Jordan–Schur Algorithms for Computing the Matrix Exponential

Abstract

In this paper, two new versions of the Schur algorithm for computing the matrix exponential of an $n\times n$ complex matrix $A$ are presented. Instead of the Schur form, these algorithms use the Jordan–Schur form of $A$ . The Jordan–Schur form is found by less computation and it is determined more reliable than the reduction to Jordan form since it is obtained using only unitary similarity transformations. In contrast to the known methods, the diagonal blocks of the matrix exponential are obtained by using finite Taylor series. This improves the accuracy and avoids the decisions made about the termination of the series expansion. The off-diagonal blocks of the exponential are determined by modifications of the Schur–Parlett or Schur–Fréchet method, which takes advantage of the Jordan–Schur form of the matrix. The numerical features of the new algorithms are discussed, revealing their advantages and disadvantages in comparison with the other methods for computing the matrix exponential. Computational experiments show that using the new algorithms, the matrix exponential is determined in certain cases with higher accuracy than some widely used methods, however, at the price of an increase in the computational cost which is of order $n^4$ . It is shown that the Jordan–Schur algorithms for computing the matrix exponential are appropriate for matrices with multiple eigenvalues and are especially efficient in cases of large Weyr characteristics.