Speaker
Description
In the usual one-precision environment the Bartels--Stewart algorithm is the standard algorithm for solving small dense Sylvester matrix equations AX+XB=C. their algorithm consists of three steps, including the computation of the Schur decompositions, the substitution for solving the Sylvester equation with quasi-triangular coefficients, and the final recovering transformation.
In the new mixed-precision algorithm we propose, the computation of the Schur decompositions, which are the most expensive part of the algorithm, is performed in a low precision. Then the algorithm refines via a stationary iteration the approximate solution obtained by solving the quasi-triangular Sylvester equation with the low-precision quasi-triangular coefficient matrices; and this is in fact an iterative refinement scheme for the quasi-triangular Sylvester equation with the coefficient matrices in a perturbed form. Finally, in order to recover the solution from the Schur decompositions we need the unitary factors to be unitary to the working precision. In order to do so, we propose two efficient approaches to orthonormalize to the working precision the low-precision unitary factors, one based on orthonormalization and the other on inversion of the unitary factors.
We test the new mixed-precision methods on various problems from the literature containing both Sylvester and Lyapunov matrix equations. Numerical experiments show that the new methods are comparable with the classical Bartels--Stewart method in terms of accuracy, and they can be faster if the employed low-precision arithmetic is sufficiently cheaper than the working-precision arithmetic.
