Preconditioning the Matrix Exponential Operator with Applications

The idea of preconditioning is usually associated with solution techniques for solving linear systems or eigenvalue problems. It refers to a general method by which the original system is transformed into one which admits the same solution but which is easier to solve. Following this principle we consider in this paper techniques for preconditioning the matrix exponential operator, e A y 0 , using diierent approximations of the matrix A. These techniques are based on using generalized Runge Kutta type methods. Preconditioners based on the sparsity structure of the matrix, such as diagonal, block diagonal, and least-squares tensor sum approximations are presented. Numerical experiments are reported to compare the quality of the schemes introduced.