Algorithm 468: algorithm for automatic numerical integration over a finite interval [D1]
TL;DR: Submittal of an algorithm for consideration for publication in Communications of the A C M implies unrestricted use of the algorithm within a computer is permissible.
Abstract: Submittal of an algorithm for consideration for publication in Communications of the A C M implies unrestricted use of the algorithm within a computer is permissible. Copyright © 1973, Association for Computing Machinery, Inc. General permission to republish, but not for profit, all or part o f this material is granted provided that A C M ' s copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery.
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TL;DR: This chapter starts with a discussion on abstract models: the basic model and associated problems, the model with selection based on features, and themodel with variable performance criteria, to explore the applicability of the approximation theory to the algorithm selection problem.
Abstract: Publisher Summary The problem of selecting an effective algorithm arises in a wide variety of situations. This chapter starts with a discussion on abstract models: the basic model and associated problems, the model with selection based on features, and the model with variable performance criteria. One objective of this chapter is to explore the applicability of the approximation theory to the algorithm selection problem. There is an intimate relationship here and that the approximation theory forms an appropriate base upon which to develop a theory of algorithm selection methods. The approximation theory currently lacks much of the necessary machinery for the algorithm selection problem. There is a need to develop new results and apply known techniques to these new circumstances. The final pages of this chapter form a sort of appendix, which lists 15 specific open problems and questions in this area. There is a close relationship between the algorithm selection problem and the general optimization theory. This is not surprising since the approximation problem is a special form of the optimization problem. Most realistic algorithm selection problems are of moderate to high dimensionality and thus one should expect them to be quite complex. One consequence of this is that most straightforward approaches (even well-conceived ones) are likely to lead to enormous computations for the best selection. The single most important part of the solution of a selection problem is the appropriate choice of the form for selection mapping. It is here that theories give the least guidance and that the art of problem solving is most crucial.
1,007 citations
TL;DR: In this paper, exact closed-form expressions for the electromagnetic induction fields produced by vertical and horizontal current sources in the conducting ocean overlying a one-dimensional earth are derived from the Maxwell equations.
Abstract: Exact closed-form expressions for the electromagnetic induction fields produced by vertical and horizontal current sources in the conducting ocean overlying a one-dimensional earth are derived from the Maxwell equations. Numerical methods for the evaluation of the solutions are given, including correction for the finite size of real sources. Simple models of the electrical conductivity structure of the ocean crust and lithosphere are deduced from geologic, petrologic, and laboratory data, and their electromagnetic response is modeled. Horizontal electric dipole sources produce much larger field amplitudes than their vertical counterparts for a given frequency and range, and the horizontal electric field offers superior received signal performance. Reflections of electromagnetic waves from the sea surface and thermocline must be considered for low enough frequencies or long ranges. Estimates of the ambient noise level from natural electromagnetic sources in the frequency range 0.01–10 Hz are presented. The ability of controlled sources to determine features of the conductivity of the ocean crust and upper mantle, especially low conductivity zones, is demonstrated. If the mantle conductivity is low enough, horizontal ranges of 50 km and conductivity estimates to over 20 km depth can be achieved.
308 citations
TL;DR: In this article, a review of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals is presented, which arise in problems involving antennas or scatterers embedded in planar multilayered media.
Abstract: A review is presented of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals (i.e. semi-infinite range integrals with Bessel function kernels), which arise in problems involving antennas or scatterers embedded in planar multilayered media. Attention is limited to partition-extrapolation procedures in which the Sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real-axis tail segment (/spl alpha/,/spl infin/) of the integration path, where /spl alpha/>0 is selected to ensure that the integrand is well behaved. An analytical form of the asymptotic truncation error (or the remainder), which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods, is derived. Several extrapolation algorithms deemed to be the most suitable for the Sommerfeld integrals are described and their performance is compared. It is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points /spl rho/ and by the choice of the subinterval break points. Furthermore, it is found that some well-known extrapolation techniques may fail for a number of values of /spl rho/ and ways to remedy this are suggested. Finally, the most effective extrapolation methods for accelerating Sommerfeld integral tails are recommended.
286 citations
01 Jan 1981
TL;DR: A survey of Gauss-Christoffel quadrature formulae can be found in this paper, with a discussion of the error and convergence theory of the quadratures.
Abstract: We present a historical survey of Gauss-Christoffel quadrature formulae, beginning with Gauss’ discovery of his well-known method of approximate integration and the early contributions of Jacobi and Christoffel, but emphasizing the more recent advances made after the emergence of powerful digital computing machinery. One group of inquiry concerns the development of the quadrature formula itself, e.g. the inclusion of preassigned nodes and the admission of multiple nodes, as well as other generalizations of the quadrature sum. Another is directed towards the widening of the class of integrals made accessible to Gauss-Christoffel quadrature. These include integrals with nonpositive measures of integration and singular principal value integrals. An account of the error and convergence theory will also be given, as well as a discussion of modern methods for generating Gauss-Christoffel formulae, and a survey of numerical tables.
224 citations
01 Jan 2015
TL;DR: In this paper, the authors proposed a method to solve the problem: see [Cas89a] and http://www.s/0098-3500/116814.html
Abstract: s/0098-3500/116814. html. See [Cas89a].
205 citations
References
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TL;DR: In this article, a new set of n-point Gauss, Lobatto and general quadrature formulae of degree 3n - 3 (n even) or 3n- 2 (n odd) is derived.
Abstract: Methods are developed for the addition of points in an optimum manner to the Gauss, Lobatto and general quadrature formulae. A new set of n-point formulae are derived of degree (3n - 1)/2. In this paper it is shown how the additional abscissae may be derived in a nu- merically stable fashion by an expansion of the equation for the abscissae in terms of Legendre polynomials. A technique is also discussed to extend the n-point Lobatto quadrature formulae by the addition of n - 1 abscissae to yield quad- rature formulae of degree 3n - 3 (n even) or 3n - 2 (n odd). Finally a method is discussed for the optimum addition of abscissae to general quadrature formulae and a new set of n-point formulae is derived of degree (3n - 1)/2.
340 citations
TL;DR: The purpose of this work is to find a method for building loopless algorithms for listing combinatorial items, like partitions, permutations, combinations, etc.
Abstract: The purpose of this work is to find a method for building loopless algorithms for listing combinatorial items, like partitions, permutations, combinations. Gray code, etc. Algorithms for the above sequence are detailed in full.
201 citations
01 Jan 1971
TL;DR: The CADRE algorithm as discussed by the authors uses the composite trapezoid sum to estimate the integral of F (x ) over suitably small subintervals of a given interval of integration, starting from the first such subinterval, the program attempts to find an acceptable estimate on a given sub-interval by cautious Romberg extrapolation.
Abstract: Publisher Summary This chapter discusses the program CADRE—an algorithm for numerical quadrature. The program employs an adaptive scheme whereby CADRE is found as the sum of estimates for the integral of F ( x ) over suitably small subintervals of a given interval of integration. Starting with the interval of integration itself as the first such subinterval, the program attempts to find an acceptable estimate on a given subinterval by cautious Romberg extrapolation. If this attempt fails, the subinterval is divided into two subintervals of equal length. For the sake of economy, values of F ( x ), once calculated, are saved until they are successfully used in estimating the integral over some subinterval to which they belong. The program CADRE uses the composite trapezoid sum. For certain classes of integrands, the composite trapezoid sum exhibits a known and characteristic convergence behavior that the algorithm attempts to detect and to exploit through cautious extrapolation.
105 citations
TL;DR: Functional procedure nielin, of the integer type, solves a system of simultaneous nonlinear algebraic or trans-cendental equations with n variables with special choice of interpolation points assures existence and uniqueness of the interpolating poly-nomials wi.
Abstract: comment Functional procedure nielin, of the integer type, solves a system of simultaneous nonlinear algebraic or trans-cendental equations. Let us consider a given system of n equations with n variables: A kth approximation of the solution of the system (1) is supposed to be given: Yo (~) = (y~), y~), ..-, y~)). (2) If for every i, Ifi(Yo(~))[ < e, (3) where e > 0 is a given number, then the approximation (2) is considered as a solution of the system (1), otherwise a further approximation is calculated. Let h (k) > 0 be given and construct the n new points: For every function of the system (1) a new interpolating polynomial of the first order is constructed on the points (2) and (4) such that: w,(y~k)) = f, A solution of the linear system: wi(y~ , y2 , \"'', Y,) = O, i = 1, 2, \"', n, (6) is used as the (kT1)-th successive approximation. The special choice of the interpolation points (2) and (4) assures existence and uniqueness of the interpolating poly-nomials wi (5). Namely, the kth approximation has for the ith function the form: n w,(~> (Y) = /,(r2)) + ~ g~(yj-y~(~)' j, (7) i-1 where g~) = (f,(y]k)) _ f,(y~))/h(k). The solution of the system (6) where w~ is given by (7) can be written in the form (see [2]): y(~+l) = y~k) _ (1/a(k))z~) X h Ck), i = 1, 2, ..., n, (9) where z (~) = (z~ k), z~ (k), ..., z(= k)) is a solution of the following linear system: n ~f~(y(k))
23 citations
TL;DR: It is shown that on the grounds of efficiency and reliability it is generally preferable to use families of common point high precision formULae directly over the whole interval of integration than to apply low precision formulae to an adaptively subdivided range.
Abstract: : A critical examination is made of two common approaches to the automatic evaluation of definite integrals It is shown that on the grounds of efficiency and reliability it is generally preferable to use families of common point high precision formulae directly over the whole interval of integration than to apply low precision formulae to an adaptively subdivided range Many examples are given to illustrate the conclusions (Author)
17 citations