Fast polar decomposition of an arbitrary matrix

Higham, Nicholas J. and Schreiber, Robert S. (1990) Fast polar decomposition of an arbitrary matrix. SIAM Journal on Scientific and Statistical Computing, 11 (4). pp. 648-655. ISSN 1095-7197

The polar decomposition of an $m \times n$ matrix $A$ of full rank, where $m \geqq n$, can be computed using a quadratically convergent algorithm of Higham [SIAM J. Sci. Statist. Comput., 7(1986), pp. 1160–1174]. The algorithm is based on a Newton iteration involving a matrix inverse. It is shown how, with the use of a preliminary complete orthogonal decomposition, the algorithm can be extended to arbitrary $A$. The use of the algorithm to compute the positive semidefinite square root of a Hermitian positive semidefinite matrix is also described. A hybrid algorithm that adaptively switches from the matrix inversion based iteration to a matrix multiplication based iteration due to Kovarik, and to Björck and Bowie, is formulated. The decision when to switch is made using a condition estimator. This “matrix multiplication rich” algorithm is shown to be more efficient on machines for which matrix multiplication can be executed 1.5 times faster than matrix inversion.