Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Bidiagonal matrix — where it appears

A matrix with entries only on the diagonal and the one above it, whose 2n − 1 numbers determine its singular values to high relative accuracy.

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

Also named here as graded matrix, relative accuracy — the same set of essays touches all of them, so they are one junction rather than several.

1234567810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, largest firstrelative errorone-sided Jacobizero-shift QRshifted QReigenvalues of BᵀBagainst a rational bisectionσ_min, exactly2.1·10⁻³⁰worst, one-sided Jacobi4.4·10⁻¹⁶worst, zero-shift QR2.2·10⁻¹⁶worst, eigenvalues of BᵀB1a relative error is a ratioand the denominator is the answer

Small compared to what

This site's own singular value routine has carried a sentence since the month it was written — that one-sided Jacobi computes the small singular values to high relative accuracy and the standard method does not. It has never been measured here, because measuring it needs a σ that is known rather than computed. A bidiagonal matrix and a Sturm count in exact rationals supply one.

spectra · relative accuracy 1234567810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, largest firstrelative errorone-sided Jacobizero-shift QRshifted QReigenvalues of BᵀBthe same four routes, reversedσ_min, exactly5.2·10⁻²⁶Jacobi's error on it0.015sweeps, zero shift400sweeps, shifted16a method is not accuratea method on a matrix is

Accurate is not a property of a method

A bidiagonal matrix whose every entry is 1 or 4096 has singular values spanning thirty decades. On it, the method recommended for small singular values loses the small one by one and a half per cent, the sweep with the theorem behind it does not converge at all, and the shift the theorem is a warning about gets every value to 5·10⁻¹⁶. Nothing there contradicts the theory.

spectra · relative accuracy

Named alongside it

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

Exact arithmeticGraded matrixJacobi's eigenvalue methodRelative accuracySingular valuesCancellationCondition numberCondition squaringConvergence rateEquilibrationSturm sequence

All concepts