Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Matrix equation — where it appears

An equation whose unknown is a matrix rather than a vector, such as AX + XB = C. Writing it as an ordinary linear system squares the dimension, so it is solved through factorisations of its coefficients instead.

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

123456710⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹order r kepterror and bounds2Σσ, and the errorσᵣ₊₁a bound that is an equalityorders7bound ÷ error, worst1spread over the sweep1error ÷ σᵣ₊₁, worst2.1computed before the modeland attained by it

The bound that is known in advance

Almost every error on this site is measured after the fact. Balanced truncation has one that is computable before the reduced model exists, in a norm of a function rather than of a residual — and on ordinary problems it is not an upper bound that is loose. It is attained.

reduction · balanced truncation 01234567810⁻⁵10⁻⁴10⁻³10⁻²10⁻¹110¹μ, the entry above the diagonalsep, and the eigenvalue gapmin |λᵢ + μⱼ|sep(A, B)the spectra never moveeigenvalue gap, throughout2sep at μ = 02sep at μ = 89.5·10⁻⁴amplification there1056solvability is the eigenvaluesand conditioning is not

An equation whose unknown is a matrix

AX + XB = C is linear in X, so it has a coefficient matrix, and writing it down is the obvious thing to do. At n = 100 that matrix has a hundred million entries for a problem with ten thousand unknowns, and the algorithm everybody uses instead never forms it. Its conditioning is not the eigenvalue gap either, which is the number a reader is invited to consult.

structure · matrix equation

Named alongside it

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

A-priori boundBack-substitutionBalanced truncationCondition numberEigenvaluesExact ground truthFlop countGramianHankel singular valuesKronecker productLyapunov equationMcMillan degree

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