Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Sequence stability — 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. 10²10³10⁴10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²steps taken‖RᵀR − AᵀA‖ ⁄ ‖AᵀA‖the bound, linear in the steps√k · uevery step safe, the chain notdrift after the run3.9·10⁻¹⁴the bound there3.3·10⁻¹³√k · u there6.1·10⁻¹⁵worst single amplification2.7worst leverage met0.69refactorisations1backward stable onceand three thousand times is a different claim

    Stable once, and three thousand times

    A sliding window adds a row and removes one at every step and never looks at the data again. No single step of it amplifies by more than 2.72, no downdate fails, and after three thousand steps the triangular factor in memory is 3.9·10⁻¹⁴ from the matrix it is supposed to be a factor of — six hundred times growth from a per-step bound that says nothing about chains.

    rung 1 · sequence

All ladders