Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Singular vectors — where it appears

The two orthonormal bases a matrix maps between, which coincide with its eigenvectors exactly when it is normal. They are what a low-rank approximation is built from, and their divergence from the eigenvectors is exactly the departure from normality.

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

10⁻⁴10⁻³10⁻²10⁻¹110¹10²10³10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹relative change in the coefficients, along the worst directionrelative increase in the residualcoefficients doubled39% change, fit unmoved in the sixth digit308×: the third digit movesκ(A) = 3.6·10⁶. Exact arithmetic would pick one point on this floor. It would not raise it.24 points, degree 9, monomial basisthe data leaves them free

The valley with no bottom

A degree-nine fit's coefficients can be moved by a third of their own size before the residual changes in the sixth significant figure. The arithmetic did not lose those digits. The data never contained them.

leastsquares · Fitting 036912151821242700.250.50.7511.25index kweight appliedonebidiagonalArnoldiis it a function of σmisfit, even fit1.6·10⁻¹²misfit, general fit0.063‖A − Aᵀ‖/‖A‖0.086one method's weights do not notice the operatorand the other's stop being a function of σ

The basis decides what a filter is

The vocabulary of regularisation is spectral — a method keeps a component or discards it, and the weights are a function of the singular value. Row-normalising a symmetric blur so that it preserves a constant makes it 8.6% asymmetric, and that is enough to move GMRES's weights from 7·10⁻¹⁴ off a function of σ to 4.4·10⁻².

regularisation · GMRES

Named alongside it

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

Arnoldi iterationBasis choiceCondition numberEigenvectorsFilter factorsGMRESIll-posed problemIll-posednessKrylov subspaceLsqrNon-normal matricesRegularisation

All concepts