A semi-conjugate gradient method for solving unsymmetric positive definite linear systems

TL;DR

This paper introduces the semi-conjugate gradient (SCG) method for unsymmetric positive definite systems, proving its equivalence to FOM, and proposes a sliding window implementation (SWI) for efficiency, demonstrating robustness and effectiveness in numerical experiments.

Contribution

The paper presents a novel semi-conjugate gradient method for unsymmetric positive definite systems and a sliding window implementation to improve computational efficiency.

Findings

SCG is theoretically equivalent to FOM.

SWI maintains semi-conjugacy despite windowing.

Numerical experiments show robustness and efficiency of SCG and SWI.

Abstract

The conjugate gradient (CG) method is a classic Krylov subspace method for solving symmetric positive definite linear systems. We introduce an analogous semi-conjugate gradient (SCG) method for unsymmetric positive definite linear systems. Unlike CG, SCG requires the solution of a lower triangular linear system to produce each semi-conjugate direction. We prove that SCG is theoretically equivalent to the full orthogonalization method (FOM), which is based on the Arnoldi process and converges in a finite number of steps. Because SCG's triangular system increases in size each iteration, we study a sliding window implementation (SWI) to improve efficiency, and show that the directions produced are still locally semi-conjugate. A counterexample illustrates that SWI is different from the direct incomplete orthogonalization method (DIOM), which is FOM with a sliding window. Numerical experiments from the convection-diffusion equation and other applications show that SCG is robust and that the sliding window implementation SWI allows SCG to solve large systems efficiently.