1986/04/01 by Nicholas J. Higham · 140 citations
Computer Science · Mathematics · Physics and Astronomy · #Applied mathematics #Eigenvalues and eigenvectors #Geometry #Iterative Methods for Nonlinear Equations #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix Theory and Algorithms #Newton's method #Nonlinear system #Root (linguistics) #Scientific Research and Discoveries #Square matrix #Square root #Symmetric matrix
paper · doi:10.2307/2007992
published in Mathematics of Computation 46(174), 537 (American Mathematical Society)
openalex publication_date 1986/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
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.