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# Interval FAQ 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.
http://solon.cma.univie.ac.at/~neum/papers.html#ode
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"}

@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",
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;
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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