Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential

Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential

The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ‖A‖ 1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the...

Generalized Polar Decompositions for the Approximation of the Matrix Exponential

In this paper we describe the use of the theory of generalized polar decompositions [H. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Found. Comput. Math., 1 (2001), pp. 297–324] to approximate a matrix exponential. The algorithms presented have the property that, if Z ∈ g, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of g. This property is very...

Recursive Algorithms for the Matrix Padé Problem

Abstract. A matrix triangularization interpretation is given for the recursive algorithms computing the Padé approximants along a certain path in the Padé table, which makes it possible to unify all known algorithms in this field [5], [6]. For the normal Padé table, all these results carry over to the matrix Padé problem in a straightforward way. Additional features, resulting from the noncommu...

On Padé Approximation of Some Practical Functions

ℋ-Matrix approximation for the operator exponential with applications

We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator [4]. In the preceding papers [12]–[17], a class of matrices (H-matrices) has been analysed which are data-sparse and allow an approximate matrix arithmetic with almost linear complexity. In particular, the matrix-vector/matrixm...