On the squared unsymmetric Lanczos method (Q1347170)

This paper deals with linear systems, (1) \(Ax = b\), where \(A\) is a large nonsymmetric \(N \times N\) matrix. The biorthogonal Lanczos method (briefly, BiL) can be used to approximate the eigenvalues of nonsymmetric matrices and to solve systems like (1). The purpose of the present paper is to improve this procedure with respect to its both branches, computing the eigenvalues and solving (1). To serve as such a starting point, BiL offers some advantages. For instance, as compared to the class of Krylov subspace methods, it has a quite modest main memory storage requirement. At the first stage, it turns out that the solution of (1), by the BiL method, can be obtained in two different ways, called briefly BiLQR and BiLMINRES. As his main step the author applies the squaring and restarting processes in terms of the matrix polynomial presentation. The first branch, computing the eigenvalues, constitutes the contents of section 4. This is in fact a review of an own work of the author, published only in a report series before, in 1989. Thus, section 5 entitled ``The restarted squared Lanczos method for linear systems'' contains the actual contribution to this article. The methods BiLQR and BiLMINRES have now got their squared and restarted forms, denoted SBiLQR(m) and SBiLMINRES(m), respectively. The corresponding algorithm falls into cycles, consisting of \(m\) iterations. In the so-called modified SBiLQR method the cycle \(m\) does not remain fixed during the iteration. Section 6 contains numerical tests in terms of two boundary value problems. In one of them the number of the equations corresponding to the system (1) is even 40 000. The author compares the number of the iteration steps needed in the three new methods to that needed in the known GMRES\((m)\) method. The comparison seems to be favorable, modified SBiLQR being the best. In section 7 there are some further conclusions.