Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Arnoldi — where it appears

The recurrence building an orthonormal basis of a Krylov space and the Hessenberg matrix of A restricted to it, from products with A alone.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

02468101210⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1iteration‖r‖ / ‖b‖Laplaciancyclic shiftno progress at allevery eigenvalue of the shift is on the unit circleand it predicts nothing

The spectrum that predicts nothing

For a symmetric matrix the eigenvalues govern how fast an iteration converges. Drop symmetry and they stop governing anything — there is a matrix whose eigenvalues are as evenly spread as eigenvalues can be, on which GMRES makes no progress at all until the last possible step.

iterative · gmres 0481216202410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹matrix–vector products, mrelative error in e^Abforming e^A: 3·10⁻¹⁶crosses at m = 18the vector, not the matrixsteps to the dense answer18dimension100Krylov megaflops0.36dense megaflops2the exponential that is computedis 18×18

The vector was what was wanted

Nobody who computes a matrix exponential wants the matrix. They want e^{At}b — one vector, the state of a system at a later time. Twenty matrix–vector products get it to sixteen digits on a hundred-by-hundred problem, without ever forming a hundred-by-hundred exponential, and the exponential that does get computed is twenty by twenty.

spectra · matrix function

Named alongside it

The objects these essays reach for when they reach for this one.

Krylov subspaceCondition numberExact ground truthFlop countGMRESJacobi's eigenvalue methodMatrix exponentialMatrix-freeMatrix functionNon-normalityOrthogonalityResidual

All concepts