Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Rayleigh quotient — where it appears

The inner product of a vector with the matrix times it, divided by the vector's squared length: the best estimate of an eigenvalue one vector gives. For a symmetric matrix its error is the square of the vector's angle from the true eigenvector, so a rough eigenvector yields a far better eigenvalue.

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

1234567891010⁻⁵⁵10⁻⁴⁹10⁻⁴³10⁻³⁷10⁻³¹10⁻²⁵10⁻¹⁹10⁻¹³10⁻⁷10⁻¹cyclesizethe residualrepairedthe reported boundwhat the stopping rule readsreported at the last cycle9.4·10⁻⁴¹the residual there5.7·10⁻⁵with the border recomputed3.8·10⁻¹⁴products, cheap and repaired9a bound with nothing under itand one product a cycle to fix it

Keeping the vectors, and losing the bound

Thick restarting keeps the Ritz vectors instead of filtering the starting vector — the same eigenvalues for a third of the products with A. Its residual bound reaches 9.4·10⁻⁴¹ while the residual it bounds sits at 5.7·10⁻⁵, and the eigenvalues are correct to 4.3·10⁻¹⁴ the whole time, so nothing reports it.

spectra · Lanczos fractions of the curvaturebounded rule's direction, every ε0.14nearest zero at ε = 10⁻⁸, ÷ λ10⁻¹⁶10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹coupling ε|quotient| ÷ |smallest eigenvalue|Bunch–Kaufman's directionthe bounded rule'seigenvalue nearest zeroinverse iteration goes where the eigenvalue nearest zero isa solve amplifies the smallest, not the most negative

A curvature direction the factors cannot refine

A direction of negative curvature read from Bunch–Kaufman's factors held 7·10⁻¹⁶ of the curvature that was there, and the bounded rule's held 14 per cent. The prediction was that a few steps of inverse iteration with the same factors would recover it from either. One step leaves both under a thousandth, and two put both on positive curvature, at the eigenvalue nearest zero — because a solve amplifies the smallest eigenvalue in magnitude, not the most negative. What recovers the curvature is the matrix, not its factors: Lanczos from either direction reaches ninety-nine per cent in six to nine products at every coupling, and power iteration from Bunch–Kaufman's direction has not reached a tenth after sixty.

elimination · Cholesky

Named alongside it

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

Arrowhead matrixBunch–KaufmanInvariant subspaceInverse iterationLanczosLanczos algorithmLDLᵀ factorisationNegative curvatureReorthogonalisationResidual boundRestartingRitz values

All concepts