Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Counterexample — where it appears

A constructed input on which a method fails, which is the only thing that distinguishes a heuristic that usually works from an algorithm with a theorem.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

the estimator maximises this quantity over the columns it visitscolumn 1 ‹visited›12column 2 ‹the answer›114column 311.4column 411.4column 511.4column 611.4column 711.4column 811.4column 911.4column 1011.4column 1111.4column 1211.4estimate 12.0a walk that stopped earlythe estimate returned12the true 1-norm114columns visited1products with the matrix5the walk's own stopping test firedand every column it could see was smaller

An estimate that can be fooled

Nobody computes a condition number, because forming an inverse costs more than the solve did. Every library estimates it instead, from four or five products with a factorisation already in hand. The estimate is exactly right on four random matrices out of five — and there is a matrix, three distinct entries wide, on which it returns a twentieth of the truth.

error · condition estimation significand bits10³10⁴10⁵10⁶10⁷10⁸10⁹10¹⁰κ(A)87892634456329625911109435761199127514162024every matrix positive definiteruns producing a false certificate14of runs in total72never above, in significand bits12first κ at eight bits10⁵the comparison was correctand what it proved was not true

Deciding that a zero has arrived

The previous tolerances were offers — accept this much error, save this much work. A detection threshold is not an offer, because both directions are failures. One matrix here has three genuinely near-invariant subspaces, and the constant somebody typed decides which of them the recurrence stops at; at eight significand bits the same kind of constant produces a proof of something false.

error · deliberate zero 171319253137434910⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110²indexmagnitude|r_kk|σ_kone factorisation, two verdicts‖AP − QR‖/‖A‖1.1·10⁻¹⁶|r_nn|8.7·10⁻⁴σ_min3.7·10⁻¹²column interchanges0|r_nn| is never below σ_minso the cheap verdict errs one way only

The cheap rank and what it cannot see

Almost nobody computes singular values to decide a rank. The standard substitute is QR with column pivoting, read off the diagonal of R — and there is a triangular matrix on which the greedy rule makes no interchange at all, has no better column available at any step, and reports a matrix eight orders of magnitude further from singular than it is.

spectra · rank -40-27-14-11225380123456789computed eigenvalueseedevery mark has a residual below 10⁻⁸a small residual, and no answercoefficients of det(A − λB)0worst residual over all seeds1.8·10⁻⁹spread of the answers66seeds drawn8the residual is small at every markand none of the marks means anything

A problem with no answer

If two matrices share a null vector then det(A − λB) is identically zero and every λ is an eigenvalue, which means none of them is. Perturb such a pencil by a ten-billionth and a solver returns six numbers with residuals below 10⁻⁹. Change the seed and it returns six different numbers, spread over forty-four, with residuals just as small.

spectra · pencil ran to the end3173lucky — a subspace closed295serious, cured by a block of two495serious, cured by a longer block29serious, incurable at any length8counted, not estimatedserious, as a fraction0.13of those, cured at two0.93incurable8matrices tried4000measure zero on the realsand an eighth of the integers

The same zero, and nothing was found

Change the recurrence by two lines and the divisor stops being a norm. It becomes an inner product of two vectors from two different sequences, and an inner product of two different vectors is zero on a whole hyperplane — with neither vector anywhere near zero, nothing invariant, and nothing converged. The arithmetic event is identical and the meaning is opposite.

iterative · breakdown

Named alongside it

The objects these essays reach for when they reach for this one.

Backward errorCondition estimationKrylov subspaceLower boundNumerical rankResidualArnoldiBiorthogonalityCertificateColumn pivotingCondition numberDeterminant

All concepts