Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Matrix free — 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. 10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor1.3·10⁻¹⁰central floor1.1·10⁻¹²truncation slope, forward1truncation slope, central2no ε reaches the roundoffand the analytic derivative is free of the choice

    An operator with no entries

    At the sizes where linear algebra is expensive the matrix does not exist. What exists is a subroutine that returns Av. Every Krylov method survives that unchanged; every algorithm that reads an entry disappears. And the derivative such a code computes is accurate to ten digits instead of sixteen, which turns out to cost nothing at all.

    rung 1 · iterative

All ladders