Naslov
Iterative Refinement of Schur decompositions

Autori
Bujanović, Zvonimir ; Kressner, Daniel ; Schröder, Christian

Izvornik
Numerical algorithms (1017-1398) 92 (2022); 247-267

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Schur decomposition ; iterative refinement ; mixed precision ; eigenvalue computation

Sažetak
The 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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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