A Continuation Method for a Right Definite Two-Parameter Eigenvalue Problem
Bor Plestenjak · SIAM Journal on Matrix Analysis and Applications · 2000
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 wegive 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.