Further Improvements in Nonsymmetric Hybrid Iterative Methods

In the past few years new methods have been proposed that can be seen as combinations of standard Krylov subspace methods, such as Bi-CG and GMRES. Such hybrid schemes include CGS, BiCGSTAB, QMRS, TFQMR, and the nested GMRESR method. These methods have been successful in solving relevant sparse nonsymmetric linear systems, but there is still a need for further improvements. In this paper we will highlight some of the recent advancements in the search for effective iterative solvers. 1. Bi-CGSTAB and variants The residual rk = b Axk in the Bi-Conjugate Gradient method, when applied to Ax = b with start $0, can be written formally as Pk(A)ro, where Pk is a k-degree polynomial. These residuals are constructed with one operation with A and one with AT per iteration step. It was pointed out in [6] that with about the same amount of computational effort one can construct residuals of the form Fk = P:(A)ro, which is the basis for the CGS method. In [7] it was shown that by a similar approach as for CGS, one can construct methods for which r k can be interpreted as rk = Pk(A)Qk(A)rO, in which Pk is the polynomial associated with BiCG and Qk can be selected free under the condition that Qk(0) = 1. In [7] it was suggested to construct Qk as the product of k linear factorr 1 :sjA, where wj was taken to minimize locally a residual. This approach leads to the BiCGSTAB method. One weak point in BiCGSTAB is that we get break-down if an wj is equal to zero. One may equally expect negative effects when wj is small. In fact, BiCGSTAB can be viewed as the combined effect of BiCG and GMRES(1) steps. As soon as the GMRES(1) part of the algorithm (nearly) stagnates, then the BiCG part in the next iteration step cannot (or only poorly) be constructed. Another dubious aspect of BiCGSTAB is that the factor Qk has only real roots by construction. It is well-known that optimal reduction polynomials for matrices with complex eigenvalues may have complex roots as well. This point of view was taken in [2] for the construction of the BiCGSTAB2 method. In the odd-numbered iteration steps the Q-polynomial is expanded by a linear factor, as in BiCGSTAB, but in the even-numbered steps this linear factor is discarded, and the Q-polynomial from the previous even-numbered step is expanded by a quadratic 1 w , ! ' ) ~ -w y ) ~ ~ . It was anticipated that the introduction of quadratic factors in Q 90 H. A. Van der Vorst et al.: Further improvements in nonsymmetric hybrid iterative methods might help to improve convergence for systems with complex eigenvalues, and, indeed, some improvement was observed in practical situations (see also [3]). However, our presentation suggests a possible weakness in the construction of BiCGSTAB2, namely in the odd-numbered steps the same problems may occur as in BiCGSTAB. Since the even-numbered steps rely on the results of the odd-numbered steps, this may equally lead to unnecessary break-downs or poor convergence. In [5] another and even simpler approach was taken to arrive at the desired even-numbered steps, without the necessity of the construction of the intermediate BiCGSTAB-type step in the odd-numbered steps. Hence, in this approach the polynomial Q is constructed straight-away as a product of quadratic factors, without ever constructing a Linear factor. As a result the new method BiCGSTAB(2) leads only to significant residuals in the even-numbered steps and the odd-numbered steps do not lead necessarily to useful approximations. In fact, it is shown in [5] that the polynomial Q can also be constructed as the product of 1-degree factors, without the construction of the intermediate lower degree factors. The main idea is that 1 successive BiCG steps are carried out, where for the sake of an AT-free construction the already available part of Q is expanded by simple powers of A. This means that after the BICG part of the algorithm vectors from the Krylov subspace s, As, A2s, ..,,A's, with s = Pk(A)Qk-L(A)rO are available, and it is then relatively easy to minimize the residual over that particular Krylov subspace. In most cases BiCGSTAB(2) will already give nice results for problems where BiCGSTAB or BiCGSTAB2 may fail. Bi-CGSTAB(2) can be represented by the following algorithm: xo is an initial guess; ro = b Axo; ?o is an arbitrary vector, such that (r , Po) # 0, e.g., Po = r; po=1 ;u=O;a=O;w2=1 ; for i=O,2,4,6 ,...