Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Lanczos — 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. λ = 105 timesλ = 9.55 timesλ = 95 timesλ = 8.55 timesλ = 2.952 timesλ = 2.92 timeseigenvalues that arrived more than once — the matrix has 40 distinct onesa spectrum with the wrong multiplicitiesextra copies, no reorthogonalisation25extra copies, full reorthogonalisation0worst relative error among the copies1.9·10⁻⁸steps taken of 80 asked for, full40no arithmetic error was madeevery one of these is right to eight digits

    An eigenvalue that arrives twice

    A matrix with forty distinct eigenvalues, handed to Lanczos for eighty steps, returns twenty-five extra copies of thirteen of them — the largest arriving five times. Every copy is accurate to 1.9·10⁻⁸ relative. No arithmetic error was made, nothing overflowed, and a caller counting eigenvalues gets the wrong multiplicity from a computation in which no individual number is wrong.

    rung 1 · spectra

All ladders