Orthogonally Accumulated Projection Methods for Linear System of Equations

TL;DR

This paper introduces orthogonally accumulated projection methods for solving linear equations, which efficiently construct orthonormal vectors to retrieve solutions in finite steps, with strategies to maintain orthogonality and demonstrated numerical efficiency.

Contribution

The paper proposes a novel class of orthogonally accumulated projection methods that improve solution retrieval efficiency and address orthogonality loss in iterative linear system solvers.

Findings

Methods can retrieve solutions in finite steps with exact arithmetic.

Strategies effectively handle loss of orthogonality.

Numerical experiments confirm improved efficiency.

Abstract

A type of iterative orthogonally accumulated projection methods for solving linear system of equations are proposed in this paper. This type of methods are applications of accumulated projection(AP) technique proposed recently by authors. Instead of searching projections in a sequence of subspaces as done in the original AP approach, these methods try to efficiently construct a sequence of orthonormal vectors while the inner-product between the solution to the system and each vector in the sequence can be easily calculated, thus the solution can be retrieved in finite number of iterations in case of exact arithmetic operations. We also discuss the strategies to handle loss-of-orthogonality during the process of constructing orthonormal vectors. Numerical experiments are provided to demonstrate the efficiency of these methods.