256 words, 0.903, 0.886 and 2.876. Halving's median at 144 words is 1.001 at all three. The refusal is fed the claim that one column over costs within a tenth of what one column under does, and required to fail.

Tested in The short leaf belongs at the bottom · the blocking series · refused by a base case read as equally forgiving either side of the edge

“An oscillatory kernel's off-diagonal rank is about twice a smooth kernel's at the same tolerance, so a code guessing its rank before compressing should double the smooth-kernel guess for an oscillation.”

Fed to the halving rule on thirty cells — six kernels at five tolerances, 512 points each — one rank per two decades of tolerance names the leaf that stores least on 23; the same guess doubled for the two oscillatory kernels names it on 18, putting the leaf at 32 where 16 is best on six cells. At a leaf of 8, cos(40r)/r's mean rank exceeds 1/r's by 0.58 to 0.65 of a rank at every tolerance, not by a factor of two. The refusal is fed the claim that the doubled guess is right at least as often, and required to fail.

Tested in One build tells the leaf · the storage growth series · refused by the oscillation's doubling read as improving the rank guess

“When an alternating least-squares fit's error stops changing, the fit has found the best approximation of that rank.”

On a random 2 × 2 × 2 tensor whose pencil has a complex pair, a rank-two fit's error settles at 0.1752 and two starts agree on it to six digits, while the largest rank-one term grows from 24.5 at sweep 1,000 to 66.8 at sweep 8,000 — about the 0.48 power of the sweep count — and has not stopped. No best rank-two approximation exists. The refusal is fed the claim that the terms settle when the error does, and required to fail.

Tested in A fit with no answer to find · the tensor rank series · refused by a settled error read as a found best approximation

“A right-hand side that hides only the faster of two negative curvatures from iterative refinement delays the verdict by the slower curvature's race — a fixed number of steps for every decade hidden.”

On a saddle with reduced curvatures −1 and −0.01 regularised by δ = 7, a right-hand side with the faster growing direction removed by anything from none to sixteen decades, or exactly, gets the verdict between step 10 and step 12 — a slope of zero steps a decade against the slower direction's race of 147.5. On −0.1 and −0.01 it gets it at step 12 at every share. The refusal is fed the claim that the slope is within a fifth of the slower direction's race, and required to fail.

Tested in One hyperplane hides nothing · the Quasi-definite series · refused by hiding the faster saddle direction read as delayed by the slower one's race

“A second negative curvature gives a right-hand side more to hide behind, so a saddle with two can delay refinement's verdict at least as long as its shallower curvature alone.”

With curvatures −0.1 and −0.01 the longest delay any share produces is 1,127 steps, against 2,907 for −0.01 alone and 1,105 for −0.1 alone; four hundred random right-hand sides' slowest verdict is at step 79 against 417 and 190. The refusal is fed the claim that the pair's worst delay is at least the shallower curvature's alone, and required to fail.

Tested in One hyperplane hides nothing · the Quasi-definite series · refused by a second negative curvature read as lengthening the worst delay

“On the eight-bit formats the subnormal band is too narrow to matter, so a recurrence that crosses it gains nothing from gradual underflow.”

Over nine two-well chains of 32 to 64 states, gradual underflow survives a barrier two octaves deeper than flush to zero in E4M3 (7 against 5) and three deeper in E5M2 (16 against 13), within an octave of the band's width of one less than the significand's bits, as at half precision. The refusal is fed the claim that the band adds at most one octave on both formats, and required to fail.

Tested in The far well in a byte · the subnormals series · refused by eight-bit subnormals read as worth nothing against the barrier

“A poor direction of negative curvature read from an LDLᵀ factorisation is recovered by a few steps of inverse iteration with the same factors, since each step is a stable solve.”

On the planted 24 × 24 family at coupling 10⁻⁸, whose smallest eigenvalue is −8.40, one step of inverse iteration with Bunch–Kaufman's factors takes its direction's Rayleigh quotient from 6.9·10⁻¹⁶ of the smallest eigenvalue to −1.5·10⁻¹⁷ of it — positive curvature — and every later step stays at the eigenvalue nearest zero. From the bounded rule's direction, which held 14 per cent, two steps reach the same place. The refusal is fed the claim that three steps reach half the curvature, and required to fail.

Tested in A curvature direction the factors cannot refine · the cholesky series · refused by the factors' own solve read as refining their curvature direction

“The critical group is a finer invariant than the spanning-tree count wherever both are defined, so on graphs that share an adjacency spectrum it separates some pairs their tree counts do not.”

Of the 733 pairs of connected eight-vertex graphs with the same adjacency polynomial, the tree count separates 717 and the critical group separates the same 717. On each of the sixteen pairs whose tree counts agree — from 1 to 545 spanning trees, one of them with the non-cyclic group ℤ₂ ⊕ ℤ₂ ⊕ ℤ₈ — both graphs have the same group. The refusal is fed the claim that the group separates at least one pair the tree count leaves together, and required to fail.

Tested in Each spectrum hears the other's pairs · the graph invariant series · refused by the critical group read as finer than the tree count on adjacency-cospectral pairs

“The stable least-squares routes fall a factor of about fifty short of the κ²uρ term because the residual in the earlier measurements was not aimed at the weak directions; a residual aimed there would bring them to the bound.”

On a 64 × 16 problem at κ = 10⁸ and a relative residual of one per cent, each route's error map from residual to solution is measured column by column against the exact rational solution and its top right singular vector used as the residual. Householder's error, over κ²uρ, is 0.113 aimed against a median of 0.014 over twenty-four random directions; block modified Gram–Schmidt with b appended, 0.082 against 0.0073; column modified Gram–Schmidt on [A b], 0.033 against 0.0031. The ratio of aimed to typical is √(m − n) at m = 32, 64 and 128 to within three per cent. The refusal is fed the claim that the aimed residual brings some route to half the bound, and required to fail.

Tested in The worst residual belongs to the route · the Gram–Schmidt series · refused by the aimed residual read as closing the bound's slack

“Block modified Gram–Schmidt with the right-hand side appended as one more block is never more than about four times Householder's error, whatever the residual.”

With the ill-conditioning between blocks of four at κ = 10⁸ and a relative residual of one per cent aimed at the appended block's own error map, the appended block is wrong by 9.3·10⁻⁴ and Householder, on the same right-hand side, by 7.5·10⁻⁵ — 12 times. Over twenty-four random residual directions the appended block's median constant is 0.0082 and Householder's 0.0016. The refusal is fed the claim that the appended block stays within four of Householder on the aimed residual, and required to fail.

Tested in The worst residual belongs to the route · the Gram–Schmidt series · refused by the appended block read as never four times behind Householder

“A caller who needs inexact Newton's answer to a relative accuracy δ can pass δ as the outer residual tolerance; the floor under the forcing rule then gives up only accuracy beyond δ.”

On five 200-unknown problems — κ(A) from 10³ to 10⁷, cubic strength from 0.3 to 30 — at δ of 10⁻⁴, 10⁻⁶, 10⁻⁸ and 10⁻¹⁰, the run stopped at relative residual δ with the floor at ½ has a forward error 7.3 to 35,000 times δ; on the standing problem at 10⁻⁸ it is 190 times. Stopped at δ times the smallest Ritz value times ‖x‖ over ‖b‖, every certifiable cell is within 0.045 to 0.18 of δ. The refusal is fed the claim that the residual tolerance δ delivers a forward error within ten of δ on the standing problem at 10⁻⁸, and required to fail.

Tested in A target the residual can promise · the inexact newton series · refused by a residual tolerance read as a forward-error target

“A circulant whose eigenvalues are the isotropic kernel's two-dimensional symbol sampled at the grid frequencies gathers the isotropic kernel's preconditioned spectrum into treads, as T. Chan's circulant does for the separable kernel.”

On a 10 × 10 grid at correlation 0.9 the symbol circulant leaves the isotropic kernel's preconditioned spectrum in 30 clusters at one per cent, against Chan's 38 and the separable kernel's 6 under the same construction; across sides 4, 6, 8 and 10 the isotropic count is 11, 17, 23 and 30, and the step count rises from 19 at side 6 to 35 at side 16. The refusal is fed the claim that the symbol circulant gathers the isotropic ramp into a dozen clusters or fewer on the 10 × 10 grid, and required to fail.

Tested in The symbol that builds the staircase · the toeplitz series · refused by the symbol circulant read as gathering the isotropic kernel's ramp into treads

“A reduced-precision conjugate-gradient run that recomputes its residual in double every few steps can stop on that residual in place of the recurrence's, at the cost of one extra product per check.”

On Laplacian grids of side 6 to 14 at working precisions of 32, 24, 16, 11 and 8 significand bits, with the preconditioner exact and the residual recomputed in double every five steps, the recomputed residual never falls below 10⁻¹² on any of the twenty-five runs; each run continues until its own recurrence residual underflows and the iteration breaks down, or the 300-step cap, at 145 to 300 steps, against 13 to 30 for the recurrence's test. The refusal is fed the claim that on the 10 × 10 grid at 24 bits the recomputed test is met within twice the recurrence's steps, and required to fail.

Tested in The honest residual is not a stopping test · the stopping test series · refused by the honest residual read as a drop-in stopping test

“Re-planning a fill-reducing order only at the steps where the numerical test refuses the planned pivot recovers the step-by-step rule's saving whatever the share of small pivots.”

On indefinite grids with a seeded share of diagonal entries set to 10⁻¹², medians over four seeds, the plan redrawn at each refusal recovers 85 to 100 per cent of the step-by-step sparsest rule's saving over the natural order on the 8 × 8 grid at every share from 5 to 30 per cent, and 93 to 101 per cent on the 10 × 10 grid up to 20 — but 66 per cent on the 10 × 10 grid at 30, and at half the diagonal small every order, step-by-step included, stores more than the natural one on the 8 × 8 grid. The refusal is fed the claim that the redrawn plan recovers 85 per cent on the 10 × 10 grid at 30, and required to fail.

Tested in A plan redrawn where it was refused · the sparse pivoting series · refused by re-planning at refusals read as recovering the saving at any share

“With more samples than unknowns, GCV can be told the noise by its own least-squares residual, and a guard that refuses a minimum whose residual per remaining degree of freedom is implausibly low removes the interior dips that oversampling leaves.”

On five grids at three noise levels and 1.25, 1.5, 2 and 4 samples per unknown, 960 draws hold 45 interior dips. With the noise estimated from the least-squares residual on m − n degrees of freedom and a minimum refused when ρ̂ falls below 1 − 2.33·√(2/(m − t) + 2/(m − n)), the guard refuses 0 dips at every ratio and at thresholds of 1.28, 2.33 and 3.09, and its pick is plain GCV's on every draw at 2.33 and above; the dip that survives at four samples per unknown still costs 877 times the oracle. The refusal is fed the claim that the guard refuses at least half of the dips at four samples per unknown, and required to fail.

Tested in An estimate that shares the dip's luck · the regularisation series · refused by a guard on ρ̂ read as removing GCV's interior dips

“The corner of the L-curve locates the stopping step of conjugate gradients as reliably as it locates Tikhonov's regularisation parameter.”

On six deconvolutions at twelve draws each, the corner of the iterates' L-curve costs more than twice the iteration's best error on 44 draws of 72, the corner of Tikhonov's on 24 and of the ramp's on 26; the worst draws are 896, 9.9 and 15.9 times their filter's best, and two, eight and thirteen draws land within a tenth of it. The refusal is fed the claim that the iteration's corner misses no more often than Tikhonov's, give or take two draws, and required to fail.

Tested in A corner chosen without the iteration · the iterative regularisation series · refused by the L-curve read as stopping the iteration as reliably as it places Tikhonov

“With the trace of the influence matrix computed exactly, generalised cross-validation cannot be drawn to a nearly unregularised solution.”

The ramp's influence matrix is diagonal in the singular basis and its trace is exact. On the narrowest blur, four draws of twenty-four put GCV's minimum at a ramp that discards 0.76 of one direction, with errors 83, 100, 607 and 696 times the ramp's best. The refusal is fed the claim that with an exact trace GCV's worst placement on the ramp is within three times its best, and required to fail.

Tested in A corner chosen without the iteration · the iterative regularisation series · refused by an exactly known trace read as making GCV safe

“Folding a recursive LU's one-column leftover into its neighbouring leaf puts that leaf past the cache's edge and so costs more words than leaving the leftover as a narrow last leaf, at 256 words of fast memory.”

At 256 words, base case fourteen columns, the one-column sizes 99, 113 and 127 move 0.947, 0.898 and 0.902 of the pure recursion's words with the leftover cut off last and 0.883, 0.846 and 0.855 with it folded — the same elimination, the same pivots. The folded cut is cheaper on every one-column size at 64, 144 and 400 words too. The refusal is fed the claim that the folded cut moves more words on every one-column size at 256, and required to fail.

Tested in One column of room past the edge · the blocking series · refused by a folded one-column leftover read as losing at 256 words

“A leaf one column wider than the cache's edge is over the cliff: the edge leaves no room above it in any dimension.”

One warm leaf product at 256 words fetches 0.143 words per multiply-add at the edge of fourteen columns, 0.138 with its inner or column dimension one over and 0.143 with its rows one or two over; with the inner or column dimension two over, or all three one over, it fetches 1.12 and 1.12 — about every operand. The refusal is fed the claim that one dimension one column over costs twice the edge leaf, and required to fail.

Tested in One column of room past the edge · the blocking series · refused by a single leaf one column over the edge read as on the cliff

“A block condition estimator with two random starting vectors is fooled by a matrix built against random starts on about the same share of seeds as the estimator with one.”

On a 48 × 48 inverse with nine steering columns, each half the hidden column's 1-norm, one random vector is fooled on 38 seeds of 100, the block of two on 16 and the block of four on 3 — close to 0.38 squared and to the fourth power. The refusal is fed the claim that the block of two's share is within a tenth of the single vector's, and required to fail.

Tested in Columns that steer a random start · the Condition-estimation series · refused by a block of two read as fooled as often as a single random start

“How badly a constructed matrix fools a random-start condition estimator is bought with how often: at a fixed factor of underestimation, the share of seeds fooled is fixed.”

With every steering column half the hidden column's norm, so that every fooled estimate is 0.51 of the truth, the single random start is fooled on 9, 38, 50, 64 and 77 seeds of 100 with 4, 9, 16, 25 and 34 steering columns. The refusal is fed the claim that the share moves by less than a fifth across the five, and required to fail.

Tested in Columns that steer a random start · the Condition-estimation series · refused by the factor a random start is fooled by read as fixing how often

“A fraction-free elimination's least-cost row order, found on one member of a family of integer matrices, does no better than the natural order on the other members.”

On twelve 8 × 8 matrices sharing power-of-two row scales of 16, 1, 8, 8, 8, 1, 2 and 8, another member's least-cost order costs a median 1.126 of each member's least, worst 1.458, where the natural order costs 1.884; on independent entries the same reuse costs 1.540 against the natural order's 1.600. The refusal is fed the claim that on the scaled family a reused order is no better than the natural order by more than a tenth, and required to fail.

Tested in An order found once knows what the rows know · the Fraction-free series · refused by a reused elimination order read as no better than the natural order on every family

“A shared pattern of zeros is the structure that lets one elimination order serve a whole family of integer matrices.”

On twelve members sharing a pattern of 40 per cent zeros, another member's least-cost order costs a median 1.592 of the least and sits at the 31st percentile of all orders; 39 of the 132 reuses meet a zero pivot and cannot be run. The refusal is fed the claim that the reused order is within a fifth of the least, and required to fail.

Tested in An order found once knows what the rows know · the Fraction-free series · refused by a shared zero pattern read as carrying the least-cost order

“A dangerous cut in a balanced truncation is one whose kept and dropped Hankel singular values agree to under about one per cent, so a rule against cutting where the gap is under one per cent avoids the failures.”

Over 1,452 reductions of twelve six-mode oscillators, ten have an error more than one and a half times twice the first dropped σ. Seven have a gap under one per cent; three have gaps of 1.85, 2.13 and 2.18 per cent at dampings 0.2 and 0.3. All ten have a gap under a fifth of the smaller gap beside it. The refusal is fed the claim that every doubled reduction has a gap under one per cent, and required to fail.

Tested in A tie is a gap smaller than its neighbours · the balanced truncation series · refused by a tie read as a gap under one per cent

“A rank-two fit to a tensor with no best rank-two approximation, stopped as soon as its error is within one per cent of the computable distance to the rank-two set's boundary, still has terms within a factor of three of the tensor's norm, because the excess error falls as the square of the term size and one per cent is reached early.”

On 23 random 2 × 2 × 2 tensors with a complex pair and a correctly computed distance, the 21 that reach one per cent within 20,000 sweeps do so with their largest term 3.81 to 12.86 times the tensor's norm, the size τ·d·size² = 0.016 to 0.044 implies. The refusal is fed the claim that every draw's terms are within three times the norm at that stop, and required to fail.

Tested in A stop that knows the distance · the tensor rank series · refused by a fit stopped at one per cent of the distance read as keeping its terms within three times the tensor's norm

“A fit with no best approximation eventually stalls, so the usual test on the relative change in error — stop when it moves by less than a small fraction of itself — is a usable stopping rule for it.”

At a threshold of 10⁻⁶ the stall test stops 22 of the 23 fits at terms 8.2 to 12.4 times the norm and excess errors from 0.12 to 1.5 per cent of the distance; at 10⁻⁹ it stops none of them in 20,000 sweeps. The refusal is fed the claim that every fit stalls at 10⁻⁹, and required to fail.

Tested in A stop that knows the distance · the tensor rank series · refused by a stall test read as a stopping rule for a fit with no best

“An oscillatory kernel's rank offset over a smooth kernel's, at a fixed leaf, rises by a fixed amount each time the frequency triples, so a term in the logarithm of the frequency corrects a rank guessed from the tolerance.”

On a 512-point grid at a leaf of 8, cos(κr)/r's mean rank above 1/r's rises, averaged over five tolerances from 10⁻⁴ to 10⁻¹², by 0.28 from κ = 13⅓ to 40, 0.95 from 40 to 120, 1.39 from 120 to 360 and 0.38 from 360 to 1,080. The refusal is fed the claim that the four steps agree within a quarter of a rank, and required to fail.

Tested in An offset that bends twice · the storage growth series · refused by an oscillatory rank offset read as rising a fixed amount per tripling of the frequency

“Once the smallest admissible blocks reach full rank, a single build at that leaf can no longer see the kernel's ranks, and stops naming the cheapest leaf.”

At κ = 360 and 1,080, where every 8-wide block is at rank 8 at the tightest tolerance, one build at a leaf of 8 fed to the halving rule names the cheapest leaf on five of the six cells, and the sixth stores 1.8 per cent over it. The refusal is fed the claim that it misses half the cells or more, and required to fail.

Tested in An offset that bends twice · the storage growth series · refused by a saturated probe read as unable to name the leaf

“An inexact inner solve turns a defective eigenvalue's square-root sensitivity into the outer iteration's cost: the extra steps grow with √τ.”

On eighteen saddle-point systems with the Hessian block solved to τ from 10⁻¹⁰ to 10⁻², the eigenvalues from the Jordan pairs sit 0.42 to 0.62 times √τ from one wherever they can be computed, and GMRES's residual after two steps grows with a slope of 0.86 to 1.27 in τ on every system, its ratio to √τ rising two hundred to a hundred thousand times across the eight decades. The refusal is fed the claim that the two-step residual follows √τ — its ratio to √τ flat to within a factor of three on every system — and required to fail.

Tested in A square root the residual does not pay · the block preconditioning series · refused by the triangular preconditioner's two-step residual read as following the square root of the inner accuracy

“Once the inner solves are inexact, the triangular preconditioner's advantage over the block-diagonal one is lost, since its eigenvalues are no longer clustered at one.”

At every inner accuracy from 10⁻¹⁴ to 10⁻², GMRES with the triangular preconditioner reaches 10⁻¹⁰ in at most as many steps as MINRES with the block diagonal of the same perturbed Hessian on every one of eighteen systems, and in fewer at the median of every τ. The refusal is fed the claim that MINRES needs no more steps than GMRES at τ = 10⁻² on any system, and required to fail.

Tested in A square root the residual does not pay · the block preconditioning series · refused by the inexact triangular preconditioner read as no better than the block diagonal

“An adaptive range finder that builds its basis from sparse probes can certify it with sparse probes too: a probe that comes back short means the basis has caught the range, whatever kind of probe it was.”

On a 64 × 64 matrix with singular values 0.8ᵏ and its ten leading directions on ten particular columns, thirty draws of the rule with probes nonzero in one entry in twenty certify eight bases whose true spectral error is above the tolerance of 0.1, the worst at ten times it. The refusal is fed the claim that none does, and required to fail.

Tested in The probe that refuses becomes the column · the randomised series · refused by sparse probes read as a safe certificate for a basis built from sparse probes

“Certifying a sparse basis with dense Gaussian probes is enough: the honest certificate keeps the rule running until the basis is right.”

On the same coherent matrix, sparse columns certified by ten Gaussian probes certify no basis falsely, but by the sixty-column cap they have certified only twelve draws of thirty; nine of the eighteen refusals still miss a direction, with errors up to ten times the tolerance. The refusal is fed the claim that all but two draws certify, and required to fail.

Tested in The probe that refuses becomes the column · the randomised series · refused by a dense certificate read as repairing a sparse basis without promoting a probe

“An infinite eigenvalue is a numerical breakdown to be filtered out of the results.”

A descriptor pencil with k algebraic constraints has exactly k infinite eigenvalues, and the exact characteristic polynomial computed in BigInt rationals is exactly k degrees short of n for k = 0, 1, 2 and 3. Each infinite eigenvalue carries a residual in the same scaling as every finite one, at most 5.5·10⁻¹⁴. The refusal is fed non-integer entries, where rounding them would answer exactly about a different pencil and report a degree that belongs to it.

Tested in An eigenvalue with no value · the pencil series · refused by an exact pencil polynomial asked for on entries that are not integers

“A random first vector buys nothing a library needs: the all-ones vector is a good guess for most matrices, so a random start loses accuracy on ordinary problems to gain immunity from a contrived one.”

On 400 random 8 × 8 matrices the single walk from random signs is exact on 86.5 per cent against 84.0 from all ones, for 4.3 products against 4.4; on 200 of size 16, 83.0 against 83.5. Over 1,600 draws the worse worst case belongs to the random start at size 8 (0.324 against 0.377) and to the all-ones start at size 16 (0.301 against 0.348). The refusal is fed the claim that the random start loses five points of exact share at size 8, and required to fail.

Tested in A first vector nobody can build against · the Condition-estimation series · refused by a random first vector read as costing the all-ones guess

“A random start can be fooled with the probability that its signs cancel the hidden column: a column on k rows is missed about as often as k random signs split evenly — half the time when k is two.”

On a size-24 matrix whose inverse hides a column on k rows, for every even k from 2 to 24, the random start is exact on all 100 seeds, where the cancellation model predicts misses on 50 to 16 per cent. At the first dual step the hidden column's pull is at least 1.80 times the strongest other column's at k = 2 and 395 times at k = 24, on every seed. The refusal is fed the claim that a two-row hidden column fools the random start on more than three seeds in ten, and required to fail.

Tested in A first vector nobody can build against · the Condition-estimation series · refused by a random start read as fooled whenever its signs cancel the hidden column

“Along a sequence of slowly changing operators, a two-stage trace estimator can run its pilot once and carry the first trace's spread forward; Student's margin covers the drift until it is large.”

Sixteen operators with eigenvalues decayᵏ, the decay falling from 0.9 by 0.0025 a trace, so one probe's relative spread grows by 1.6 per cent a trace, from 0.264 to 0.332. With the first trace's pilot of eight frozen, the sixteenth estimate is inside a 3% target at one standard error in 52.0 per cent of 400 sequences against the table's 68.3; at 1.96, 89.0 against 95. The refusal is fed the claim that the frozen rule's last trace covers within five points of the table at that drift, and required to fail.

Tested in A spread carried from the trace before · the trace estimation series · refused by a frozen pilot read as covering a slowly drifting sequence

“A recurrence that multiplies its running value by an exact power of two whenever the value nears the bottom or the top of the format removes every range failure, leaving only the precision's own error, even in an eight-bit format.”

In E4M3, rescaling the value to between one and two whenever it falls below the smallest normal times two to the margin, or rises above a quarter of the largest normal, keeps the far well past sixty octaves on 0, 2, 1, 1 and 2 of nine chains at margins of 0, 2, 4, 6 and 8 octaves, where the same significand with no exponent limit keeps it on eight; at a margin of six only 396 of 531 runs reproduce the unbounded exponent. The refusal is fed the claim that the floor-and-ceiling rule fails exactly where the unbounded exponent does, and required to fail.

Tested in Room for the step, not the value · the subnormals series · refused by a rescale on the value read as removing every range failure in a byte

“An LDLᵀ factorisation's most negative pivot, pushed a little further — by a tenth — is a usable shift for inverse iteration towards the smallest eigenvalue.”

On thirty random symmetric 24 × 24 matrices with standard normal entries, Bunch–Kaufman's most negative pivot pushed by 1.1 is nearer the smallest eigenvalue than the second on 13 and the bounded rule's on 9; a push of 1.25 serves 16 and 13, of 1.5 serves 25 and 22, and only 2 serves 29 for both. The refusal is fed the claim that Bunch–Kaufman's pivot pushed a tenth serves at least 27 of the thirty, and required to fail.

Tested in The push a pivot needs belongs to the matrix · the cholesky series · refused by a pivot pushed a tenth read as a reliable shift

“The push a factorisation's most negative pivot needs to become a usable shift varies by about a factor of two from matrix to matrix.”

On the same thirty matrices the push that makes Bunch–Kaufman's pivot nearest the smallest eigenvalue runs from 0.59 to 2.27, a factor of 3.8: its pivot is more negative than every eigenvalue on eight matrices, by up to 1.58 times, and as little as 0.38 of the smallest on another. The bounded rule's push runs from 0.85 to 2.01. The refusal is fed the claim that Bunch–Kaufman's largest needed push is within 2.2 times its smallest, and required to fail.

Tested in The push a pivot needs belongs to the matrix · the cholesky series · refused by the push a pivot needs read as varying by a factor of two

“Among cospectral regular graphs, where the tree count is shared, the critical group separates pairs about as often as it separates Laplacian-cospectral pairs in general — two in five at eight vertices.”

On 705 distinct cospectral pairs of regular graphs made by switching, the critical group separates 181 of 587 pairs of 4-regular graphs on 12 to 16 vertices and none of 118 pairs of 3-regular graphs on 14 to 20 — 26 per cent overall, against 435 of 1,022 Laplacian-cospectral pairs at eight vertices. The refusal is fed the claim that it separates more than 38 per cent of the 705, and required to fail.

Tested in The group hears a switch only at two · the graph invariant series · refused by regular cospectral pairs read as separated by the group as often as the census's

“A switching that preserves both spectra can change any part of the critical group.”

On every one of the 705 pairs, the odd parts of the two groups' invariant factors agree exactly; every separation is in the powers of two, most often three factors of two split as ℤ2 × ℤ2 × ℤ2 against ℤ8 (74 pairs). The refusal is fed the claim that some pair differs in an odd part, and required to fail.

Tested in The group hears a switch only at two · the graph invariant series · refused by a switched pair read as separable away from the prime two

“Scaling the constraint rows of the saddle-point system just past the point where partial pivoting changes its row order makes that route as accurate as eliminating the constraints first.”

On forty-nine constrained least-squares problems, the best of five scales from 1.01 to 10⁵ times the switch is within three times the constraints-first error on forty-five; on the four problems at coupling 10⁻⁶ that are consistent or barely strained it is 30 to 39 times worse, 1.1·10⁻¹⁰ against 3.5·10⁻¹². The refusal is fed the claim that a scale 1.01 times the switch is within three times the constraints-first error on every problem, and required to fail.

Tested in The switch is read before the solve · the constrained least squares series · refused by a scale just past partial pivoting's switch read as matching the constraints-first order

“The switching scale is the ratio of the largest normal-matrix entry to the largest constraint entry in the first column, read before any elimination step.”

At coupling 10⁻⁸ and strain 10⁻⁵ the unreduced first column's ratio is 0.076, and the switch is at 3.198 — the ratio at the second step, after the first step's update, to within 4.4·10⁻¹⁶ on all forty-nine problems. The refusal is fed the claim that the first column's ratio is within a tenth of the switch, and required to fail.

Tested in The switch is read before the solve · the constrained least squares series · refused by the switching scale read off the unreduced first column

“On the unpreconditioned Laplacian, which converges several times more slowly than the preconditioned one, a stagnation check every five steps stops some reduced-precision runs early, above twice the attainable error.”

On Laplacian grids of side 6 to 14 at 24, 16, 11 and 8 significand bits without a preconditioner, the clause that stops when two checks five steps apart fail to halve the double residual stops all twenty runs within 1.10 times the least energy-norm error in the run. The refusal is fed the claim that at least one run stops above twice it, and required to fail.

Tested in Five steps is a rate · the stopping test series · refused by a fixed five-step stagnation check read as failing on the unpreconditioned Laplacian

“A stagnation check that stops reliably on one slowly converging problem is safe on any problem, since slowness is what it was tested against.”

With the same Laplacians' rows and columns scaled by a seeded diagonal spanning a decade and a half, so that the condition number rises from 19–91 to 1.8·10³–8.3·10³, the five-step clause stops nineteen of twenty reduced-precision runs at step 15 and one at step 35, at 8.6 to 775,000 times the least error; half of them are more than 500 times. The refusal is fed the claim that every run stops within twice the least error, and required to fail.

Tested in Five steps is a rate · the stopping test series · refused by a fixed five-step stagnation check read as safe on a slowly converging run

“In a tree of Householder factorisations, appending the right-hand side as an extra column at every leaf inherits Householder's accuracy at each leaf and so makes the tree's least-squares solve more accurate than carrying Qᵀb up from the leaves.”

On a 64 × 16 problem at κ = 10⁸ and relative residual 10⁻², with the ill-conditioning between column blocks, the residual aimed at the tree's weakest direction draws 0.04239 of κ²uρ from a two-leaf tree carrying Qᵀb and 0.04239 from the same tree with b appended; the two aimed directions agree to a cosine of 1 − 4·10⁻¹¹. Four leaves and the spread placement agree to five figures as well. The refusal is fed the claim that the appended tree's constant is under nine tenths of the carried one's, and required to fail.

Tested in A tree leaks along its leaves · the Gram–Schmidt series · refused by b appended at a tree's leaves read as inheriting a better accuracy

“A tree of Householder factorisations is backward stable in the same way as the sequential factorisation, so its least-squares error leaks along the same direction of the residual.”

With the ill-conditioning spread through the columns, the aimed residual of a four-leaf tree lies at a cosine of 0.40 from the sequential Householder route's and of a two-leaf tree at 0.58; between column blocks, at 0.13 and 0.18. The refusal is fed the claim that the four-leaf tree's aimed direction is within a cosine of 0.9 of the sequential one's, and required to fail.

Tested in A tree leaks along its leaves · the Gram–Schmidt series · refused by a tree of factorisations read as leaking where the sequential one does

“In a cubic matrix polynomial with three groups of eigenvalues, the middle group is inverted by neither the leading nor the trailing reduction of the companion pencil, so both lose it and it needs a shift-and-invert at the middle tropical root.”

On five generic order-four cubics at each separation of 2, 4, 6 and 8 decades between groups, the leading reduction's worst backward error on the middle group is 8.5·10⁻¹⁴, 3.3·10⁻¹⁴, 2.0·10⁻¹¹ and 6.4·10⁻¹⁴. The refusal is fed the claim that both reductions leave the middle group above 10⁻⁶ at every separation, and required to fail.

Tested in The middle group needs no reduction of its own · the polynomial scaling series · refused by the middle group of a three-group cubic read as lost by both reductions

“A tropical scaling at each root rescues the group that root predicts, so scaling at the small root and running the leading reduction keeps the small group.”

When the four coefficients share one diagonal form the scaling at the small root keeps the small group under the leading reduction to 3.9·10⁻¹³ at eight decades. Perturbed out of that form by three tenths of their norm, the same scaling leaves it at 3.0·10⁻¹ — worse than not scaling at all, 7.3·10⁻². The refusal is fed the claim that the scaling brings the generic family's small group below 10⁻¹⁰ at eight decades, and required to fail.

Tested in The middle group needs no reduction of its own · the polynomial scaling series · refused by a tropical scaling at each root read as rescuing its own group

“A guard on GCV's local minima told a noise level estimated from held-out samples refuses GCV's interior dips about as often as the same guard told the true noise.”

Over 960 draws at 1.25, 1.5, 2 and 4 samples per unknown holding 45 interior dips, the guard at its 1% threshold told the true noise refuses 3 dips; told the noise estimated from every fourth sample held out of the fit, it refuses none, and its floor sits below every dip's ρ̂ by at least 0.14. The refusal is fed the claim that the held-out guard refuses at least as many dips as the true-noise guard, and required to fail.

Tested in An estimate that shares nothing with the fit · the regularisation series · refused by a held-out guard read as refusing the dips the true noise refuses

“Told a held-out noise estimate, the discrepancy principle loses its catastrophes only where the held-out set has more degrees of freedom than the least-squares residual, m − n.”

At two samples per unknown the held-out set has m/4 samples against the least-squares residual's m/2 degrees of freedom. Told the least-squares estimate there, the discrepancy principle's worst draw is 5,500 times the oracle's error; told the held-out estimate, 7.5. At 1.5 samples per unknown, where m/4 is also smaller than m − n, the held-out worst is 18. The refusal is fed the claim that at two samples per unknown the held-out discrepancy principle still misses by more than a thousand, and required to fail.

Tested in An estimate that shares nothing with the fit · the regularisation series · refused by the held-out estimate read as rescuing the discrepancy principle only where it has more degrees of freedom

“When a symmetric indefinite matrix's small diagonal entries are gathered into one region rather than scattered, a fill-reducing plan fixed from the pattern does worse, because the whole region's ranking is wrong at once and the pivoting test refuses it there.”

On 8 × 8 and 10 × 10 indefinite grids with 5 to 50 per cent of the diagonal set to 10⁻¹², medians over four seeds, the minimum-degree plan fixed once is refused 0, 2, 2, 3 and 2 times on the 8 × 8 grid with the small entries clustered, against 7, 10, 14, 17 and 13 scattered; at three in ten it saves 10.2 per cent of the natural order's factor clustered against 3.0 scattered, and on the 10 × 10 grid 23.3 against 8.9. The refusal is fed the claim that the clustered layout refuses the fixed plan at least as often as the scattered one at every share, and required to fail.

Tested in A small pivot with a small neighbour · the sparse pivoting series · refused by a cluster of small pivots read as refusing the fixed plan more

“Choosing the sparsest pivot at every step, re-reading every degree, is the best ordering a symmetric indefinite factorisation can follow wherever its small pivots sit; a plan computed less often can only approach it.”

With the small entries clustered, the sparsest pivot at every step stores 504 entries on the 8 × 8 grid at three in ten against the redrawn plan's 448, and 512 against 491 at half; on the 10 × 10 grid 898 against 840 and 1,047 against 916. The refusal is fed the claim that the sparsest rule stores no more than the redrawn plan at three in ten and at half on both grids, and required to fail.

Tested in A small pivot with a small neighbour · the sparse pivoting series · refused by the step-by-step sparsest rule read as the best order wherever the small pivots sit

“Eisenstat and Walker's first forcing term, which sets each step's tolerance from how far the linear model's prediction of the residual was wrong, does not solve its last step past what the stopping test needs, so it has no use for the floor that the ratio rule needs.”

On five problems at four forward targets, nineteen certifiable cells, the first forcing choice run without the floor solves its last step to a residual a median of 34 times smaller than the stopping test asks, and up to 19,000 times. The floor removes 17.5 per cent of its inner work in total and still meets every target within a fifth. The refusal is fed the claim that the unfloored first choice's median over-solve is under three, and required to fail.

Tested in Two forcing terms, one floor · the inexact newton series · refused by the first forcing choice read as needing no floor

“The first forcing choice is cheaper than the second because it reads the model's actual error rather than a proxy for it, and so it is the cheaper choice on every problem.”

Floored and stopped by the same forward rule, the first choice spends 0.684 of the second's inner iterations on the problem with λ = 30, 0.891 on the standing problem, 0.988 and 0.995 on two others, and 1.045 on the problem with κ(A) = 10³, where it is the dearer; it takes more Newton steps on 15 of 19 cells. The refusal is fed the claim that the first choice is cheaper on every problem, and required to fail.

Tested in Two forcing terms, one floor · the inexact newton series · refused by the first forcing choice read as cheaper on every problem

“Adding a small isotropic part to a separable kernel broadens the six treads of the symbol-preconditioned spectrum into six clusters, and the conjugate-gradient count stays near eight while those clusters stay under the one-per-cent threshold.”

With ε = 10⁻⁴ of the isotropic kernel at ρ = 0.9, the preconditioned spectrum on a 10 × 10 grid still has six clusters at the one-per-cent rule, its widest tread 7.0·10⁻⁴ wide; conjugate gradients to 10⁻¹⁰ takes 15, 13 and 15 steps on the 6 × 6, 10 × 10 and 16 × 16 grids, against 8 without it. The refusal is fed the claim that the count stays at nine or fewer on all three grids, and required to fail.

Tested in Six steps were six eigenvalues · the toeplitz series · refused by a small isotropic part read as harmless while the clusters stay under one per cent

“As the isotropic fraction grows, the step count under the symbol circulant climbs towards the isotropic kernel's count.”

On the 16 × 16 grid to 10⁻¹⁰, the separable kernel takes 8 steps and the isotropic kernel 35; with ε = 0.3 the mixture takes 37, and on the 10 × 10 grid 29 against the isotropic kernel's 26 at ε = 0.1 and 0.3. The preconditioned condition number at ε = 0.3 is 197, below both kernels' 361 and 250. The refusal is fed the claim that at ε = 0.3 the 16 × 16 count is no more than the isotropic kernel's, and required to fail.

Tested in Six steps were six eigenvalues · the toeplitz series · refused by the mixed kernel's step count read as lying between its two kernels'

“An eigenvalue on the far side of a circle's centre contributes to a contour count's error with its sign flipped, so pairs across the centre cancel differently from pairs along one ray.”

About centre 1 with radius √(0.178 × 0.587), holding the eigenvalue 0.822 behind the centre and leaving 1.587 outside in front at the same ratio 0.551, the count's error at even point counts is under 10⁻¹⁰ from 14 points and falls at the next eigenvalue's rate, 0.192 a point — the two terms cancel as two on one ray do — while at odd counts it falls at 0.551 and stays above 10⁻⁸ through 29. The refusal is fed the claim that the even counts' error is within a factor of ten of the odd counts' on every pair from 10 to 28, and required to fail.

Tested in A centre that chooses which pair cancels · the nonlinear cost series · refused by a far-ray eigenvalue read as flipping sign at every point count

“Moving a contour's centre off the origin buys about another factor of two on a count of real eigenvalues.”

On the delay problem, the cheapest circle about any real centre holding exactly the first two, three and four real eigenvalues costs 22, 21 and 32 points for ten digits against 41, 78 and 341 about the origin — gains of 1.9, 3.7 and 10.7. The refusal is fed the claim that every gain lies between 1.5 and 2.5, and required to fail.

Tested in A centre that chooses which pair cancels · the nonlinear cost series · refused by an off-origin centre read as buying a factor of two on every real count

“The Loewner pencil's next singular value, σₖ₊₁/σ₁, is the error of the reduced model of order k, so a code can choose the order by reading the singular values against a tolerance.”

On four systems sampled at 48 points left of their spectra, the true error of the order-k Loewner model over σₖ₊₁/σ₁ is between 0.73 and 1.53 at the first two orders, and by the fifth order 109, 181 and 457 on the three systems still above the arithmetic floor, reaching 6,680 at the sixth. The refusal is fed the claim that the ratio stays under ten at every order above the floor, and required to fail.

Tested in An error estimate made of samples · the Data-driven realisation series · refused by the Loewner pencil's next singular value read as the error of the reduced model

“A reduced model that reproduces its own samples to a given accuracy reproduces the function between them to about the same accuracy.”

On the same four systems the largest error between the samples is 40.7, 64.2 and 182 times the largest misfit at the samples at order five, and 751 times on the eight-pole system at order six. The refusal is fed the claim that it stays within ten, and required to fail.

Tested in An error estimate made of samples · the Data-driven realisation series · refused by a reduced model's misfit at its own samples read as its error between them

“The power conjugate gradients' filter carries past the answer is set by the decay rate of the singular values, so for a Gaussian blur it can be computed from the blur width and the corner alone.”

At the iterate nearest the answer, on six blurred-signal problems over twelve draws, the iteration's m = θmin·Σ1/θ against the sum θmin·Σ1/σ² over the squared singular values at or above θmin: the closed-form Gaussian decay misses by more than 0.1 on 62 draws of 72 and the computed singular values on 65, by up to 3.2. The refusal is fed the claim that the blur width's sum lands within 0.1 of the iteration's median on every problem, and required to fail.

Tested in A power the spectrum does not set · the iterative regularisation series · refused by the power ramp's m read off the blur width

“After k steps the Ritz values of a Krylov method approximate the k largest eigenvalues, one each.”

At 1.3 of the answer the iteration holds more Ritz values than there are squared singular values above its smallest one on 67 draws of 72 — 139 against 51 on one draw of the narrowest blur, the leading values found up to six times — and fewer at half the answer, 4 against 18. The refusal is fed the claim that the count matches one for one on every problem at 1.3, and required to fail.

Tested in A power the spectrum does not set · the iterative regularisation series · refused by the iteration's Ritz values read as the leading squared singular values past the answer

“Bounding every intermediate by the determinant makes the Hermite form cheaper to compute.”

Counted in schoolbook bit operations, the determinant included, the route modulo the determinant costs 3.8 times plain elimination at n = 4, 2.6 times at 24 and 1.2 times at 64; the two are within three per cent at n = 72 and the modular route is a sixth cheaper at 80. The refusal is fed the claim that the modular route's cost is at most the plain route's at every size, and required to fail.

Tested in Every entry under the determinant · the normal forms series · refused by a bound on the intermediates' size read as a saving in work

“A CP decomposition that satisfies Kruskal's condition is determined by its tensor: a small change to the tensor makes a small change to the factors.”

A 4 × 4 × 4 rank-three tensor whose third factor has two columns at 10⁻³ radians has k-ranks 3, 3 and 3, satisfying the condition with one to spare; perturbed by one part in 10⁸, its exact decomposition's terms move by 200 parts in 10⁸ on the first of three random tensors and by 311 and 422 on the others. The refusal is fed the claim that the move is at most ten times the perturbation, and required to fail.

Tested in Unique, and not determined · the uniqueness series · refused by a decomposition that satisfies Kruskal's condition read as well conditioned

“An alternating least-squares fit whose residual has reached the noise in the tensor has found the decomposition, when the decomposition is unique.”

Started from the exact factors on the perturbed tensor, ALS's relative residual is within 0.6 per cent of the exact decomposition's at every angle on every tensor; at θ = 10⁻³ its terms are 200 to 423 perturbations from the decomposition's, after its own stopping test has fired. The refusal is fed the claim that it lands within one perturbation at every angle, and required to fail.

Tested in Unique, and not determined · the uniqueness series · refused by an ALS fit whose residual reaches the noise read as having found the decomposition

“A probe that compresses only the near-diagonal blocks of one compression at a leaf of 8 names the cheapest leaf at least as often as the whole compression, because the far blocks the whole compression averages in are ones the leaf decision does not depend on.”

On cos(κr)/r over a 512-point grid at κ = 13⅓, 40, 120, 360 and 1,080 and tolerances 10⁻⁴, 10⁻⁸ and 10⁻¹², the mean rank of the 186 admissible 8-wide blocks fed to the halving rule names the cheapest of five leaves on 12 cells, the whole compression at 8 on 13. The refusal is fed the claim that the near probe does at least as well, and required to fail.

Tested in Each halving reads its own width · the storage growth series · refused by a probe of the 8-wide blocks read as naming the leaf as often as the full compression

“The wider blocks of a compression carry nothing the choice of leaf depends on, so dropping them from the probe changes no leaf.”

On the five cells whose cheapest leaf is 32, the 8-wide blocks alone and the whole compression at 8, each fed to the halving rule, name different leaves on two. The refusal is fed the claim that they agree on all five, and required to fail.

Tested in Each halving reads its own width · the storage growth series · refused by the wider blocks read as irrelevant to the leaf decision

“An inner solve stopped at a relative residual of τ delivers an answer accurate to about τ, so the tolerance is the accuracy an outer iteration receives.”

Conjugate gradients on the Hessian blocks of eighteen saddle-point systems, stopped at relative residuals from 10⁻¹² to 10⁻²: on the six blocks with κ(H) = 1 the first step is exact and every solve is correct to rounding at every τ, and over all 198 runs the median residual delivered is at least ten times below τ on 140 and the energy error on 163. The refusal is fed the claim that every run delivers more than a tenth of its tolerance, and required to fail.

Tested in A tolerance is not an accuracy · the block preconditioning series · refused by an inner tolerance read as the accuracy the inner solve delivers

“An inner solve accurate to τ costs an outer Krylov iteration what a fixed perturbation of the same size would, so a perturbed-operator model predicts the inexact iteration's convergence.”

Flexible GMRES's residual after two steps, divided by the fixed model's constant times the energy error the inner solves delivered, has a median of 12 on the systems with κ(A) = 10 and 99 with κ(A) = 10⁴, and exceeds three on 42 runs of 66 even on the better-conditioned constraints — though the slope in the error stays near one. The refusal is fed the claim that the ratio is at most three on every run, and required to fail.

Tested in A tolerance is not an accuracy · the block preconditioning series · refused by a real inner solve read as the fixed perturbation of the same accuracy

“Multiplying a carried spread by the ratio of the last two carried spreads costs nothing in coverage on a sequence that does not drift, but inflates its probe counts by the noise in that ratio.”

Along sixteen identical 60 × 60 operators over 400 sequences, the spread the trend carry uses scatters with an interquartile range of 0.271 of its median against the plain carry's 0.105, and it spends 1,225 products for the sixteen traces against the plain carry's 1,246, covering 68.1 per cent against 68.0. The refusal is fed the claim that the trend carry spends more than two per cent more products there, and required to fail.

Tested in A drift put back at the table's price · the trace estimation series · refused by the trend factor's noise read as inflating the no-drift sequence's products

“Extrapolating the drift restores the carried spread's coverage while keeping what carrying saves over a fresh pilot every trace.”

With the decay falling 0.02 a trace, the trend carry covers 68.9 per cent averaged over the third to sixteenth traces and spends 4,412 products for the sixteen; the plain carry covers 65.6 for 3,994, and a fresh pilot every trace 66.5 for 4,377. The refusal is fed the claim that the trend carry spends less than a fresh pilot at that drift, and required to fail.

Tested in A drift put back at the table's price · the trace estimation series · refused by a trend carry read as keeping the carried rule's saving at the fastest drift

“A Sylvester solver whose relative residual is at the rounding level has a small backward error for the equation: the computed X exactly solves AX + XB = C with A, B and C perturbed at the rounding level.”

On equations of order 8 with κ(X) = 10⁸ and C = 10⁻⁶X, Bartels and Stewart's solution has relative residual 1.8·10⁻¹⁶ and structured backward error 3.9·10⁻¹¹, a ratio of 2.1·10⁵; with X well conditioned the ratio is 4. The refusal is fed the claim that the ratio stays under ten on every equation, and required to fail.

Tested in A backward error the answer does not feel · the matrix equation series · refused by a Sylvester residual at the rounding level read as a small backward error for the equation

“Of two routes to the same equation, the one with the smaller residual has the smaller backward error.”

The Kronecker system's relative residual is about four times smaller than Bartels and Stewart's on every one of the twenty equations, and its structured backward error is larger wherever κ(X) ≥ 10⁶ and C ≤ 10⁻⁴X — 1.0·10⁻¹⁰ against 3.9·10⁻¹¹ at the extreme. The refusal is fed the claim that the smaller residual always comes with the smaller backward error, and required to fail.

Tested in A backward error the answer does not feel · the matrix equation series · refused by the route with the smaller residual read as the one with the smaller backward error

“A polynomial expressed in Chebyshev coefficients on an interval containing its roots, with the colleague matrix, has root conditioning that does not degrade exponentially with degree.”

Wilkinson's polynomial expanded exactly in the Chebyshev basis on [1, n], rounded once and solved by the colleague matrix: worst relative root error 1.3·10⁻¹⁵ at degree 6, 4.4·10⁻¹³ at 20, 6.8·10⁻⁸ at 32 and 2.8·10⁻⁴ at 40, a third of a decade a degree from 12 to 32. The refusal is fed the claim that the error at degree 40 is under 10⁻¹⁰, and required to fail.

Tested in A basis that has to know where the roots are · the conditioning series · refused by the colleague matrix's root accuracy read as not degrading exponentially with degree

“Because the power iteration on a nonnegative matrix never subtracts, it computes every entry of the Perron vector to high relative accuracy — the one property that decides which method to use.”

On a 24-state chain of two clusters joined by a coupling ε, the renormalised power iteration from the uniform vector reaches a relative 10⁻¹² on every entry after 680, 6,900 and 69,370 steps at ε = 10⁻², 10⁻³ and 10⁻⁴ — about 7/ε — and then stalls at 0.05 to 1.5 times u/(4ε) from the answer; from ε = 10⁻⁶ it is more than ten per cent wrong after 200,000 steps. The refusal is fed the claim that it reaches 10⁻¹² within a thousand steps at every coupling from 10⁻⁴ up, and required to fail.

Tested in No subtraction, and no waiting · the perron frobenius series · refused by the power iteration read as converging in steps independent of how slowly the chain mixes

“The digits lost in a subtraction are about log₁₀ of the operands' size over the result's, so the one-pass variance loses log₁₀ of the mean square over the variance and no more.”

On 1,000 Gaussian values at every κ² from 10⁴ to 3·10¹⁴ in double precision, and at 16, 24 and 32 bits, the one-pass formula with running sums loses between 2 and 60 times κ²u, median about 0.4√n — so the rule under-predicts by the square root of the number of values. The refusal is fed the claim that the error stays within κ²u at every κ² from 10⁴, and required to fail.

Tested in The formula sets the power, the sum sets the constant · the cancellation series · refused by the running-sum one-pass variance read as meeting the digits-lost rule

“The one-pass variance is repaired by summing more accurately — Kahan's compensated summation or a pairwise tree.”

Pairwise and compensated sums bring the one-pass error to within three times κ²u at every precision and length measured, and no lower: at κ² = 10¹² the compensated formula is still 4.7·10¹¹ u from the exact variance, where two passes are 9 u. The refusal is fed the claim that compensated sums hold the error under 100 u at every κ² from 10⁸, and required to fail.

Tested in The formula sets the power, the sum sets the constant · the cancellation series · refused by a better summation read as removing the square

“The floor the double residual of a reduced-precision conjugate-gradient run reaches is predictable from the working precision's unit roundoff and the grid to within a factor of four.”

On sixty runs — grids of 6 to 14 points a side, 24, 16, 11 and 8 bits, three problems — the floor over u spreads by a factor of 18.1, which no constant covers within four, and at 16 bits it rises and falls across the grids rather than following them: on the preconditioned Laplacian 2.3, 7.7, 5.8, 2.5 and 1.5. u·n needs a factor of 6.7 either side and u·κ(A) 23.9. The refusal is fed the claim that one constant times u fits within four, and required to fail.

Tested in A floor the error has not reached · the stopping test series · refused by a residual floor read as predictable from the precision and the grid alone

“A run whose residual has come within a small factor of its predicted floor has reached the error its precision allows, so stopping there keeps a stagnation clause's safety without its overrun.”

Stopping at the first check where the double residual is within four times u·‖|A||x|‖/‖b‖ stops some run on each of three problems at 5.03, 3.46 and 5.01 times the least error the run reaches, against at most 1.37 for the rate-set clause. The refusal is fed the claim that it stays within two fifths of the least, and required to fail.

Tested in A floor the error has not reached · the stopping test series · refused by a residual near its predicted floor read as a signal to stop

“Small pivots along a line, each with two small neighbours, refuse a fixed fill-reducing plan a number of times between a disc's and a scatter's at every share.”

On 8 × 8 and 10 × 10 grids at 5, 10, 20, 30 and 50 per cent small, median over four seeds, the fixed plan is refused along lines between the disc's count and the scatter's on seven cells of ten; on the 10 × 10 grid it is refused less than in the disc at 5 and 20 per cent (2 against 4, 5 against 6) and more than scattered at 50 (20 against 19). The refusal is fed the claim that the line lies between them on every cell, and required to fail.

Tested in A refusal has no fixed price · the sparse pivoting series · refused by small pivots along a line read as refused between a disc and a scatter at every share

“The number of times the pivoting test refuses a fixed plan measures what a layout of small pivots costs it: the layout refused most adds the most fill.”

At 20, 30 and 50 per cent small on both grids the three layouts add within a factor of 1.6 of the same number of entries to the fixed plan's factor, while their refusal counts differ by more than three times; on the 8 × 8 grid at 30 per cent the scatter, refused 17 times, adds 122 entries and the disc, refused 3 times, adds 146. The refusal is fed the claim that the order of the layouts by refusals is their order by added entries on every such cell, and required to fail.

Tested in A refusal has no fixed price · the sparse pivoting series · refused by a refusal count read as the fill a layout costs the fixed plan

“A spanning tree that balances short loops against the cycle property — reading the resistances only coarsely — buys the least-resistance tree's conditioning at a fraction of its density.”

On a 10 × 10 grid at resistance spreads of two, four and six decades, five draws each, Kruskal on resistances rounded down into bands of 0.25 to 3 decades never keeps the median conditioning within three times the least tree's while paying under half of its extra factor work over the breadth-first tree. The refusal is fed the claim that some band width does, and required to fail.

Tested in Long loops pay before the factor starts · the null space basis series · refused by a tree reading the resistances coarsely read as buying the conditioning at a fraction of the fill

“The loop formulation of a network flow is the cheaper one to factor, since it has fewer unknowns than the node formulation once the grid has many loops.”

On a 12 × 12 grid with resistances over four decades, the least tree's loop equations — 121 unknowns against the node equations' 143 — need a median 3.8 times the node equations' factor work under minimum-degree ordering, across five draws from 3.21 to 4.86. The refusal is fed the claim that they need no more, and required to fail.

Tested in Long loops pay before the factor starts · the null space basis series · refused by the loop equations' factor read as no denser than the node equations'

“A structure-preserving method is the more accurate one: computing a hyperbolic quadratic's spectrum through a definite linearisation, which keeps every eigenvalue real, beats the unstructured companion route near the boundary of the class.”

On damped chains of 4, 8 and 12 masses at damping β*(1 + 10⁻¹²), the definite route's worst relative error against the closed-form spectrum is 5.64·10⁻⁴, 2.00·10⁻³ and 1.14·10⁻³, and the companion route's is 9.0·10⁻¹⁰, 2.69·10⁻⁹ and 2.45·10⁻⁹. The refusal is fed the claim that the definite route is at least as accurate on all three, and required to fail.

Tested in A guarantee paid for in digits · the hyperbolic quadratic series · refused by a definite linearisation read as more accurate than the unstructured route near the boundary

“An unstructured eigensolver applied to a hyperbolic quadratic near the boundary of its class returns complex eigenvalue pairs, which is the reason to use a structured one.”

On twenty eight-mass chains rotated by random orthogonal congruences at margins of 10⁻⁴, 10⁻⁸ and 10⁻¹², the companion route returns no complex value on any run, reading every 2 × 2 Schur block for what it is; at a margin of exactly zero it returns a complex pair on five of twenty. The refusal is fed the claim that it returns one inside the class, and required to fail.

Tested in A guarantee paid for in digits · the hyperbolic quadratic series · refused by an unstructured solver read as returning complex eigenvalues for a hyperbolic quadratic near its boundary

“A sliding window's correction can be read from the rows most likely to carry the error — the quarter of highest leverage — and recover most of what a correction against every row recovers.”

On a collinear stream at κ(A) = 3·10⁶ and noise one, a window of 24 rows corrected each step against its 6 rows of highest leverage ends with a median relative error of 2.80 against exact rational coefficients; corrected against all 24 it ends at 3.30·10⁻⁹. The refusal is fed the claim that the leverage quarter comes within a factor of a hundred of the full correction, and required to fail.

Tested in A correction that reads every row · the sequence stability series · refused by a correction against the rows of highest leverage read as recovering the full correction

“Leaving the least important row out of a correction loses a little accuracy; at worst the correction does nothing.”

On the same stream a correction that reads 23 of the 24 rows, leaving out the one of least leverage, ends with a median relative error of 0.286; with no correction at all the window ends at 9.39·10⁻⁴. The refusal is fed the claim that the 23-row correction is at least as good as none, and required to fail.

Tested in A correction that reads every row · the sequence stability series · refused by a correction missing one row read as no worse than none

“GCV's guard failed for want of an accurate noise estimate: given one built from all the samples by four-fold cross-validation, it refuses the dips about as often as when told the true noise.”

Over 960 draws holding 45 interior dips, the guard told the four-fold estimate refuses no dip at the 1% threshold, where told the true noise it refuses three; at 10% it refuses 44 dips and 520 real minima against the true noise's 14 and 78. The refusal is fed the claim that it refuses at least as many dips at 1% as the true-noise guard, and required to fail.

Tested in Four folds feed one rule · the regularisation series · refused by a four-fold noise estimate read as rescuing the guard

“The discrepancy principle needs the noise level itself; any estimate made from the data costs it a large factor somewhere.”

Told the four-fold estimate, the discrepancy principle's worst draw over the oracle is 1.194, 1.193, 1.184 and 1.193 at 1.25, 1.5, 2 and 4 samples per unknown, against 1.163, 1.162, 1.155 and 1.164 told the true noise. The refusal is fed the claim that it misses by more than four at every ratio, and required to fail.

Tested in Four folds feed one rule · the regularisation series · refused by the discrepancy principle read as needing the true noise to stay near the oracle

“A pivot rule that looks two eliminations ahead avoids the exponential growth Wilkinson's construction forces on the greedy and one-step rules.”

On W ⊗ I₂ — Wilkinson's matrix with every entry replaced by that entry times the two-by-two identity, ties broken by 10⁻¹² — the rule looking two eliminations ahead attains the greedy rule's growth 2^(n/2 − 1) to a millionth at every n from 6 to 14: 4, 8, 16, 32 and 64. The refusal is fed the claim that its growth is under 2.01 on every member of the family, and required to fail.

Tested in A doubling that arrives two steps late · the elimination series · refused by the two-step look-ahead read as escaping a construction that defers by two

“A pivot rule that looks a fixed number of eliminations ahead bounds the growth factor below exponential.”

On W ⊗ I₃ the rules looking one, two and three steps ahead all attain 2^(n/3 − 1) — 4, 8 and 16 at n = 9, 12 and 15 — and only the rule looking four ahead gives 2; in general d interleaved copies hold the rule looking d ahead to 2^(n/d − 1). The refusal is fed the claim that the three-step rule gives growth under 2.01 on W ⊗ I₃, and required to fail.

Tested in A doubling that arrives two steps late · the elimination series · refused by a fixed look-ahead depth read as bounding the growth below exponential

“Total least squares, like ordinary least squares, gives an answer that belongs to the data: written in other units, the columns produce the same fitted relationship.”

On forty 60 × 2 fits with equal noise in every column, rewriting the first column in units from 10⁻⁴ to 10⁴ times the recorded ones moves total least squares' median error, read in the recorded units, from 0.1139 to 0.1716, while ordinary least squares stays at 0.3171 to nine digits. The refusal is fed the claim that the error moves by under one per cent, and required to fail.

Tested in A unit is a statement about the noise · the total least squares series · refused by a total least-squares answer read as independent of the units of the columns

“Written in units far enough from the right ones, total least squares can become arbitrarily wrong, worse than ignoring the noise in the matrix altogether.”

On the same fits, with the noise the same in every column, the median error of total least squares at every unit factor from 10⁻⁴ to 10⁴ lies between 0.1139 and 0.1716, and ordinary least squares is at 0.3171. The two ends are the reverse regression, 0.1607, and the fit with that column exact, 0.1716. The refusal is fed the claim that some unit makes it worse than least squares, and required to fail.

Tested in A unit is a statement about the noise · the total least squares series · refused by a wrong unit read as unbounded damage to total least squares

“Without loss of orthogonality the Ritz values of a Krylov run converge one to each extreme eigenvalue, so a reorthogonalised run past its answer holds one Ritz value for each squared singular value above its smallest.”

On six deconvolutions over twelve draws at 1.3 of the answer, with every basis vector reorthogonalised, the count of Ritz values is within two of the squared singular values above the smallest on 49 draws of 72 and more than two short on 23 — nine short on one draw of the widest blur — and never more than one over. The refusal is fed the claim that it is within two on every draw, and required to fail.

Tested in The copies cost steps, not the filter · the iterative regularisation series · refused by the reorthogonalised Ritz values read as one per singular value above the corner

“Plain elimination to the Smith normal form suffers the same coefficient swell as the Hermite form, so the Smith form of anything beyond a small integer matrix needs a modular or Hermite-first algorithm.”

On random integer matrices with entries to ±9 at sizes 4 to 64 and to ±999 at sizes 4 to 32, plain two-sided elimination with the smallest entry as pivot never writes an entry wider than the determinant by more than one bit, where plain Hermite elimination on the same matrices reaches fifteen times the determinant's width by n = 48. The refusal is fed the claim that the plain Smith routine exceeds the determinant's width by more than eight bits on every matrix, and required to fail.

Tested in The swell was one-sided · the normal forms series · refused by the plain Smith elimination read as swelling past the determinant

“Working modulo the determinant makes a normal-form computation cheaper as well as narrower, because bounded intermediates mean cheaper arithmetic.”

Counted in schoolbook bit operations with the determinant's cost included, the modular Smith route costs 25 to 69 times as much as plain two-sided elimination on random matrices with entries to ±9 from n = 8 to 64, and 5.6 to 14 times on structured lattices where the plain route does write entries wider than the determinant. The refusal is fed the claim that the modular route is the cheaper on every random matrix from n = 8, and required to fail.

Tested in The swell was one-sided · the normal forms series · refused by the Smith form modulo the determinant read as cheaper than plain elimination on random matrices

“On scattered points, admissible blocks of one width and one offset from the diagonal have ranks within one of each other, so one class still behaves as one matrix.”

On two draws of 512 uniformly random points, cos(κr)/r at κ = 13⅓, 40, 120, 360 and 1,080 and tolerances 10⁻⁴, 10⁻⁸ and 10⁻¹², the ranks inside one 16-wide class of the leaf-4 partition spread by more than one on 18 of 30 cells and by up to four; 8-wide classes by up to three and 32-wide by up to four. The refusal is fed the claim that every 16-wide class stays within one rank, and required to fail.

Tested in A class is no longer one matrix · the storage growth series · refused by the 16-wide classes on scattered points read as spreading by at most one rank

“A probe that samples a few blocks per class fails first at the highest frequency, where a wavelength approaches the gaps between scattered points.”

At κ = 1,080 a wavelength is 0.0058, three times the mean gap of 0.0019 and shorter than the widest gaps of 0.014 and 0.018; the three-block probe names the cheapest leaf at all three tolerances on both draws, as it does on every other cell. The refusal is fed the claim that it misses at least one cell at κ = 1,080, and required to fail.

Tested in A class is no longer one matrix · the storage growth series · refused by the three-block probe read as failing first at the highest frequency

“Checking a reduced model at points between its samples — the midpoints, or points the model chooses for itself where the next order disagrees with it — measures its error, so a model whose checks are small is accurate between the samples.”

With relative noise of 10⁻⁸ and 10⁻⁶ on the samples of two systems, 17 of the 32 models of orders one to eight have a real pole inside the sampled interval, where the error is infinite. One ten-millionth of the pole's distance from the origin away from it, the error is between 1.0·10⁻³ and 0.66, and 133 to 48,600 times the largest error at the 47 midpoints and the four chosen points. The refusal is fed the claim that the error there stays within ten times what the checks report, and required to fail.

Tested in A pole between the samples · the Data-driven realisation series · refused by a finite set of check points read as the error of a model with a pole between the samples

“The disagreement between a Loewner model and the model one order higher, built from the same samples, is the error of the lower one, so a code can estimate the error without sampling again.”

With exact samples the disagreement is the error within three per cent at every order whose successor is twenty times better. With noise of 10⁻⁶ it overstates the error of the twelve-mode system's order-3 model by a factor of 579, because the order-4 model is the one with the pole, and on a model with a pole between the samples it reports a finite number for an infinite error. The refusal is fed the claim that it stays within a factor of two at every noisy order, and required to fail.

Tested in A pole between the samples · the Data-driven realisation series · refused by the disagreement with the next order read as the error once the samples are noisy

“One circle about the best real centre is the cheapest way to count a set of eigenvalues closed under conjugation.”

For the delay problem's four real eigenvalues and its first one to six bands of complex clusters, one circle about the best real centre costs 52, 97, 150, 192, 234 and 275 points for ten digits. The real eigenvalues on one circle and each upper cluster on its own, the lower clusters read from the upper by conjugation and the ten digits shared among the pieces, cost 55, 68, 83, 92, 104 and 115. The refusal is fed the claim that one circle is never dearer, and required to fail.

Tested in Count the upper half, read the lower · the nonlinear cost series · refused by one real-centred circle read as the cheapest count of a conjugate-closed set

“A circle about a complex centre has to be paid for twice when the set it counts is closed under conjugation: once about the centre and once about its mirror.”

For the delay problem, whose coefficients are real, the quadrature about each upper cluster's mirrored centre is the conjugate of the quadrature about the centre to 1.6·10⁻¹⁵ on all six clusters, each at its own point count. The refusal is fed the claim that the two differ by more than 10⁻⁶, and required to fail.

Tested in Count the upper half, read the lower · the nonlinear cost series · refused by the conjugate circle read as costing its own evaluations

“Alternating least squares started at a good point on a tensor whose decomposition Kruskal's condition makes unique converges within a few thousand sweeps, so a cap of three thousand is generous.”

On three 4 × 4 × 4 rank-three tensors whose third factor has two columns at θ = 0.01, perturbed by one part in 10⁸ and started from the exact factors, ALS needs 35,720, 73,020 and 40,640 sweeps to come within one perturbation of the moved decomposition. The refusal is fed the claim that three thousand is enough on every tensor, and required to fail.

Tested in The iteration pays the square · the uniqueness series · refused by a few thousand sweeps read as enough for a unique decomposition

“When alternating least squares' residual has stopped changing, the factors have reached the decomposition.”

At θ = 0.01 the relative residual stops changing to the last bit at sweeps 14,227, 26,859 and 18,446 on the three tensors, under half the sweeps needed to arrive, with the factors 3.3, 9.3 and 6.2 perturbations from the decomposition. The refusal is fed the claim that the factors are within one perturbation when the residual stops, and required to fail.

Tested in The iteration pays the square · the uniqueness series · refused by an ALS run whose residual has stopped changing read as having reached the decomposition

“A singular value computed to full machine precision is accurate; a routine that returns 10⁻³⁰ has resolved a quantity of size 10⁻³⁰.”

On an 8×8 bidiagonal graded over 29.5 decades, one-sided Jacobi returns every singular value to 4.4·10⁻¹⁶ relative against an exact Sturm bisection in BigInt rationals, while the eigenvalues of BᵀB return σₘᵢₙ = 2.1·10⁻³⁰ as zero — a relative error of exactly one. The refusal is fed a family asked for at less than a bit of grading per row, where the matrix is not graded and the whole comparison is about nothing.

Tested in Small compared to what · the relative accuracy series · refused by a graded matrix asked for at less than a bit of grading per row

“A 31-bit linear congruential generator can be written in double-precision arithmetic as s = (a·s + c) mod 2³¹, since every state and the multiplier fit in a double.”

With a = 1103515245 and c = 12345 the product a·s reaches 2⁶¹, where a double's spacing is 256, so the product and the constant added to it are rounded before the modulus. From seed 1 the recipe and the exact recurrence part at the second state; of a thousand rounded states 718 are multiples of 256 and only fourteen distinct low bytes occur, where the exact recurrence visits all 256. The refusal is fed the claim that the two sequences agree for a thousand states, and required to fail.

Tested in The generator that rounded its own product · the cancellation series · refused by the double-precision recipe read as the congruential recurrence

“Rook pivoting costs at most about three times partial pivoting's search, since it scans one column and one row a step.”

On Wilkinson's matrix interleaved one to four times at orders four to ten, sixteen matrices from n = 4 to 40, rook pivoting compares between 3.16 and 3.31 times as many entries as partial pivoting: a search that moves at all ends on two scans that find nothing larger, so three scans a step is its floor and a move costs a fourth. The refusal is fed the claim that it stays within three times on every matrix, and required to fail.

Tested in The doubling is in the row already · the elimination series · refused by rook pivoting's search read as at most three times partial pivoting's

“Balancing a companion-type matrix before its eigenvalues are computed repairs the exponential loss of accuracy in polynomial roots.”

Balanced, the colleague route's worst root error still grows from 9.1·10⁻¹⁴ at degree 20 to 7.9·10⁻⁸ at degree 40, with the roots' conditioning, and the balanced monomial companion matrix still loses 1.7·10⁻² at degree 20 after its norm falls from 2.3·10¹⁹ to 380. The refusal is fed the claim that the balanced colleague route keeps every root to 10⁻¹² at degree 40, and required to fail.

Tested in The second factor was the last row · the conditioning series · refused by balancing the colleague matrix read as removing the exponential loss

“Aggregation–disaggregation accelerates the power iteration on a nearly decomposable chain by a constant factor, so its step count still grows like the inverse of the coupling.”

With one power step as its smoother, aggregation–disaggregation reaches every stationary entry to a relative 10⁻¹² in twelve steps at ε = 10⁻¹ and fourteen at every coupling from 10⁻² to 10⁻¹², where the power iteration needs 60, 680, 6,900 and 69,000 steps from 10⁻¹ to 10⁻⁴ and does not get there within 200,000 below. The refusal is fed the claim that the count at 10⁻¹² is more than ten times the count at 10⁻¹, and required to fail.

Tested in The stall was in the block · the perron frobenius series · refused by aggregation's step count read as growing with the inverse coupling

“Total least squares can estimate its own column weights: fit, take each column's root-mean-square correction as its noise level, reweight, and repeat until the weights settle.”

On forty sixty-row fits at each of three placements of the noise, started from five unit factors from 10⁻² to 10², not one of six hundred runs converges: every one leaves a millionfold range of weights within two to seven steps, ending at the reverse regression from every start with c ≤ 1 and at ordinary least squares from every start with c ≥ 10. The refusal is fed the claim that more than half the runs from each start converge, and required to fail.

Tested in The weights the fit chooses for itself · the total least squares series · refused by reweighting total least squares by its own corrections read as converging

“The weights at which total least squares' corrections are consistent with themselves are the columns' noise levels.”

The self-consistent weights, found by running the reweighting backwards, have medians of 0.48 to 0.53 on the first column and 0.71 to 0.78 on b relative to the second column, on placements whose true ratios are 1 and 1, 4 and 2, and 1 and 12. The refusal is fed the claim that the self-consistent weight on b is within a factor of two of the true ratio on every placement, and required to fail.

Tested in The weights the fit chooses for itself · the total least squares series · refused by the self-consistent weights read as the noise levels

“A spanning tree built by Kruskal on resistances, refusing any arc whose fundamental loop would exceed a length budget, is a middle tree: it keeps the least-resistance tree's conditioning while shedding the work its long loops cost a direct solver.”

On five draws of a 10 by 10 network at four decades, a budget of 24 arcs keeps the median conditioning of the loop equations at 1.04 times the least tree's but pays all of its extra factor work over the breadth-first tree, a median share of 1.00; no budget from 8 to 32 pays less than 1.00 at the median, and budgets of 12 and 16 pay 1.55 and 1.31. The refusal is fed the claim that the budget of 24 keeps the conditioning within twice and pays at most two thirds of the work, and required to fail.

Tested in The budget cannot finish the tree · the null space basis series · refused by a loop-length budget read as a middle tree that saves the long loops' work

“Below the margin at which an unstructured solver's rounding can split a hyperbolic quadratic's colliding pair, it returns that pair complex: a solver that does not know the class cannot keep the spectrum real once the class's margin is below its noise.”

On a hundred eight-mass chains rotated by random orthogonal congruences, the companion route returns the colliding pair complex on 45 to 47 of them at every margin from 10⁻¹⁵ down to the boundary itself, never more, because its error in the pair's squared half-separation takes either sign with about equal odds. The refusal is fed the claim that more than ninety runs in a hundred return a complex pair below 10⁻¹⁵, and required to fail.

Tested in Real or complex by the sign of an error · the hyperbolic quadratic series · refused by an unstructured solver read as returning complex pairs whenever the margin is below its noise

“An unstructured eigensolver applied to a hyperbolic quadratic close to its boundary returns complex eigenvalues at margins well above the unit roundoff, since the colliding pair's sensitivity is a square root.”

No run of a hundred returns a complex pair at any margin from 10⁻¹² to 2·10⁻¹⁴; the first appear at 10⁻¹⁴, on three runs, where the exact squared half-separation 2.3·10⁻¹⁵ first falls within reach of the computation's error, whose median size is 7 to 10·10⁻¹⁶ at every margin. The refusal is fed the claim that complex pairs appear at 10⁻¹³ or above, and required to fail.

Tested in Real or complex by the sign of an error · the hyperbolic quadratic series · refused by the companion route read as returning complex pairs at a margin of 10⁻¹³

“An adaptive range finder that builds its basis from sparse probes needs dense probes promoted into the basis whenever the matrix is noticeably coherent, since sparse probes miss directions concentrated on a few columns.”

At mixture angles of 0.05, 0.1 and 0.2, where the leading subspace's coherence is still 6.39, 6.35 and 6.20 against 6.40 at the exactly coherent end, a sparse basis certified by ten dense probes without any promotion certifies all thirty draws with its error under the tolerance; the unpromoted rule refuses — nineteen draws of thirty — only at the exactly coherent matrix. The refusal is fed the claim that it refuses somewhere at each of the three angles, and required to fail.

Tested in Promotion buys rank, not safety · the randomised series · refused by a sparse basis read as needing promoted probes to certify on a nearly coherent matrix

“Promotions in a sparse-probe range finder track coherence: once a matrix is mixed away from its coherent extreme the dense probes stop being promoted.”

At the threshold ρ = 8 the median draw promotes ten dense probes at a mixture angle of 0.05 and nine at 0.1, exactly as at the coherent end, though from 0.05 none is needed for a correct certificate; promotions fall below five only past 0.2 and to none only at 0.8. The refusal is fed the claim that fewer than five are promoted at 0.1, and required to fail.

Tested in Promotion buys rank, not safety · the randomised series · refused by promotions read as vanishing once the matrix is mixed away from exact coherence

“The upward bias of a k-fold cross-validation noise estimate comes from each fit training on (k−1)/k of the samples, so leave-one-out, which trains on all but one, removes most of it.”

Over 960 draws at 1.25, 1.5, 2 and 4 samples per unknown and noise 10⁻² to 10⁻⁴, leave-one-out's median over the true noise is within four per cent of four folds' on every cell and above 1.1 on ten of twelve, reaching 1.52. The refusal is fed the claim that it is under 1.1 on every cell, and required to fail.

Tested in The excess is what the fit spent · the regularisation series · refused by leave-one-out read as removing the folds' bias

“A noise estimate handed to the discrepancy principle must never be below the true noise, since a level too low costs the rule everything.”

With the fit's degrees of freedom taken out, the leave-one-out estimate is below the true noise on 344 of 960 draws and as low as 0.645 of it, and the discrepancy principle told it misses the oracle by at most 1.163, 1.162, 1.160 and 1.164 at the four ratios — against 1.163, 1.162, 1.155 and 1.164 told the true noise. The refusal is fed the claim that it misses by a factor of two somewhere, and required to fail.

Tested in The excess is what the fit spent · the regularisation series · refused by an estimate that is sometimes low read as unusable by the discrepancy principle

“A sliding window's correction can form its residual in single precision where the rows are mildly conditioned: at κ(A) = 3·10⁴ it ends within a factor of ten of a correction with a double-precision residual.”

At κ(A) about 2·10⁴, every step corrected against all 24 rows, the window's median relative error against exact rational coefficients is 3.1·10⁻⁴, 2.6·10⁻⁴, 6.1·10⁻⁵, 8.4·10⁻⁵ and 7.7·10⁻⁵ at noise 1 to none with a single-precision residual, against 2.2·10⁻¹² to 1.4·10⁻¹³ with a double one. The refusal is fed the claim that the two are within ten at every noise level, and required to fail.

Tested in The sum can be single, the residual cannot · the sequence stability series · refused by a single-precision residual read as costing little where the problem is mildly conditioned

“A correction's residual rounding is amplified by the inverse of the Gram matrix and so grows with the square of the rows' condition number.”

At noise 1 the window's median error with a single-precision residual is 3.7·10⁻², 3.6·10⁻³, 3.1·10⁻⁴ and 3.4·10⁻⁵ at κ(A) about 2·10⁶, 2·10⁵, 2·10⁴ and 2·10³: a factor of 9 to 12 for each decade of κ, and a fraction 0.04 to 0.5 of 2⁻²⁴κ(A) on every noisy stream. The refusal is fed the claim that it falls by fifty or more a decade, and required to fail.

Tested in The sum can be single, the residual cannot · the sequence stability series · refused by the residual's rounding read as amplified by the square of the conditioning

“The natural order's large factor on small pivots along grid rows comes from meeting the lines along its own direction, so lines turned to run along columns leave the fixed plan within 3 per cent of its 907 entries and bring the natural order below 1,000.”

On the 10 × 10 grid at three tenths small, median of four seeds, lines along columns give the natural order 1,008 entries and the fixed plan 977 — 7.7 per cent above the row lines' 907 — for a saving of 3.1 per cent against the scatter's 8.9. The refusal is fed the claim that the natural order falls below 1,000 and the plan stays within 3 per cent, and required to fail.

Tested in The baseline was the interchanges · the sparse pivoting series · refused by lines across the grain read as removing the natural order's penalty and leaving the plan alone

“The natural order is a fixed yardstick: the factor it stores is the factor of eliminating the grid in index order, whatever pivots the test interchanges along the way.”

Offered by position, the natural order stores more than when kept to the grid's index on every layout at most shares; on row lines by 70, 59, 125 and 191 entries at 10, 20, 30 and 50 per cent small on the 10 × 10 grid, the most of the four layouts at every share from a tenth on both grids. The refusal is fed the claim that the two agree within 2 per cent everywhere, and required to fail.

Tested in The baseline was the interchanges · the sparse pivoting series · refused by the natural order read as a fixed baseline for a saving

“Where a Sylvester solve's structured backward error rises and its forward error stays put, the structured condition number for perturbations of A, B and C falls by the same factor, so that their product still matches the error.”

At C = 10⁻⁶X the structured condition number, computed from the exact X, is 7.1·10⁶ at κ(X) = 1 and 2.0·10⁷ at κ(X) = 10⁸, a rise of 2.8 that is the same at every size of C, while the backward error rises by 26,000. The refusal is fed the claim that it falls by ten thousand, and required to fail.

Tested in The residual knew the error · the matrix equation series · refused by the structured condition number read as falling where the backward error rises

“A structured backward error times its structured condition number estimates the forward error of a Sylvester solve, as a backward error times a condition number does for a linear system.”

Over 120 solves the product is between 1.1 and 3.2·10⁶ times the forward error for Bartels and Stewart and up to 9.3·10⁶ for the Kronecker system, growing with κ(X) as C shrinks; the relative residual times the unstructured condition number is between 0.96 and 9.7 times it on every solve. The refusal is fed the claim that the product is within ten of the error on every solve, and required to fail.

Tested in The residual knew the error · the matrix equation series · refused by the structured backward error times its condition number read as the forward error

“Conjugate gradients' window estimate of the energy error, summed over twenty steps, stops a reduced-precision run within thirty steps of its least error on the median run.”

On the scaled Laplacian at grids 6 to 14 and 24, 16, 11 and 8 bits, the clause stopping when the twenty-step sum of α‖r‖² falls below u times the iterate's energy norm stops the median run 33 steps after its least error — safely, within 1.22 times the least on all twenty — and totals 601 steps past against the rate-set residual clause's 1,952. The refusal is fed the claim that the median is thirty or fewer, and required to fail.

Tested in A later look is a worse stop · the stopping test series · refused by the window-twenty error clause read as stopping within thirty steps of the least on the median run

“A residual-based stagnation clause, with its checking interval set from the observed rate, stops a reduced-precision conjugate-gradient run promptly once the error stops improving.”

On the same twenty runs the rate-set clause stops a median 97 steps after the least error and up to 163, 1,952 steps past in total, because the residual it reads has flattened long before and its checks are spaced by a rate measured while it was still falling. The refusal is fed the claim that it stops within thirty steps on every run, and required to fail.

Tested in A later look is a worse stop · the stopping test series · refused by the rate-set residual clause read as stopping promptly after the least error

“A Ritz value accurate to eight digits is evidence of an eigenvalue that is there.”

On a 40×40 matrix with forty distinct eigenvalues by construction, eighty Lanczos steps return twenty-five extra copies across thirteen of them — the largest arriving five times, all accurate to 1.9·10⁻⁸ relative. A hundred and twenty steps return seventy-one extra copies with the largest arriving eight times. The accuracy is what makes them undetectable: nothing but the true spectrum separates a copy from a discovery.

Tested in An eigenvalue that arrives twice · the lanczos series · refused by the claim that an accurate Ritz value is a distinct eigenvalue

“The residual a Krylov method reports is the residual of the vector it is about to return, so a stopping test written in it is a statement about the answer.”

On a 50×50 diagonal matrix with one eigenvalue at 10⁻¹⁴, the recurrence reports 6.9·10⁻²¹ — below the unit roundoff of 1.1·10⁻¹⁶ — while the answer it holds has a relative residual of 5.1·10⁻¹⁰. The refusal is fed the same claim on a well-conditioned matrix where the two agree to the last bit, and required to fail.

Tested in The residual the method reports · the residual gap series · refused by a residual gap, claimed where the iterates do not grow

“A Krylov method's reported residual cannot be trusted, because the arithmetic that produces it drifts.”

On the diagonal matrix with an eigenvalue at 10⁻¹⁴, on a bidiagonal with a superdiagonal thirty times its diagonal, and on a single Jordan block, GMRES's reported residual is never more than 1.80, 2.86 and 2.00 times its answer's. On two of the three, ‖VᵀV − I‖ for the basis is 1.41. The refusal is fed the claim on the conjugate gradient recurrence, where the factor is 7.3·10¹⁰, and required to fail.

Tested in The number that is re-derived · the residual gap series · refused by the recurrence's failure, claimed of the projection

“A count computed in floating point is right, because a count is an integer and integers do not have rounding errors.”

A hundred shifts are placed strictly between two eigenvalues separated by 10⁻¹⁵ of the norm, where the count must read three. The claim that it does is fed the result: it reads something else at every one of the hundred.

Tested in An eigenvalue count that cannot be slightly wrong · the inertia series · refused by a pair of eigenvalues 10⁻¹⁵ apart

“The diagonal of a pivoted R disagrees with the singular values in both directions, so its error is scatter that averages out along the curve.”

One hundred and twenty Gaussian 12×9 matrices are factorised with column pivoting and the last diagonal entry compared with σₘᵢₙ. The claim that some of them report a matrix as closer to singular than it is must fail, and it does — none of the 120, with the smallest ratio in the sample at 1.087.

Tested in A good curve and a bad verdict · the rank series · refused by a one-sided rank test claimed to err in both directions

“A rank budget large enough to store the answer's leading structure will get an iteration to the answer, so the budget is a cost rather than an accuracy.”

The same operator is solved with a random right-hand side under a rank budget of eight out of a possible ten, and the assertion that the residual falls below 10⁻⁶ is fed the result. It stalls at 0.47, because the solution has no low-rank structure and eight columns of a ten-column answer buy a factor of two.

Tested in An iterate that must be made smaller · the Low-rank iteration series · refused by a budget is not a solver

“Central differencing fails on a convection–diffusion problem whose layer the grid already resolves, so stabilising every element is free.”

At ε = 0.5 and a flow at 45° on a fifteen-point grid, central differencing's error is 5.62·10⁻⁵, the tuned scheme's is 5.63·10⁻⁵ and upwinding's is 3.61·10⁻³. The assertion that central differencing's error exceeds 10⁻² there is fed exactly that case and required to fail: where there is nothing to stabilise, the unstabilised scheme is the accurate one and the tuning is charged at ξ times upwinding's error for nothing.

Tested in A parameter that is also a price · the convection series · refused by a scheme comparison drawn on a problem with no layer in it

“A rank budget is a memory setting, so raising it can only help, and it should be set as high as the machine allows.”

The budgeted solver is asked for a rank of forty on a tensor with ten points a side, and the assertion that a budget is a free parameter is fed that request. No cut of a 10×10×10 array has rank above ten, so the request names a rank that cannot exist and is refused.

Tested in A run that is over at step five · the Low-rank iteration series · refused by a budget above the cut's own ceiling

“The failure of smoothed aggregation at 45° is a property of the method, so it costs about the same wherever the anisotropy is.”

The claim implies the failure survives the anisotropy being removed. Fed an operator with ε = 1, where the rotation changes nothing and there is no direction to lose, the rotation figure must refuse to be drawn rather than report a failure. It is fed exactly that and required to reject it.

Tested in How much direction there was to lose · the anisotropy series · refused by a rotation figure on an operator with no direction in it

“The gap between a carried residual and a computed one accumulates in proportion to what it accumulates against, so a bound linear in that quantity describes the behaviour and a run with more roundings in it carries a larger gap.”

The sweep is run at seven sizes from n = 20 to n = 80 and the fitted slope of the gap against the largest iterate comes out between 0.443 and 0.507 at every one of them. The assertion that the slope is 1 to within 0.06 is fed the same sweep and required to fail. At the near end of that sweep those seven sizes take between 39 and 96 iterations at an identical largest iterate of 9,308.66, and the gap moves from 5.04·10⁻¹⁵ to 5.33·10⁻¹⁵.

Tested in A walk needs a length · the residual gap series · refused by the bound's slope, claimed as the behaviour's

“The negative-curvature direction is a descent direction for the model, so following it is what produces the decrease and the radius is an implementation detail.”

Along the certificate direction the quadratic model is unbounded below, so there is no decrease for a cheap step to be a share of until a radius bounds it. Asked for the exact subproblem answer at Δ = 0, the eigen-route returns a model decrease of −8.27·10⁻¹³ at a solution of norm 8.27·10⁻¹³ and a secular shift of 1.21·10¹², while the truncated iteration returns exactly zero — a ratio of zero to a rounding. The refusal is fed that subproblem, asked for a model decrease, and required to fail.

Tested in The certificate that arrives soonest is worth least · the Trust-region series · refused by a trust-region subproblem posed with no region