Add–sub pivoting triangular factorization for symmetric matrix

Abstract

Commonly used diagonal pivoting strategies for general symmetric matrix would bring about triangular factorization PAP’=LDL’, in which D is block diagonal matrix with either 1 × 1 or 2 × 2 diagonal blocks. This paper proposed a modified add–sub pivoting strategy resulting in D with no 2 × 2 diagonal block and a corresponding symmetric triangular factorization algorithm for solving symmetric linear system Ax=b. The factorization algorithm is based on symmetric Gaussian elimination; The pivoting strategy involves searching two suitable columns in the trailing submatrix A k by rook method, interchanging the first one of them with column k, and then adding the second one multiplied by a scalar γ to column k. This strategy is similar to an earlier one proposed by Dax; Its comparisons required are still O(n 2 ) ∼ O(n 3 ), but the bound on elements of L is lower, and especially the growth factor is bounded by 3. The results from numerical experiments with both ill-conditioned and good-conditioned symmetric indefinite linear systems show that the new algorithm is competitive with the B–K algorithm, the BBK algorithm, and the LU factorization with partial pivoting.