Shifted LOPBiCG( <i>ℓ</i> for Solving Nonsymmetric Shifted Linear Systems

ABSTRACT The shifted LOPBiCG method can circumvent a problem of the optimal choice for the initial seed system through a seed‐switching technique when solving shifted linear systems, thanks to its natural collinearity of residual vectors. However, LOPBiCG, employing the same one‐degree accelerating polynomial as that in the Bi‐CGSTAB method, may not perform well for linear systems whose real coefficient matrix has complex eigenvalues with relatively large imaginary parts. Here, we utilize an ‐degree accelerating polynomial to extend LOPBiCG to LOPBiCG() and apply LOPBiCG() to solve shifted linear systems, which is referred to as shifted LOPBiCG (). Since the natural collinearity for LOPBiCG() is also satisfied, we can incorporate the seed‐switching technique into shifted LOPBiCG(). Numerical experiments demonstrate that our method effectively solves linear systems with a complex spectrum and is free from the issue associated with choosing the initial seed when solving shifted linear systems.