Numerical approximation of the product of the square root of a matrix with a vector Original Research Article

Given an n×n symmetric positive definite matrix A and a vector image, two numerical methods for approximating image are developed, analyzed, and computationally tested. The first method applies a Newton iteration to a specific nonlinear system to approximate image while the second method applies a step-control method to numerically solve a specific initial-value problem to approximate image. Assuming that A is first reduced to tridiagonal form, the first method requires O(n2) operations per iteration while the second method requires O(n) operations per iteration. In contrast, numerical methods that first approximate A1/2 and then compute image generally require O(n3) operations per iteration.