117077618
A Singular Multi-Grid Iteration Method for Bifurcation Problems
We propose an efficient technique for the numerical computation of bifurcating branches of solutions of large sparse systems of nonlinear, parameter-dependent equations. The algorithm consists of a nested iteration procedure employing a multi-grid method for singular problems. The basic iteration scheme is related to the Lyapounov-Schmidt method and is widely used for proving the existence of bifurcating solutions. We present numerical examples which confirm the efficiency of the algorithm.
122551767
Numerical computation of the fourier transform using Laguerre functions and the Fast Fourier Transform
SummaryIn this paper we propose a numerical technique for the computation of Fourier transforms. It uses a bilateral expansion of the unknown transformed function with respect to Laguarre functions. The expansion coefficients are obtained via trigonometric interpolation and may be computed very efficiently by means of the Fast Fourier Transform. The convergence of the algorithm is analyzed and numerical results are presented which confirm that it works well.
121375499
An Efficient Technique for the Computation of Stable Bifurcation Branches
This paper deals with an application of the multi-grid iteration for large sparse linear systems to the problem of computing stable branches of solutions of nonlinear eigenvalue problems which bifurcate from simple eigenvalues. The theoretical background of the algorithm considered here is the selective Picard iteration. We present numerical results for difference approximations of nonlinear elliptic eigenvalue problems in one and two space dimensions. They confirm the efficiency of the algorithm proposed here.
120892864
Zur Verzweigung bei einfachen Eigenwerten
Let W and Y be real Banach spaces and F:W → Y be a smooth mapping. Let xɛW be a solution of F(z)=0, Bifurcation theory deals with the problem of determining the structure of the zeros of F near x. We give a modification of a well-known theorem of Crandall and Rabinowitz which is useful for applications, i.e. for the computation of bifurcation points. A numerical example is included.
120311575
Multigrid Bifurcation Iteration
We analyze a multigrid technique for the numerical computation of bifurcating solutions of large, sparse systems of nonlinear parameter-dependent equations which arise from discretizations of nonlinear elliptic eigenvalue problems by finite differences or finite elements. The algorithm consists of a nested iteration procedure which employs a multigrid method for singular problems. The underlying iteration scheme related to the Lyapunov–Schmidt method has been widely used for proving the existence of bifurcating branches and also sometimes for numerical calculations. Our multi-grid technique is shown to be a very efficient and theoretically sound tool for computing bifurcating branches of solutions of nonlinear elliptic eigenvalue problems on arbitrarily bounded domains in $\mathbb{R}^2 $. Several numerical examples confirm this.
7501088
Zur numerischen Behandlung gestörter Verzweigungsprobleme bei gewöhnlichen Differentialgleichungen
ZusammenfassungWir geben eine Möglichkeit der numerischen Behandlung von gestörten Verzweigungsproblemen bei Randwertaufgaben gewöhnlicher Differentialgleichungen mit Standardverfahren an. Dies gelingt mit Hilfe von Umparametrisierungen nach Störungsparametern und durch Aufstellung geeigneter regulärer Randwertaufgaben. Ein numerisches Beispiel — ein spezielles Stabknickungsproblem —belegt die Wirksamkeit unserer Methode.AbstractWe present a method for the numerical solution of perturbed bifurcation problems of boundary-value problems for ordinary differential equations by standard techniques. This is achieved by reparametrisation by perturbation parameters and definition of appropriate boundary-value problems which have isolated solutions. The efficiency of the method is demonstrated by a numerical example —a special rod buckling problem.
16938020
EULER: a generalization of ALGOL and it formal definition: Part 1
In this section the algorithmic language EULER is described first informally and then formally by its syntax and semantics. An attempt has been made to generalize and extend some of the concepts of ALGOL, thus creating a language which is simpler and yet more flexible than ALGOL 60. A second objective in developing this language was to show that a useful programming language which can be processed with reasonable efficiency can be defined in rigorous formality.
814779
A microprogrammed implementation of EULER on IBM system/360 model 30
An experimental processing system for the algorithmic ~anguage EULER has been implemented in microprogramming on an IBM System/360 Model 30 using a second Read-Only Storage unit. The system consists of a mlcroprogrammed compiler and a microprogrammed String Language Interpreter, and of an I/O control program written in 360 machine language. The system is described and results are given in terms of microprogram and main storage space required and compiler and interpreter performance obtained. The role of microprogramming is stressed, which opens a new dimension in the processing of interpretive code. The structure and content of a higher level language can be matched by an appropriate interpretive language which can be executed efficiently by microprograms on existing computer hardware.
118567473
Compact convex sets in non-locally-convex linear spaces
In the theorems about compact convex sets it is usually assumed that the compact convex set is contained in a locally convex linear space. Examples for such theorems are the Schauder-Tychonoff fixed point theorem and the Krein-Milman theorem.The problem whether the Krein-Milman theorem remains true also in non locally convex linear spaces, was solved by J.W.Roberts; he gave examples for absolutely convex compact sets without extreme points ([R 76], [R 77], [Ro 84]).It is, however, still an open problem, whether the Schauder-Tychonoff theorem remains true in non locally convex linear spaces [M 81, Problem 54 of Schauder]. A sufficient condition fora compact convex set to have e.g. the fixed point property or to have extreme points is that it can be affinely embedded in a Hausdorff locally convex linear space.This observation, which was the starting point for [JOT 76] and [R 76], was left out of account in various newer publications about Schauder-Tychonoff's fixed point theorem in non locally convex spaces, e.g. [H 84].