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JournalISSN: 0272-4960

Ima Journal of Applied Mathematics 

Oxford University Press
About: Ima Journal of Applied Mathematics is an academic journal published by Oxford University Press. The journal publishes majorly in the area(s): Boundary value problem & Partial differential equation. It has an ISSN identifier of 0272-4960. Over the lifetime, 2436 publications have been published receiving 54334 citations. The journal is also known as: IMA journal of applied mathematics - online services & Institute of Mathematics and Its Applications journal of applied mathematics.


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TL;DR: In this article, a more detailed analysis of a class of minimization algorithms, which includes as a special case the DFP (Davidon-Fenton-Powell) method, has been presented.
Abstract: This paper presents a more detailed analysis of a class of minimization algorithms, which includes as a special case the DFP (Davidon-Fletcher-Powell) method, than has previously appeared. Only quadratic functions are considered but particular attention is paid to the magnitude of successive errors and their dependence upon the initial matrix. On the basis of this a possible explanation of some of the observed characteristics of the class is tentatively suggested. PROBABLY the best-known algorithm for determining the unconstrained minimum of a function of many variables, where explicit expressions are available for the first partial derivatives, is that of Davidon (1959) as modified by Fletcher & Powell (1963). This algorithm has many virtues. It is simple and does not require at any stage the solution of linear equations. It minimizes a quadratic function exactly in a finite number of steps and this property makes convergence of this algorithm rapid, when applied to more general functions, in the neighbourhood of the solution. It is, at least in theory, stable since the iteration matrix H,, which transforms the jth gradient into the /th step direction, may be shown to be positive definite. In practice the algorithm has been generally successful, but it has exhibited some puzzling behaviour. Broyden (1967) noted that H, does not always remain positive definite, and attributed this to rounding errors. Pearson (1968) found that for some problems the solution was obtained more efficiently if H, was reset to a positive definite matrix, often the unit matrix, at intervals during the computation. Bard (1968) noted that H, could become singular, attributed this to rounding error and suggested the use of suitably chosen scaling factors as a remedy. In this paper we analyse the more general algorithm given by Broyden (1967), of which the DFP algorithm is a special case, and determine how for quadratic functions the choice of an arbitrary parameter affects convergence. We investigate how the successive errors depend, again for quadratic functions, upon the initial choice of iteration matrix paying particular attention to the cases where this is either the unit matrix or a good approximation to the inverse Hessian. We finally give a tentative explanation of some of the observed experimental behaviour in the case where the function to be minimized is not quadratic.

2,306 citations

Journal ArticleDOI
Victor Namias1
TL;DR: In this article, a generalized operational calculus is developed, paralleling the familiar one for the ordinary transform, which provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians.
Abstract: We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field.

1,523 citations

Journal ArticleDOI
TL;DR: It is shown theoretically that the new algorithm is stable and it is proved is the only member of the class considered for which a certain matrix error is reduced strictly monotonically when minimizing quadratic functions.
Abstract: The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm d where q and ql are uniquely determined orthonormal vectors. The parameter 1/ is . ntially arbitrary in that it depends upon p. It was suggested in Part 1 that a suitable ice for I] would be zero since if it were negative, or large and positive, the matrix KI hence HI might become needlessly badly conditioned. It was noted moreover that osing I] in this way gives rise to a new algorithm. the two algorithms in this class already published, that due to Davidon (1959) modified by Fletcher & Powell (1963) is obtained by putting P equal to zero and s shown in Part I that this led, in general, to negative values of 1]. We thus expect quence of matrices {HI} obtained by that algorithm to exhibit a tendency to arity and this tendency has been noted by, among others, Broyden (1967) and on (1969). In a more recent algorithm, due to Greenstadt (1967), if H is positive ite the values of 1] are even more negative than those occurring in the DFP ithm. One result of this is that for this algorithm the matrices H cannot, unlike for the DFP algorithm, be proved to be positive definite and this has serious tions when considering numerical stability. this paper we show theoretically that the new algorithm is stable and we prove is the only member of the class considered for which a certain matrix error is reduced strictly monotonically when minimizing quadratic functions. We the effect of rounding and of poor conditioning of H on the attainable accuracy solution and conclude by presenting the results of a numerical survey in he performance of the new algorithm for a variety of test problem is compared t of the DFP algorithm. C. G. BROYDEN Computing Centre, University of Essex, Wivenhoe Park, Colchester, Essex

1,414 citations

Journal ArticleDOI
TL;DR: Inversion of almost arbitrary Laplace transforms is effected by trapezoidal integration along a special contour as mentioned in this paper, in which the number n of points to be used is one of several parameters, in most cases yielding absolute errors of order 10 for n = 10.
Abstract: Inversion of almost arbitrary Laplace transforms is effected by trapezoidal integration along a special contour. The number n of points to be used is one of several parameters, in most cases yielding absolute errors of order 10 for n = 10, 10 for n = 20, 10 for n = 40 (with double precision working), and so on, for all values of the argument from 0+ up to some large maximum. The extreme accuracy of which the method is capable means that it has many possible applications of various kinds, and some of these are indicated.

750 citations

Performance
Metrics
No. of papers from the Journal in previous years
YearPapers
202314
202231
202155
202039
201950
201846