MINRES — where it appears
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
Three eigenvalues, and two are the golden ratio
Precondition a saddle-point system by the block diagonal of its own two definite pieces and the preconditioned matrix has exactly three distinct eigenvalues — 1, and the two roots of λ² − λ − 1. A minimal polynomial of degree three means three steps, at every conditioning, and the preconditioner nobody can afford turns out to be the statement the affordable ones are measured against.
A preconditioner that need not know the constraint
Keep the constraint block exactly and replace the objective block by anything positive definite on the null space. The preconditioned matrix then has 2m eigenvalues at exactly one, and its remaining n − m are the generalised eigenvalues of a pencil in which the constraint does not appear. Sweep its condition number over six decades and they do not move in six digits.
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.
Block preconditionerIndefinite matrixKrylov subspacePreconditioningSaddle-point systemsCertificateCholeskyCondition numberConjugate gradientsConstraint preconditionerEigenvaluesExact ground truth