The right measurement, blamed on the wrong thing

“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 right-hand side with little of the saddle's growing direction delays refinement's verdict by the time the saddle's own growth takes to recover the missing share.”

On a saddle of curvature −0.01 regularised by δ = 7, the verdict moves 147.3 steps for each decade of the growing direction removed from the right-hand side. The growth of 1.0014 a step alone would take 1,611 steps a decade; against the minimum's slowest contraction of 0.986 a step it predicts 147.5. The verdict comes when the growing direction overtakes the decaying one, and the decay does nine tenths of the work. The refusal is fed the claim that the delay a decade is within a fifth of the growth's own recovery time, and required to fail.

Tested in The right-hand side that hides the saddle · the Quasi-definite series · refused by a hidden saddle's delay read as the growth's recovery time alone

“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

“Past the answer conjugate gradients falls behind fixed filters at the same count of directions because its factors overshoot one.”

At 1.3 of the answer the ramp's error is between 0.85 and 0.95 of the iteration's on six problems. Cutting the iterate's factors back to one recovers between 28 and 58 per cent of that gain — on the narrowest blur at 0.1% noise, 0.074 of 0.139. The rest is the tail's weight against the corner, which a filter built from the iterate's smallest Ritz value and its tail's slope reproduces to within 0.056 without any overshoot at all. The refusal is fed the claim that clipping recovers over four fifths of the fixed filter's gain there, and required to fail.

Tested in The first step's filter, kept for the whole run · the iterative regularisation series · refused by the iteration's deficit past the top blamed on its overshoot

“A random rank-three tensor fitted at rank two is the border-rank swamp again, with error traded one for one against the size of the cancelling terms.”

The border-rank tensor is at distance zero from the rank-two set and its error falls as one over the term size. A random rank-three 2 × 2 × 2 tensor is at a positive distance, found here by a constrained Newton iteration to the discriminant's zero set; the fit's error minus that distance falls as one over the term size squared, its product with the size changing by under a factor of two from 100 sweeps to 8,000 while its product with the size alone falls more than fivefold. The refusal is fed the claim that the excess trades one for one against the size, and required to fail.

Tested in A fit with no answer to find · the tensor rank series · refused by a random rank-three tensor's swamp read as the border-rank exchange rate

“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

“When an eight-bit stationary vector loses its far well, the loss is the narrow exponent range's, and a format with the same significand and an unlimited exponent would keep it.”

With E5M2's three significant bits and no limit on the exponent at all, eight of nine chains return a far well wrong by half its value at some depth before forty octaves, the first at 21; the median error over 531 runs is 0.146 and a quarter of runs are wrong by more than a quarter. The refusal is fed the claim that an unbounded exponent keeps the far well on every chain, and required to fail.

Tested in The far well in a byte · the subnormals series · refused by an eight-bit far well lost only to the range

“The saddle-point route to a constrained least-squares fit loses accuracy when the multipliers are large because it solves for the answer and the multipliers as one vector, and a backward-stable elimination commits an error proportional to that vector's norm.”

On the near-repeated constraint at ε = 10⁻⁸, eliminated in the row order partial pivoting takes on the unscaled matrix, the error is 1.2·10⁻³ with the constraints as given and 4.3·10⁻³ with them multiplied by 10⁶, which makes the multipliers a million times smaller; eliminated in the order partial pivoting takes once the constraint rows are larger, it is 5.0·10⁻⁹ with the constraints as given, multipliers 1.6·10⁶ times the solution. Partial pivoting's own error falls from 1.2·10⁻³ to 5.6·10⁻⁸ between a scale of one and of ten, the step at which its second pivot changes row. The refusal is fed the claim that at a fixed order multipliers a million times smaller cut the error tenfold, and required to fail.

Tested in The scale that only moved a pivot · the constrained least squares series · refused by the saddle-point loss read as set by the multipliers' size

“When a quadratic's tropical roots are far apart, the small group of eigenvalues is lost because the coefficients are badly scaled, and scaling at the small tropical root restores it.”

On an eight-mass overdamped chain with damping 10⁶, ‖C‖²/(‖M‖‖K‖) = 4.2·10¹¹, the companion pencil reduced by inverting its leading coefficient returns the small group with a worst backward error of 2.7·10⁻⁷ as given, 7.1·10⁻⁷ after Fan–Lin–Van Dooren scaling and 1.0·10⁻⁵ after scaling at the small tropical root; reduced by inverting its constant coefficient, 7.1·10⁻¹⁶. The loss follows the reduced matrix's norm over the eigenvalue, which a two-sided scaling of a pencil reduced to one matrix does not change. The refusal is fed the claim that tropical scaling at the small corner brings the small group's backward error below 10⁻¹³ at that damping, and required to fail.

Tested in Two groups need two reductions · the polynomial scaling series · refused by tropical scaling read as separating the two groups under a standard reduction

“A balanced truncation that cuts inside a tie fails because it leaves a near-integrator, a pole much closer to the imaginary axis than any of the full model's, and that pole is what doubles the error.”

Thirty-two of the 1,452 reductions have a pole closer to the axis than a tenth of the damping. Ten of them are doubled; the other twenty-two have errors between 0.75 and 1.42 times the reading, and one of those has its slowest pole at 4.4·10⁻⁴ of the damping, slower than seven of the ten doubled. The refusal is fed the claim that every near-integrator doubles the error, and required to fail.

Tested in A tie is a gap smaller than its neighbours · the balanced truncation series · refused by a near-integrator read as doubling the error

“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

“When a rescaled recurrence still loses its far well, the floor is too close to the subnormal band, and raising it cures the loss.”

With the floor eight octaves above E4M3's smallest normal and the ceiling at a quarter of its largest, two of nine chains keep the far well; 145 of 531 runs contain a quotient clipped at 448 and only 18 products in all fall into the band. Lowering the ceiling to a sixty-fourth of the largest normal with the floor at four octaves keeps eight of nine, with 27 products in the band and no quotient clipped. The refusal is fed the claim that a floor eight octaves up keeps every far well the unbounded exponent keeps, and required to fail.

Tested in Room for the step, not the value · the subnormals series · refused by the floor rule's losses read as underflow

“The trailing reduction loses the middle group because it inverts the constant coefficient, which is the wrong end for the middle eigenvalues.”

Reversing the polynomial and taking the leading reduction of its companion pencil inverts the same constant coefficient. On the generic family at eight decades it keeps the middle group at 2.2·10⁻¹⁴, where the trailing reduction of the original pencil reads 4.5·10⁻⁹, and keeps the small group at rounding. The refusal is fed the claim that this route also leaves the middle group above 10⁻¹⁰, and required to fail.

Tested in The middle group needs no reduction of its own · the polynomial scaling series · refused by the middle group's loss read as belonging to the inverted coefficient

“The Hermite form's entry growth belongs to its transform — the coefficients of the combination that reports how the answer was reached — while the form's own entries stay bounded.”

Plain integer Hermite elimination, instrumented to record the form's widest intermediate and the transform's separately, on random matrices to n = 80: the form reaches 21, 307, 3,371 and 13,835 bits at n = 4, 16, 48 and 80 against the transform's 18, 304, 3,366 and 13,832 — within five per cent at every size. The refusal is fed the claim that the form stays under half the transform's width on every matrix, and required to fail.

Tested in Every entry under the determinant · the normal forms series · refused by the Hermite form's swell read as belonging to its transform

“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

“The monomial basis fails on Wilkinson's polynomial because its roots are integers spread evenly — a property of the roots, not of the basis.”

With the same number of roots placed at the Chebyshev points of [1, n] instead of the integers, the monomial route's worst error at degree 20 is 4.2·10⁻³ against the integers' 7.6·10⁻³, and the Chebyshev route's is 2.6·10⁻¹⁴. The refusal is fed the claim that the monomial route keeps well-placed roots to 10⁻⁸ at degree 20, and required to fail.

Tested in A basis that has to know where the roots are · the conditioning series · refused by the monomial route's loss read as a property of the integer roots

“A nearly decomposable Markov chain has an ill-conditioned stationary vector, so no method can compute it accurately when the clusters are weakly coupled.”

Every off-diagonal probability of the two-cluster chain multiplied by 1 + 10⁻¹⁰z moves the exact stationary vector by at most 0.92 × 10⁻¹⁰ in relative terms at every coupling from 10⁻¹ to 10⁻¹²; moved by 10⁻¹⁰z times the largest probability instead, it moves by about 0.008/ε times that. The refusal is fed the claim that the relative sensitivity exceeds ten at ε = 10⁻¹⁰, and required to fail.

Tested in No subtraction, and no waiting · the perron frobenius series · refused by a slowly mixing chain read as having an ill-conditioned stationary vector

“The extra Ritz values a Krylov run acquires through loss of orthogonality distort what is read off its Ritz values, so the filter's power past the answer is an artefact of the copies.”

Reorthogonalised, the power m at 1.3 of the answer has the same median to 0.1 on five problems of six, while draw by draw it moves by up to 0.49 because the two runs reach that position at different iterates, and the operator's sum misses it by more than 0.1 on 62 draws of 72. The refusal is fed the claim that m moves by less than 0.1 on every draw, and required to fail.

Tested in The copies cost steps, not the filter · the iterative regularisation series · refused by the copies read as carrying none of the power on any draw

“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

“A congruential generator's lowest bits are its weakest, so a bias in sums of its values comes from them and is removed by replacing them.”

On forty-one seeds of a thousand values near 10⁶, replacing the lowest five bits of every state with unrelated bits raises the one-pass constant from a median of 62 to 107, every seed still over-estimating; the lean ends only at eight bits replaced, the number the running sum discards. The refusal is fed the claim that five replaced bits bring the median within 1.5 times the well-mixed generator's and give both signs, and required to fail.

Tested in The generator that rounded its own product · the cancellation series · refused by replacing the lowest five bits read as removing the lean

“Coupling the copies of an interleaved construction lets the copies' doublings reach a shallow look-ahead's horizon early, so any coupling above the growth they are about to suffer breaks the deferral.”

An entry ε in every off-copy position of the first copy's last column of W5 ⊗ I2 and W6 ⊗ I2 leaves both the greedy and the two-step rule at 2^(m−1) to a millionth at every ε from 2⁻³⁰ to one, because that column takes part in no pivot choice until the last step. The refusal is fed the claim that the two-step rule escapes for every ε above 2^(−m), and required to fail.

Tested in The doubling is in the row already · the elimination series · refused by a leak in the first copy's last column read as letting the two-step rule escape

“The colleague matrix's roots for a polynomial with badly placed roots are as accurate as the roots' conditioning allows, since the eigenvalue computation is backward stable.”

On Wilkinson's polynomial in the Chebyshev basis on [1, n], the colleague matrix's eigenvalues are more than ten times the normwise conditioning times u from degree 24 on — 60, 90, 300 and 7,000 times at 24, 28, 32 and 40 — because the backward error is relative to the matrix's norm, which is the coefficient ratio, up to 4.4·10⁴. Balanced, the same roots are within 0.5 to 2.4 times the conditioning at every degree from 14 to 40. The refusal is fed the claim that the unbalanced error is within ten times the conditioning at every degree from 14, and required to fail.

Tested in The second factor was the last row · the conditioning series · refused by the unbalanced colleague matrix read as accurate to its roots' conditioning

“Iterative aggregation–disaggregation with a block Gauss–Seidel step removes the difficulty of nearly decomposable Markov chains, since the coupling enters only through a small aggregated chain that is solved directly.”

On two clusters of twelve states joined by a coupling ε, Koury–McAllister–Stewart aggregation with block Gauss–Seidel by ordinary elimination stalls at a worst relative entry error of 1.8·10⁻¹² at ε = 10⁻⁴, 7.1·10⁻⁹ at 10⁻⁸ and 1.7·10⁻⁴ at 10⁻¹² — about u/ε — and never reaches 10⁻¹² from ε = 10⁻⁴ down; one block solved alone with the exact right-hand side does the same. The refusal is fed the claim that it stalls below 10⁻¹² at every coupling, and required to fail.

Tested in The stall was in the block · the perron frobenius series · refused by aggregation with a block Gauss–Seidel step read as free of the coupling's stall

“The least-resistance tree's loops are long because a minimum spanning tree is chosen without regard to loop length, so they can be shortened substantially without losing the property that conditions the loop equations.”

For every loop of the least tree on five draws, the shortest loop its off-tree arc could close through lighter arcs — the length any tree keeping the cycle property must give it: 33 to 42 of the 81 loops are already that short, the median loop is 1.0 to 1.5 times its forced length, and the total is 1.8 to 2.2 times the forced total. The refusal is fed the claim that the median loop is twice its forced length or more on every draw, and required to fail.

Tested in The budget cannot finish the tree · the null space basis series · refused by the least tree's loops read as far longer than the cycle property forces

“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