Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

LDLᵀ factorisation — where it appears

Cholesky with the square roots pulled out, which extends to indefinite matrices once the diagonal D is allowed blocks of size two.

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

Also named here as symmetric indefinite — the same set of essays touches all of them, so they are one junction rather than several.

10¹10³10⁵10⁷10⁹10¹¹110¹10²10³10⁴condition number of the matrixgrowth factorbound 2^11partial pivotingCholeskyno pivot to gain fromCholesky growth, every κ1Cholesky interchanges0partial pivoting, worst9the bound, 2^112048both eliminations reach the same growthand only one of them had to swap to get there

A factorisation with nothing to pivot for

Cholesky's growth factor is not bounded by one. It is equal to one, at every size and every condition number, and the two-line reason is why the algorithm needs no pivoting at all — not "usually gets away without it". Its only failure is the square root of a non-positive number, which is exactly the test for definiteness, and in floating point that test moves with the precision.

elimination · cholesky D from PAPᵀ = LDLᵀ — the shaded pairs are 2×2 pivots10⁻⁶0.749······0.749·········1.2·10⁻⁶1.4······1.4·········2.1·10⁻⁶0.549······0.549·········3.6·10⁻⁶0.614······0.614·three rules, one matrix‖PAPᵀ − LDLᵀ‖, blocks5.8·10⁻¹⁷‖PAPᵀ − LDLᵀ‖, diagonal3.1·10⁻¹¹growth, blocks1.3growth, diagonal5·10⁵the zero block is what the problem saysand one rule does not need it to be nonzero

When symmetry is not enough

The matrix [[0, 1], [1, 0]] is symmetric, nonsingular and perfectly conditioned, and there is no diagonal entry to pivot on. Every factorisation restricted to symmetric interchanges and one-by-one pivots fails on it, at any depth of searching, because every entry it could search is zero. The repair is to take two variables at once.

elimination · cholesky

Named alongside it

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

CholeskyGrowth factorPartial pivotingSymmetric indefiniteBunch–KaufmanComplete pivotingConstrained minimisationFactorisationGaussian eliminationPositive definiteSaddle-point systemsUnit roundoff

All concepts