Continuation method for a right definite two-parameter eigenvalue problem

The continuation method has been successfully applied to the classical Ax = lambda x and to the generalized Ax = lambda Bx eigenvalue problems. Shimasaki applied the continuation method to the right definite two-parameter problem, which resulted in a discretization of a two-parameter Sturm-Liouville problem. We show that the continuation method can be used for a general right definite two-parameter problem and we give a sketch of the algorithm. For a local convergent method we use the tensor Rayleigh quotient iteration (TRQI), which is a generalization of the Rayleigh iterative method to two-parameter problems. We show its convergence and compare it with Newton's method and with the generalized Rayleigh quotient iteration (GRQI), studied by Ji, Jiang, and Lee.