Verified computation of the matrix exponential
- PDF / 1,581,639 Bytes
- 16 Pages / 439.642 x 666.49 pts Page_size
- 66 Downloads / 176 Views
Verified computation of the matrix exponential Shinya Miyajima1
Received: 20 July 2017 / Accepted: 16 April 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Two numerical algorithms for computing interval matrices containing the matrix exponential are proposed. The first algorithm is based on a numerical spectral decomposition and requires only cubic complexity under some assumptions. The second algorithm is based on a numerical Jordan decomposition and applicable even for defective matrices. Numerical results show the effectiveness and robustness of the algorithms. Keywords Matrix exponential · Numerical Jordan decomposition · Verified numerical computation Mathematics Subject Classification (2010) 15A16 · 65F60 · 65G20
1 Introduction The matrix exponential of A ∈ Cn×n , denoted by eA , is defined by eA := I + A +
1 2 1 A + A3 + · · · , 2! 3!
Communicated by: Peter Benner This work was partially supported by JSPS KAKENHI Grant Number JP16K05270. Shinya Miyajima
[email protected] 1
Faculty of Science and Engineering, Iwate University, 4-3-5 Ueda, Morioka-shi, Iwate 020-8551, Japan
(1.1)
S. Miyajima
where I is the identity matrix. It is known (see [2], e.g.) that the series converges for any A. The interest in the matrix exponential stems from its key role in the solution to differential equations. Depending on the application, the problem may be to compute eA for a given A, to compute eAt for a fixed A and many t ∈ R, or to apply eA or eAt to a vector. The target problem in this paper is the first one. Many numerical algorithms for computing eA have been proposed (see [2, 3, 6, 7], e.g.). The work presented in this paper addresses the problem of verified computations for eA , specifically, numerically computing an interval matrix which is guaranteed to contain eA . As reported in [1, Section 6], even when we increase the order of approximation, which theoretically has to improve the accuracy of the result, we can not assert that the obtained result is accurate. One way for asserting the accuracy is to compute an error bound together with the approximation. The radius of the interval obtained by the verified computation can be regarded as the error bound. Therefore, if the radius is small, then we can assert the accuracy. The pioneering work seems to be the algorithm in [1]. This algorithm is based on a Pad´e approximation with scaling and squaring and safe error monitoring. For realizing the safe error monitoring, a staggered correction format is effectively used. The VERSOFT [8] routine VERMATFUN is also applicable. This routine is applicable not only to eA but also to other matrix functions, and usually requires O(n4 ) operations. If A is upper triangular and all the diagonal elements are equal, the approach in [5, Section 4] is applicable. On the other hand, this literature does not mention the other cases. The purpose of this paper is to propose two algorithms for computing intervals containing eA . The first algorithm is based on a numerical spectral decompos
Data Loading...
