Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Structured perturbation — where it appears

A perturbation restricted to matrices of the same kind as the one perturbed, which makes a problem better conditioned and its backward error worse. Restricting the perturbation makes the problem better conditioned and the backward error larger, so the nearby problem a general bound promises may be of the wrong kind.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

the smallest perturbation of any kind — 1.89·10⁻¹⁷the smallest Toeplitz one — 2.65·10⁻¹³both exact for the same x̂smallest of any kind1.9·10⁻¹⁷smallest Toeplitz one2.7·10⁻¹³the price of the constraint1.4·10⁴diagonal defect, unconstrained0.97an exact answer to a nearby problemof a kind nobody posed

A nearby problem of the wrong kind

A good algorithm returns the exact answer to a nearby problem. A hundred and eighteen essays have measured the distance and not one has asked what the nearby problem looks like. On a Toeplitz system it is a rank-one matrix that is constant along none of its diagonals — and the smallest one that is Toeplitz is two and a half million times larger.

structure · Structured backward error 110¹10²10³10⁻¹⁸10⁻¹⁴10⁻¹⁰10⁻⁶10⁻²10²10⁶numbers that describe the matrixcondition number, and backward errordensetoeplitzsymmetricρ aloneno such problemcondition numberbackward errorone matrix, four descriptionsκ, all n² entries7.9·10⁴as one number, ρ62backward error, unconstrained2·10⁻¹⁷as a symmetric Toeplitz matrix5.6·10⁻¹²fewer numbers, better conditionedand no nearby problem left

The condition number of the model

Describe a 40×40 Toeplitz matrix by its 1,600 entries and its condition number is 78,800. Describe it by the one number it actually contains and the condition number is 61.9. The three decades in between are not an approximation or a bound — they are what κ has been over-stating, and the drop is not where the linear algebra is.

structure · Structured backward error 10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³relative compression error, the representation‖b − Ax‖ ⁄ ‖A‖‖x‖, the solveequala backward error, chosenslope0.99representation at 10⁻⁸1.2·10⁻⁹backward error there1.3·10⁻¹⁰representation at 10⁻¹²1.2·10⁻¹³backward error there2.5·10⁻¹⁴the compression is not an approximationit is a perturbation of the problem

An accuracy that is a backward error

Every backward error on this site is something an algorithm produced and somebody then measured. This one is a line in the program. Solving with a compressed matrix gives a residual that is the compression's own error, at a slope of 1.000 over ten decades, so the knob that sets the storage sets the backward error directly.

error · Backward error 10⁻³10⁻²10⁻¹110⁻¹⁹10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹1 − ρbackward errorLevinson, Toeplitzelimination, ToeplitzLevinson, any kindelimination, any kindat ρ = 0.999Levinson, any perturbation2.9·10⁻¹⁷Levinson, Toeplitz only6·10⁻¹²the ratio between them2·10⁵diagonal defect of the first0.97the number that is reportedand the number that is asked about

The number that cannot rank them

Levinson and Gaussian elimination are indistinguishable on the backward error a library reports — every one of ninety-six measurements between 1.16·10⁻¹⁷ and 5.73·10⁻¹⁷. The structured backward error separates them by up to a hundredfold, in whichever direction the point happens to give. Only the forward error ranks them, and only because this family's exact answer is known.

structure · Structured backward error

Named alongside it

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

Backward errorCondition numberBackward stabilityExact inverseToeplitz matrixForward errorLevinson recursionAdmissibilityComponentwise condition numberEquilibrationFast algorithmGaussian elimination

All concepts