Analysis of the Cholesky method with iterative refinement for solving the symmetric definite generalized eigenproblem (Q2784358)

The authors consider the Cholesky-Jacobi method for solving the symmetric definite generalized eigenvalue problem \(Ax=\lambda Bx\): the Cholesky factorization \(\Pi^T B\Pi= LD^2 L^T\) is computed to solve the equivalent standard symmetric eigenvalue problem by applying the Jacobi method. A detailed rounding error analysis is given. A fixed precision iterative refinement is investigated as a means for improving the backward errors of the approximate eigenpairs. Theoretical results and numerical experiments show that the Cholesky-Jacobi method has better numerical stability properties than the previous backward error bound suggests.