Title: Krylov methods and determinants for detecting bifurcations in one
       parameter dependent partial differential equations

Authors: B. Garc\'{\i}a-Archilla (1), J. S\'anchez (2) and C. Sim\'o (3),

(1) Depto. de Matem\'atica Aplicada II, Univ. de Sevilla, Escuela Superior de
    Ingenieros, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain.
(2) Dept. de F\'{\i}sica Aplicada, Univ. Polit\`ecnica de Catalunya,
    Jordi Girona, 1--3, m\`odul B4--B5, Campus Nord, 08034 Barcelona, Spain
(3) Dept. de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona,
    Gran Via, 585, 08071 Barcelona, Spain
E-mail: bosco.garcia@esi.us.es, sanchez@fa.upc.es, carles@maia.ub.es

Abstract

In this paper, we study the computation of the sign of the determinant of a
large matrix as a byproduct of the preconditioned GMRES method when applied to
solve the linear systems arising in the discretization of partial differential
equations. Numerical experiments are presented where the technique is applied
to detect and locate pitchfork and transcritical bifurcation points on a one
parameter dependent system. With an appropriate selection of the initial guess
in the GMRES method, the technique is shown to detect and accurately locate
bifurcation points. We present an analysis that helps to explain the good
performance observed in practice.