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Mathematics of Computation

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Newton's method for the matrix square root


Author: Nicholas J. Higham
Journal: Math. Comp. 46 (1986), 537-549
MSC: Primary 65F30; Secondary 65H10
DOI: https://doi.org/10.1090/S0025-5718-1986-0829624-5
MathSciNet review: 829624
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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) \equiv {X^2} - 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.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1986-0829624-5
Keywords: Matrix square root, Newton's method, numerical stability
Article copyright: © Copyright 1986 American Mathematical Society