NUMERICAL-METHODS FOR FINDING MULTIPLE-EIGENVALUES OF MATRICES DEPENDING ON PARAMETERS

Let A(alpha, lambda) be a square matrix dependent on parameters alpha and lambda, of which we choose lambda as the eigenvalue parameter. Man y computational problems are equivalent to finding a point (alpha, la mbda) such that A(alpha, lambda) has a multiple eigenvalue lambda = l ambda at alpha = alpha*. An incomplete QR decomposition of a matrix d ependent on several parameters is proposed. Based on the developed the ory two new algorithms are presented for computing multiple eigenvalue s of A(alpha, lambda) with geometric multiplicity m greater than or eq ual to 2. A third algorithm is designed for the computation of multipl e eigenvalues with geometric multiplicity m = 1 but which also appears to have local quadratic convergence to semi-simple eigenvalues. Conve rgence analyses of these methods are given. Several numerical examples are presented which illustrate the behaviour and applications of our methods.