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## 2006.161: Fast polar decomposition of an arbitrary matrix

2006.161: Nicholas J. Higham and Robert S. Schreiber (1990) Fast polar decomposition of an arbitrary matrix. SIAM Journal on Scientific and Statistical Computing, 11 (4). pp. 648-655. ISSN 1095-7197

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## Abstract

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.

Item Type: Article polar decomposition, complete orthogonal decomposition, matrix square root, matrix multiplication, Schulz iteration, condition estimator MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2006.161 Miss Louise Stait 30 June 2006