Iterative Refinement of Schur decompositions
Abstract
The Schur decomposition of a square matrix $A$ is an important intermediate step of stateoftheart numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of $A$ from a given approximate Schur decomposition. This task arises, for example, in the context of parameterdependent eigenvalue problems and mixed precision computations. We have developed a Newtonlike algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed lowhigh precision environment, our algorithm essentially reduces to only four highprecision matrixmatrix multiplications per iteration. When refining double to quadruple precision, it often needs only 34 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 1020.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.10879
 arXiv:
 arXiv:2203.10879
 Bibcode:
 2022arXiv220310879B
 Keywords:

 Mathematics  Numerical Analysis