. Numerical simulations of lattice gauge theories with fermions rely heavily on the iterative solution of huge sparse linear systems of equations. Due to short recurrences, which mean small memory requirement, Lanczos-type methods (including suitable versions of the conjugate gradient method when applicable) are best suited for this type of problem. The Wilson formulation of the lattice Dirac operator leads to a matrix with special symmetry properties that makes the application of the classical biconjugate gradient (BiCG) particularly attractive, but other methods, for example BiCGStab and BiCGStab2 have also been widely used. We discuss some of the pros and cons of these methods. In particular, we review the specic simplication of BiCG, clarify some details, and discuss general results on the roundo behavior. 1 The symmetry properties of the Wilson fermion matrix In the Wilson formulation of the lattice Dirac operator, where the Green's function of a single quark with bare mass ...

Among the Lanczos-type product methods, which are characterized by residual polynomials pntn that are the product of the Lanczos polynomial Pn and another polynomial tn of exact degree n with tn(O) = 1, Zhang's algorithm GPBICG has the feature that the polynomials tn are implicitly built up by a pair of coupled two- term recurrences whose coefficients are chosen so that the new residual is minimized in a 2-dimensional space. There are several ways to achieve this. We discuss here alternative algorithms that are mathematically equivalent (that is, produce in exact arithmetic the same results). The goal is to find one where the ultimate accuracy of the iterates Xn is guaranteed to be high and the cost is at most slightly increased. Key words: Krylov space method, biconjugate gradients, Lanczos-type product method, BiCGxMR2, GPBi-CG I

. Many iterative methods for solving linear systems, in particular the biconjugate gradient (BiCG) method and its \squared" version CGS (or BiCGS), produce often residuals whose norms decrease far from monotonously, but uctuate rather strongly. Large intermediate residuals are known to reduce the ultimately attainable accuracy of the method, unless special measures are taken to counteract this eect. One measure that has been suggested is residual smoothing: by application of simple recurrences, the iterates xn and the corresponding residuals rn : b Axn are replaced by smoothed iterates yn and corresponding residuals sn : b Ayn . We address the question whether the smoothed residuals can ultimately become markedly smaller than the primary ones. To investigate this, we present a roundo error analysis of the smoothing algorithms. It shows that the ultimately attainable accuracy of the smoothed iterates, measured in the norm of the corresponding residuals, is, in general, not higher t...

IMPLEMENTATION MAN-CHUNG YEUNG ∗ Abstract. With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in [35] in a more systematic way. It turns out that there are n ways to define the ML(n)BiCGStab residual vector. Each definition will lead to a different ML(n)BiCGStab algorithm. We demonstrate this by deriving a second algorithm which requires less storage. We also analyze the breakdown situations from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab. Implementation issues are also addressed. We discuss in detail the choices of the parameters in ML(n)BiCGStab and their effects on the performance of the algorithm. Key words. CGS, BiCGStab, ML(n)BiCGStab, multiple starting Lanczos, Krylov subspace, iterative methods, linear systems

IMPLEMENTATION MAN-CHUNG YEUNG ∗ Abstract. With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in [35] in a more systematic way. It turns out that there are n ways to define the ML(n)BiCGStab residual vector. Each definition will lead to a different ML(n)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG a bridge connecting BiCG and FOM. We also analyze the convergence of ML(n)BiCG/ML(n)BiCGStab from the probabilistic point of view when a singular system is solved, and summarize some of their useful properties. Implementation issues are also addressed. Key words. CGS, BiCGStab, ML(n)BiCGStab, multiple starting Lanczos, Krylov subspace, iterative methods, linear systems

IMPLEMENTATION MAN-CHUNG YEUNG ∗ Abstract. With the help of index functions, we re-derive the ML(n)BiCGStab algorithm in [35] in a more systematic way. There are n ways to define the ML(n)BiCGStab residual vector. Each different definition will lead to a different ML(n)BiCGStab algorithm. We demonstrate this by deriving a second algorithm which requires less storage. We also analyze the breakdown situations and summarize some useful properties about ML(n)BiCGStab. Implementation issues are also addressed. In particular, we discuss in details on the choices of the parameters in ML(n)BiCGStab. Key words. CGS, BiCGStab, ML(n)BiCGStab, multiple starting Lanczos, Krylov subspace, iterative methods, linear systems

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