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. Its 2n − 1 numbers determine its singular values to high relative accuracy, which is why every accurate singular value algorithm reduces to this form first.

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

deepest barrier, far well right to within halfno exponent limit40gradual, started at 102433fp16, gradual underflow23fp16, flush to zero13051015202530354010⁻⁵10⁻⁴10⁻³10⁻²10⁻¹110¹barrier depth, octaves below the near wellrelative error in the far well's masssmallest normalsmallest subnormalno exponent limitgradual, started at 1024fp16, gradual underflowfp16, flush to zeroan error of one: the far well is emptysubnormals bought nine octaves

The well on the far side of the band

Gradual underflow was said to buy a predicate and not an answer, because a quantity that has decayed into the subnormal range is already lost. A quantity that passes through the band on its way somewhere else is not. The stationary distribution of a two-well chain, computed in half precision across a barrier whose top is two to the minus sixteen of the near well, keeps its far well's probability of 0.2454 to three digits with subnormals and returns exactly zero without them. The normwise backward error calls both answers exact.

arithmetic · Subnormals 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.

spectra · Relative accuracy 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.

spectra · Relative accuracy 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.

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 valuesCondition numberCondition squaringSturm sequenceBackward errorCancellationComponentwise condition numberConvergence rate

All concepts