Iterative refinement of Schur decompositions

The Schur decomposition of a square matrix A is an important intermediate step state-of-the-art numerical algorithms for addressing eigenvalue problems, functions, and equations. This work concerned with the following task: Compute (more) accurate from given approximate decomposition. task arises, example, in context parameter-dependent problems mixed precision computations. We have developed Newton-like algorithm that requires solution triangular equation orthogonalization every iteration. prove local quadratic convergence matrices mutually distinct eigenvalues observe fast practice. In low-high environment, our essentially reduces to only four high-precision matrix-matrix multiplications per When refining double quadruple precision, it often needs 3–4 iterations, which time computing by up factor 10–20.