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