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.
123580420
Multi-Grid Solution of Bifurcation Problems
A continuation strategy is developed for a method of inverse iteration type to compute solutions of nonlinear eigenvalue problems. The solution curves are parametrized by a norm of the solution. Thus continuation near turning points does not represent any difficulties. A multi-grid version of the algorithm is presented and successfully applied to solve problems on general two-dimensional domains. Numerical results including a comparison with a more standard approach are also given for the continuation strategy proposed for the basic algorithm.
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.
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.
117989535
Numerical Methods for Bifurcation Problems — A Survey and Classification
The purpose of this paper is to give an account of recent developments in numerical methods for the solution of bifurcation problems. For readers not too familiar with our subject we shall first summarize important applications of bifurcation and dicuss some of the basic ideas, problems and tools of bifurcation theory.
123517753
On the numerical approximation of secondary bifurcation problems
We discuss stable numerical methods for the approximation of the solutions of a nonlinear parameter - dependent equation near a non-trivial bifurcation point. The problem of finding the bifurcation point is reformulated as a well-posed equation of higher dimension. The nearby branches can be calculated in a stable manner after applying a certain transformation having its origin in the Lyapunov — Schmidt theory. We also treat the perturbed bifurcation problem and present numerical results.
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.
12124100
EULER: a generalization of ALGOL, and its formal definition: Part II
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.
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.