Test matrices with specified eigenpairs for the symmetric positive definite generalized eigenvalue problem

Abstract The generalized eigenvalue problem is a significant topic with numerous applications in scientific and technological computing. Therefore, verifying the reliability of the results produced by numerical solvers is crucial. This article presents a method for generating test problems for generalized eigenvalue problems such that the exact eigenvalues and eigenvectors are known in advance, thereby enabling precise error monitoring. The proposed method generates a matrix pair that can be represented exactly in floating-point numbers. Ozaki and Ogita previously proposed a method for generating test problems for standard eigenvalue problems. We extend their approach to generalized eigenvalue problems, proposing a method that allows users to specify target eigenvalues and a condition number of the matrix defining the generalized inner product in advance. Furthermore, iterative refinement methods to approach the expected eigenvalues as closely as possible for the user are also discussed. Numerical examples are presented to evaluate the accuracy of the computed results and to assess potential overestimation in the error bounds produced by a method for given approximate eigenvalues.