Showing papers in "Mathematics of Computation in 1967"

Calculation of Gauss quadrature rules

Abstract: Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Quasi-Newton methods and their application to function minimisation

Abstract: can in general only be found by an iterative process in which successively better, in some sense, approximations to the solution are computed. Of the methods available most rely on evaluating at each stage of the calculation a set of residuals and from these obtaining a correction to each element of the approximate solution. The most common way of doing this is to take each correction to be a suitable linear combination of the residuals. There is, of course, no reason in principle why more elaborate schemes should not be used but they are difficult both to analyse theoretically and to implement in practice. The minimisation of a function of n variables, for which it is possible to obtain analytic expressions for the n first partial derivatives, is a particular example of this type of problem. Any technique used to solve nonlinear equations may be applied to the expressions for the partial derivatives but, because it is known in this case that the residuals form the gradient of some function, it is possible to introduce refinements into the method of solution to take account of this extra information. Since, in addition, the value of the function itself is known, further refinements are possible. The best-known method of solving a general set of simultaneous nonlinear equations, in which the corrections are computed as linear combinations of the residuals, is the Newton-Raphson method. The principal disadvantage of this method lies in the necessity of evaluating and inverting the Jacobian matrix at each stage of the iteration and so a number of methods have arisen, e.g. [1], [2], [4] and [8] in which the inverse Jacobian matrix is replaced by an approximation which is modified in some simple manner at each iteration. Although each method has its own peculiarities certain properties are common to a large class of these methods, and several of these are discussed here. In particular, if it is known that the functions to be zeroed are the first partial derivatives of a function F, then it is possible, if F is quadratic, to modify the approximating matrix in such a way that F is minimised in a finite number of steps. This method of modification is not unique and leads to a subclass of methods of which one example is the method of Davidon [3] as amended by Fletcher and Powell [4]. Since in the methods under discussion the corrections are computed as linear combinations of the residuals, it is natural to introduce matrix notation. Thus a function fj of the variables X\\, x2, • ■ ■, x„, may be regarded as a function of the nth order vector x, and each fj in turn may be treated as the jth element of the nth

Parallel methods for the numerical integration of ordinary differential equations

Abstract: . In this paper we derive a class of numerical integration formulas of a parallel type for ordinary differential equations. These formulas may be used simultaneously on a set of arithmetic processors to increase the integration speed. Conditions for the convergence of such formulas are formulated. Explicit examples for two and four processor cases are derived. Results of numerical experiments are given which show an effective improvement in computation speed.

On the relative efficiencies of gradient methods

Abstract: Summary. A comparison is made among various gradient methods for maximizing a function, based on a characterization by Crockett and Chernoff of the class of these methods. By defining the "efficiency" of a gradient step in a certain way, it becomes easy to compare the efficiencies of different schemes with that of Newton's method, which can be regarded as a particular gradient scheme. For quadratic functions, it is shown that Newton's method is the most efficient (a conclusion which may be approximately true for nonquadratic functions). For functions which are not concave (downward), it is shown that the Newton direction may be just the opposite of the most desirable one. A simple way of correcting this is explained. In trying to maximize a function f of the N variables {X1, X2, * XN , iterative techniques of the class known as gradient methods have proved of great utility [1], [2]. For general functions, the analysis usually begins with an expansion of f in a

Computation of tangent, Euler, and Bernoulli numbers

Abstract: Some elementary methods are described which may be used to cal- culate tangent numbers, Euler numbers, and Bernoulli numbers much more easily and rapidly on electronic computers than the traditional recurrence relations which have been used for over a century. These methods have been used to prepare an accompanying table which extends the existing tables of these numbers. Some the- orems about the periodicity of the tangent numbers, which were suggested by the tables, are also proved.

On the distribution of parity in the partition function

Abstract: Kolbcrg [1] proved, but by contradiction and without identifying the arguments n, that i nitely many p(n) are even, and infinitely many are odd. His proof is almost as simple as Euclid's proof that there are infinitely many primes, but like that proof it offers only very little more in the way of exact information concerning questions of distribution. From Gupta's tables [2], [3] we find the following cumulative distribution into odds and evens for 0 ^ n :£ 499.

Rational Chebyshev approximations for Fermi-Dirac integrals of orders -1/2, 1/2 and 3/2

Abstract: Rational Chebyshev approximations are given for the complete Fermi-Dirac integrals of orders -a, 2 and 2. Maximal relative errors vary with the function and interval considered, but generally range down to 10-9 or less.

A Construction of Nonnegative Approximate Quadratures

Abstract: Tchakaloff's demonstration is a very beautiful one, involving the theory of convex bodies. A separating hyperplane is employed and a nonconstructive proof is obtained. The theorem is valid for weighted integrals of dimension d _ 1. Equivalent results on finite moment spaces were obtained earlier by various authors. See, e.g., Karlin and Studden [2, Chapter II]. Tchakaloff's independent work appears to be the first to formulate the result explicitly as in (1.1), thereby stressing its numerical analysis aspect. This result is interesting for numerical analysis because: (1) Quadrature rules with nonnegative weights are more favorable than rules with mixed weights in that they lead to more stable computations; (2) Interpolating quadrature formulas determined by brute force methods do not often yield weights that are of one sign. The purpose of the present paper is to give an alternative proof of Tchakaloff's theorem which is constructive in its nature. The present proof is also a more "elementary" one than Tchakaloff 's in that it makes use only of the familiar raw materials of elementary numerical analysis. Extensions and numerical applications will be published subsequently by the author and by M. W. Wilson.

A survey of equal sums of like powers

Abstract: has been studied by numerous mathematicians for many years and by various methods [1], [2]. We recently conducted a series of computer searches using the CDC 6600 to identify the sets of parameters k, m, n for which solutions exist and to find the least solutions for certain sets. This paper outlines the results of the computation, notes some previously published results, and concludes with a table showing, for various values of k and m, the least n for which a solution to (1) is known. We restrict our attention to fc < 10. We assume that the Xi and y¡ are positive integers and x, ^ y¡. We do not distinguish between solutions which differ only in that the a\\or y¡ are rearranged. We will refer to (1) as (fc. m. n) and say that a primitive solution to (fc. m. n) is one in which no integer > 1 divides all the numbers xi, Xi, • • •, xm, yi, ?/2, • • •, yn. Putting

The programming language lisp: its operation and applications,

Abstract: : An introduction to LISP is given on an elementary level. Topics covered include the programming system, 240 exercises with solutions, debugging of LISP programs, and styles of programming. More advanced discussions are contained in the following articles: Techniques using LISP for automatically discovering interesting relations in data; Automation, using LISP, of inductive inference on sequences; Application of LISP to machine checking of mathematical proofs; METEOR: A LISP interpreter for string transformations; Notes on implementing LISP for the M-460 computer; LISP as the language for an incremental computer; The LISP system for the Q-2 computer; An auxiliary language for more natural expression -- the A-language. Some applications of the utilization of the LISP programming language are given in the appendices.

Introduction to the Theory of Finite Automata

Abstract: In what case do you like reading so much? What about the type of the introduction to the theory of finite automata book? The needs to read? Well, everybody has their own reason why should read some books. Mostly, it will relate to their necessity to get knowledge from the book and want to read just to get entertainment. Novels, story book, and other entertaining books become so popular this day. Besides, the scientific books will also be the best reason to choose, especially for the students, teachers, doctors, businessman, and other professions who are fond of reading.

A Simple “Filon-trapezoidal” rule

Abstract: which retains uniform accuracy even when co is so large that many oscillations of the integrand occur within a given element bt of the range of integration. The original Filon formula [1] was derived on the assumption that f(t), rather than the complete integrand, may be approximated stepwise by parabolas, so that it may be called a 'Filon-Simpson' rule. More sophisticated 'Filon' rules have appeared (e.g. [2], and the references quoted in [2]), but in fact with fast computers it is more useful to go in the other direction, towards the least sophisticated integration formula of all. The ordinary trapezoidal rule gives as an approximation

Generations of triangulations of the sphere

Abstract: It is easily seen that there is only one triangulation of the sphere with four vertices and one with five. This paper concerns an algorithm for finding all (nonisomorphic) triangulations of the 2-sphere with N vertices from those with N - 1. "Triangulation" shall always refer to a triangulation of the 2-sphere. First we develop a method for generating all triangulations with N vertices which may yield several triangulations of the same isomorphism type, and then we describe an isomorphism routine for elimninating these duplications. Let T be a triangulation with N _ 5 vertices, E edges, and F faces. Let Xk denote the number of vertices of T of valency k. Then 3F = 2E as each face is a triangle and each edge is on two faces, and 2E =E kXk as each edge is incident to two vertices. Hence 6F - 6E -2E = - kXk and by Euler's fornmula we have

The cyclotomic numbers of order eighteen with applications to difference sets.

Abstract: where the integers z, y are chosen from 0,1, • • • , / — 1. Eq. (1.1) shows that there are at most e distinct cyclotomic numbers (k, h)e of order e. This paper is concerned mainly with determining the (18)2 different (ft, A)is associated with a fixed primitive root of a prime of the form p = 18/ + 1. We also tabulate the cyclotomic numbers of order 9. Complete solutions to this cyclotomic number problem have been computed for e = 2 6, 8, 10, 12, 14, 16, 20. For e = 2 6 see L. E. Dickson |2], for e = 8 see E. Lehmer [9], for e = 10, 12, 16 see A. L. Whiteman [13], [14], [15] and for e = 14 see J. B. Muskat [11]. The case e = 20 is due to Muskat and Whiteman jointly and is as yet unpublished. Cyclotomic numbers play an important role in many number theoretical investigations. The difference sets of M. Hall, Jr. [6] and E. Lehmer [8] provided the impetus for this computation (see Section 5). Before we turn to the actual calculation, a word about the nature of the problem is in order. Eq. (1.1) shows that (fc, h)e depends not only on the prime p but also on which of the