TITLE:
Krylov methods and test functions for detecting bifurcations in one 
parameter-dependent partial differential equations

AUTHORS:
Bosco Garcia-Archilla (bosco.garcia@esi.us.es)
Departamento de Matematica Aplicada II, Universidad de Sevilla
Escuela Superior de Ingenieros, Camino de los Descubrimintos, s/n, 41092 Sevilla

Juan Sanchez (sanchez@fa.upc.es)
Departament de Fisica Aplicada, Universitat Politecnica de Catalunya
Jordi Girona, 1-3, modul B4-B5, Campus Nord, 08034 Barcelona

Carles Simo (carles@maia.ub.es)
Departament de Matematica Aplicada i Analisi, Universitat de Barcelona
Gran Via de les Corts Catalanes, 585, 08071 Barcelona

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(PDEs). Convergence is proved using not the eigenvalues but the singular 
values of the PDE operator, when preconditioned by a fast solver. Numerical
experiments are presented where the technique is applied to locate pitchfork 
and transcritical bifurcations on a one parameter dependent system. Experiments 
reveal that some extra precautions may have to be taken in the presence of 
symmetries.

KEYWORDS AND PHRASES: 
Determinants, Arnoldi decomposition, compact operators in Hilbert spaces, 
spectral methods for PDEs, continuation methods, bifurcation location.