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About Sun's f95 with Interval Support

Giovanni Spagnuolo -- Interval matrix exponential?

I am searching for papers describing new approaches to the calculation of the exponential of an interval matrix. I have A as an interval matrix (not diagonal), and I need to calculate exp(A). Of course, by using the classical approach of the Taylor expansion, even using a nested form, a great overestimation of the result can often be verified. On the other side, an interval extension of the Pade approach is quite difficult, I think.

Many thanks in advance.

Regards.
Giovanni

You may find

A. Neumaier,useful in this respect. (Specialize the general case to a constant coefficient differential equation.)

Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory, Computing 52 (1994), 315-336.

http://solon.cma.univie.ac.at/~neum/papers.html#ode

Arnold Neumaier

While this paper doesn't address interval arithmetic, if you weren't aware of it already, you might find it a useful guide into the older literature:

@String{j-SIAM-REVIEW = "SIAM Review"} @Article{Moler:1978:NDW, author = "C. B. Moler and C. F. {Van Loan}", title = "Nineteen dubious ways to compute the exponential of a matrix", journal = j-SIAM-REVIEW, volume = "20", number = "4", pages = "801--836", month = oct, year = "1978", CODEN = "SIREAD", ISSN = "0036-1445", bibdate = "Fri Mar 21 15:55:27 MST 1997", acknowledgement = ack-nhfb, classcodes = "C4140 (Linear algebra)", corpsource = "Dept. of Math., Univ. of New Mexico, Albuquerque, NM, USA", keywords = "eig; matrix algebra; matrix exponential; matrix function; nla", treatment = "A Application; G General Review", }Here are two more from my files:

@String{j-IEEE-TRANS-CIRCUITS-SYST = "IEEE Transactions on Circuits and Systems"} @Article{Oppenheimer:1988:AIAb, author = "E. P. Oppenheimer and Anthony N. Michel", title = "Application of interval analysis techniques to linear systems. {II}. The interval matrix exponential function", journal = j-IEEE-TRANS-CIRCUITS-SYST, volume = "35", number = "10", pages = "1230--1242", month = oct, year = "1988", CODEN = "ICSYBT", ISSN = "0098-4094", bibdate = "Thu Dec 14 17:19:38 MST 1995", abstract = "For pt.I see ibid., vol.35, no.9, p.1129-38 (1988). In part I the authors established new results for continuous and rational interval functions which are of interest in their own right. The authors use these results to study interval matrix exponential functions and to devise a method of constructing augmented partial sums which approximate interval matrix exponential functions as closely as desired. The authors introduce and study `scalar' and matrix interval exponential functions. These functions are represented as infinite power series and their properties are studied in terms of rational functions obtained from truncations. To determine optimal estimates of error bounds for the truncated series representation of the exponential matrix function, the authors establish appropriate results dealing with Householder norms. In order to reduce the conservativeness for interval arithmetic operations, they consider the nested form for interval polynomials and the centered form for interval arithmetic representations. They also discuss briefly machine bounding arithmetic in digital computers. Finally, the authors present an algorithm for the computation of the interval matrix exponential function which yields prespecified error bounds.", acknowledgement = ack-nhfb, affiliation = "Appl. Phys. Lab., Johns Hopkins Univ., Laurel, MD, USA", classification = "C1210 (General system theory); C4140 (Linear algebra); C4130 (Interpolation and function approximation)", keywords = "Interval analysis techniques; Linear systems; Interval matrix exponential function; Augmented partial sums; Infinite power series; Rational functions; Optimal estimates; Error bounds; Truncated series representation; Householder norms; Nested form; Interval polynomials; Centered form; Interval arithmetic representations; Machine bounding arithmetic; Digital computers", language = "English", pubcountry = "USA", thesaurus = "Linear systems; Matrix algebra; Polynomials", } @String{j-COMPUTING = "Computing"} @Article{Bochev:1989:SVN, author = "P. Bochev and S. Markov", title = "A self-validating numerical method for the matrix exponential", journal = j-COMPUTING, volume = "43", number = "1", pages = "59--72", month = "", year = "1989", CODEN = "CMPTA2", ISSN = "0010-485X", bibdate = "Thu Dec 14 17:20:22 MST 1995", abstract = "An algorithm is presented, which produces highly accurate and automatically verified bounds for the matrix exponential function. The computational approach involves iterative defect correction, interval analysis and advanced computer arithmetic. The algorithm presented is based on the `scaling and squaring' scheme, utilizing Pade approximations and safe error monitoring. A PASCAL-SC program is reported and numerical results are discussed.", acknowledgement = ack-nhfb, affiliation = "Inst. of Math., Bulgarian Acad. of Sci., Sofia, Bulgaria", classification = "B0290H (Linear algebra); C4140 (Linear algebra)", keywords = "Self-validating numerical method; Matrix exponential; Computational approach; Iterative defect correction; Interval analysis; Advanced computer arithmetic; Pade approximations; Safe error monitoring; PASCAL-SC program; Numerical results", language = "English", pubcountry = "Austria", thesaurus = "Iterative methods; Matrix algebra", }Nelson H. F. Beebe

Interval FAQ: [ Entry page | Contents | Search ]

About Sun's f95 with Interval Support

If *you* have a question related to validated computing, interval
analysis, or related matters, I recommend

- Vladik Kreinovich's Interval Computation web site, or
- Submit it to the interval computations listserv: reliable_computing@interval.louisiana.edu

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