Adaptive selection of parameters for precise computation of matrix exponential based on padé approximation

Adaptive selection is discussed for scaling parametersN and expanded series q in precise integration method (PIM) of matrixexponential based on Pad\'e approximation. In general, scaling parametersN and expanded series q play important roles in the numerical accuracyand computational efficiency of matrix exponential. Using theapproximation theory of matrix functions, influences of parameters N andq on the computational accuracy and efficiency are firstly studied, andthen the iterative adaptive selection algorithm for optimal combination ofparameters (N,q) is presented. Appropriate parameters (N,q) can beselected automatically depending on the characteristics of the matrix, andthe computation amount of adaptive selection can be neglected compared withthat of matrix exponential. So it is very important for enhancing theadaptations and increasing the computation efficiencies of matrixexponential. In addition, computational examples are carried out to testifythe correctness and validity of the present method.