Numerical Approximation of the Product of the Square Root of a Matrix With a Vector

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