Practical Use of Some Krylov Subspace Methods for Solving Indefinite and Nonsymmetric Linear Systems
Youcef Saad · SIAM Journal on Scientific and Statistical Computing · 1984
The main purpose of this paper is to develop stable versions of some Krylov subspace methods for solving linear systems of equations $Ax = b$. As in the case of Paige and Saunders’s SYMMLQ [SIAM J. Numer. Anal., 12 (1975), pp. 617–624], our algorithms are based on stable factorizations of the banded Hessenberg matrix representing the restriction of the linear application A to a Krylov subspace. We will show how an algorithm similar to the SYMMLQ can be derived for nonsymmetric problems and we will describe a more economical algorithm based upon the $LU$ factorization with partial pivoting. In the particular case where A is symmetric indefinite the new algorithm is theoretically equivalent to SYMMLQ but slightly more economical. As a consequence, an advantage of the new approach is that nonsymmetric or symmetric indefinite or both nonsymmetric and indefinite systems of linear equations can be handled by a single algorithm.