Matrices with Rank Deficiency Two in Eigenvalue Problems and Dynamical Systems

• Published in: SIAM journal on numerical analysis 1994-04, Vol.31 (2), p.524-539
• Main Authors: K.-W. E. Chu, Govaerts, W., Spence, A.
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic conjugates; Applied mathematics; Dynamical systems; Eigenvalues; Exact sciences and technology; Jacobians; Linear algebra; Linear systems; Mathematical vectors; Mathematics; Matrices; Neighborhoods; Newtons method; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/0731028

Let K(α, β) (α, β real) be be a family of real square matrices. Several computational problems are equivalent to the calculation of a pair (α0, β0) of parameter values for which K(α0, β0) has rank deficiency 2. Among these problems are the computation of a conjugate pair of complex eigenvalues of a given real matrix, the computation of a Hopf bifurcation point on a branch of stationary solutions to a parametrized differential equation, and the computation of a Takens-Bogdanov point in the two-dimensional solution manifold of a set of nonlinear equations. Developing ideas of Griewank and Reddien, the authors define scalar functions g1(α, β), g2(α, β) that vanish in (α0, β0). A nondegeneracy condition NDC, which expresses the fact that (α0, β0) is in a natural sense an isolated point in (α, β)-space, is introduced. It is proved that under certain conditions on the family K(α, β) the Jacobian of g1, g2 with respect to α, β is nonsingular if and only if NDC holds. A Newton method to compute (α0, β0) is then described. The problems mentioned above and some related ones are analysed in detail. The derived algorithms for the dynamical systems problems have the following features: (1) they are simple and natural, being based on linear algebra concepts only; (2) they treat the two parameters in a symmetric way; (3) they do not lead to formally large systems; and (4) NDC is expressed in terms independent of the particular problem. Numerical results are given which illustrate the quadratic convergence of the Newton algorithm during the computation of a Hopf bifurcation point arising in a model of a tubular reactor.