An error analysis of the modified scaling and squaring method

As a time discretization scheme for an ordinary differential equation with a stiff linear term, there is a class of methods that utilize the exponential or related functions of the coefficient matrix of the linear term. To implement these methods, we must compute a set of matrix functions called “φφ-function”, that includes the exponential itself, and it is important to compute these functions efficiently and accurately. In this paper, we consider the modified scaling and squaring method for the computation of φφ-function. An algorithm based on Higham’s method is defined, and the bounding parameter θmθm appropriate for φφ-function is determined from an analysis of the truncation error under the assumption of the exact arithmetic. We also consider the propagation of the rounding error in the squaring process, and show that the error of φφ-function is expected to be less than or roughly equal to that of the matrix exponential. Several evaluations are performed for famous test matrices, and the result shows that when the matrix exponential is computed accurately, the other φφ-functions can also be obtained with the same level of accuracy.