Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems

We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix–vector products per iteration without accessing AT. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix–vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm.

[1]  D. P. Young,et al.  GMRES acceleration of computational fluid dynamics codes , 1985 .

[2]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[3]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[4]  C. Brezinski,et al.  Hybrid procedures for solving linear systems , 1994 .

[5]  Martin H. Gutknecht,et al.  Variants of BICGSTAB for Matrices with Complex Spectrum , 1993, SIAM J. Sci. Comput..

[6]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[7]  C. Brezinski,et al.  A breakdown-free Lanczos type algorithm for solving linear systems , 1992 .

[8]  G. Golub,et al.  Gmres: a Generalized Minimum Residual Algorithm for Solving , 2022 .

[9]  D. R. Fokkema,et al.  BiCGstab(ell) for Linear Equations involving Unsymmetric Matrices with Complex Spectrum , 1993 .

[10]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[11]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[12]  Roland W. Freund,et al.  An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices , 1993, SIAM J. Sci. Comput..

[13]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[14]  Gerard L. G. Sleijpen,et al.  Reliable updated residuals in hybrid Bi-CG methods , 1996, Computing.

[15]  Tony F. Chan,et al.  A Quasi-Minimal Residual Variant of the Bi-CGSTAB Algorithm for Nonsymmetric Systems , 1994, SIAM J. Sci. Comput..

[16]  T. Manteuffel,et al.  Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method , 1984 .

[17]  Claude Brezinski,et al.  Avoiding breakdown and near-breakdown in Lanczos type algorithms , 1991, Numerical Algorithms.

[18]  Anthony T. Chronopoulos,et al.  On Squaring Krylov Subspace Iterative Methods for Nonsymmetric Linear Systems on Squaring Krylov Subspace Iterative Methods for Nonsymmetric Linear Systems , 2022 .

[19]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[20]  M. Gutknecht.  The Unsymmetric Lanczos Algorithms And Their Relations To Pad ' E Approximation, Continued Fractions , 1990 .

[21]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[22]  Claude Brezinski,et al.  Transpose-free Lanczos-type algorithms for nonsymmetric linear systems , 1998, Numerical Algorithms.

[23]  A. Greenbaum.  Estimating the Attainable Accuracy of Recursively Computed Residual Methods , 1997, SIAM J. Matrix Anal. Appl..

[24]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[25]  Roland W. Freund,et al.  Conjugate Gradient-Type Methods for Linear Systems with Complex Symmetric Coefficient Matrices , 1992, SIAM J. Sci. Comput..

[26]  H. V. D. Vorst,et al.  Generalized conjugate gradient squared , 1996 .

[27]  Gene H. Golub,et al.  Matrix computations , 1983 .

[28]  M. Redivo.  Look{ahead in Bi{cgstab and Other Product Methods for Linear Systems , 1995 .

[29]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

[30]  T. Chan,et al.  The Use of Iterative Linear-Equation Solvers in Codes for Large Systems of Stiff IVPs for ODEs , 1986 .