Transpose{free Implementations of Lanczos' Method for Nonsymmetric Linear Systems

The method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coeecients of these recurrence relationships requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coeecients is proposed. It is based on the qd{algorithm and it does not need the transpose of the matrix. Lanczos method 23] for solving a system of linear equations is based on formal orthogonal polynomials. Such polynomials satisfy, under certain assumptions discussed below, some recurrence relationships. However, the computation of the coeecients of these recurrence relationships involves products of the form A T v where v is an arbitrary vector. When the matrix is large and sparse, such products are diicult to compute due to the indirect addressing needed by the structure of A. So, it was interesting to derive transpose{free algorithms for the implementation of the method of Lanczos. In 9], a transpose{free Lanczos/Orthodir algorithm was presented. It is based on a very simple idea known for a long time (see, for example, 7]): the computation of the coeecients of the recurrence relationships of the formal orthogonal polynomials can be performed by the qd{algorithm of Rutishauser 25]. It was also mentioned, in 9], that the same technique could be use for avoiding the use of A T in the other algorithms for the implementation of Lanczos method, that a more stable variant of the qd{algorithm has to be used and that the questions of breakdown (division by zero) and near{breakdown (division by a number close to zero which produces numerical instability) have to be studied. It is the purpose of this paper to address these questions. The method of Lanczos will be presented in Section 1. Section 2 is devoted to formal orthogonal polynomials. Recursive algorithms for the implementation of Lanczos method are given in Section 3 and their transpose{free versions are discussed in Section 4.

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