Optimal Preconditioner for the Biconjugate Gradient Method

This article discusses various modifications of the conjugate gradient method preconditioned by the peripheral incomplete factorization (PIF) scheme using incomplete factorization with peripheral compensation of the fill elements. The PIF scheme is written for nine-diagonal matrices approximating 2D elliptical equations using nine-point stencils. Results are presented for testing of the PIF scheme numerically for convergence when the scheme is used as a preconditioner in the conjugate gradient (CG) method for symmetrical initial matrices and in two modifications of the biconjugate gradient method (BI-CG and BI-CGSTAB) for unsymmetrical initial matrices. It is demonstrated that the highest convergence rate of the proposed method can be achieved at an optimal value of the iteration parameter described by an empirical formula given in the article.