Numerical Linear Algebra
where the algebra stops being the arithmetic
Series

Relative accuracy — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 1234567810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, largest firstrelative errorone-sided Jacobizero-shift QRshifted QReigenvalues of BᵀBagainst a rational bisectionσₘᵢₙ, 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.

    part 1 · spectra
  2. 1234567810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹singular value, largest firstrelative errorone-sided Jacobizero-shift QRshifted QReigenvalues of BᵀBthe same four routes, reversedσₘᵢₙ, 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.

    part 2 · spectra
  3. 0102030405010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹10⁴10⁷decades of gradingworst relative errorthe answer is gonevia BᵀBone-sided Jacobizero-shift QRone axis, four routesBᵀB at the narrowest grading3.6·10⁻¹⁴and at the widest1.9·10⁷Jacobi, worst over the sweep1.5·10⁻¹⁵zero shift, worst1.1·10⁻¹⁵the definition is not a methodand squaring buries what it squares

    A threshold the matrix does not set

    Two numbers come out of a relative-accuracy comparison and they belong to different things. The size of the matrix moves the constant of the routes that never fail, by a factor of 2.7 between n = 4 and n = 10; it does not move the point where the route through BᵀB stops returning an answer, which sits between ten and eleven decades of grading at every size drawn.

    part 3 · spectra

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