Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Perturbation theory — where it appears

The study of how much the answer to a problem moves when its data move by a small amount. It supplies condition numbers — first-order sensitivities of eigenvalues, solutions or subspaces — which combined with a backward error bound the forward error of a computation.

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

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 largest eigenvalue erroras given, order 960.043balanced, order 964.4·10⁻⁴symmetrised, order 962.2·10⁻¹²016324864809611210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹order nlargest |computed − exact| eigenvaluean error of one: the integers are no longer told apartas givenbalancedsymmetrisedopen dots: complex pairs returned for a real spectrumevery entry is an integer, stored exactly

Balanced is not symmetric

The Sylvester–Kac matrix is made of small integers, so a double holds it exactly, and its eigenvalues are the integers from −(n − 1) to n − 1 in steps of two. Every digit an eigensolver loses on it is therefore the solver's own, and it loses them at exactly the rate first-order perturbation theory predicts: the median error is half the prediction across 1,568 eigenvalues. Balancing, the preprocessing libraries apply for this kind of matrix, divides every condition number by about sixty and leaves their growth untouched, and at order 112 the unbalanced solver returns eighteen complex eigenvalues for a spectrum of integers.

error · Exact ground truth

Named alongside it

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

Backward errorComponentwise condition numberCondition numberEigenvalue condition numberEquilibrationExact ground truthExact inverseFrancis's double shiftNon-normalityParametrisationPseudospectrumSimilarity transformation

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