AN IMPROVED SCHUR-PADE ALGORITHM FOR FRACTIONAL POWERS OF A MATRIX AND THEIR FRECHET DERIVATIVES

作者：Higham Nicholas J^*; Lin Lijing

来源：SIAM Journal on Matrix Analysis and Applications, 2013, 34(3): 1341-1360.

DOI：10.1137/130906118

• matrix power
• fractional power
• matrix root
• Frechet derivative
• condition number
• condition estimate
• Schur decomposition
• Pade approximation
• Pade approximant
• matrix logarithm
• matrix exponential
• MATLAB

The Schur-Pade algorithm [N. J. Higham and L. Lin, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1056-1078] computes arbitrary real powers A(t) of a matrix A is an element of C-nxn using the building blocks of Schur decomposition, matrix square roots, and Pade approximants. We improve the algorithm by basing the underlying error analysis on the quantities parallel to(I - A)(k) parallel to(1/k), for several small k, instead of parallel to I - A parallel to. We extend the algorithm so that it computes along with A(t) one or more Frechet derivatives, with reuse of information when more than one Frechet derivative is required, as is the case in condition number estimation. We also derive a version of the extended algorithm that works entirely in real arithmetic when the data is real. Our numerical experiments show the new algorithms to be superior in accuracy to, and often faster than, the original Schur-Pade algorithm for computing matrix powers and more accurate than several alternative methods for computing the Frechet derivative. They also show that reliable estimates of the condition number of A(t) are obtained by combining the algorithms with a matrix norm estimator.

• 出版日期2013