An efficient algorithm for the minimal least squares solution of linear system with indefinite symmetric matrices

TL;DR

This paper introduces a new efficient algorithm for solving symmetric indefinite linear systems that reduces error and computational cost while also providing null space basis and minimal solutions.

Contribution

The paper presents a novel LDLt factorization-based algorithm with improved accuracy and efficiency, capable of computing null space bases and minimal least squares solutions.

Findings

Error approximately 50% smaller than Bunch-Kaufman method

Computational cost at least 20% smaller than LAPACK's Complete Orthogonal Decomposition

Calculates null space basis with little additional cost

Abstract

In this work, a new algorithm for solving symmetric indefinite systems of linear equations is presented. It factorizes the matrix into the form LDLt using Jacobi rotations in order to increase the pivot's absolute value. Furthermore, Rook's pivoting strategy is also adapted and implemented. In determinate compatible systems, the computational cost of the algorithm was similar to the cost of the Bunch-Kaufman method, but the error was approximately 50 % smaller for intermediate and large matrices, regardless of the condition number of the coefficient matrix. Furthermore, unlike Bunch-Kaufman, the new algorithm calculates with little additional cost the fundamental basis of the null space, and obtains the minimal least squares and minimum norm solutions. In minimal least squares with minimum norm problems, the new algorithm was compared with the LAPACK Complete Orthogonal Decomposition algorithm, among others. The obtained error with both algorithms was similar but the computational cost was at least 20 % smaller with the new algorithm, even though the Complete Orthogonal Decomposition is implemented in a blocked form.