Table Of ContentComputing Supplementum 9
R. Albrecht, G. Alefeld, H.1. Stetter (eds.)
Validation Numerics
Theory and Applications
Springer-Verlag Wien New York
Prof. Dr. R. Albrecht
Institut flir Informatik
Universitat Innsbruck
Austria
Prof. Dr. G. Alefeld
Institut fur Angewandte Mathematik
Universitat Karlsruhe
Federal Republic of Germany
Prof. Dr. H. J. Stetter
Institut fur Angewandte und Numerische Mathematik
Technische Universitat Wien
Austria
This work is subject to copyright.
All rights are reserved, whether the whole or part ofthe material is concerned, specifically those of translation,
reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means,
and storage in data banks.
© 1993 Springer-VerlaglWien
Reprint of the original edition 1993
Typesetting: Asco Trade Typesetting Limited, Hong Kong
Printed on acid-free paper
With 23 Figures
ISBN-13: 978-3-211-82451-1 e-ISBN-13: 978-3-7091-6918-6
001: 10.1007/978-3-7091-6918-6
ISSN 0344-8029
ISBN 978-3-211-82451-1 Springer-Verlag Wien New York
ISBN 978-3-211-82451-2 Springer-Verlag New York Wien
Preface
This volume is dedicated to Professor Dr. Ulrich Kulisch, Director of the Institute for Applied
Mathematics of Karlsruhe University, on the occassion of his 60th birthday.
The contributions give a comprehensive overview of methods which can be effectively used
for the validation of the solutions of a variety of problems. These include simultaneous
systems of linear and nonlinear equations, differential and integral equations, and certain
applications from the technical sciences. Furthermore there are some papers which present
improvements of the tools of Validation Numerics.
All papers have been refereed. We wish to thank the contributors for their manuscripts and the
referees for their prompt and careful reviews. We are also grateful to the Publisher, Springer
Verlag, Wien for an efficient and quick production of this volume.
Last but not least we want to thank Professor Kulisch to whom we owe so much in the field
of Validation Numerics. Without his pioneering work, most of the papers in this volume
would not have been written.
InnsbrucklKarlsruheNienna, R. Albrecht
February 1993 G. Alefeld
H. J. Stetter
Contents
Apostolatos, No: On a Unified Concept of Mathematics
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Atanassova, L., Herzberger, Jo: A General Approach to a Class of Single-Step Methods
for the Simultaneous Inclusion of Polynomial Zeros 11
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Dimitrova, No: On Some Properties of an Interval Newton Type Method and its
Modification 21
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Dobner, Ho Jo: Verified Solution of the Integral Equations for the Two-Dimensional
Dirichlet and Neumann Problem 33
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Frommer, A, Mayer, Go: Two-Stage Interval Iterative Methods 45
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Han, Wo, Potra, F. A: Convergence Acceleration for Some Rootfinding Methods 67
00 0
Heindl, Go: A Program for Enclosing Cumulative Binomial Probabilities 79
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Jahn, K.-Uo: Effective Evaluation of Hausdorff Distances for Finite Unions of Hyper-
rectangles 93
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Kaucher, Eo, Baumhof, Co: A Verified Computation of Fourier-Representations of
Solutions for Functional Equations 101
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Kearfott, Bo, Du, K.: The Cluster Problem in Global Optimization: The Univariate Case 117
Konig, So, Ullrich, Co Po: Developing Expert Systems for Validating Numerics 129
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Kramer, Wo, Barth, Bo: Computation of Interval Bounds for Weierstrass' Elliptic
Function fp (z) 147
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Nakao, Mo To: Solving Nonlinear Elliptic Problems with Result Verification Using an
H-1 Type Residual Iteration 161
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Neumaier, A: The Wrapping Effect, Ellipsoid Arithmetic, Stability and Confidence
Regions 175
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Rump, So Mo: Validated Solution of Large Linear Systems 191
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Schwandt, Ho: The Interval Buneman Algorithm for Arbitrary Block Dimension 213
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Spreuer, Ho, Adams, E.: On the Existence and the Verified Determination of Homoc linic
and Heteroclinic Orbits of the Origin for the Lorenz Equations 233
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Stetter, Ho Jo: Verification in Computer Algebra Systems 247
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Walter, Wo Vo: FORTRAN-XSC. A Portable Fortran 90 Module Library for Accurate
and Reliable Scientific Computing 265
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Wolff von Gudenberg, Jo: Implementation of Accurate Matrix Multiplication on the
CM-2 287
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Listed in Current Contents
Computing, Suppl. 9,1-10 (1993)
Computing
© Springer-Verlag 1993
On a Unified Concept of Mathematics*
N. Apostolatos, Athens
Dedicated to Professor U. Kulisch on the occasion of his 60th birthday
Abstract - Zusammenfassung
On a Unified Concept of Mathematics. Mathematics is not simply a branch of science, but the unique and
conclusive means of expressing and solving any problem in any field whatever. In this paper I wish to
submit some thoughts justifing my belief in the unification of Mathematics. Besides that, I make a
proposal for replacement of the structureblock for questions (with two or more answers) in struc
turegrams, which not only makes the structuregrams more representative and comprehensive but
also corrects certain shortcomings and makes them more adapted to the structure of programming
languages.
AMS Subject Classification: 68Q20, 65GlO, 65D99
Key words: Algorithms, finite numerical mathematics, finite approximation theory, iterative methods,
interval mathematics, structuregrams.
Dber eine einheitliche Auffassung der Mathematik. Mathematik ist nicht einfach ein Zweig der Wissen
schaften, sondern das einzige und entscheidende Mittel zum Beschreiben und Losen von Problemen zu
allen Bereichen. In dieser Arbeit mochte ich einigen Gedanken vorlegen, welche meinen Glauben an die
einheitliche Mathematik rechtfertigen. Nebenher unterbreite ich einen Vorschlag wie der Strukturblock
fiir Fragen (mit Zwei- bzw. Mehrfacherverzweigungsblocken) durch Struktogramme ersetzt werden
kann, urn nicht nur Struktogramme repriisentativer und umfassender zu machen, sondern auch manche
Miingel zu beseitigen und diese mehr der Struktur von Programmiersprachen anzupassen.
1. Mathematics
The language, in the broadest sense of the word, is the predominant factor in our
life. The need for a strict definition led to its mathematization. Mathematics became
the basic instrument for this purpose. In reality Mathematics is a super-language,
since it is not simply a language, but constitutes the only language for the strict
definition of other languages. In order to define a language we need an alphabet.
So far, no one has succeeded in recording the entire alphabet of the language of
Mathematics and the rules which govern it. Thus, if we are to study Mathematics
we must divide it into sub-fields or, rather, sub-languages.
Given, therefore, that Mathematics is a language, we can say that a mathematical
problem is nothing but a word (text) in the language of Mathematics. And because
of the need here for restriction to certain strictly defined spaces, we are obliged to
* Received July 20,1992; revised December, 9,1992.
2 N. Apostoiatos
express problems in corresponding sub-languages. From the theory of languages
we know that if we have one word, then we can use the rules of the language to
produce equivalent words. Consequently, each problem has as equivalents all the
words equivalent to that which was used to express the problem. We call one of
these words the solution to the problem. So when we say that a problem has no
solution, then that problem is the solution or rather that equivalent word (problem)
which evidently proves that there is no solution.
The introduction of infinity into Mathematics was a decisive turning-point in the
history of Mathematics. Without this concept a strict definition of the most funda
mental concepts would be impossible. And whereas the finite appears accessible to
us, in practice it often leads to an insuperable situation. Let us just try to conceive
the finite number: 1(4,4) where 1 is the function of Ackermann.
The tremendous development of computers enabled us to process a great deal of
information, always however, in a finite space, the power of which is almost zero
compared with the above number 1(4, 4). The very need for accessible finite spaces
was the motivation behind the development and establishment of Interval Mathe
matics. In an attempt to use modern computing methods an indispensable prerequi
site was that the input, processing and output of data through a computer system
should be carried out automatically with the help of that computer. This attempt
resulted in the modern field of Data Structures. Thus the algorithm became predom
inantly important. In my opinion, the Theory of Algorithms, as founded in our time,
is the basic and principal achievement of the 20th century.
Let us say that we have a problem p which has a solution s and we are searching
for an algorithm a, which when applied to the problem will lead to the solution. We
symbolize this with the mapping:
a
p-+s
or simply: ap = s
In Mathematics, the probability of having a problem p with a solution s and having
a finite algorithm a to solve it is almost zero. So usually, and not only because we
are in finite spaces, the following model applies:
p --a s
j
--
Ii
P S
where p is an "approximation" of p, s is an "approximation" of s and a is a feasible
algorithm which, when applied to p gives the solution s. Ultimately, whereas we are
interested for s, we calculate s, so that our final problem is to check how good sis.
(N.B. Obviously some algorithm is also required for the mapping p -+ p)
The whole of the above model is incorporated into Mathematics. More specifically:
To define p, s, a we need what is known simply as Mathematics.
On a Unified Concept of Mathematics 3
r l
MATHEMATICAL LANGUAGE
MATHEMATICS
NUMERICAL MATHEMATICS
APPROXIMATION THEORY
I
L J
Figure 1. Mathematical language
To realize the mapping p ~ S we need what is known as Numerical Analysis or better
still, Numerical Mathematics.
Finally, for the comparison of P and p, a and a, sand S we need the Approximation
Theory.
Thus, we essentially have the above overlapping scheme (Fig. 1):
In addition, however, all these must become accessible finite and here we see that
we need:
a) The Finite Numerical Mathematics
b) The Finite Approximation Theory
The basic modern means for achieving the above is the computer.
2. The Iterative Method
In Numerical Mathematics we divide methods for solving problems into two large
classes:
a) Direct methods
b) Iterative methods
The distinction between the two methods is made as follows:
Definition 2.1. A method (algorithm) M is called a direct method iff it is split into n
methods M1, M2, ... , Mn, so that when applied to a problem p, which has the
solution s, it transforms it equivalently into n successive problems Pi' i = 1( l)n, so
that:
Mi+1Pi = Pi+l, for i = O(I)n - 1, where Po = p, Pn = s
Definition 2.2. A method M is called iterative iff, when applied to a problem p, which
has the solution s, it transforms it equivalently into a sequence of problems Pi' i = 1,
4 N. Apostolatos
2, ... so that:
Mpi = Pi+l, i = 0, 1,2, ... , where Po = P and lim Pi = s
;-+00
Hence on the basis of the above two definitions 2.1 and 2.2 the Elimination
method of Gauss is a direct method, while the Gauss-Seidel method is an iterative
one.
However, in a direct method, if n is large and the methods Mi, i = 1(1)n are
independent, then it would be impossible to apply it, since a large number of
independent algorithms would have to be created and put into the computer sys
tem. As we know, basically every direct method has a few independent steps
(independent algorithms), each of which is probably repeated several times, each
time, of course, with different data. Thus, here too, we actually have an "iterative"
method. Moreover, in an iterative method too, we interrupt after a finite number
of iterations.
As we know, each sequence of elements produced by an iterative method, since an
finite automaton, such as the computer, is used to produce them, is a cyclical
sequence. This cyclic property is not easy to ascertain, because apart from the
finiteness of each computer system the number of its elements is too large. But even
if we do ascertain it, this only leads to an impasse, since the only profit that we
receive is that with this specific computer system we have, and with the method we
applied, we cannot expect any further information. While the results do not provide
us with elements capable of utilizing them reliably.
Let us consider a problem p, with the solution s which is an element of a topological
space S. We use T to symbolize all the closed subsets of S.
Definition 2.3. Each element t E T with the property SEt is called an admissible
solution of the problem p.
Let A be a set of algorithms for the solution of the problem p, so that for every a E A
we have ap = t, where t is an admissible solution.
Definition 2.4. An algorithm a E A is called optimal in relation to the problem p, iff
ap = t, where t = {s}.
Definition 2.5. Two elements a, bE A are called equal (a = b), iff ta = tb, where
ta = ap and tb = bp.
Definition 2.6. For two elements a, bE A we introduce an order relation (::;;), as
following:
a::;; b,
Definition 2.7. For a, bE A we define union and intersection as following:
a u b = C E A: tc = ta u tb
a n b = C E A: tc = ta n tb
N.B. We assume that A is closed under union and intersection.
