Indefinite matrix — where it appears
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
The division that cannot be done
Conjugate gradients divides by pᵀAp at every step, and on a matrix that is not positive definite that number can be zero or negative. This site has guarded against it since its first commit and described it as a failure. In the method that made conjugate gradients famous it is the single most valuable object the iteration can produce, and it costs six matrix–vector products.
Named alongside it
The objects these essays reach for when they reach for this one.
CholeskyBunch–KaufmanCertificateComplete pivotingConjugate gradientsConstrained minimisationEigenvaluesGrowth factorKrylov subspaceLDLᵀ factorisationMinresNegative curvature