CR Variants of Hybrid Bi-CG Methods for Solving Linear Systems with Nonsymmetric Matrices

s at ICCAM 2008 CR Variants of Hybrid Bi-CG Methods for Solving Linear Systems with Nonsymmetric Matrices Kuniyoshi Abe Gifu Shotoku University 1-38, Nakauzura, Gifu, 500-8288 Japan [email protected] Joint work with: S. Fujino By Krylov subspace methods, we are solving a large sparse linear system Ax = b, where A stand for an n-by-n matrix, and x and b are n-vectors, respectively. The Bi-Conjugate Gradient (Bi-CG) method is a well-known generic Krylov subspace method for this problem, and a number of hybrid Bi-CG methods such as the Conjugate Gradient Squared method (CGS), the Bi-Conjugate Gradient STABilized method (Bi-CGSTAB), the BiCGStab2 method, the Generalized Product-type method based on Bi-CG (GPBi-CG) and the BiCGstab(l) method have been developed as faster and smoother modifications of Bi-CG. Moreover, the Conjugate Residual (CR) method has been known as a Krylov subspace method based on the minimum residual approach. Also the Bi-Conjugate Residual (Bi-CR) method has been proposed as CR for nonsymmetric matrices. It is reported that the residual norm of CR decreases more smoothly than that of Bi-CG, and that the residual norm of Bi-CR converges faster than that of Bi-CG. However, the hybrid Bi-CG methods based on CR for nonsymmetric matrices have not previously been proposed. Therefore, we propose CR variants of hybrid Bi-CG methods for solving linear systems with nonsymmetric matrices. In other words, the Bi-CG part of the residual polynomials of the hybrid Bi-CG methods is replaced by CR for nonsymmetric matrices. The recurrence formulas for updating an approximation and a residual vector are the same as those of the original hybrid Bi-CG methods, while the recurrence coefficients αk and βk are determined so as to compute the coefficients of the residual polynomial of CR for nonsymmetric matrices. Numerical experiments show that our proposed CR variants of hybrid Bi-CG methods are more effective than the original hybrid Bi-CG methods.