SIAM J. So. COMJT. Vol. 14, No. 5, pp. 1020-1033, September 1993 () 1993 Society for Industrial and Applied Mathematics 0O2 VARIANTS OF BICGSTAB FOR MATRICES WITH COMPLEX SPECTRUM* MARTIN H. GUTKNECHTt Abstract. Recently Van der Vorst [S/AM Y. Sci. Statist. Comput., 13 (1992), pp. 631--644] proposed for solving nonsymmetric linear systems Az b a biconjugate gradient (BICG)-based Krylov space method called BICGSTAB that, like the biconjugate gradient squared (BICGS) method of Sonneveld, does not require matrix- vector multiplications with the transposed matrix AT, and that has typically a much smoother convergence behavior than BCG and BICGS. Its nth residual polynomial is the product of the one of BICG (i.e., the nth Lanczos polynomial) with a polynomial of the same degree with real zeros. Therefore, nonreal eigenvalues of A are not approximated well by the second polynomial factor. Here, the author presents for real nonsymmetric matrices a method BCGSTAB2 in which the second factor may have complex conjugate zeros. Moreover, versions suitable for complex matrices are given for both methods. Key words. Lanczos algorithm, biconjugate gradient algorithm, conjugate gradient squared algorithm, BICGSTAB, formal orthogonal polynomial, nonsymmetric linear system, Krylov space method AMS subject classification. 65F10 1. From BICG to complex BICGSTAI. The biconjugate gradient method (BICG) of Lanczos [7] and Fletcher [1] is a Krylov space method for solving (real or complex) non- Hermitian linear system Az b, where A is, say, a nonsingular N N matrix. (Typically, this matrix will be the result of applying a preconditioner to the original system matrix.) Starting from some initial guess z0 for the solution, BICG generates a sequence z, with the property that the nth residual r, := b Az lies in the Krylov space generated by A from r0, i.e., (1) rn e K:n+I := span (to, Aro,..., Anro), and is orthogonal to another Krylov space generated from some other initial vector g0 by the Hermitian transpose A n (2) r, 2_ , := span (Y0, AHyo,..., (AH)n-lyo). The sequence of residual polynomials p,, which are implicitly defined by (3) rn pn(A)ro, is in view of (2) a sequence of formal orthogonal polynomials: if we define a linear functional ,I on the space of polynomials with complex coefficients by setting ,I(k) := goAkzo, the formal orthogonality relation ,/,(Trkp,) 0 holds for every polynomial 7rk of degree k < n; see [5], [3] for further details and references. As a consequence of the consistency condition for polynomial acceleration methods, these residual polynomials are normalized by p,(0) 1. They are often called Lanczos polynomials. In general, neither these polynomials nor the residuals satisfy a minimality condition, in contrast to the case in which A is Hermitian positive definite and g0 r0, where the method reduces to the classical conjugate gradient method. Theoretically, the BICG algorithm terminates in at most v(A, r0) steps if this number denotes the degree of the minimal polynomial of the restriction of A to the maximum Krylov space generated by A from Received by the editors September 9, 1991; accepted for publication (in revised form) August 17, 1992. tlnterdiseiplinary Project Center for Supercomputing (IPS), ETH Ziidch, ETH-Zentrum, CH-8092 Ziirich, Switzerland (mhg@Lp. ethz. oh). 1020 Downloaded 11/04/14 to 129.132.208.91. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php