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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 ofconvergence and it is shownhow reliable estimates ofthe optimal parametersmaybe computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermitian polar factorHof 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. Key words, polar decomposition, singular value decomposition, Newton’s method, matrix square root AMS(MOS) subject classifications. 65F25, 65F30, 65F35 1. Introduction. The

A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rank-deficiency and instability. Theoretical results and computational methods are described. Applications of nearness problems in areas including control theory, numerical analysis and statistics are outlined.

This paper provides an overview of methods to detect, locate, and characterize damage in structural and mechanical systems by examining changes in measured vibration response. Research in vibration-based damage identification has been rapidly expanding over the last few years. The basic idea behind this technology is that modal parameters (notably frequencies, mode shapes, and modal damping) are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties will cause detectable changes in the modal properties. The motivation for the development of this technology is presented. The methods are categorized according to various criteria such as the level of damage detection provided, model-based vs. non-model-based methods and linear vs. nonlinear methods. The methods are also described in general terms including difficulties associated with their implementation and their fidelity. Past, current and future-planned applications of this technology to actual engineering systems are summarized. The paper concludes with a discussion of critical issues for future research in the area of vibration-based damage identification.

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. Key words, polar decomposition, singular value decomposition, Newton’s method, matrix square root AMS(MOS) subject classifications. 65F25, 65F30, 65F35 1. Introduction. The