The Snap-Back Pivoting Method for Symmetric Banded Indefinite Matrices

The four existing stable factorization methods for symmetric indefinite matrices suffer serious defects when applied to banded matrices. Partial pivoting (row or column exchanges) maintains a band structure in the reduced matrix and the factors, but destroys symmetry completely once an off-diagonal pivot is used. Two-by-two block pivoting and Gaussian reduction to tridiagonal (Aasen's algorithm) maintain symmetry at all times, but quickly destroy the band structure in the reduced matrices. Orthogonal reductions to tridiagonal maintain both symmetry and the band structure, but are too expensive for linear-equation solvers. We propose a new pivoting method, which we call snap-back pivoting. When applied to banded symmetric matrices, it maintains the band structure (like partial pivoting does), it keeps the reduced matrix symmetric (like 2-by-2 pivoting and reductions to tridiagonal), and it is fast. Snap-back pivoting reduces the matrix to a diagonal form using a sequence of elementary elimination steps, most of which are applied symmetrically from the left and from the right (but some are applied unsymmetrically). In snap-back pivoting, if the next diagonal element is too small, the next pivoting step might be unsymmetric, leading to asymmetry in the next row and column of the factors. But the reduced matrix snaps back to symmetry once the next step is completed.