SOLUTION OF BORDERED SINGULAR SYSTEMS IN NUMERICAL CONTINUATION AND BIFURCATION

[摘要] In numerical continuation and bifurcation problems linear systems with coefficient matrices in the block form [GRAPHICS] arise naturally. Here A is-an-element-of R(nxm), B,C is-an-element-of R(nxm), D is-an-element-of R(mxm) and n may be large but m is small. A usually has a special structure (banded, block banded, sparse, ...) and B, C, D are dense, so that it is advisable to use a specialized solver for A and to solve with M by some block method. Unfortunately, A is often also a nearly singular matrix (in fact, made nonsingular only by roundoff and truncation errors). On the other hand, M is usually nonsingular but can be ill-conditioned and in certain situations will degenerate to singularity as well. We describe numerical tests for this problem using the mixed block elimination method of Govaerts and Pryce (1993) for solving bordered linear systems with possibly nearly singular blocks A. To this end, we compute by Newton's method a triple-point bifurcation point in a parametrized reaction-diffusion equation (the Brusselator). The numerical tests show that the linear systems are solved in a stable way, in spite of the use of a black-box solver (SGBTRS from LAPACK) for a nearly singular matrix.

[发布日期] 1994-05-20

[效力级别]  Proceedings Paper