Fast enclosure of matrix eigenvalues and singular values via rounding mode controlled computation

Modifications of Bauer-Fike type and Weyl type perturbation theorems are presented for matrix eigenvalue and singular value problems. It is shown that the conditions of the presented theorems can be rigorously checked by floating point computation with rounding mode control. It is stressed that verification programs can be easily constructed on usual numerical softwares like MATLAB. Computational cost of obtaining rigorous error bounds for computed eigenvalues is shown to be 6n³ flops for a real symmetric n×n matrix.