Journal ArticleDOI
Newton's method for the matrix square root
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In this paper, it was shown that these simplified iterations are numerically unstable and a further variant of Newton's method for the matrix square root, recently proposed in the literature, is shown to be numerically stable.Abstract:
One approach to computing a square root of a matrix A is to apply Newton's method to the quadratic matrix equation F(X) _ x2 - A = 0. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simplified iterations are numerically unstable. A further variant of Newton's method for the matrix square root, recently proposed in the literature, is shown to be, for practical purposes, numerically stable.read more
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Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory
TL;DR: This book brings together a vast body of results on matrix theory for easy reference and immediate application with hundreds of identities, inequalities, and matrix facts stated rigorously and clearly.
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Functions of matrices
TL;DR: The most common matrix function is the matrix inverse, which is not treated specifically in this chapter, but is covered in Section~1.5 and Section~51.3 as discussed by the authors.
Journal ArticleDOI
Computing the polar decomposition with applications
TL;DR: 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.
Journal ArticleDOI
The matrix sign function
Charles Kenney,Alan J. Laub +1 more
TL;DR: A survey of the matrix sign function is presented including some historical background, definitions and properties, approximation theory and computational methods, and condition theory and estimation procedures.
Journal ArticleDOI
Computing real square roots of a real matrix
TL;DR: An extension of the Schur method is presented which enables real arithmetic to be used throughout when computing a real square root of a real matrix.
References
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The Theory of Matrices
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Book
Numerical Analysis
TL;DR: This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis, and describes the accuracy, efficiency and robustness of these algorithms.
Book
Introduction to matrix computations
TL;DR: Rounding-Error Analysis of Solution of Triangular Systems and of Gaussian Elimination.
Journal ArticleDOI
Solution of the matrix equation AX + XB = C [F4]
R. H. Bartels,G. W. Stewart +1 more
TL;DR: The algorithm is supplied as one file of BCD 80 character card images at 556 B.P.I., even parity, on seven ~rack tape, and the user sends a small tape (wt. less than 1 lb.) the algorithm will be copied on it and returned to him at a charge of $10.O0 (U.S.and Canada) or $18.00 (elsewhere).