1986/10/01 by Nicholas J. Higham · 11 citations
Earth and Planetary Sciences · Engineering · Mathematics · #Aerospace Engineering and Energy Systems #Applied mathematics #Cholesky decomposition #Computer science #Geophysics and Gravity Measurements #Hermitian matrix #Inertial Sensor and Navigation #Invertible matrix #Materials science #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix decomposition #Physics #Polar #Polar decomposition #Positive-definite matrix #Pure mathematics #Quantum mechanics #Rate of convergence
paper · doi:10.1137/0907079
openalex publication_date 1986/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 Aand (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.