Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Inversion — the ladder

One essay so far against this idea. A ladder is the distinct arguments that stand against one idea, and this one has room to grow.
  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.

    rung 1 · elimination

All ladders