New Scaling-Squaring Taylor Algorithms for Computing the Matrix Exponential

The matrix exponential plays a fundamental role in linear differential equations arising in engineering, mechanics, and control theory. The most widely used, and the most generally efficient, technique for calculating the matrix exponential is a combination of “scaling and squaring” with a Padé approximation. For alternative scaling and squaring methods based on Taylor series, we present two modifications that provably reduce the number of matrix multiplications needed to satisfy the required accuracy bounds, and a detailed comparison of the several algorithmic variants is provided.