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Giovanni Spagnuolo -- Interval matrix exponential?

From Giovanni Spagnuolo (spanish@ieee.org), Dipartimento di Ingegneria dell'Informazione ed Ingegneria Elettrica, University of Salerno

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

Response by Arnold Neumaier:

(neum@cma.univie.ac.atm), University of Vienna:

You may find

A. Neumaier,
Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory, Computing 52 (1994), 315-336.
useful in this respect. (Specialize the general case to a constant coefficient differential equation.)

Arnold Neumaier

Response by Nelson H. F. Beebe:

(beebe@math.utah.edu), University of Utah:

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"}

  author =       "C. B. Moler and C. F. {Van Loan}",
  title =        "Nineteen dubious ways to compute the exponential of a
  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,
  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"}

  author =       "E. P. Oppenheimer and Anthony N. Michel",
  title =        "Application of interval analysis techniques to linear
                 systems. {II}. The interval matrix exponential
  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,
  classification = "C1210 (General system theory); C4140 (Linear
                 algebra); C4130 (Interpolation and function
  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"}

  author =       "P. Bochev and S. Markov",
  title =        "A self-validating numerical method for the matrix
  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,
  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

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