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2003 | Book

On Symmetric Solution Sets Read chapter Read first chapter

Inclusion Methods for Nonlinear Problems

With Applications in Engineering, Economics and Physics

Editor: Univ.-Prof. Dr. Jürgen Herzberger

Publisher: Springer Vienna

Book Series : Computing Supplementum

Included in: Professional Book Archive

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About this book

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". GiHz Alefeld Karlsruhe, Dezember 2001 (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.

Table of Contents

Frontmatter
On Symmetric Solution Sets
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 [Al. 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 Es y m.
Götz Alefeld, Vladik Kreinovich, Güunter Mayer
Methods for Computing All Roots of a Polynomial Simultaneously ­ Known Results and Open Problems
Abstract
Let f(x)=xn+an−1xn−1+ ... +a1x+a0,a0,...,an−1 ∈ C, be a polynomial with the simple real or complex zeros x i,i=1,2,...,n, and x i 0 ,i=1,2,...,n be distinct reasonably close approximations of these zeros.
Lydia Atanassova, N. Kyurkchiev, Tetsuro Yamamoto
Narrow Bounds for the Effective Rate of Return Concerning a Special Problem for Annuities
Abstract
We consider the calculation of the effective rate of an annuity which was derived by changing the conditions of a given annuity in a certain manner. In this note we will derive narrow bounds for the effective rate of the changed annuity in terms of the (given) rate of the original annuity and the coefficients of the polynomial involved. It is demonstrated by some numerical examples that the bounds achieved for this purpose are of sufficient accuracy for practical applications and thus an application of an iterative method for solving the equation can be avoided.
F. Detmers, J. Herzberger
Algorithmic Differentiation with Intervals
Abstract
The computational solution of many mathematical problems involves derivatives. Furthermore, in search of verified computer results, it is necessary to construct algorithms for interval inclusions of derivatives. For “small” functions hand-coded formulas for derivatives and appropriate algorithms for interval inclusions are possible, but for “large” functions algorithmic methods have to be employed. In this lecture we screen the complexity of algorithmic differentiation methods, and we investigate the quality of corresponding interval inclusions of derivatives.
H. Fischer
Computation of a Family of Non-cosymmetrical Equilibria in a System of Nonlinear Parabolic Equations
Abstract
We study a system of two nonlinear parabolic equations with cosymmetry and find a continuous family of equilibria with nonconstant spectrum. Basing on the finite-difference approach, we develop a numerical method for solving the partial differential equations and calculating the continuous family of non-cosymmetrical equilibria. An application of the proposed system to population kinetics is demonstrated. Numerically computed families are presented. The dependence on model parameters is discussed.
K. Frischmuth, V. G. Tsybulin
Quadratic Convergence of Scaled Iterates by Kogbetliantz Method
Abstract
A sharp quadratic convergence bound of scaled iterates for the serial Kogbetliantz method is derived. Iterates are symmetrically scaled by diagonal matrices so that diagonal elements are ones. The result is obtained for a scaled diagonally dominant complex triangular matrix with multiple singular values. The estimate depends on the relative separation of the singular values.
Vjeran Hari, Josip Matejaš
On a Method for Computing Inclusions of Solutions of the Basic GPS Equations
Abstract
A main step in the computation of the coordinates of a GPS navigator from received satellite signals is to solve a system of four equations
Pk=∥xk−x∥2+b, k=1,...,4
for the unknowns x ∈ R3 and b ∈ R, where ∥·∥2 denotes the standard Euclidean norm on R3 and Pk ∈ R and xk ∈ R3, k=1,...,4, are given.
The aim of the paper is to present conditions for Pk and xk, k = 1,...4, guaranteeing existence and local uniqueness of solutions of (E) and to develop a method for computing inclusions of solutions of (E).
Mathematics Subject Classification: 86A30, 86-08, 65G20. Keywords: GPS-equations, automatic result verification.
G. Heindl
Construction of Bounds for the Positive Root of a General Class of Polynomials with Applications
Abstract
We consider classes or polynomials arising in numerical analysis when determining the order of convergence and in classical mathematics of finance when calculating the effective rate of interest. Those polynomials always have a unique positive root according to Descartes’ rule of signs. In the presence of parameters, for example, we want to give a priori bounds for this root with an accuracy satisfying the demands of the applications. Starting with a review of the cases of polynomials treated in the literature and the bounds given there, we propose a kind of pertubation method using the monotonicity principle first established in Herzberger [5] using some slightly improved bounds for these cases. Thus we are able to construct a priori bounds for classes of polynomials with more general coefficients compared with those treated in earlier papers. Some numerical examples demonstrate how good our method works out in practice.
J. Herzberger, A. Langer
Rounding Near Zero
Abstract
This paper deals with arithmetic on a discrete subset S of the real numbers IT and with floating-point arithmetic in particular. We assume that arithmetic on S is defined by semimorphism. Then for any element a E S the element ¡ªa E S is an additive inverse of a, i.e. a0 (¡ªa) = 0. The first part of the paper describes a necessary and sufficient condition under which ¡ªa is the unique additive inverse of a in S. In the second part this result is generalized. We consider algebraic structures M which carry a certain metric, and their semimorphic images on a discrete subset N of M. Again, a necessary and sufficient condition is given under which elements of N have an unique additive inverse. This result can be applied to complex floating-point numbers, real and complex floating-point intervals, real and complex floating-point matrices, and real and complex floating-point interval matrices.
Ulrich W. Kulisch
a Note on the Convergence of the SOR-like Weierstrass Method
Abstract
Convergence properties of the SOR-like Weierstrass method for the simultaneous approximation of polynomial roots are considered. Then the choice of the acceleration parameter is discussed.
N. Kyurkchiev
Boundary Regularity Aspects in Solving Contact Problems
Abstract
The nonlinear contact problem consists in finding the unknown contact zone. Predictor/corrector iterations over the partitions of the boundary approximate the contact zone. Each step contains the solution of a mixed problem of linear elasticity, whose behaviour is well understood. In particular square-root singularities occur in the collision points between regions with boundary conditions of different types, except for the correct partition including the searched contact region.
This property is used to construct a numerical algorithm providing the solution of a quasi-stationary contact problem with dry friction. A relation between macroscopical input parameters of the contact problem is stated. It is checked with known theoretical results and applied ­ as an example ­ for discussing the influence forces to the deformation behaviour within the contact zone.
Dirk Langemann
Convex-decomposable Operators and Inclusive Algorithms
Abstract
If an operator has the nature Fy − Fx ≥ D 1 (x, y) + D2 (x, y)) (y − x) for x ≥ y, where D 1 and D 2 are isotone and antitone respectively, we denote it a convex-decomposable operator. We will show that it is a rather large class of operators. For these operator equations, some monotone inclusive iterations are given. Under some usual assumptions, the iteration converges to the minimum and maximum solution.
In this paper, we discuss some algorithm for solving a class of nonlinear systems Fx=0, x ∈ Ω, where Ω = x ∈ Rn| a≥x≥b is a finite ordered interval.
Yong-xiang Ling, Hua-lin Cao, Han-fang Sheng
Fast Inclusion and Residual Iteration for Solutions of Matrix Equations
Abstract
This paper is concerned with the problem of verifying an accuracy of numerical solutions of matrix equations. In this paper, a fast algorithm of obtaining very sharp verified error bound is proposed. The new verification method is based on combination of the rounding mode controlled double precision computation and the high precision computation. The method can also be used to improve numerical solutions by the residual iteration. Numerical results have also been presented for illustrating that a very sharp verified error bound of numerical solutions can be obtained with a computational cost comparable to that for calculating an approximate solution. Moreover, it is shown using an example that a few residual iterations give extremely improved solutions with little computational cost.
Takeshi Ogita, Shin’ichi Oishi, Yasunori Ushiro
Schröder-like Methods for the Simultaneous Inclusion of Polynomial Zeros
Abstract
The subject of this paper is the study of Schröder-like methods for the simultaneous inclusion of multiple complex zeros of a polynomial in circular complex arithmetic. These methods are based on the fundamental work of Gargantini and Henrici [7] and belong to the class of the most efficient inclusion methods. Apart from a review of the existing methods of Schröder’s type, together with their convergence properties, a new, accelerated method in serial (single-step) mode with Schröder’s correction is proposed. The comparison of the considered methods based on their computational efficiency and numerical results are given.
L. D. Petković, M. S. Petković
Interval Root-finding Methods of Laguerre’s Type*
Abstract
In this paper, several algorithms of Laguerre’s type for the inclusion of simple complex zeros of a polynomial are presented. These methods are realized in complex circular arithmetic and have the convergence order > 4. The proposed algorithms possess a great computational efficiency since the acceleration of the convergence is attained with few additional calculations. High convergence speed is demonstrated on numerical examples.
L. D. Petković, M. S. Petković, D. Živković
Exact Behavior of Singularities of Protter’s Problem for the 3-D Wave Equation
Abstract
For the wave equation we study boundary value problems, which are three-dimensional analogues of Darboux-problems on the plane. It is shown that for n in h there exists a right hand side smooth function from C*(Ω), for which the corresponding unique generalized solution belongs to C*(Ω\O), but it has a strong power-type singularity at the point O. This singularity is isolated at the vertex O of the characteristic cone and does not propagate along the cone. The present article describes the exact behavior of the singular solutions at the point O. It states some exact a priori estimates for the solution.
Nedyu Popivanov, Todor Popov
Construction of Shortest Line of Restricted Curvature in a Non-singly-connected Polygonal Area
Abstract
In a computer aided laying-out of roads and railway lines there arises the problem of the modeling of optimal routes with restricted curvature in a given non-singly-connected polygonal (NSCP) area. However, since the triangulating network models are used now for the NSCP areas, the development of the respective AutoCADs is sufficiently hampered due to the lack of an efficient approach allowing to simulate the routes of continuous curvature with the required accuracy. Therefore, optimization of routes cannot be carried out in a proper way. For solving this nonlinear problem in compliance with the requirements of the engineering standards, a model is presented that comprises a set of basic optimization problems for the construction of the shortest route composed of segments and circular arcs. In order to provide smooth growth of acceleration for eliminating overload at high speeds, these elements are mated with the easement curves — clothoid (Euler spiral) or cubic parabola fragments. For these problems the condition of optimality and respective optimization methods are presented that show linear memory and time complexity.
S. V. Smelyakov
Backmatter
Metadata
Title
Inclusion Methods for Nonlinear Problems
Editor
Univ.-Prof. Dr. Jürgen Herzberger
Copyright Year
2003
Publisher
Springer Vienna
Electronic ISBN
978-3-7091-6033-6
Print ISBN
978-3-211-83852-5
DOI
https://doi.org/10.1007/978-3-7091-6033-6