Shifted LOPBiCG: A locally orthogonal product‐type method for solving nonsymmetric shifted linear systems based on Bi‐CGSTAB

Abstract

When solving shifted linear systems using shifted Krylov subspace methods, selecting a seed system is necessary, and an unsuitable seed may result in many shifted systems being unsolved. To avoid this problem, a seed‐switching technique has been proposed to help switch the seed system to another linear system as a new seed system without losing the dimension of the constructed Krylov subspace. Nevertheless, this technique requires collinear residual vectors when applying Krylov subspace methods to the seed and shifted systems. Since the product‐type shifted Krylov subspace methods cannot provide such collinearity, these methods cannot use this technique. In this article, we propose a variant of the shifted BiCGstab method, which possesses the collinearity of residuals, and apply the seed‐switching technique to it. Some numerical experiments show that the problem of choosing the initial seed system is circumvented.