Name
Abstract | ||
|---|---|---|
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 parallel to A parallel to >> 1 and a subdiagonal Pade approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the same magnitude as that of the standard algorithm. The new algorithm performs well if a rough estimate of the rightmost eigenvalue of A is available and the rightmost eigenvalues do not have widely varying imaginary parts, and it achieves significant speedup over the conventional algorithm especially when A is of large norm. Our algorithm uses the partial fraction form to evaluate the Pade approximant, which makes it suitable for parallelization and directly applicable to computing the action of the matrix exponential exp(A)b, where b is a vector or a tall skinny matrix. For this problem the significantly smaller squaring factor has an even more pronounced benefit for efficiency when evaluating the action of the Pade approximant. |
Year | DOI | Venue |
|---|---|---|
2016 | 10.1137/15M1027553 | SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS |
Keywords | Field | DocType |
matrix exponential,scaling and squaring,subdiagonal Pade approximation,stable matrix,conditioning,matrix function times a vector | Pascal matrix,Mathematical optimization,Square (algebra),Padé approximant,Mathematical analysis,Matrix (mathematics),Square matrix,Shift matrix,Matrix exponential,Mathematics,Eigenvalues and eigenvectors | Journal |
Volume | Issue | ISSN |
37 | 1 | 0895-4798 |
Citations | PageRank | References |
2 | 0.43 | 12 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Stefan Güttel | 1 | 109 | 9.68 |
| Yuji Nakatsukasa | 2 | 97 | 17.74 |