Rigorous eigenvalue computation for matrices plays an important role in verifying solutions to nonlinear partial differential equations (PDEs), including the Navier-Stokes equation. Utilizing the finite element method for PDE discretization, refining mesh partitions results in the generation of large-scale matrices. Often, instead of evaluating all eigenvalues, it’s imperative to ascertain rigorous upper and lower bounds for specific eigenvalues. Our research focuses on developing an algorithm specifically designed for the rigorous computation of eigenvalues in large-scale matrices, with dimensions reaching up to 100,000,000, using the Fugaku supercomputer. This presentation will report the developmental process of this algorithm, along with computational examples.