What is taught, and what the arithmetic does instead

“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

“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

“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 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

“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

“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 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 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

“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 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 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

“Gram–Schmidt produces an orthonormal basis for the column space.”

It does, over the reals, and the derivation is three lines. On H₈ the classical process returns a Q with an off-diagonal inner product of 1.0 — two computed basis vectors that are parallel — while every column is still a unit vector to 10⁻¹². Orthogonality is lost like κ² classically and like κ in the modified order; the same algebra written in a different sequence, and eight decades between them.

Tested in Two Gram–Schmidts · the Gram–Schmidt series · refused by the orthogonality check

“Replacing λ by γμ is an exact change of variable, so it cannot change the answer — the two problems have the same eigenvalues up to a factor of γ and any solver will return the same digits.”

The same overdamped chain is solved before and after the substitution and both are compared against the closed form. The claim that the forward error is unchanged is fed the pair and fails: 7.5·10⁻¹⁴ becomes 1.3·10⁻³, and the substitution is exact in both directions at every stop.

Tested in The scaling that buys ten orders · the polynomial scaling series · refused by an exact change of variable that costs eleven orders

“Adding a small number to a running total moves the total.”

Below the spacing of the total it does not arrive at all, and it does not arrive again for every subsequent term of the same size — which is how a sum of a million equal terms stops growing before it is half computed. The refusal is fed a sum that has stagnated and required to report that the addend was lost.

Tested in The order they are added in · the summation series · refused by the claim that a small addend always arrives

“Reassociating a floating-point computation changes its answer, so a reordering has to be paid for in accuracy.”

It has to be paid for whenever the operation being reassociated amplifies. An orthogonal transformation does not amplify — that is what orthogonal means — and a product of orthogonal transformations is orthogonal however it is bracketed. Measured on a 512×12 matrix at κ = 7,151: the tree at depths 1, 2, 3 and 4 returns 3.38, 4.26, 1.65 and 1.48·10⁻¹⁵, which does not grow with the depth. Classical Gram–Schmidt, whose steps are not orthogonal, returns 4.6·10⁻¹⁰ on the same matrix.

Tested in A reduction that changes the order · the communication series · refused by a reduction tree with under-determined leaves

“Use partial pivoting and Gaussian elimination is stable.”

The same 2×2 is solved before and after its first row is multiplied by 1/ε. Partial pivoting makes one interchange on the first and none on the second, and its forward error goes from 0 to 1. The refusal is fed the claim that it makes the same choices on both, and required to fail.

Tested in The pivot that reads the units · the pivoting series · refused by partial pivoting claimed to be invariant under a row scaling

“The moment basis and an orthogonalised basis span the same subspace, so the interpolation conditions are identical and the choice between them is about numerical hygiene rather than about what can be computed.”

Eight moments taken at one point reach κ₂ = 7.7·10⁹, multiplying by about 66 per vector at the far end of the range, while the same eight solves spent at eight spread points and orthogonalised as they are built stay at 1.0. The subspaces agree in exact arithmetic; only one of the two admits a projection.

Tested in A basis that is the same subspace and not the same thing · the moment matching series · refused by the condition number of eight moments at a single interpolation point

“Projecting an ill-posed problem onto a small Krylov subspace makes it a small well-conditioned problem, which is why the iteration regularises.”

The projected condition number climbs from 1.38 at two steps to 7.8·10⁶ at forty-eight. The refusal is fed the claim that it stays under 10⁴ at forty-eight steps and required to fail.

Tested in The step that stops mattering · the iterative regularisation series · refused by a projected problem claimed to be well conditioned because it is small

“Every square matrix is orthogonally similar to a triangular one.”

Over the complex numbers, yes — that is Schur's theorem. Over the reals, an orthogonal similarity cannot separate a conjugate pair, and the reachable form carries one 2×2 block per pair. Checked at four prescribed spectra: exactly one block per pair every time, and a real spectrum does come out genuinely triangular. The surviving subdiagonal entry is 2.0 against a matrix norm of 5.8 — the size of the matrix, not of a tolerance.

Tested in The form a real matrix can reach · the real schur series · refused by triangular form claimed for a real matrix with a conjugate pair

“Two formats with the same number of significand bits have the same precision, so the same computation gives the same answer.”

They have the same unit roundoff and their largest finite numbers are 33 decades apart. The naive norm of a vector of sixteen 1,000s is an infinity in fp16 and 4,000 in tf32, at identical precision — and bfloat16, the least precise of the three, returns 3,968. This site reported fp16 and tf32 as one format for two phases, because its arithmetic modelled the significand alone.

Tested in The other half of a format · the Floating-point series · refused by fp16 and tf32 treated as one format because their mantissas match

“A symmetry of the problem is a symmetry of the computed answer — if the eigenvalues come in reciprocal pairs then the numbers a solver returns come in reciprocal pairs.”

A palindromic quadratic whose spectrum spans twenty decades is solved by a general eigensolver, and the computed set is tested for closure under λ ↦ 1/λ. The claim fails: the departure is 1.0·10⁻⁷, and it is exactly the error in the small eigenvalues.

Tested in A spectrum that comes in reciprocal pairs · the structured spectrum series · refused by an exact reciprocal pairing from an unstructured solver

“There is one condition number of a matrix, and κ(A) = ‖A‖‖A⁻¹‖ is it.”

Two condition numbers of the same matrix are computed and one is invariant under a row scaling to fourteen digits while the other moves by seven orders. The refusal is fed the claim that the componentwise number notices the scaling, and required to fail.

Tested in A condition number scaling cannot move · the scaling series · refused by the componentwise condition number claimed to move with the units

“Cramer's rule solves a linear system. It is closed form, it is exact, and for small systems it is the obvious thing to write.”

Forty thousand 2×2 systems with nearly parallel rows, at 24 bits. Elimination's normwise backward error never exceeds 1.3 units of roundoff; Cramer's reaches 458. The refusal is fed the claim that Cramer's rule returns the exact answer to a nearby problem, and required to fail.

Tested in A rule that is correct and unusable · the determinant series · refused by Cramer's rule offered as a numerically sound way to solve a system

“A shift accelerates the QR iteration, so a good shift is enough.”

A real single shift cannot converge onto a complex conjugate pair at all — there is no real number to shift towards. The double shift is not a faster single shift; it is the construction that makes the problem addressable in real arithmetic, and it is performed without ever forming the complex matrix it is derived from. The implicit step and the explicitly formed product agree to 2.2·10⁻¹⁵ entry by entry in absolute value.

Tested in Two shifts that are never formed · the francis series · refused by convergence claimed for the single-shift algorithm on a conjugate pair

“The filter-factor identity is a theorem about the iterate, so the measured and predicted factors agree at whatever step the method is best at.”

The identity is exact over the reals at every step and the recurrence is not run over the reals. Asked for the filter at step 30 — past the step the Lanczos basis loses orthogonality at — the figure's own agreement assertion is required to fail, and it does.

Tested in An expiry date the noise does not move · the iterative regularisation series · refused by the filter identity drawn past the step orthogonality is lost at

“If x − y computes to zero then x and y are the same number.”

With gradual underflow it is true, and it is one of the reasons subnormals exist. With flush-to-zero it fails on ten of ten distinct neighbouring pairs immediately above fp16's smallest normal: the difference is representable only as a subnormal, the subnormal is flushed, and the test for equality returns true for two different numbers.

Tested in The numbers below the smallest one · the subnormals series · refused by flush-to-zero treated as leaving subtraction faithful

“The Galerkin coarse operator RAP is the discretisation of the same problem on the coarser grid.”

This site asserted it, and still does — in one dimension it holds to a relative difference of exactly zero at every level of a six-level hierarchy. In two, a five-point operator produces a nine-point coarse one, with four diagonal couplings carrying a sixth of the row, so every level below the first solves a different discretisation. The method converges at 0.20 a cycle regardless, which is the interesting half.

Tested in The coarse problem is a different problem · the multigrid series · refused by the one-dimensional Galerkin identity carried into two dimensions

“A reduced model that matches the transfer function of a stable system at r points is a model of that system.”

Four of nineteen shift placements return a model with a pole at up to +24.1 from a system whose rightmost eigenvalue is −3.49, and every one of those models interpolates to 1.3·10⁻¹⁴. The refusal is the other half: fed the claim that every placement destabilises, the assertion must reject, because most do not.

Tested in A model that cannot be run · the reduced stability series · refused by the claim that interpolation always loses stability

“An even power is non-negative, so a computed even power can be compared with zero.”

(x − 1)⁶ expanded and evaluated by Horner at 401 points within 10⁻³ of 1 is negative at 179 of them. Away from the root it is not: the refusal is fed the assertion that a sixth power evaluates negative over a window of width 1 and must reject it, which locates the failure in the neighbourhood rather than in the polynomial.

Tested in A square that evaluates negative · the fma contraction series · refused by the claim that the fusion matters away from a cancellation

“The basis a polynomial is fitted in changes its coefficients but not the fitted curve, so any basis will do when only the curve is wanted.”

On exact data at degree 40 the monomial and orthogonal-basis curves agree to 2.7·10⁻¹⁴ despite κ = 7.4·10¹⁴. With 0.1% noise at the same degree they differ by 1.7·10⁻⁵, within a factor of ten of κ·u·‖r‖ at every degree measured. The refusal is fed the claim that the two curves agree to rounding on noisy data and required to fail.

Tested in A basis built from the points · the fitting series · refused by a fitted curve claimed independent of the basis on noisy data

“A test that is performed correctly on a correctly computed number gives a correct verdict.”

Every matrix in a nine-by-eight grid is positive definite by construction. At twelve significand bits and below, conjugate gradients produces a direction of negative curvature on some of them — a proof of something false, from a comparison performed exactly as written on a number computed as accurately as the format allows. The refusal is fed the claim that the same happens at double precision over the same range of conditioning, where it does not.

Tested in Deciding that a zero has arrived · the deliberate zero series · refused by a false certificate of indefiniteness claimed at double precision

“Scaling a set of numbers by a power of two changes no value, so it changes no result.”

The scaling is exact and what it changes is which numbers share a block. Sorting the same values — which alters no value at all and no scale factor's exactness — takes the count of entries quantised to zero from 310 to 22. The order is not a presentational choice once one exponent is shared across a group.

Tested in One exponent for thirty-two numbers · the block formats series · refused by the claim that a block format is order-independent

“The higher moments of a contour integral cost one multiplication each, so raising K is free and the number of moments can be increased until the ceiling stops binding.”

Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ^2K. Measured: the unscaled matrix is 19.6 times worse conditioned than the scaled one at K = 3 and radius 6, 53.6 at K = 4, and 19,440 at K = 5 and radius 15. The arithmetic is free and the rank decision is not.

Tested in The conditioning that rises with the ceiling · the nonlinear eigenvalue series · refused by the condition number of the block Hankel at five moments on a contour of radius fifteen

“Interval arithmetic gives rigorous bounds, so carrying intervals through an algorithm gives a rigorous answer.”

It gives rigorous bounds and they become useless. Rotating an interval box by 45° twenty times — an isometry, so the true set does not change size at all — leaves an enclosure 1,024 times too wide, growing by exactly √2 a step: measured at 1.4142137 against √2 = 1.4142136. No rounding error is responsible for any of it and a higher precision does not touch it.

Tested in A bound that is proved · the interval series · refused by the claim that interval arithmetic can be carried through an algorithm

“Regularising the subproblems of an alternating fit removes the swamp, so a ridge is a repair with no cost on problems that do not have one.”

On a tensor built from three rank-one terms, a fit with a ridge of λ reaches a relative error of 9.73·10⁻¹¹, 9.71·10⁻⁷, 9.45·10⁻⁵, 9.74·10⁻⁴ and 9.66·10⁻³ at λ = 10⁻¹⁰, 10⁻⁶, 10⁻⁴, 10⁻³ and 10⁻² — the error is the ridge. The refusal is the other half: fed the claim that the ridge also moves the terms, the assertion must reject, because the congruence against the planted factors stays above 0.9997.

Tested in The repair that costs exactly itself · the Alternating least-squares series · refused by a ridge that is free

“QR with column pivoting is rank-revealing: the diagonal of R shows you where the rank is.”

On a 50×50 Kahan matrix at c = 0.5 the smallest diagonal entry of R is 2.3·10⁸ times σₘᵢₙ, with no interchange made anywhere. The refusal is fed the claim that the two agree within a factor of 100, and required to fail.

Tested in The cheap rank and what it cannot see · the rank series · refused by a pivoted diagonal read as the singular values

“The eigenvalues describe the matrix. Two matrices with the same spectrum behave the same way.”

A normal matrix and a bidiagonal one, both with every eigenvalue at 0.8, are perturbed by the same twenty matrices of norm 10⁻⁸. One set of eigenvalues moves by 10⁻⁸ and the other by 0.07. The refusal is fed the claim that their powers are comparable, and required to fail.

Tested in The eigenvalues that are not there · the Non-normality series · refused by a spectrum read as a description of a matrix

“f(A) = V f(Λ) V⁻¹. Diagonalise, apply f to the eigenvalues, undiagonalise — that is what a function of a matrix is.”

An upper bidiagonal matrix with eigenvalues 7·10⁻⁸ apart, whose eigenvalues are its diagonal and whose eigenvectors have a closed form. The route is handed both, exactly, and returns a matrix with a relative error of 5.5·10³⁷; the refusal is fed the claim that it is accurate, and required to fail.

Tested in A function of a matrix is not a function of its entries · the matrix function series · refused by the eigendecomposition offered as a method for a matrix function

“Scaling is an accuracy device: it improves the digits in an answer, so a code that does not care about the last few digits can skip it.”

The coefficients of a quadratic are formed at a change of units of 10¹⁰ in fp16 and every entry is rounded to the format. The claim that the problem can still be written down fails: γ²M is past 65504 and the entries are infinities, while the scaled coefficients are representable at every stop of the same sweep.

Tested in The units that overflow before the answer does · the overflow series · refused by a change of units past the format's largest number

“CGLS and LSQR are the same method, so the choice between them is a matter of taste.”

Both minimise the residual over the same Krylov space, so their iterates are equal in exact arithmetic. Reaching a relative error of 10⁻⁶ on a κ = 10¹⁰ problem costs 110 steps one way and 209 the other, at four seeds each.

Tested in One sequence and two recurrences · the krylov series · refused by the symmetric operator used as the non-normal test case

“The block-circulant preconditioner makes the conjugate gradient step count on a two-dimensional Toeplitz system independent of the correlation that makes the system ill-conditioned.”

On a 10 × 10 grid the separable kernel's preconditioned count is 18 at ρ = 0.5 and 18 at 0.98, while its unpreconditioned count goes from 43 to 178. The isotropic kernel with the same correlation along each axis goes from 17 to 36 preconditioned, rising at every step of ρ. The refusal is fed the claim that the isotropic kernel's preconditioned count moves by at most three across the same range, and required to fail.

Tested in The staircase a separable kernel builds · the toeplitz series · refused by the circulant preconditioner's step count read as independent of the correlation whatever the kernel

“Balanced truncation's a-priori bound, twice the sum of the discarded Hankel singular values, is attained on ordinary problems, so it can be read as a formula for the error.”

On the heat model the bound over the measured H∞ error is 1.0000 at every order. On twelve states of six oscillators at damping 0.1, it is 3.23, 4.53, 3.52, 4.47, 4.55, 1.75, 2.43, 3.16, 2.01 and 2.13 at orders 1 to 10, and 1.000 only at order 11, where one state is removed. The refusal is fed the claim that at damping 0.1 the bound is within five per cent of the error at every order, and required to fail.

Tested in A mode that rings is counted twice · the balanced truncation series · refused by the balanced-truncation bound read as attained on any stable system

“Streamline diffusion with ξ = coth(Pe) − 1/Pe makes the discrete solution exact at the nodes.”

At fifteen degrees to the grid the same scheme's worst nodal error is 3.2·10⁻², against 2.4·10⁻¹⁷ at zero. The refusal is fed the claim that it is below 10⁻¹⁰ there and required to fail.

Tested in Exact along one axis · the convection series · refused by nodal exactness claimed for a flow that is not along a grid line

“A method that has driven its residual to 10⁻¹² has produced an answer good to about 10⁻¹², so the reported residual is the accuracy achieved.”

The same preconditioned conjugate gradient solve with its working arithmetic at 24 significand bits stops with a relative residual of 1.10·10⁻¹³ and a relative error of 1.62·10⁻⁷. The assertion that the error is within a factor of ten of the residual is fed that run and required to fail.

Tested in The reading that never moves · the stopping test series · refused by a convergence test read as an accuracy guarantee

“A basis orthogonalised by modified Gram–Schmidt at every step stays orthogonal, so a Krylov method built on it can be run for as many steps as memory allows.”

A second-order Krylov basis is built to twenty vectors with the same modified Gram–Schmidt sweep the linearised one uses, and ‖QᵀQ − I‖ is measured. The claim fails: it is 1.35, orthogonality gone entirely, while the linearised basis on the same problem is at 2.4·10⁻¹³.

Tested in A Krylov space for a problem that is not linear · the krylov series · refused by an orthogonal second-order basis at twenty vectors

“Squaring is exact in the algebra, so taking more squarings than the smallest number that works cannot make the answer worse.”

The squaring count is swept from zero to fifteen on an 8×8 whose exponential is known in closed form, and the best value is taken. The assertion that the last value is no worse than the best is fed that sweep and must reject it: the best is 6.88·10⁻¹⁶ at s = 2 and the last is 3.16·10⁻¹² at s = 15, a factor of 4,593 paid for thirteen matrix products.

Tested in The error the method already knows · the matrix function series · refused by extra squarings claimed to be harmless

“Weighted Jacobi at ω = 2/3 damps the oscillatory half of the error by a factor of three.”

Exactly true of the Laplacian, where the smoothing factor is max(|1 − ω|, |1 − 2ω|) and is ⅓ at ω = 2/3, asserted here to 10⁻¹². On the central-difference convection operator at 63 points and ε = 0.005 the same sweep returns 1.0937 — above one, so the modes it is supposed to remove grow. The refusal is fed the claim that the two agree to 10⁻³ and requires it to fail.

Tested in A smoother that stops being one · the multigrid series · refused by the symmetric smoothing factor carried onto a non-symmetric operator

“Two implementations with the same operation count cost the same.”

At n = 48 with 144 words of fast memory both orderings perform exactly 72,568 operations, choose exactly the same pivots and return a factorisation whose residual agrees to the last bit. One moves 41,332 words between fast and slow memory and the other 19,476 — a factor of 2.12, and the ratio does not fall with size. The refusal is fed the claim that equal flop counts mean equal data movement and requires it to fail.

Tested in The same arithmetic at a different price · the blocking series · refused by the claim that a flop count predicts the cost

“The answer came back wrong, so the algorithm was unstable.”

A Householder least-squares solve of a 48-point deconvolution returns a relative error of 5.5·10⁸ with a residual at the level of rounding. The algorithm did everything that can be asked of it. κ is 5.7·10¹², the singular values decay exponentially with no gap anywhere, and the data is consistent with a range of answers differing by orders of magnitude. The refusal is fed the claim that a backward-stable method answers an ill-posed question accurately and requires it to fail.

Tested in When the answer is a choice · the regularisation series · refused by the claim that a stable algorithm answers an ill-posed question

“Run an iterative method to convergence; stopping early is a compromise you make when you are short of time.”

On the 64-point deconvolution at 1% noise the error is least at step 20, at 0.1426, and by step 120 it is 6.02 — forty-two times worse. The residual falls at every one of those 120 steps. The refusal is fed the claim that the last iterate is within a factor of 1.5 of the best one and requires it to fail.

Tested in A parameter that counts steps · the iterative regularisation series · refused by the claim that an iterative method should be run to convergence on an ill-posed problem

“An answer this inaccurate means the arithmetic was not accurate enough — compute it in higher precision.”

The same deconvolution solved at eight precisions gives 4.6·10² at eight significand bits and 5.5·10⁹ at fifty-three. The sweep runs the wrong way: more precision is a more faithful amplification of the noise. The refusal is fed the claim that the double-precision solve beats the eight-bit one and requires it to fail.

Tested in Four knobs and one floor · the Parameter choice series · refused by precision offered as a remedy for ill-posedness

“A Newton step that lands short of the root was solved too loosely; tightening the inner tolerance is what fixes it.”

At an iterate 3.719·10⁻² from the root, tolerances of 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰, 10⁻¹² and 10⁻¹⁴ land at 2.4966·10⁻³ to 2.4967·10⁻³ — within four parts in ten thousand of each other — at 354 and 1,126 conjugate gradient iterations. The refusal is fed the claim that the shortfall is the inner solve's, and required to fail.

Tested in The accuracy that is thrown away · the inexact newton series · refused by the plateau, claimed at a tolerance above it

“An error bound tells you what a computation will return, to within the bound.”

Two reductions of one vector return different numbers and one bound covers both. The refusal is fed the assertion that the bound excludes one of the two answers, and must reject it — which it does, because the bound contains n and the sum of magnitudes and nothing about the order.

Tested in A bound every answer satisfies · the reduction order series · refused by the claim that an error bound identifies an ordering

“The residual came back at 10⁻¹⁶, so the answer is accurate.”

A 13×13 Hilbert system whose exact answer is the integers 1 to 13 solves to a backward error of 2.2·10⁻¹⁷ and returns 0.049 where 8 belongs and −2.58 where 10 belongs. The algorithm is blameless and κ is 1.7·10¹⁸. The residual measures the algorithm; the error is the residual times the problem's own sensitivity, and only the second is what anybody sees.

Tested in A small residual is not a small error · the backward error series · refused by a usable answer inferred from ‖PA − LU‖ alone

“A backward-stable eigenvalue routine returns an answer that is exact for a nearby problem, so a small residual from the solver means the eigenpair is right for a problem close to the one that was posed.”

The claim is fed the eigenpairs of a quadratic whose coefficients have been rescaled by an exact change of variable. The linearised matrix's backward error is 7.6·10⁻¹³ and the quadratic's is 1.2·10⁻⁴, so the assertion that the pair is backward stable for the problem posed fails by eight orders of magnitude.

Tested in A backward-stable answer to a problem nobody asked · the linearisation backward error series · refused by a stable solve of a linearisation at γ = 10⁸

“A computation is as accurate as its least accurate part, so every component of a solver needs the working precision.”

The preconditioner of a conjugate gradient solve, computed and applied with a four-bit significand, returns an answer whose relative error is 7.6·10⁻¹³ — the same as the double-precision run's. The refusal is fed the claim that a four-bit preconditioner produces four bits of accuracy and requires it to fail.

Tested in The part of a solver that may be rounded · the Mixed-precision series · refused by the claim that a solver is as accurate as its least accurate part

“Pick the regularisation parameter from the corner of the L-curve.”

Scored against the λ that actually minimises the error, on a 64-point deconvolution at 0.1% noise: the L-curve's corner gives a relative error of 0.2406 against the oracle's 0.1051 — 2.29 times. Generalised cross-validation on the same problem lands on the oracle's λ exactly and the discrepancy principle costs 6%. The corner is a real feature and picking it is not the same as picking the best parameter.

Tested in Choosing without knowing · the Parameter choice series · refused by a rule that beats the truth it is scored against

“Cholesky QR is the communication-optimal way to factorise a tall matrix: one reduction, and the rest is local.”

It is the fewest rounds, and its loss of orthogonality grows like κ² — a fitted slope of 1.97 across five decades. At κ = 10⁷ the implied Q has ‖QᵀQ − I‖ = 1.5·10⁻³ where the reduction tree's is 10⁻¹¹. The refusal is fed the claim that the one-round method returns an orthogonal factor there and requires it to fail.

Tested in The message and the word · the communication series · refused by a factorisation chosen by its communication cost alone

“Refinement has not converged yet — run more steps.”

Past its threshold the method is inert rather than slow. With the residual computed in single precision the same code ends at 8.80·10⁻⁵ having improved by a total factor of 1.51 across six steps; each step is doing exactly as much as the first, which is almost nothing. The step count is not the parameter and the precision the residual is computed in is.

Tested in Buying the accuracy back · the Mixed-precision series · refused by more refinement steps claimed to help past the threshold

“The estimator is unbiased, so with enough samples it is accurate — and the choice of random vector is a detail.”

On a diagonal matrix the ±1 probe has variance exactly zero and the normal probe has variance 545. The refusal is fed the claim that the two distributions have the same variance, and required to fail.

Tested in Counting what cannot be looked at · the trace estimation series · refused by the probe distribution treated as a detail

“Circulant preconditioning of a positive definite Toeplitz system can produce an indefinite preconditioner at moderate size.”

Strang's circulant has smallest eigenvalues of −0.655, −0.426, −0.173 and −0.013 at n = 16 to 128 with ρ = 0.95. The averaged circulant on the same matrices has 0.0431, 0.0382, 0.0332 and 0.0295 — falling, and never through zero. The refusal is fed the claim that every circulant approximation goes indefinite there and requires it to fail.

Tested in The circulant that cannot be indefinite · the toeplitz series · refused by indefiniteness blamed on circulant preconditioning rather than on which circulant

“Whichever fit has the smaller residual is the better fit.”

With all the noise in the matrix, total least squares is 2.5 times closer to the true coefficients and its residual is 3.6% larger. The refusal is fed the claim that the residual and the error order the two methods the same way, and required to fail.

Tested in When the matrix is wrong too · the Total least-squares series · refused by a residual comparison read as an accuracy comparison

“An update formula's accuracy is governed by how well conditioned the matrix it produces is. If A + uvᵀ is well conditioned, the update will be accurate.”

A is QΛQᵀ with Λ = (1, …, 1, 10⁻¹⁴) and the rank-one update takes the last eigenvalue back to 1, so A + uvᵀ is the identity. The refusal is fed the claim that the update's error is governed by κ(A + uvᵀ) = 1, and required to fail.

Tested in A correction cheaper than the problem · the Low-rank update series · refused by an update formula's accuracy read off the conditioning of its answer

“A factorisation that reconstructs its matrix to the level of rounding has produced an orthogonal Q — the residual is the check.”

At κ = 10⁸ Cholesky QR reconstructs A to 1.2·10⁻¹⁶ and its implied Q is 0.37 from orthogonal. The refusal is fed the claim that the orthogonality is within a factor of a thousand of the residual and required to fail.

Tested in Doing it twice · the communication series · refused by orthogonality inferred from a residual

“A sketched least-squares solve is approximate — that is the price of the speed.”

The same sketch, of the same width, on the same problem: used as an answer it is orders of magnitude from the exact solution and moves with the seed; used as a preconditioner it is within κ(A)·u of it and does not. The refusal is fed the claim that a sketched solve is a per cent off, and required to fail.

Tested in The sketch that is not the answer · the sketching series · refused by the accuracy cost of sketch-and-solve attributed to the sketch rather than to the use

“An answer computed to only half the available digits means something in the problem was ill conditioned — find the large condition number and you have found the cause.”

A chain is set at exactly its critical damping, where two eigenvalues coincide, and the computed spectrum is compared with a closed form. The claim that a well-conditioned problem is computed to the rounding level fails: the error is 3.7·10⁻⁸ while κ(K) is 32.16 and the coefficients are integers.

Tested in Every eigenvalue real, and a test that says so · the hyperbolic quadratic series · refused by sixteen digits at a double root

“A large condition number means the problem is sensitive. If κ(A) is 10⁸, expect to lose eight digits.”

The same system is written twice with its rows in different units. The solution is unchanged to fourteen digits and κ₂ moves by seven orders of magnitude; the refusal is fed the claim that two matrices with the same solution set have comparable condition numbers, and required to fail.

Tested in The units the matrix is measured in · the scaling series · refused by a condition number read as a property of the problem rather than of its units

“Iterative refinement recovers double-precision accuracy from a low-precision factorisation.”

It recovers it below κu ≈ 1 and not above, and the three hardware formats straddle that threshold at κ = 10³. bfloat16 at κu = 3.91 ends at 1.04·10⁻³ having recovered a factor of 412 — which looks like the method working hardest, and is the confounding this figure exists to remove. The shortfall against a full double solve of the same system is not confounded.

Tested in Where the hardware went · the Mixed-precision series · refused by refinement claimed to converge past its threshold

“A Sylvester equation is well conditioned when the two spectra are well separated — the difficulty is how close λᵢ(A) comes to −μⱼ(B).”

Two 6×6 triangular matrices whose eigenvalue sums are all at least 2 at every point of a sweep, and whose sep falls from 2 to 9.5·10⁻⁴. The refusal is fed the claim that sep is within a factor of ten of the gap, and required to fail.

Tested in An equation whose unknown is a matrix · the matrix equation series · refused by a Sylvester equation's conditioning read off its eigenvalue gap

“Generalised cross-validation needs no noise estimate, so it can be applied to whatever problem is in front of it — including the projected one inside an iterative method.”

The trace in GCV's denominator cannot exceed k + 1, and at eight steps of a sixty-four-row problem it is 1.03. The refusal is fed the claim that it exceeds thirty-two there and required to fail.

Tested in A parameter chosen on a smaller problem · the Parameter choice series · refused by a projected GCV trace claimed to count the problem's degrees of freedom

“A leverage near one means the reduced problem is ill conditioned — that is why removing a high-leverage point is numerically hard.”

At h = 1 − 10⁻⁷ the downdated Gram matrix has a condition number of 4.3 and a fresh factorisation of it is exact to 10⁻¹⁶, while the downdate's residual is 3.5·10⁻¹⁰. The refusal is fed the claim that the downdated matrix is ill conditioned, and required to fail.

Tested in The observation that cannot be removed · the Low-rank update series · refused by a leverage near one read as ill-conditioning of the matrix it produces

“Balanced truncation costs O(n³) and comes with a proved bound, so where both methods can be run it should be substantially the more accurate of the two.”

Both methods are run at the same order on four models. In H₂ the interpolatory model wins on every one, by 0.4 to 1.9 per cent. In H∞ balanced truncation wins on every one, by 12 to 25 per cent. Each method wins the norm it was designed for, and the margins are small enough that the field's choice is about cost.

Tested in Interpolating at the model’s own poles · the moment matching series · refused by the two errors on four systems, in both norms

“Multigrid works because the smoother removes the error.”

The smoother removes the oscillatory part and leaves the rest almost untouched: weighted Jacobi's own convergence factor climbs 0.981 → 0.998 as the grid refines, each value checked against cos(πh). The V-cycle's stays at 0.1002 ± 0.001 over the same range. A rate belonging to two components attributed to one of them is a claim that predicts the wrong number by a factor of ten.

Tested in The error smoothing cannot reach · the multigrid series · refused by the multigrid rate claimed for the smoother on its own

“Overflow is a very large rounding error, so a more careful algorithm loses a few more digits.”

It is the loss of the number rather than of digits, and no subsequent arithmetic recovers from it. The naive √(Σx²) is correct over 18 of fp16's 30 normal exponents, because squaring doubles the exponent; the scaled form is correct at every normal exponent of all three formats tested. Same precision, same algebra, and one of them returns an infinity where the answer is 4,000.

Tested in A norm that overflows before it is a norm · the overflow series · refused by overflow described as a rounding error

“Each update is backward stable, so a chain of them is backward stable and the factor can be carried indefinitely.”

No step amplifies by more than 2.72 and no downdate fails, yet ‖RᵀR − AᵀA‖/‖AᵀA‖ climbs from 3.5·10⁻¹⁵ to 3.9·10⁻¹⁴ over the run — a fitted slope of 0.554 in the step count. The refusal is fed the same claim about a factor rebuilt from the rows at every step, where there is nothing to accumulate, and required to fail.

Tested in Stable once, and three thousand times · the sequence stability series · refused by a drift, claimed for a factor that is never carried

“Static pivoting is safe because iterative refinement repairs whatever the replaced pivots cost.”

At the last member, two pivots are replaced, the factorisation's own residual is 1.22·10⁻⁹ and the solve's backward error is 4.8·10⁻⁹. Six steps of refinement reach 5.5·10⁻¹⁰ — a factor of 8.8 — and stop. The refusal is fed the same claim after the rows have been equilibrated, where no pivot is replaced and there is nothing to repair, and required to fail.

Tested in The order that was right last time · the sparse pivoting series · refused by the failure of refinement, claimed at the member the order was chosen for

“rcond reported the matrix was fine, so the matrix was fine.”

A 16×16 matrix is constructed on which the estimator converges — its own stopping test fires — and reports 2.7·10² against a true κ₁ of 3.5·10³. The refusal is fed the claim that the estimate and the truth agree to within a factor of 1.5, and required to fail.

Tested in An estimate that can be fooled · the Condition-estimation series · refused by a condition estimate read as a condition number

“The eigenvalues of a Gramian decay because the systems people build are smooth, so the decay is an empirical fact about physical models rather than something predictable in advance.”

λₖ₊₁(P)/λ₁(P) is measured for a thirty-state model and compared with Zₖ², the squared Zolotarev number for the interval the spectrum of −A spans. Every ratio is under the bound at every k, and nothing about the model enters the bound except the two ends of that interval — so the decay is predictable before the Gramian exists.

Tested in Why a Gramian can be truncated at all · the gramian decay series · refused by the decay of a Gramian against a bound computed from two numbers

“κ(A) says how many digits a linear system can lose.”

A 40×40 Kac–Murdock–Szegő matrix at ρ = 0.999 has κ = 7.88·10⁴. Restricted to perturbations that are symmetric Toeplitz matrices, its condition number is 3.18·10⁴; restricted to perturbations of the one number it contains, 61.9 — three decades below κ. The refusal is fed the one-parameter basis asked for without its parameter, where a zero matrix in its place would report the condition number of a perturbation set containing nothing.

Tested in The condition number of the model · the structured backward error series · refused by the one-parameter basis of a family, asked for without the parameter

“Multiplying by a computed inverse is as accurate as solving; the difference is speed, and if you need many right-hand sides it is worth forming.”

One 30×30 system at κ = 10¹⁴, solved twice from the same factorisation. The LU solve's backward error is 2.2·10⁻¹⁷ and the inverse route's is 4.5·10⁻⁵. The refusal is fed the claim that the two are within a hundred of each other, and required to fail.

Tested in The inverse that is never formed · the inversion series · refused by the inverse-and-multiply route claimed to be backward stable

“Fusing the multiply and the add is more accurate, so a build that fuses gives the right answer where one that does not gives the wrong one.”

Over 200 Gram matrices whose definiteness is settled exactly by Sylvester's criterion in BigInt rationals, the two forms disagree 26 times, and the fused form is the correct one on 9 of them. The refusal is fed the assertion that the unfused form is never the right one and must reject it.

Tested in A matrix that is definite on one machine · the fma contraction series · refused by the claim that fusing the multiply-add makes an answer right

“The forward error says which of two routes was the unstable one: the route whose answer is more wrong is the route to blame.”

One 30×30 system at κ = 10¹⁴ solved both ways. The forward errors are 2.8·10⁻⁴ and 1.5·10⁻², a factor of 54, while the backward errors are 2.2·10⁻¹⁷ and 4.5·10⁻⁵. The refusal is fed the claim that the forward errors differ by a factor of a million — that the instability is legible in how wrong the answer is — and required to fail.

Tested in The gap refinement can close · the inversion series · refused by a forward error offered as evidence of which method was unstable

“A kept incomplete factorisation ages because it is old, so the penalty for keeping it is a property of how long it has been held.”

The same nineteen-member sweep is run with the drift rate set to zero, so every member is the same operator. On 144 unknowns the kept factorisation costs 18 preconditioned iterations at the first member and 18 at the nineteenth, and the ratio against a freshly built factorisation is 1.000 at every one of them. The assertion that a kept factorisation more than doubles its count over the sequence is fed that run and required to fail.

Tested in The penalty for keeping it is a ratio · the reuse series · refused by ageing, claimed where nothing changes

“κ(A) is the conditioning of the problem, so a well-conditioned A means a fit whose answer can be trusted.”

On a 24×3 fit the gap σₙ(A) − σₙ₊₁([A b]) closes from 1.4002 to 0.2389 as the noise rises, the total least-squares error rises from 0.0117 to 0.722 — a factor of 62 — and κ(A) moves from 3.54 to 2.46, downward and monotonically, across the same sweep. The refusal is fed the claim that the error grows with κ(A) and required to fail.

Tested in The two numbers a caller has · the Total least-squares series · refused by κ(A) offered as the conditioning of a total least-squares problem

“Multigrid is grid-independent, so it converges at the same rate on any second-order elliptic problem.”

Give the operator a strong direction and the V-cycle's convergence factor goes from 0.10 to 0.9565 while the grid-independence claim stays true. And the problem did not get harder: the anisotropic operator's condition number is cot²(πh/2) for every ε, measured at 103.08686891981742 across four decades with a relative spread of 2.8·10⁻¹⁶. Only the method is worse.

Tested in A direction the smoother cannot see · the anisotropy series · refused by the claim that a point smoother copes with anisotropy

“The computed eigenvector is badly wrong, so the computation failed.”

At a gap of 10⁻⁹ the returned eigenvector is decided by the perturbation rather than by the matrix — four seeds give 0.483, 1.466, 0.536 and 0.167 radians — and every one of them satisfies Ax = λx to 5·10⁻¹⁵. Nothing failed. The quantity that is a property of the matrix is the plane the two vectors span, and it moves by 7.63·10⁻⁸ and does not move with the gap at all.

Tested in The plane survives what its vectors do not · the invariant subspace series · refused by measuring a subspace by one of its vectors

“An augmented Lagrangian preconditioner with a large enough γ makes the Schur complement trivial and the saddle-point system easy, at no cost to the answer.”

At γ = 10⁸ MINRES needs 6 steps instead of 21, but each solve with H + γAᵀA needs 43 conjugate gradient steps instead of 14, κ(H + γAᵀA) is 4.0·10⁹, and the answer's forward error against the exact rational solution is 2.4·10⁻⁸ where at γ = 0.01 it is 6.6·10⁻¹⁶. The refusal is fed the claim that the answer keeps twelve digits at γ = 10⁸ and required to fail.

Tested in Where the augmentation puts the cost · the block preconditioning series · refused by an augmented Lagrangian at γ = 10⁸

“A carried factor's drift is what costs a sliding-window least-squares solver its accuracy, so recomputing the factor periodically is the repair.”

On a stream of nearly parallel integer columns, medians over three seeds, the recomputed factor's coefficient error is within a factor of five of the carried one's at every κ(A) from 2·10² to 2·10⁶, both growing like κ(A)²u, while one correction from the window's rows brings the carried factor below 10·κ(A)·u through κ(A) = 2·10⁵. The refusal is fed the claim that refactorising recovers the lost coefficients and required to fail.

Tested in The repair the drift did not need · the sequence stability series · refused by a refactorised window claimed to recover the coefficients a carried one lost

“Spurious eigenvalues from a linearised approximation can be identified by their residual, since a genuine eigenpair has a small one and a spurious one does not.”

All thirty-six eigenvalues of the linearisation have a residual at rounding against the matrix that produced them — that is what an eigenvalue is. The residual that separates them is the one against the original problem, and for the spurious ones it does not merely fail to be small: they lie past the branch point, where the function is not real, and it does not exist.

Tested in The eigenvalues that are answers to nothing · the spurious spectrum series · refused by the residual of every returned eigenvalue against the linearised approximant

“A Gramian's condition number measures how far a model is from being reducible, so a model that is barely reducible has better-conditioned Gramians and all of its states are there to be kept.”

A twenty-state model of McMillan degree four is asked for a reduction of order nineteen. Its Gramians have exact rank four, the nineteenth Hankel singular value comes back as exactly zero, and the assertion that a truncation sits above the numerical rank of the Gramians is fed that case and must reject it. The directions past the rank are not there to be kept, whatever the condition number is read as saying about them.

Tested in A condition number that is not the model's · the gramian conditioning series · refused by a truncation below the numerical rank of the Gramians

“Compressing a matrix to ε gives an answer accurate to ε.”

At a compression tolerance of 10⁻⁶ on a matrix at κ = 1.07·10⁴, the answer's forward error is 8.5·10⁻⁵ — a hundred times the tolerance, and exactly what κ times the backward error predicts. The refusal is fed that number against the tolerance and required to fail.

Tested in An accuracy that is a backward error · the backward error series · refused by the answer is accurate to the tolerance

“A block Toeplitz matrix with Toeplitz blocks conditions like the one-dimensional family it is built from.”

At ρ = 0.8 the 8×8 grid has κ = 1793.81 and the 64-row one-dimensional section of exactly the same kernel has κ = 78.05, a factor of 23. The assertion that the two agree within 50% is fed that pair and must reject it.

Tested in Four orders of conditioning, and four steps · the toeplitz series · refused by a two-dimensional condition number read off the one-dimensional family

“A block format loses accuracy because its significand is narrower than a per-element format's, so the repair is a wider significand.”

The loss is real and the width is not what causes it. On data spanning one octave inside a block, a six-bit block format at 6.25 bits a value is more accurate than E4M3 at 8, and the assertion that it cannot be is fed exactly that comparison and must reject it. Across the widths, a bit moves the survivable outlier by one octave and no further, while reordering the same values moves it past every width at once.

Tested in A bit buys an octave · the block formats series · refused by the claim that the wider format is the more accurate one

“A residual bound computed from the projected matrix is a bound on the residual — that is what makes it usable as a stopping rule without an extra product with A.”

After eight cycles the reported bound is 1.1·10⁻³¹ and the residual it names is 5.7·10⁻⁵. The refusal is fed the claim that the bound is within a factor of ten of the residual and required to fail.

Tested in Keeping the vectors, and losing the bound · the lanczos series · refused by a projected residual bound claimed as a bound on the residual

“A model's behaviour is the sum of its modes, so the state balanced truncation removes is the least important mode and the σ it pays for is that mode's own size, |r| ÷ 2|p|.”

The assertion that a modal sum still equals the transfer function after one residue has been set to zero is fed exactly that: a sixteen-state model with its fourth residue deleted, compared against the factorised solve at a frequency where both are large. It refuses. Dropping a term from the sum is the operation the modal reading assumes a truncation performs, and it is not the operation a truncation performs — measured on five systems, the cheapest available mode deletion costs 1.21 to 7.46 times what balanced truncation's single removal costs.

Tested in The state that is removed is not a mode · the balanced truncation series · refused by a truncated modal sum read as the whole function

“Root-finding by the eigenvalues of a companion matrix is backward stable, so the roots it returns are the exact roots of a nearby polynomial and are therefore accurate.”

The polynomial with roots 1 to 20 is expanded exactly, its companion matrix is factorised, and the computed roots are compared with the integers. The claim that a backward-stable routine returns accurate roots fails: the worst is out by 7.6·10⁻³, and the computation is exact for a companion matrix a rounding away.

Tested in The roots are not the coefficients · the conditioning series · refused by accurate roots from a companion matrix at degree twenty

“Newton converged and the residual is at the level of rounding, so the root has been found.”

On a problem with two roots 10⁻³ apart, Newton converges from either side with a residual of 10⁻¹⁹ and reports nothing about the second. The operator verifies a box of half-width 5·10⁻⁴ — exactly half the separation — and refuses any box wide enough to contain both. The refusal is fed a box containing two roots and required to reject the claim that it contains one.

Tested in Proving the answer is in the box · the interval series · refused by a uniqueness claim for a box that contains two roots

“A wrong answer has two possible authors — the problem's conditioning and the algorithm's backward error — so measuring both accounts for the whole of it.”

A quadratic eigenvalue problem in badly chosen units has κ = 4.98 at every stop of a sweep and a linearised solve at 7.6·10⁻¹³, and loses eleven orders. A nonlinear eigenvalue problem has a residual of 8·10⁻¹⁵ against the object its solver factorised and an answer wrong in the fourth digit. Neither loss is in either factor.

Tested in Three errors and one number · the backward error series · refused by a problem whose condition number is 4.98 and whose algorithm is backward stable at 10⁻¹⁵, returning four digits

“The solution came back oscillating, so the solver did not converge.”

The oscillating vector is the exact solution of the linear system: ‖Ax − b‖/‖b‖ = 4.6·10⁻¹⁸. It alternates at every point and leaves [0, 1] at 16 of 31 grid points, by as much as 0.515, while the continuous problem it discretises is monotone from 0 to 1. No iteration, no preconditioner and no precision changes it, because nothing about the solve is wrong.

Tested in The stencil that is not symmetric · the convection series · refused by an oscillation blamed on the solver

“A computed quantity at the rounding level is a property of the matrix, since nothing larger than a rounding is involved.”

‖QᵀQ − I‖ moves by 2.4% across ten partitionings and 125 of the factor's 1,128 entries change sign. The refusal is fed the assertion that a single reduction is as stable as an aggregate of many, and must reject it.

Tested in An inner product with no fixed sign · the orthogonality series · refused by the claim that a norm is as unstable as its entries

“Once the conditioning, the backward error, the reformulation and the approximation are all accounted for, the answer is determined.”

Twenty-six runs of one sum, with one condition number, one backward error, no reformulation and no approximation, return twenty-one distinct answers. The refusal is fed the assertion that two backward-stable reductions of one vector return one answer, and must reject it.

Tested in The fifth author · the backward error series · refused by the claim that two backward-stable runs return one answer

“The shifted fast-transform preconditioner fails early because shifting is the wrong way to make an approximation invertible.”

The same shift applied to the exact operator leaves the path where its own effective dimension reaches the answer's — 1.05 of it on the collection's blur and 1.02 to 1.05 across three blurs at two noise levels — while the shift on the fast approximation leaves at 0.66, and between 0.49 and 0.77 on the six. The refusal is fed the claim that the two leave at the same share of the answer's dimension to within 0.15, and required to fail.

Tested in The shift had an edge, and the approximation moved it · the iterative regularisation series · refused by a fast shift's early failure blamed on the shift rather than on the approximation

“A residual scan that misplaces the step on every draw, even at 0.01% noise, shows that the data do not contain the step's position.”

At 0.01% noise on 48 points the same scan run at λ = 10⁻⁴ puts the box on the step on 2 draws of 16 and leaves an error of 0.1508; run at λ = 10^−1.5 it puts the box on the step on all sixteen and the error is 0.0050, the hand-placed box's. The refusal is fed the claim that the scan at λ = 10⁻⁴ finds the step on at least half the draws, and required to fail.

Tested in A corner the penalty can afford · the regularisation series · refused by a breakpoint located at an overfitting λ read as found

“Given more digits than a double has, a relation search returns a relation among the rounding rather than among the numbers.”

On the relation essay's own numbers — numerators of about a million over one denominator of about a million — the vector returned from doubles scaled to 30 digits is an exact relation among the numbers on 16 runs of 16, and the planted one on none. Those numbers satisfy integer relations of their own, and the vector returned is the one relation they and their rounding share. On numbers of sixty random digits the vector returned at 25 and 30 digits is exact on none of 32. The refusal is fed the claim that fewer than half the narrow family's answers at 30 digits are exact relations, and required to fail.

Tested in The room a relation has to stand out · the lattice reduction series · refused by a relation found past a double's digits read as false on numbers with relations of their own

“Solving tridiag(−1, 2 + σ, −1) through its periodic circulant and a rank-two correction loses the answer because the correction multiplies the capacitance matrix's condition number onto the cancellation.”

At n = 64 and σ = 10⁻⁸, with the two-by-two capacitance system solved by Cramer's rule, the corrected solve's forward error is 1.2·10⁻⁴. The same correction with that system solved by elimination with a row interchange has an error of 2.9·10⁻¹⁰, against a cancellation times u of 1.7·10⁻¹⁰ and elimination on the band matrix at 1.1·10⁻¹⁴. The refusal is fed the claim that the error does not depend on how the two-by-two system is solved, and required to fail.

Tested in The correction lost to its own two-by-two solve · the circulant series · refused by the corrected circulant's loss on the Laplacian attributed to the correction rather than to Cramer's rule

“The optimal artificial diffusion is the right amount to add to a convection-dominated problem.”

On the boundary-layer problem the tuned diffusion gives a nodal error of 2.4·10⁻¹⁷. On a smooth manufactured solution of the same operator at the same ε and the same grid it gives 6.7·10⁻², against 1.6·10⁻³ for adding nothing at all. The refusal is fed the claim that the tuned scheme improves the answer there and requires it to fail.

Tested in The diffusion that makes the answer exact · the convection series · refused by the optimal artificial diffusion claimed optimal for a problem it was not derived from

“Recording the seed makes a randomised computation reproducible.”

At a fixed seed, five partitionings of the sketch's products return more than one error. The refusal is fed the assertion that a recorded seed makes a randomised method return one answer, and must reject it — the seed fixes the draw and not the arithmetic.

Tested in The variation that comes with a seed · the Run-to-run variation series · refused by the claim that a seed makes a randomised method reproducible

“The three deliberate tolerances agree to within a factor of 5.7, and that factor measures how alike the trade is in three fields sharing no arithmetic.”

The factor of 5.744 is the truncation slope over the deflation slope, and neither of those curves has a problem attached: sweeping the only problem the comparison contains, a Poisson grid from 36 to 196 unknowns, moves the drop slope through 0.188, 0.040, 0.051, 0.054 and 0.057 while the reported spread reads 5.744 at all five. The truncation slope is 9.0152 divided by the size of a test matrix, exactly, at every size tried, so the spread is 11.49 at forty unknowns fewer and 2.87 at forty more. The refusal is fed a truncation tolerance of two, which keeps no singular values at all and is reported as a rank-one point whose accepted error is 0.600 — the decay constant the test matrix was built from, standing in for a measurement.

Tested in A tolerance is priced by the problem · the deliberate zero series · refused by a low-rank approximation whose tolerance keeps nothing

“The tropical roots locate both ends of a matrix polynomial's spectrum from three norms, and at the small end the estimate simply degrades as the problem grows — losing a factor like n², the way a first-order bound loses accuracy.”

The assertion that the small tropical root lies within fifty per cent of the smallest modulus is handed the overdamped chain at thirty-two masses, and returns 229.6. The measurement is right and the cause named is not: across n = 4 to 64 the estimate moves from 0.333755 to 0.347275, four per cent, while the modulus it is supposed to locate falls from 6.2325·10⁻² to 3.8921·10⁻⁴.

Tested in An estimate that does not move · the polynomial scaling series · refused by a norm-based estimate of the smallest modulus

“A route to 1 − h through the orthogonal factor loses a digit for every decade of the condition number, whatever else it does.”

A straight-line fit of thirty-one points on [−1, 1] with two of them weighted by 4²⁵ and 4²⁴ has κ(A) = 3.1, so κ(A)·u is 3.4·10⁻¹⁶. Placed after the twenty-nine light rows, the heavier row's 1 − h from the complementary block is wrong by 2.4·10⁻¹⁰; placed first, it is exact. The refusal is fed the claim that the error is within ten times κ(A)·u in the first order, and required to fail.

Tested in The weight the factor met first · the leverage series · refused by the complement's error read as κ(A)·u under weights

“A kernel whose singular values decay more slowly carries higher ranks, and a higher rank pushes the smallest blocks out of the range in which compression pays, so the leaf size that stores least is larger on a rougher kernel.”

At n = 512 and eight digits the largest rank runs 5, 5, 9 and 10 across four kernels and the leaf storing least is 16, 8 and 16 tied, 16 and 16. It does not move. Swept over accuracy, it moves once — the roughest kernel prefers 32 at twelve digits, where its largest rank is 14. The refusal is fed the claim that the roughest kernel wants a larger leaf at eight digits, and required to fail.

Tested in A prediction that arrives a decade late · the storage growth series · refused by a rougher kernel claimed to want a larger leaf

“Sorting a weighted least-squares problem's rows by decreasing weight makes its deletion diagnostics accurate.”

A straight-line fit of thirty-one noisy points with one row weighted by 4²⁴ and placed first. The deleted residual formed the textbook way, as b − Ax over one minus the thin factor's row norm, is wrong by 2.8·10⁻², against 1.4·10⁻¹⁵ for the same quotient taken from the complementary block. The refusal is fed the claim that the textbook quotient keeps its digits once the rows are sorted, and required to fail.

Tested in The residual the solution cannot hold · the leverage series · refused by the row sort read as a repair of the deletion numerator

“Past its optimum, conjugate gradients' error rises more slowly than Tikhonov's at the same effective dimension because the iteration spends its dimension where the data has content.”

On the 64-point deconvolution with a blur 1.5 points wide at 0.1% noise, over twelve draws, Tikhonov's error at 1.3 of the answer's effective dimension is 2.35 times the iteration's. Matched instead on the sum of the factors with each clipped to one, it is 0.96 times, and on all six problems Tikhonov's noise part at 1.3 is 0.85 to 0.99 of the iteration's. The refusal is fed the claim that the lead survives the clipped count at more than 1.5, and required to fail.

Tested in The overshoot was the lead · the iterative regularisation series · refused by the iteration's lead past the top read as a property of the filter

“The smallest-pivot rule keeps a fraction-free elimination cheap because it is a proxy for choosing the shortest rows, which is what bounds the minors.”

Over twenty 10 × 10 integer matrices with entries in ±6, the shortest-original-row rule keeps the area under the bit-length profile at 0.966 of the natural order's against the smallest pivot's 0.969, but its schoolbook cost of every product and division is 0.841 against 0.704: the smallest pivot entry is cheaper for a reason the row lengths do not carry. The refusal is fed the claim that the two rules' median costs are within five per cent, and required to fail.

Tested in The pivot is in every product · the Fraction-free series · refused by the shortest-row rule claimed to cost what the smallest-pivot rule costs

“The forward error is the residual a solver prints multiplied by a condition number, so an eigenvalue whose condition number is about five and whose reported residual sits at the rounding level has been computed to nearly full precision.”

The claim is fed the eigenpairs of a quadratic whose coefficients have been rescaled by an exact change of variable. At γ = 10⁸ the forward error is 1.26·10⁻³ and the linearisation's own backward error is 7.59·10⁻¹³, so the multiplier the claim needs is 1.66·10⁹ and the eigenvalue's condition number is 4.98. The assertion that the two are within four orders of each other fails by five.

Tested in The number that moves when the problem does · the linearisation backward error series · refused by a forward error predicted from the linearisation's own residual

“A nearly dependent pair of equality constraints costs a constrained fit κ(B) times the unit roundoff, and that loss is what large multipliers warn of.”

At ε = 10⁻¹², κ(B) = 4.64·10¹² and κ(B)·u = 5.2·10⁻⁴ for two right-hand sides. When the third constraint's datum asks for a new condition the multipliers reach 3.8·10¹⁰ and the null-space route is wrong by 1.16·10⁻⁴; when it asks for nothing they are 2.0·10⁶ and the error is 2.1·10⁻⁵. Across twelve decades of strain at ε = 10⁻⁸ the error stays between 7.8·10⁻⁹ and 1.8·10⁻⁸ while the multipliers go from 1.4 to 1.3·10¹². The refusal is fed the two cases at ε = 10⁻¹² and the claim that the error rises with the multipliers to within a factor of a hundred — the multipliers are 1.9·10⁴ times apart and the errors 5.5 — and required to fail.

Tested in A multiplier is a force · the Constrained least-squares series · refused by multipliers read as the size of a constrained fit's error

“A sparsity-first pivot rule does well on a saddle-point matrix with a zero constraint block because it defers the constraint rows to the end, by which time elimination has filled their diagonal.”

On the saddle-point matrix of 24 variables and 8 constraints each touching 3 of them, a minimum-degree order computed from the pattern alone holds 80 entries in its factor against the natural order's 161 and the step-by-step sparsest rule's 81, and eliminates the constraint rows at steps 5, 8, 12, 16, 19, 23, 27 and 31 of 32 — one of them in the last eight. The refusal is fed the claim that the ordering puts every constraint row last, and required to fail.

Tested in An order fixed before the numbers · the sparse pivoting series · refused by the sparsest pivot's advantage read as deferring the constraint rows

“A stationary vector that loses its far well in half precision was lost to the balance matrix's ill-conditioning, which is past the reciprocal of the unit roundoff.”

At a barrier thirty octaves deep the balance matrix's normwise condition number is 1.45·10⁹, over seven hundred thousand times the reciprocal of half precision's unit roundoff, and the same eleven-bit significand with no limit on its exponent returns the far well with a relative error of 7.0·10⁻⁴; every state's componentwise condition number is 95 at every depth. The refusal is fed the claim that no eleven-bit arithmetic determines the far well there, and required to fail.

Tested in The well on the far side of the band · the subnormals series · refused by the empty far well blamed on the balance matrix's normwise condition number

“An oscillatory kernel block's rank grows with the size of the block, so the larger of its two clusters decides what it costs.”

At separation ratio one half and κ = 40, a target of length a quarter against a source of length four needs 10 columns at 10⁻⁸, and the square block of side four needs 33; with the target held at a half, sources from one to four all need 13. The refusal is fed the claim that the quarter-by-four block needs at least four fifths of the square block's rank, and required to fail.

Tested in The smaller cluster sets the rank · the Off-diagonal rank series · refused by the oscillatory rank read as belonging to the larger of the two clusters

“The parameter-choice rules that miss a Tikhonov parameter by orders of magnitude are unreliable rules, and they will miss a polynomial degree the same way.”

Over a hundred draws at each of three noise levels, generalised cross-validation, leave-one-out, the Bayesian information criterion and the discrepancy principle choose the degree of a polynomial fit in an orthonormal basis, and the worst draw of any rule costs 2.69 times the best degree's error. The refusal is fed the claim that some rule misses by a hundredfold on some draw, and required to fail.

Tested in The degree that is safe to overshoot · the fitting series · refused by a degree rule read as missing the way a Tikhonov parameter rule misses

“Bunch–Kaufman's unbounded multipliers make the factors unsafe for anything that later uses them — the solve, its refinement and a condition estimate all inherit the size of L.”

On a 24 × 24 symmetric matrix whose coupling of 10⁻¹⁰ gives Bunch–Kaufman multipliers of 1.2·10¹⁰, the solve's normwise backward error is 1.6·10⁻¹⁶ and after one step of refinement 6.3·10⁻¹⁷, and ‖|L||D||Lᵀ|‖ is 6.72 times ‖A‖ at every coupling from 10⁻¹ down. The refusal is fed the claim that the refined backward error is a thousand times worse than the unrefined one or above 10⁻¹⁰, and required to fail.

Tested in Where the multipliers go · the cholesky series · refused by the solve and its refinement read as inheriting the size of L

“A one-nonzero sketch's error on a nearly coherent matrix fades by a fixed fraction for every factor of ten of mixing, so each draw improves steadily as the matrix becomes less coherent.”

At singular-value decay 0.8, the median over eighty draws falls by a factor of 0.57 and then 0.62 per decade from t = 10⁻³ to 10⁻¹, but the median draw goes from a fifth to four fifths of its own fall in 1.9 decades, and the draws' falls are centred anywhere from 10⁻³ to 10⁻⁰·⁸. At decay 0.9 the median holds at 1.83 until t = 10⁻² and then falls within a decade and a half. The refusal is fed the claim that a typical draw fades across three decades or more, and required to fail.

Tested in A fade made of drops · the randomised series · refused by the median's constant fraction per decade read as each draw's rate

“A contour count converges at a rate set by the distance from the contour to the nearest eigenvalue.”

Two circles each 0.2 from their nearest eigenvalue — radius 1.022 beside 0.822, radius 2.886 beside 2.686, each with the next eigenvalue far off — buy 0.0948 and 0.0312 digits a point, a factor of three apart. Both agree with log₁₀ of the ratio of moduli to the third figure. The claim that equal distances give rates within a fifth of each other is fed the two circles and fails.

Tested in The circle between two eigenvalues · the nonlinear cost series · refused by the rate set by the distance to the nearest eigenvalue

“Blocking a Householder factorisation makes its orthogonal factor more nearly orthogonal, by about a factor of two over one reflector at a time.”

Formed, the factor departs from orthogonality by 7.5·10⁻¹⁵ one reflector at a time and 3.6·10⁻¹⁵ in blocks of sixteen. Applied through its stored blocks, it departs by 3.4·10⁻¹⁵ and 3.2·10⁻¹⁵, and after eight blocks of eight by no more than after the first. The refusal is fed the claim that the applied factor's departure grows past twice its first block's over the eight blocks, and required to fail.

Tested in The factor nobody forms · the householder series · refused by the applied factor read as accumulating its blocks' departures

“One level of nested dissection lengthens the factorisation's critical path because of the shape of the halves it leaves, on which minimum degree does worse than on the whole grid.”

On the 24 × 24 grid the depth-one path is 53,508 with each half ordered by minimum degree on its own, and 41,050 with the same halves ordered by minimum degree counting the separator in every degree — within 0.1% of minimum degree's 41,072 on the whole grid — while the path through the halves' own rows is 17,813 either way. The refusal is fed the claim that the depth-one path stays at least a fifth above minimum degree's when the halves are ordered with the separator counted, and required to fail.

Tested in Pieces ordered blind · the ordering series · refused by the depth-one penalty read as the shape of the halves

“Reducing Ax = λBx through a Cholesky factor of B rather than through B⁻¹A is what keeps the eigenvalues accurate.”

On a pencil whose eigenvalues are exactly known by construction, the two reductions have fitted error slopes of 0.92 and 0.98 against κ(B) and stay within a factor of 2.3 of each other over fifteen decades. The refusal is fed a pencil whose second matrix is negative definite, where the Cholesky factor the reduction is defined in terms of does not exist and a routine that continued would be reducing by a matrix that is not there.

Tested in Two matrices and one problem · the pencil series · refused by a symmetric-definite reduction of a pencil whose second matrix is negative definite

“Streamline diffusion damps the oscillation without smearing the layer, because the diffusion it adds acts along the flow.”

τbbᵀ applied to any vector perpendicular to b is exactly zero. The refusal is fed the claim that the streamline tensor acts across the flow and required to fail.

Tested in The direction the diffusion does not go · the convection series · refused by a rank-one tensor claimed to damp the direction it annihilates

“A Jacobian accurate to only ten digits limits how far a Newton method can converge.”

Newton on the Bratu problem with an analytic Jacobian and with one differenced at ε = 10⁻⁶ — accurate to 1.3·10⁻¹⁰ — produce the same residual sequence and stop at the same 4·10⁻¹³. The refusal is fed the claim that the differenced run stops a hundred times short, and required to fail.

Tested in An operator with no entries · the Matrix-free series · refused by an approximate derivative claimed to limit the accuracy of the answer

“A small residual on every computed eigenvalue means the eigenvalues are right.”

A pencil whose two matrices share a null vector has a characteristic polynomial with zero coefficients — computed exactly in BigInt rationals, not judged small. Perturbed by 10⁻⁸ and handed to a solver, it returns six eigenvalues per seed with residuals below 1.8·10⁻⁹, spread across seeds by up to forty-four. The refusal is fed a request for the eigenvalues of such a pencil, where returning a finite set of them is the failure the whole essay is about.

Tested in A problem with no answer · the pencil series · refused by the eigenvalues of a singular pencil, which has no finite set of them

“A breakdown in a Krylov method means the method has found something, so it is safe to treat it as convergence.”

The two-sided recurrence divides by ⟨w̃, ṽ⟩, which on 13.3 per cent of random 5×5 matrices with entries in {−2,…,2} vanishes with both vectors of full size and no subspace invariant. The refusal is fed a starting pair that is already bi-orthogonal to itself, so the divisor is zero before any step has been taken — a badly chosen shadow vector, not a fact about the matrix, and a case a routine that treats every zero as convergence reports as an answer.

Tested in The same zero, and nothing was found · the breakdown series · refused by a two-sided recurrence whose starting pair is not bi-orthogonal

“One-sided Jacobi computes small singular values to high relative accuracy, so it is the method to use whenever the small ones matter.”

On a bidiagonal with unit diagonal and superdiagonal 4096, whose smallest singular value is 5.2·10⁻²⁶, one-sided Jacobi returns the seven large values to 4.4·10⁻¹⁶ and σₘᵢₙ wrong by 1.45·10⁻², while a shifted sweep gets all eight to 5.6·10⁻¹⁶ in sixteen sweeps. The refusal is fed an infinite entry to the exact converter, where a reference that is not the matrix in question makes every relative error in the comparison a number about something else.

Tested in Accurate is not a property of a method · the relative accuracy series · refused by an infinite entry handed to the exact converter

“A conjugate gradient iteration that meets a non-positive curvature has failed and should report an error.”

On a 40×40 matrix that is positive definite apart from one eigenvalue at −0.1, the iteration meets a non-positive curvature after six products and stops with a unit direction whose Rayleigh quotient is −2.66·10⁻². That vector is checkable from outside in one further product, and it is the output a trust-region method wants. The refusal is fed the claim that such a direction exists on a matrix built to be positive definite, where a routine reporting one would be reporting a rounding error as a proof.

Tested in The division that cannot be done · the breakdown series · refused by a certificate of negative curvature claimed on a positive definite matrix

“Forming BᵀB is a large-matrix hazard. On a small problem the smallest singular value survives the squaring, and the risk grows with the size of the matrix.”

Across n = 4, 6, 8, 10 and 12 the grading at which the route through BᵀB stops returning a single correct digit of the smallest singular value falls in one window between 8.43 and 9.03 decades, shared by every size. Over the same range one-sided Jacobi's worst error rises from 6.4 to 17 units of roundoff. The size moves the constant of the routes that hold and not the point at which the squaring route fails, so the failure is real and the size is not what sets it. The refusal is fed n = 20, where the exact reference the whole comparison is measured against costs a hundred times what it costs at n = 4, and must refuse rather than let a comparison of two floating-point routes stand in for it.

Tested in A threshold the matrix does not set · the relative accuracy series · refused by a rational Sturm bisection asked for at a size the build cannot afford

“The Péclet number is a property of the equation, so a convection-dominated problem is one that cannot be discretised rather than one that has not been discretised finely enough.”

The cell Péclet number is h/2ε. The assertion that it does not depend on the mesh is fed its value on 15 points and on 255 points at the same ε — 6.25 against 0.390625 — with a tolerance of 10⁻⁹, and must fail. The factor of sixteen between them is exactly the factor by which the artificial diffusion falls across this essay's sweep.

Tested in A different equation on every grid · the convection series · refused by a Péclet number with no mesh in it

“A rank rule that miscounts is miscounting because the data has been perturbed.”

The largest-gap rule is run on twenty-five descriptor pencils whose entries are integers, whose rank deficiency is exact, and to which nothing has been added. It returns the wrong count on nine of them, so the miscount is real and the perturbation named as its cause is absent. The refusal is the degenerate end of the same axis: a pencil with no constraints has no infinite eigenvalues and no null block for a gap to hide in, both rules are trivially right, and the figure is fed that case and required to refuse it rather than draw the comparison at its easiest.

Tested in The largest gap is inside the null space · the pencil series · refused by a deficit figure drawn on a pencil that has none

“The residual bound a restarted method reports says how good its answer is, so the setting whose convergence curve ends lower has the better answer — and the way to a lower curve is a longer basis.”

Five splits of one budget of about a hundred and forty products with A end with reported bounds of 6.11·10⁻¹¹, 4.03·10⁻¹², 1.08·10⁻¹³, 2.37·10⁻¹⁷ and 1.42·10⁻²¹, and with true errors of 1.24·10⁻¹⁴, 7.11·10⁻¹⁵, 7.11·10⁻¹⁵, 1.24·10⁻¹⁴ and 1.78·10⁻¹⁵. The reported quantity moves by ten orders and the answer by a factor of seven, with no trend in the split. The far end of the same axis is fed to the assertion that a restart basis must be smaller than the problem it restarts on, as a basis of twenty-four vectors on a twenty-by-twenty matrix, and is required to reject it.

Tested in The same budget, spent five ways · the lanczos series · refused by a restart basis larger than the problem

“θ = 0.25 is a safe default because it is calibrated to separate the strong direction from the weak one.”

The strength measure is built on the anisotropic operator at ε = 10⁻³ and required to find no horizontal coupling strong at θ = 0.25. The assertion that it finds some is fed that matrix and must reject it — so the constant does separate the directions, exactly as advertised, and separating them is not the same as choosing the faster method.

Tested in The switch does not know which side is better · the algebraic multigrid series · refused by the weak couplings counted as strong at θ = 0.25

“Finding a direction of negative curvature is an eigenvalue computation, so it costs what an eigenvalue computation costs and gets steadily worse as the matrix grows.”

Across n = 20, 30, 40, 60 and 80 the curvature test fires after three products at λ = 3 — four at one of the five sizes — and after nine to eleven at λ = 10⁻³, while a third of n³ runs from 2,667 to 170,667. The cost is set by where the negative eigenvalue sits in the spectrum and not by how many other eigenvalues there are. What cannot be afforded at n = 200 is the sweep that measures this, which builds eight dense matrices from prescribed spectra at about 8n³ multiply–adds; the certificate it measures costs eleven products there. The figure family is fed a sweep at that size and required to refuse it.

Tested in A proof that does not ask how large the matrix is · the breakdown series · refused by a detection sweep drawn at a size it cannot afford

“A Krylov method that reaches machine precision has converged, and the number of steps it took is a measurement of its rate.”

The second-order recurrence's actual rate on the chain of forty masses is measured at six vectors, where the distance to the dominant eigenvalue is 1.4·10⁻². The assertion that six vectors put it below 10⁻⁸ is fed that number and must throw. It does — and the same run reads 4.7·10⁻⁵ at thirty-nine vectors and 1.9·10⁻¹⁵ at forty, which is the chain's own length. The machine-precision reading is the space ending, not the rate arriving.

Tested in The answer that arrives when the space runs out · the krylov series · refused by convergence claimed at six Krylov steps

“A dense matrix costs n³ to factorise.”

It costs n³ if the only thing known about it is that it is dense. A circulant of size n has no zero entry and is described by n numbers, and its factorisation is a transform: A = F* Λ F, with F known in advance and Λ one transform away. The residual printed on the figure — 4.7·10⁻¹⁶ — is the same quantity an elimination's badge carries, from a factorisation that never formed a factor.

Tested in The matrix that is one row · the circulant series · refused by a circulant with a zero eigenvalue

“The number of conjugate gradient iterations is set by √κ.”

Asymptotically. At step 20 a κ = 10⁶ problem sits inside the bound belonging to κ = 100 — 1.7·10⁻² against 3.6·10⁻² — because early convergence is governed by how the eigenvalues cluster and only the tail by κ. The bound holds at every step and is loose by an order of magnitude: 400 iterations permitted on the model problem, 40 taken, closest approach 43%.

Tested in The rate the condition number predicts · the krylov series · refused by the bound belonging to κ = 1

“Blocking makes an elimination communication-optimal.”

It buys a constant, not an exponent. Fitted across n = 16 to 64, the blocked traffic grows at n^3.182 and the unblocked at n^3.185 — the same exponent to within the fit's own noise — and the ratio between them is 2.19, 2.11, 1.98, 2.17, 2.17. Against the Hong–Kung floor of n³/√M the blocked traffic sits at 1.50, 2.03, 2.25, 2.05, 2.04 times, which is a constant above the floor rather than a match to it.

Tested in A block size is a property of the machine · the blocking series · refused by a saving claimed for a matrix that fits in fast memory

“The condition number of this Toeplitz family is ((1+ρ)/(1−ρ))².”

That is the limit as the size grows, and every section is strictly below it. At ρ = 0.8 the limit is 81.000 and the measured κ is 42.35 at n = 8, 59.98 at n = 16, 72.20 at 32, 78.05 at 64 and 80.14 at 128 — 52%, 74%, 89%, 96% and 98.9% of it. The refusal is fed the claim that the 16×16 section attains its own asymptotic value and requires it to fail.

Tested in A limit the matrix never reaches · the toeplitz series · refused by Szegő's limit claimed as a value at a finite size

“Nested dissection is the ordering to use; it has the better asymptotics.”

It loses to minimum degree at every size this site draws — 1,413 entries against 1,026 on the 12×12 grid — and minimum degree is the cheap heuristic. A complexity class is a statement about a limit, and the fill exponent fitted across grid sizes 6 to 12 is 1.488 for the factor against 1.044 for the matrix, which is the quantity the asymptotics are about.

Tested in The order decides the memory · the ordering series · refused by the claim that a good ordering removes fill rather than limiting it

“Circulant preconditioning makes Toeplitz conjugate gradients converge in a number of steps independent of n.”

It does, once the preconditioner is positive definite, and at ρ = 0.95 that has not happened by n = 128. Measured on the same systems: 21 steps against 21 at n = 16, 55 against 40 at n = 32, 109 against 66 at n = 64, 134 against 111 at n = 128, and then 10 against 179 at n = 256. Four of those five rows are the method costing steps rather than saving them.

Tested in A preconditioner that changes sign · the preconditioning series · refused by a preconditioner claimed to help at every size

“Circulant preconditioning of a Toeplitz system gives a step count independent of the size.”

On square grids of 16, 36, 64 and 100 unknowns the preconditioned count is 10, 18, 20, 21 — climbing — while the one-dimensional matrices of exactly those sizes give 7, 10, 10, 10. The refusal is fed the claim that the two-dimensional count does not depend on the grid and requires it to fail.

Tested in Two dimensions, and the cluster that thins · the toeplitz series · refused by the one-dimensional clustering result claimed in two dimensions

“Cholesky's growth factor is bounded by one, so it is stable.”

The growth factor is measured at nine combinations of size and condition number and is exactly 1 at every one of them. The refusal is fed the claim that an SPD elimination shrinks the largest entry, and required to fail.

Tested in A factorisation with nothing to pivot for · the cholesky series · refused by Cholesky's growth factor read as a strict inequality

“Gauss–Seidel converges twice as fast as Jacobi.”

Its spectral radius is the square of Jacobi's, which is a statement about the rate rather than about the count, and at n = 24 the counts are 1,574 against more than 3,000 — a ratio that is not two and would not be two at any other size either. Optimally relaxed SOR takes 128, and its iteration matrix is defective at ωₒₚₜ, so even its rate holds only in the mean.

Tested in A rate that is known in advance · the stationary series · refused by Jacobi's rate against Gauss–Seidel's

“Newton's iteration converges quadratically, so a handful of steps is enough.”

At κ = 10⁶ the unscaled iteration's error falls by a factor of 0.5000 per step for the first several steps — linear, not quadratic — and is still 10³ away after twelve. The refusal is fed the claim that it falls faster than a half a step, and required to fail.

Tested in An iteration that only multiplies · the polar decomposition series · refused by a quadratically convergent iteration claimed to be quadratic before it is close

“Replicating the data c times reduces a matrix multiplication's communication volume by √c — that is what the 2.5D algorithms buy.”

At 576 processors four layers save a factor of 1.44 against the law's 2, and at 64 they cost a factor of 1.14. The refusal is fed the claim that the saving reaches √4 at 576 processors and required to fail.

Tested in Memory bought with messages · the communication series · refused by the asymptotic saving claimed at a size anybody runs

“Bounding the rank of every block makes the representation cheaper.”

At 256 unknowns the strong partition has 112 blocks and a worst rank of 5 and stores 27,008 numbers; the weak partition has 46 blocks and a worst rank of 12 and stores 24,064. The refusal is fed the assertion that the strong one stores fewer and required to fail.

Tested in The test that costs what it saves · the admissibility series · refused by the finer partition is the cheaper one

“The symbolic phase computes the structure in advance, so the memory can be allocated before any arithmetic runs.”

Without interchanges the symbolic count is exact as an integer — 233 and 233 on the same grid. With partial pivoting it is 233 predicted against 242 measured, and the George–Ng structural bound that does hold gets looser as the problem grows: 1.39× at 4×4 to 1.68× at 7×7. A bound that loosens with size cannot be allocated from, which is the practical conclusion.

Tested in What the symbolic phase can only bound · the sparse pivoting series · refused by a symbolic fill count offered as a prediction under pivoting

“A contour integral of an analytic function counts the zeros inside the contour, so evaluating it numerically gives an integer and the integer is the count.”

The counting integral is taken on a circle of radius four with an eight-point trapezoidal rule and rounded. The claim that a contour integral is an integer however coarsely it is taken is fed the result and fails: the answer is minus two, where four eigenvalues lie inside.

Tested in A problem with infinitely many eigenvalues · the nonlinear eigenvalue series · refused by an eight-point count of a contour of radius four

“A hierarchical solver is faster than a dense factorisation.”

At 64 unknowns the recursion performs 1.29·10⁵ multiplications against the dense factorisation's 8.74·10⁴ — 1.48 times as many, for an answer at the same backward error. The refusal is fed the assertion that the ratio is below one at every size on the sweep and required to fail.

Tested in Where the format starts paying · the hierarchical solve series · refused by the hierarchical solve is cheaper at every size

“Accuracy gets more expensive as you demand more of it: the first digits are cheap and each further one costs at least as much as the last.”

The counting integral is taken at 8, 16, 32, 64 and 128 quadrature points and the factor by which the error falls at each doubling is computed. The claim that a doubling buys no more than the one before it is fed the four factors and fails at every one of them: the digits gained per doubling are 0.6, 1.5, 3.2 and 6.4.

Tested in The last digit is the cheapest · the nonlinear cost series · refused by a doubling of the quadrature that buys more than the one before it

“A randomised contour method returns the eigenvalues inside its contour, so what it returns is what is there.”

Beyn's moment is formed with a two-column random probe block on a contour containing four eigenvalues, and its rank is measured. The claim that the method returns every eigenvalue inside the contour fails: the rank is two, both of the returned pairs are genuine, and nothing in the result mentions the other two.

Tested in Counting what is inside a circle · the trace estimation series · refused by two probes for four eigenvalues

“The saving from circulant preconditioning grows with the problem size.”

The saving is a ratio whose numerator stops. At ρ = 0.5 the unpreconditioned count reads 15, 22, 29, 28, 30, 30 across n = 16 to 512 and then 29, 28, 26, 27, 25 out to n = 16,384, against an averaged-circulant count of 8, 8, 7, 6, 6, 5 — so the ratio peaks at 6.0 and falls to 5.6. The refusal is fed the comparison at ρ = 0.05, where the family has no conditioning in it for a preconditioner to remove and the ratio of two step counts is a ratio of two constants, and requires it to fail.

Tested in A speedup with a ceiling of its own · the preconditioning series · refused by a preconditioner comparison on a family with no conditioning in it

“Rook pivoting costs a small constant multiple of partial pivoting's search, so it is always far cheaper than complete pivoting.”

On Gaussian matrices the rook compares 3.2 to 3.4 times partial pivoting's entries at every size from 8 to 80. On a matrix whose entries rise along a staircase it compares 1,976 entries at n = 16, about 14,700 at 32 and 113,376 at 64, against complete pivoting's 1,496, 11,440 and 89,440. The refusal is fed the claim that the rook never compares more than complete pivoting and required to fail.

Tested in A pivot that searches one row and one column · the pivoting series · refused by a rook search claimed never to cost more than a complete one

“A stationary iteration with spectral radius below one converges, and the radius tells you how fast.”

A 6×6 with ρ = 0.8 has ‖Aᵏ‖ rising to 1.98·10⁴ at step 24 before it turns over. The refusal is fed the claim that the powers decay from the start, and required to fail.

Tested in A spectral radius that grows first · the Non-normality series · refused by a spectral radius read as a statement about every power

“The exponential series converges for every matrix, so summing it until the terms are negligible computes the exponential.”

On the 2×2 with eigenvalues −1 and −17 and a norm of 64, the largest term of the series has norm 1.4·10⁷ and the sum has norm 2.6 — a cancellation of 5.4·10⁶ — and the computed answer has a relative error of 5.2·10⁻⁹ where scaling and squaring returns 8.5·10⁻¹⁴. The refusal is fed the claim that the cancellation is negligible, and required to fail.

Tested in The series that has to be squared back · the matrix function series · refused by the Taylor series offered as a method for the matrix exponential

“A random real tensor with two typical ranks takes each of them a substantial share of the time, as a 2 × 2 × 2 tensor takes rank two with probability π/4 and rank three with probability 1 − π/4.”

For real n × n × 2 tensors with standard normal entries the typical ranks are n and n + 1 at every n. Over 10,000 draws at n = 6 the tensor has rank 6 on 385, 3.85 per cent; over 4,000 at n = 8, on 16; at n = 10, on none. Every draw counted as rank n was decomposed into n rank-one terms with a relative residual below 10⁻¹¹. The refusal is fed the claim that rank n is taken on more than a fifth of draws at n = 6, and required to fail.

Tested in The rank that stops being typical · the tensor rank series · refused by the lower typical rank read as a substantial share at every size

“Computing eᴬᵗb means computing eᴬᵗ and multiplying. The exponential is the hard part and the multiply is free.”

On a 100×100 matrix whose exponential is known in closed form, twenty matrix–vector products give the vector to 4.3·10⁻¹⁶ — the accuracy of the full dense exponential — at 0.40 against 2.00 megaflops. The refusal is fed a dimension at which the dense route cannot be run at all, so the comparison the claim assumes does not exist.

Tested in The vector was what was wanted · the matrix function series · refused by a Krylov comparison drawn where the dense route cannot be run

“GMRES converges in at most n steps, and that is the bound worth knowing.”

On a 30×30 matrix with m distinct eigenvalues, the recurrence stops at step m for every m from 2 to 16 — never at 30 — with a relative residual below 6·10⁻¹⁶ there and at least 3·10¹⁰ times that one step earlier. The refusal is fed a recurrence with no first basis vector at all, where every subsequent step would divide by zero and the basis would orthogonalise perfectly against itself.

Tested in The zero that means it is finished · the breakdown series · refused by an Arnoldi recurrence started from the zero vector