In this talk, we focus on estimation of the eigenvalue using floating-point arithmetic. We extend the algorithm of the Rump–Yamamoto theorem on perturbation analysis of the ma- trix eigenvalue problem to a generalized problem. By then applying the Rehmann-Georiesh theorem, which gives bounds for the eigenvalues of the matrix-generalized eigenvalue prob- lem, we develop an algorithm to verify the eigenvalues of the matrix. Our algorithm can take advantage of the sparsity of the matrix to verify the eigenvalues in an effective way and can be applied to clusters of eigenvalues. Numerical experiments show that we can obtain sharp bounds for eigenvalues, even with ill-conditioned problems.