Computing the Polar Decomposition—with Applications

Abstract

Abstract. A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice.To add to the known best approximation property of the unitary polar factor, the Hermitian polar factor H of a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to A and 1/2(A / H) is shown to be a best Hermitian positive semi-definite approximation to A. Perturbation bounds for the polar factors are derived.Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix.

A toolkit for computational molecular biology. II. On the optimal superposition of two sets of coordinates

“…Higham (1984) has proposed a fast algorithm to determine U and If an orthogonal starting approximation V to the orthogonal matrix U is available, when we may apply the decomposition to VrA to produce WH, where W-1, and then U = VW.…”

Iterative Algorithms to Approximate Canonical Gabor Windows: Computational Aspects

“…However, this observation has little practical relevance, because the computed canonical windows g d and g t will have a bad time-frequency localization. Higham [12] uses a scaling strategy for algorithm I that approximates the optimal scaling. This requires an estimate for the smallest eigenvalue of the matrix, but this is easy to obtain since the matrix is inverted as part of the iteration step.…”

“…We refer to [6, and [11,[11][12][13] for recent and comprehensive treatments of the theory of Gabor systems and frames; to fix notations and conventions we briefly give here the main features. We denote for g ∈ L 2 (R) and a > 0, b > 0 by (g, a, b) the collection of time-frequency shifted windows g na,mb , m,n ∈ Z , (1.1)…”

“…γ k+1 = 3δ k0 γ k − 3δ k1 S k g + δ k2 SS k γ k ; 12) for the approximation of g d . The algorithms II-V are, contrary to algorithm I, conditionally convergent in the sense that the frame bound ratio In the algorithms just presented, all computed windows are normalized.…”

Orthogonal and projected orthogonal matched filter detection

“…Using (24) and (25), the OMF signals may be computed very e ciently exploiting the many known e cient algorithms for computing the polar decomposition and the SVD (see, e.g., [19,21,29]). …”