Numerical Linear Algebra
where the algebra stops being the arithmetic
Series

Inversion — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 10²10⁴10⁶10⁸10¹⁰10¹²10¹⁴10⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1κ₂(A)relative errorforward, A⁻¹bforward, A\bbackward, A⁻¹bbackward, A\bat κ = 10¹⁴η, LU solve2.2·10⁻¹⁷η, via the inverse4.5·10⁻⁵forward, LU solve2.8·10⁻⁴forward, via the inverse0.015one factorisation, two ways to use itand one of them forfeits the backward error

    The inverse that is never formed

    x = A⁻¹b is how the solution of a linear system is written and it is not how it is computed. The usual reason given is cost — three times the arithmetic. The real reason is that one of the two routes is backward stable and the other is not, and at κ = 10¹⁴ they differ by twelve orders of magnitude in the number that says whose fault a wrong answer is.

    part 1 · elimination
  2. 01234510⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1correction steprelative errorLU route: η = 2.2·10⁻¹⁷LU route: forward 2.8·10⁻⁴forward errorbackward errorwhat a correction buysη before refinement4.5·10⁻⁵η after four steps2.8·10⁻¹⁷forward, unchanged7.1·10⁻⁴cost of a step, flops1800the residual is repairableand the accuracy floor is the problem's

    The gap refinement can close

    Multiplying by a computed inverse is not backward stable, and refinement at the working precision repairs it. That much is settled. The claim beside it — that the forward error does not move — was read at one conditioning and four corrections too late. Swept over ten, it moves at every one, and it lands on the LU route's own number after a single correction.

    part 2 · elimination

All series