On reduction schemes and the symmetry of a matrix

In this paper, Lanczos and Arnoldi reduction methods as the special cases o f the generalized Hessenberg method are briefly reviewed. Attention is paid to the effect of symmetry of matrices on the behaviour of the reduction sc hemes, such as serious numerical breakdown. Based on the summation decompos ition of matrices, two structures of the upper Hessenberg form of a general unsymmetric matrix and their relationship are revealed, in terms of which, Arnoldi reduction schemes for unsymmetric matrices can be reformulated in two respective forms. The relationship between the reformulated reduction s cheme and the current Lanczos schemes for skew and symmetric matrices are a lso discussed. Copyright (C) 1999 John Wiley & Sons, Ltd.