Table Of ContentJ. Herzberger (ed.)
Inclusion Methods
for Nonlinear Problems
With Applications in Engineering,
Economics and Physics
Computing
Supplement 16
Springer-Verlag Wien GmbH
Univ.-Prof. Dr. Jiirgen Herzberger
Faculty of Mathematics, University of Oldenbourg, Oldenbourg, Germany
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Foreword
This workshop was organized with the support of GAMM, the International
Association of Applied Mathematics and Mechanics, on the occasion of
J. Herzberger's 60th birthday. GAMM is thankful to him for all the time and
work he spent in the preparation and holding of the meeting.
The talks presented during the workshop and the papers published in this volume
are part of the field of Verification Numerics. The important subject is fostered by
GAMM already since a number of years, especially also by the GAMM
FachausschuB (special interest group) "Rechnerarithmetik und Wissenschaft
liches Rechnen".
Karlsruhe, Dezember 2001 GiHz Alefeld
(President of GAMM)
Preface
At the end of the year 2000, about 23 scientists from many countries gathered in
the beautiful city of Munich on the occasion of the International GAMM
Workshop on "Inclusion Methods for Nonlinear Problems with Applications in
Engineering, Economics and Physics" from December 15 to 18. The purpose of
this meeting was to bring together representatives of research groups from
Austria, Bulgaria, China, Croatia, Germany, Japan, Russia, Ukraine and
Yugoslavia who in a wider sense work in the field of calculating numerical
solutions with error-bounds. Most of those participants have already known each
other from earlier occasions or closely cooperated in the past. Representatives
from three Academies of Sciences were among the speakers of this conference:
from the Bulgarian Academy, the Russian Academy and the Ukrainian Academy
of Sciences.
The contributions covered a wide field of topics with emphasis on nonlinear
problems and possible implications to the solution of practical problems. The
pleasant atmosphere of the location of the conference, the Adolf-Kolping-House
in the very center of the city, was stimulating many direct contacts among the
participants and gave them the opportunity for intensive discussions and scientific
exchange. This volume, extended by some contributions from authors who belong
to these research groups but who were not able to attend the meeting due to the
restricted number of invited guests, is intended to inform a broader audience
about the result and scientific progress made in this area by a number of com
petent members of the international scientific community.
I would like to express my sincere thanks to all of those who have contributed
to this volume and to all the numerous referees who helped to keep this volume
on a high level. I am also grateful to my colleague Professor Dr. R. Albrecht,
Innsbruck, for drawing the attention of Springer-Verlag in Vienna to this con
ference proceedings. My special thanks go to Ms. Silvia Schilgerius and
Mag. Wolfgang Dollhaubl from the Springer-Verlag in Vienna who helped to
make the appearance of this special volume possible and showed patience for this
project.
Last but not least, I would like to thank Ms. Theresia Meyer from the University
of Olden bourg for promptly handling all the correspondence connected with my
editorial work on this volume.
Oldenbourg, October 2002 Jurgen llerzberger
List of Participants of the International GAMM-Workshop
on "Inclusion Methods for Nonlinear Problems
with Applications in Engineering, Economics and Physics"
Munich and OberschleiBheim, December 15-18, 2000
Prof. Dr. R. Albrecht, Innsbruck (Austria)
Prof. Dr. G. Alefeld, Karlsruhe (Germany)
Prof. Dr. A. Andreev, Sofia (Bulgaria)
Dr. L. Atanassova, Munich (Germany)
Acad. Dr. F.L. Chernousko, Moscow (Russia)
Dr. H. Fischer, Munich (Germany)
Prof. Dr. V. Hari, Zagreb (Croatia)
Prof. Dr. G. Heindl, Wuppertal (Germany)
Prof. Dr. J. Herzberger, Oldenburg (Germany)
Prof. Dr. V. Hristov, Sofia (Bulgaria)
Prof. Dr. G. Iliev, Sofia (Bulgaria)
Prof. Dr. U. Kulisch, Karlsruhe (Germany)
Prof. Yong-xiang Ling, Xi'an (PR China)
Prof. Dr. G. Mayer, Rostock (Germany)
Prof. Shin'ichi Oishi, Tokyo (Japan)
Prof. K. Okomura, Kyoto (Japan)
Prof. Dr. M. Petkovic, Nis (Yugoslavia)
Prof. Dr. Lj. Petkovic, Nis (Yugoslavia)
Prof. Dr. N. Popivanov, Sofia (Bulgaria)
Acad. Dr. V.L. Rvachev, Kharkov (Ukraine)
Prof. Dr. S. Smelyakov, Kharkov (Ukraine)
Prof. Dr. Yu. Stoyan, Kharkov (Ukraine)
Prof. Dr. Shen Zuhe, Nanjing (PR China)
Contents
Alefeld, G., Kreinovich, V., Mayer, G.: On Symmetric Solution Sets
Atanassova, L., Kyurkchiev, N., Yamamoto, T.: Methods for Computing
all Roots of a Polynomial Simultaneously - Known Results and Open
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Detmers, F., Herzberger, J.: Narrow Bounds for the Effective Rate of
Return Concerning a Special Problem for Annuities . . . . . . . . . . . . . . . 43
Fischer, H.: Algorithmic Differentiation with Intervals . . . . . . . . . . . . . 45
Frischmuth, K., Tsybulin, V. G.: Computation of a Family of Non-
co symmetrical Equilibria in a System of Nonlinear Parabolic Equations. 67
Hari, V., Matejas, J.: Quadratic Convergence of Scaled Iterates by
Kogbetliantz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Heindl, G.: On a Method for Computing Inclusions of Solutions of the
Basic GPS Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 107
Herzberger, J., Langer, A.: Construction of Bounds for the Positive Root
of a General Class of Polynomials with Applications. . . . . . . . . . . . . . . 121
Kulisch, D.: Rounding Near Zero. . . . . . . . . . . . . . . . . . . . . . . . . . .. 135
Kyurkchiev, N.: A Note on the Convergence of the SOR-like Weierstrass
Method .............................................. 143
Langemann, D.: Boundary Regularity Aspects in Solving Contact Problems 151
Ling, Y.-x., Cao, H.-I., Sheng, H.-f.: The Convex-decomposable Operator
and Inclusive Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165
Ogita, T., Oishi, S., Ushiro, Y.: Fast Inclusion and Residual Iteration for
Solutions of Matrix Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Petkovic, L. D., Petkovic, M. S.: Schroder-like Methods for the Simulta-
neous Inclusion of Polynomial Zeros. . . . . . . . . . . . . . . . . . . . . . . . .. 185
XII Contents
Petkovic, L. D., Petkovic, M. S., Zivkovic, D.: Interval Root-finding
Methods of Laguerre's Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Pop ivanov, N., Popov, T.: Exact Behaviour of Singularities of Protter's
Problem for the 3-D Wave Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 213
Smelyakov, S. V.: Construction of Shortest Line of Restricted Curvature in
a Non-singly-connected Polygonal Area. . . . . . . . . . . . . . . . . . . . . . .. 237
Computing [Suppl] 16, 1-22 (2002)
Computing
© Springer-Verlag 2002
On Symmetric Solution Sets
G. Alefeld, V. Kreinovich and G. Mayer
Abstract
Given an n x n interval matrix [A] and an interval vector [b] with n components we present an overview
on existing results on the solution set Ssym of linear systems of equations Ax = b with symmetric
matrices A E [A] and vectors b E [b]. Similarly we consider the set Esym of eigenpairs associated with the
symmetric matrices A E [A]. We report on characterizations of Ssym by means of inequalities, by means
of intersection of sets, and by an approach which is generalizable to more general dependencies of the
entries. We also recall two methods for enclosing Ssym by means of interval vectors, and we mention a
characterization of Esym.
AMS Subject Classifications: 65GIO.
Keywords: Solution set of linear systems, symmetric solution set of linear systems, eigenpair set,
symmetric eigenpair set, linear interval systems, Oettli-Prager theorem.
1. Introduction
With this paper we intend to give an overview on existing results for the sym
metric solution set
Ssym := {x E IRnl Ax = b, A = AT E [AJ, b E [b]}, (1)
where [A] is a given n x n interval matrix with [A] = [Af, and [b] is a given interval
vector with n components. This set obviously is a subset of the general solution set
S := {x E IRnl Ax = b, A E [AJ, bE [b]}, (2)
where the restriction A = AT on A E IRnxn is not required. Knowing Sand Ssyrn is
particularly interesting in the following situations:
(a) Assume that one has to solve a linear system Ax = h on a computer using
x
floating point arithmetic. Due to rounding errors, the computed result nor
mally will not fulfill Ax = h. If L1A E IRnxn, L1b E IRnare given nonnegative tol
erances one may view x as an acceptable solution whenever xES with S formed
as in (2) with respect to [A] := A + [-L1A, L1AJ, [b] := h + [-L1b, L1b]; in this case x
can be interpreted as exact solution of a linear system Ax = b with some
A E [AJ, bE [b].
2 G. Alefeld et al.
(b) In contrast to (a), where the linear system is known we assume now that one
has to solve a linear system Ax = b where A, b are not given exactly, but they are
known to differ from some A E IRnxn, b E IRn by at most LiA E IRnxn and Lib E IRn,
respectively (LiA, Lib nonnegative). Then A E [A] := A + [-LiA, LiAJ, bE
[b] := b + [-Lib, Lib]. Compute a solution x* of Ax = b. Since x* E S one can accept
x* as a good approximation for the unknown solution of Ax = b. This situation
can occur due to
- conversion errors (from decimal to binary or vice versa),
- errors in measurements,
- errors in adjusting the technical devices.
As we shall see Sand Ssym are not so easy to handle. Therefore, enclosures of S
and Ssym are important. For S such enclosures can be computed by means of
interval arithmetic. Since such methods are contained in textbooks like [1], [20],
e.g., we will omit them here. They trivially deliver also enclosures for Ssym s::; S.
But there are also methods to enclose Ssym without bounding S at the same time.
We will study such methods later on. Although we shall concentrate on Ssym in
this paper we will give a short glance at S in order to work out particularities of
Ssym. So we start in Sect. 3 with several equivalent statements for xES and list
some properties of S. In Sect. 4, we characterize the boundary 8Ssym of Ssym by
means of parts of hyperplanes and quadrics. In Sect. 5, we introduce two methods
for enclosing Ssym and in Sect. 6, we report on the eigenpair set
E := {(xT,.-1f E IRn+11 Ax = Ax, x =f. 0, A E [A]} (3)
and the symmetric eigenpair set
It turns out that quadrics are needed in order to describe E and algebraic in
equalities of order at most three in order to describe Esym.
2. Notations
In the sequel we denote intervals in square brackets, i.e., [aJ = [g, aJ, and identify
point intervals [a, a] by their unique element omitting the brackets. We assume
that the reader is familiar with the elementary rules and basic facts of interval
arithmetic as introduced in the first chapters of [1] or [20], e.g. We will write m,
IlRn, IlRmxn for the set of real compact intervals, interval vectors with n compo
nents and m x n interval matrices, respectively. We apply the notation
[AJ = [4,A] = ([aL) = ([gij' aij]) simultaneously for interval matrices and have a
similar notation for interval vectors, real vectors and real matrices. An interval
matrix [A] is called regular if each A E [A] has this property. By A we denote the
midpoint of [AJ, i.e., A := ! (4 + A), and by rad[AJ := ! (A - 4) its radius which we
