Finding solution of linear systems via new forms of BiCG, BiCGstab and CGS algorithms

Abstract

This paper introduces some Krylov subspace methods utilizing the s-step technique. The variable s-step technique is applied to CGS and BiCG algorithms, and extended to the BiCGstab algorithm as an intermediate state between this two algorithms. By proposing the use of the s parameter as a variable, these algorithms become adaptable. To enhance stability, a regularization technique is incorporated. Through the integration of these techniques, stable algorithms are developed. Numerical examples are provided to demonstrate the efficacy and quality of the proposed algorithms.