## Abstract

We consider the problem of computing the square root of a perturbation of the scaled identity matrix, A = αI

Key words. matrix pth root, matrix square root, low-rank update, matrix iteration, Newton iteration, MATLAB

AMS subject classifications. 15A16, 65F60, 65F99

_{n}+^{UV ∗}, where U and V are n × k matrices with k ≤ n. This problem arises in various applications, including computer vision and optimization methods for machine learning. We derive a new formula for the pth root of A that involves a weighted sum of powers of the pth root of the k × k matrix αI_{k}+^{V ∗U}. This formula is particularly attractive for the square root, since the sum has just one term when p = 2. We also derive a new class of Newton iterations for computing the square root that exploit the low-rank structure. We test these new methods on random matrices and on positive definite matrices arising in applications. Numerical experiments show that the new approaches can yield a much smaller residual than existing alternatives and can be significantly faster when the perturbation UV^{∗}has low rank.Key words. matrix pth root, matrix square root, low-rank update, matrix iteration, Newton iteration, MATLAB

AMS subject classifications. 15A16, 65F60, 65F99

Original language | English |
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Journal | SIAM Journal on Matrix Analysis and Applications |

Publication status | Accepted/In press - 18 Aug 2022 |