Numerical computation of branch points in nonlinear equations

Summary The numerical computation of branch points in systems of nonlinear equations is considered. A direct method is presented which requires the solution of one equation only. The branch points are indicated by suitable testfunctions. Numerical results of three examples are given.     

Numerical computation of branch points in ordinary differential equations

Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.     

Numerical computation of least constants for the Sobolev inequality

Summary Least constantsc for the well-known Sobolev inequality fcf _{m, G} ,fH ^m (G) are obtained in closed form by a reproducing kernel technique, where the Sobolev spaceH ^m (G) for a domainG in ⁿ is defined as the completion ofC ^m (G) with respect to the Sobolev norm given by , where is the norm ofL ₂ (G) and is the supremum norm onG. Numerical values for the case whereG is the ⁿ are given.     

The approximation of generalized turning points by projection methods with superconvergence to the critical parameter

Summary A procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between Banach spaces. This characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization. Applications are given to parameter dependent two-point boundary value problems. In particular, collocation at Gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points.     

Regular splittings and monotone iteration functions

Summary In this paper we introduce the set of so-called monotone iteration functions (MI-functions) belonging to a given function. We prove necessary and sufficient conditions in order that a given MI-function is (in a precisely defined sense) at least as fast as a second one.Regular splittings of a function which were initially introduced for linear functions by R.S. Varga in 1960 are generating MI-functions in a natural manner.For linear functions every MI-function is generated by a regular splitting. For nonlinear functions, however, this is generally not the case.     

Enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations

Summary A solution of a nonlinear equation in Hilbert spaces is said to be a simple singular solution if the Fréchet derivative at the solution has one-dimensional kernel and cokernel. In this paper we present the enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations. Once singularities are resolved, we can compute accurately the singular solution by Newton's method. Conditions for which the procedure terminates in finite steps are given. In particular, if the equation defined in ⁿ is analytic and the simple singular solution is geometrically isolated, the procedure stops in finite steps, and we obtain the enlarged problem with an isolated solution. Numerical examples are given.This research is partially supported by Grant-in-Aid for Encouragment of Young Scientist No. 60740119, the Ministry of EducationDedicated to Professor Seiiti Huzino on his 60th birthday     

Zur Konvergenz des SSOR-Verfahrens für nichtlineare Gleichungssysteme

Zusammenfassung In der vorliegenden Arbeit wird für die Konvergenz des symmetrischen Relaxationsverfahrens (SSOR-Verfahren) bei Anwendung auf eine Klassenichtlinearer Gleichungssysteme die globale Konvergenz bewiesen. Diese Gleichungssysteme treten z.B. bei der Diskretisierung nichtlinearer partieller Differentialgleichungen auf.

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Convergence of the SSOR-method for nonlinear systems of simultaneous equations

Summary In this paper we prove the convergence of the symmetric successive overrelaxation method if it is applied to certain nonlinear systems of simultaneous equations. These equations are obtained, for example, by discretizing nonlinear elliptic partial differential equations.

Herrn Helmut Brakhage, Kaiserslautern, anläßlich seines sechzigsten Geburtstages am 8. 7. 1986 gewidmet     

Numerical stability of descent methods for solving linear equations

Summary In this paper we perform a round-off error analysis of descent methods for solving a liner systemAx=b, whereA is supposed to be symmetric and positive definite. This leads to a general result on the attainable accuracy of the computed sequence {x _i } when the method is performed in floating point arithmetic. The general theory is applied to the Gauss-Southwell method and the gradient method. Both methods appear to be well-behaved which means that these methods compute an approximationx _i to the exact solutionA ^–1 b which is the exact solution of a slightly perturbed linear system, i.e. (A+A)x _i =b, A of order A, where is the relative machine precision and · denotes the spectral norm.     

A priori-Schranken für diskrete Lösungen monotoner Operatorgleichungen und Anwendungen

Summary Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsu _h in the norm of a suitable space . Sufficient conditions for the uniform boundedness u _{h Z} =O(1) ash0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain . The approximating subspaces are finite element spaces. Especially the caseX=W ₀ ^{m, p} (), 1<p<,Y=W ₀ ^{m. 2} (),Z=W ₀ ^{m. max (2,p)} ()W^m, () is treated. Some examples for 1<p<2 are considered. Forp2 a refined technique is introduced in the author's paper [7].

     

A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions

Summary This paper gives a method for finding sharpa posteriori error bounds for Newton's method under the assumptions of Kantorovich's theorem. On the basis of this method, new error bounds are derived, and comparison is made among the known bounds of Dennis [2], Döring [4], Gragg-Tapia [5], Kantorovich [6, 7], Kornstaedt [9], Lancaster [10], Miel [11–13], Moret [14], Ostrowski [17, 18], Potra [19], and Potra-Pták [20].This paper was written while the author was visiting the Mathematics Research Center, University of Wisconsin-Madison, U.S.A. from March 29, 1985 to October 21, 1985Sponsored by the Ministry of Education in Japan and the United States Army under Contract No. DAAG 29-80-C-0041     

Composite sequence transformations

Summary The aim of this work is to introduce the new concept of composite sequence transformations and to show, by very simple examples and theorems, that it can be useful in accelerating the convergence of sequences. Generalizations of classical transformations and results are obtained.Work performed under the Nato Research Grant 027.81.Presented at the International Conference on Numerical Analysis, Munich, March 19–21, 1984     

Optimal transformations for scalar sequences

Summary We consider transformations which accelerate convergence in some specified classes of convergent sequences. As an asymptotic measure of acceleration we introduce the order of transformation. We find a sharp upper bound on the order and show the explicit form of transformations of maximal order. We consider also the efficiency of transformations for fast convergent sequences. As a special case we find that the Germain-Bonne version of Richardson extropolation has maximal order for linearly convergent sequences.     

Numerical methods for reaction-diffusion problems with non-differentiable kinetics

Summary We consider a class of steady-state semilinear reaction-diffusion problems with non-differentiable kinetics. The analytical properties of these problems have received considerable attention in the literature. We take a first step in analyzing their numerical approximation. We present a finite element method and establish error bounds which are optimal for some of the problems. In addition, we also discuss a finite difference approach. Numerical experiments for one- and two-dimensional problems are reported.Dedicated to Ivo Babuka on his sixtieth birthdayResearch partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant Number AFOSR 85-0322     

A divide and conquer method for unitary and orthogonal eigenproblems

Summary LetH ^{n xn} be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrixH is split into two smaller unitary upper Hessenberg matricesH ₁ andH ₂ by a rank-one modification ofH. The eigenproblems forH ₁ andH ₂ can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues ofH. The eigenvector ofH can be determined from the eigenvalues ofH and the eigenvectors ofH ₁ andH ₂. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer.WhenH ^{n xn} is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paper.Research supported in part by the NSF under Grant DMS-8704196 and by funds administered by the Naval Postgraduate School Research Council     

The numerical computation of compressed states of nonlinearly elastic anisotropic plates

Summary In this paper we study the numerical computation of the compressed states of nonlinearly elastic anisotropic circular plates. The singular boundary value problem giving the compressed states depend parametrically on the applied pressure at the edge of the plate. We give a finite difference approximation of this problem and derive bounds for the global error by using the techniques of Brezzi, Rappaz and Raviart for the finite dimensional approximation of nonlinear problems. Some numerical results are given for a class of materials whose constitutive functions reflect the standard Poisson ratio effects.     

On the computation of multi-dimensional solution manifolds of parametrized equations

Summary A new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromR ⁿ toR ^m ,p=n–m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.Dedicated to Professor Ivo Babuka on the occasion of his sixtieth birthdayThis work was supported in part by the National Science Foundation under Grant DCR-8309926, the Office of Naval Research under contract N-00014-80-C-9455, and the Air Force Office of Scientific Research under Grant 84-0131     

Das modifizierte Newton-Verfahren in verallgemeinerten Banach-Räumen

Summary A Newton-Kantorowitsch-analysis of the modified Newton's method in generalized Banach-spaces is given. The application of generalized norms — mappings from a linear space into an ordered Banach-space — improves convergence- and existence results as well as error estimates compared with the real-norm theory.

     

Multivariate Padé approximation

Summary Multivariate Padé approximation is considered in the framework of the theory developed by A. Cuyt (1982). The denominators are determined from determinants of matrices whose entries are homogeneous polynomials. The main difference to the univariate case is a typical shift of the degree of the polynomials. Singularities atx=0 are analyzed since it is the rule rather than the exception that simultaneous zeros of the numerator and the denominator cannot be removed.Lecture delivered on March 20^th, 1984, at the International Conference on Numerical Analysis, at Munich     

Numerical solution of eigentuple-eigenvector problems in Hilbert spaces by a gradient method

Summary A gradient technique previously developed for computing the eigenvalues and eigenvectors of the general eigenproblemAx=Bx is generalized to the eigentuple-eigenvector problem . Among the applications of the latter are (1) the determination of complex (,x) forAx=Bx using only real arithmetic, (2) a 2-parameter Sturm-Liouville equation and (3) -matrices. The use of complex arithmetic in the gradient method is also discussed. Computational results are presented.This research was partially supported by NSF Grants MPS74-13332 and MCS76-09172     

Numerical analysis of cavitational flow-theory

Summary We present theoretical results on the numerical approximation of ideal two dimensional flows of jets and cavities.