Bujanovic, ZvonimirKressner, DanielSchroeder, Christian2022-06-202022-06-202022-06-20202310.1007/s11075-022-01327-6https://infoscience.epfl.ch/handle/20.500.14299/188615WOS:000806675300002The Schur decomposition of a square matrix A is an important intermediate step of state-of-the-art 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 parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like 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 low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3-4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10-20.Mathematics, AppliedMathematicsschur decompositioniterative refinementmixed precisioneigenvalue computationrecursive blocked algorithmssolving triangular systemsmultishift qr algorithmpart iiinvariantsylvestertext::journal::journal article::research article