An overview of approaches for the stable computation of hybrid BiCG methods Original Research Article

It is well known that BiCG can be adapted so that the operations with AT can be avoided, and hybrid methods with computational complexity almost similar to BiCG can be constructed in a further attempt to improve the convergence behavior. Examples of this are CGS, Bi-CGSTAB, and BiCGstab(l). In many applications, the speed of convergence of these methods is very dependent on the incorporated BiCG process. The accuracy of the iteration coefficients of BiCG depends on the particular choice of the hybrid method. We will discuss the accuracy of these coefficients and how this affects the speed of convergence. We will show that hybrid methods exist which have better accuracy properties. This may lead to faster convergence and more accurate approximations. We also discuss look-ahead strategies for the determination of appropriate values for l in BiCGstab(l). These strategies are easily applied for the hybrid part, in contrast to similar techniques for the BiCG part (but of course they do not solve the breakdown problems of the BiCG part).