Title:

Numerical solution of linear and nonlinear eigenvalue problems

Given a real parameterdependent matrix, we obtain an algorithm for computing the value of the parameter and corresponding eigenvalue for which two eigenvalues of the matrix coalesce to form a 2dimensional Jordan block. Our algorithms are based on extended versions of the implicit determinant method of Spence and Poulton [55]. We consider when the eigenvalue is both real and complex, which results in solving systems of nonlinear equations by Newton’s or the GaussNewton method. Our algorithms rely on good initial guesses, but if these are available, we obtain quadratic convergence. Next, we describe two quadratically convergent algorithms for computing a nearby defective matrix which are cheaper than already known ones. The first approach extends the implicit determinant method in [55] to find parameter values for which a certain Hermitian matrix is singular subject to a constraint. This results in using Newton’s method to solve a real system of three nonlinear equations. The second approach involves simply writing down all the nonlinear equations and solving a real overdetermined system using the GaussNewton method. We only consider the case where the nearest defective matrix is real. Finally, we consider the computation of an algebraically simple complex eigenpair of a nonsymmetric matrix where the eigenvector is normalised using the natural 2norm, which produces only a single real normalising equation. We obtain an underdetermined system of nonlinear equations which is solved by the GaussNewton method. We show how to obtain an equivalent square linear system of equations for the computation of the desired eigenpairs. This square system is exactly what would have been obtained if we had ignored the non uniqueness and nondifferentiability of the normalisation.
