Showing papers in "Mathematics of Computation in 1963"

Generation of permutations by adjacent transposition

Abstract: 1. RICHARD S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Inc., 1962. 2. E. G. D'YAKONOV, "On a Method of Solving the Poisson Equation," Dokl. Akad. Nauk SSSR 143 (1962), 21-24, the same paper appears in English in Soviet Math.-Doklady 3, 1962, p. 320-323. 3. R. B. KELLOGG, "Another Alternating-Direction-Implicit Method," to appear in J. Soc. Ind. Appi. Math. 4. A. M. OsTRowshI "Iterative Solution of Linear Systems of Functional Equations," J. Math. Anal. Apple , 2, 1961, p. 351-369. 5. L. COLLATZ, "Fehlerabschatzung fur das Iterationsverfahren zur Aufl6sung linearer Gleichungssysteme," Z. Angew. Math. Mech., 22, 1942, p. 357-361. 6. DAVID G. FEINGOLD, & R. S. VARGA, "Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem," to appear in the Pacific J. Math. 7. J. DOUGLAS, JR., and H. H. RACHFORD, JR., "On the numerical solution of heat conduction problems in two or three space variables," Trans. Amer. Math. Soc., 82, 1956, p. 421-439. 8. J. DOUGLAS, JR., "Alternating direction methods for three space variables," Numer. Math., 4, 1962, p. 41-63.

Optimal numerical integration on a sphere

Abstract: Little has been published on this subject or on its extension to the solid sphere. The literature is surveyed briefly in Section 7. Most of our space is devoted to formulas invariant with respect to a finite group of rotations of the sphere. We study such formulas by means of the group characters, as does Sobolev [12, 13]. The criterion by which integration formulas are usually judged is that of efficiency. It is defined like this. Consider a system of functions over the domain of integration such as polynomials in Euclidean space or surface harmonics on the sphere. They have properties of completeness and they are ordered in a natural way. Suppose that the integration formula is exact for the first L independent functions and therefore for all linear combinations of them. The efficiency E is the ratio of L to the number of arbitrary constants in the formula. The latter is a fixed multiple (one more than the dimensionality of the domain of integration) of the number N of points at which the integrand is evaluated. A linear combination of surface harmonics (of degree not more than p) will be called a spherical polynomial (of degree p). If we choose to embed the surface of the sphere in Euclidean space of three dimensions, we find that the trace left on the surface by an ordinary polynomial in x, y and z is a spherical polynomial of the same degree. For the surface of the sphere a pth degree integration formula (exact for spherical polynomials of degree p) has

Polynomial approximation—a new computational technique in dynamic programming: Allocation processes

Abstract: In principle, this equation can be solved computationally using the same technique that applies so well to (1.3). In practice (see [1] for a discussion), questions of time and accuracy arise. There are a number of ways of circumventing these difficulties, among which the Lagrange multiplier plays a significant role. In this series of papers, we wish to present a number of applications of a new, simple and quite powerful method, that of polynomial approximation. We shall begin with a discussion of the allocation process posed in the foregoing paragraphs and continue, in subsequent papers, with a treatment of realistic trajectory and

Deterministic simulation of random processes

Abstract: and the sciences we are required to simulate random processes. The simulation is usually effected by a computer program which generates a non-random, deterministic sequence of numbers xl, x2, I * which is supposed to resemble a sequence of independent, random samples from the uniform probability distribution on the interval 0 ? x < 1. The purpose of this paper is to define some general properties of random sequences and to investigate certain deterministic sequences which have some or all of these properties. We shall ignore the limitation that a digital computer with a finite word-length and a finite memory, operating under a single stored program, can produce only sequences of limited precision which are ultimately periodic. This limitation is a kind of round-off error. We shall take as a model of a deterministic mechanism any of the stored-program digital computers now commonly used for scientific computation modified in a single respect: let the wordlength be infinite; let rational and irrational numbers x be recorded and computed with perfect precision. The fundamental problem approached in this paper is to construct an infinite, deterministic sequence xi which has every property shared by all infinite, random sequences of independent samples from the uniform distribution. Equidistribution is a first requirement of randomness. The sequence x" is equidistributed in 0 < x < 1 if, for 0 < a < b < 1,

Second-Order Correct Boundary Conditions for the Numerical Solution of the Mixed Boundary Problem for Parabolic Equations

Abstract: Received June 21, 1962. This research was supported by the Air Force Office of Scientific Research. In May, 1961 the author submitted a thesis containing the principle part of this paper to the Rice University in partial fulfillment of the requirements for a degree of Master of Arts. * ,p(x, t) E Ca'd(R) if and only if sp is continuously differentiable a times with respect to x and a times with respect to t in the region R. 405

Approximate integration formulas for certain spherically symmetric regions

Abstract: where the As are constants and the Pi(Pil , Pin) are points in the, space. Each formula is exact for all polynomials up to and including a specified degree k where, as usual, k is called the degree of the formula. In Section 2 we give formulas of degrees 2, 3, 5 and 7 which are similar to previously developed formulas for other regions. The formulas of degree 2 are discussed by Stroud [14] for an arbitrary region; the formulas of degrees 3, 5 and 7 are similar to formulas for spheres given by Hammer and Stroud [4] and Ditkin [3]. In Sections 3, 4 and 5 we develop spherical product type formulas of arbitrarily high degree for U and V. These formulas have degree 2h 1 (h 1, 2, . ) and use hn points for even h and h' h -1+ 1 points for odd h. These formulas are obtained by products of one-dimensional formulas and are similar to formulas for the circle and 3-sphere given by Peirce [10, 11] and recently extended to the nsphere by Hetherington [6]. The spherical product formulas are most useful in 2 and 3 dimensions since in higher dimensions the numbers of points in the formulas become very large. In Section 4 we tabulate one-dimensional formulas, which are particular to the integrals U and V, from which product formulas for n = 2, 3, 4 may be constructed. From the formulas we give for U and V we can obtain formulas for any integrals of the form

Tables for the Evaluation of ∞ 0 x β e -x f(x) dx by Gauss-Laguerre Quadrature

Abstract: Tables of abscissae and weight coefficients to fifteen places are prexße~xf(x) dx ~ 23 Hkfiak)

Two High-Order Correct Difference Analogues for the Equation of Multidimensional Heat Flow

Abstract: 1. Introduction. Several high-order accuracy difference equations for the heat equation in one space variable [1] have been proposed, but most do not extend to several space variables with any ease, if at all. Two three-level difference equations are discussed here, each of which is fourth-order correct in space and second in time. One is stable and convergent in 22 for as many as four space variables but is limited essentially to the heat equation itself and in the four space variables case to bounded r = At( Ax) -2. The other is stable and convergent in 22 for three space variables and is adapted to extension to more complicated differential equations. Alternating direction techniques based on the two three-level formulas are developed. These methods retain the accuracy of the original procedures and require much less arithmetic to complete a problem. Only the results will be given here; their analyses will be presented in another paper [3] as examples of a general approach to alternating direction methods.

Quadrature formulas with simple Gaussian nodes and multiple fixed nodes

Abstract: where fix) is an integrable function on the finite or infinite interval (a b), and wix) is a given, fixed function such that its moments ck = /(w; xk) (fc = 0, 1, 2, • • • ) exist and c0 > 0. In this paper we consider quadrature formulas which use multiple nodes chosen in advance and other simple nodes which we choose to increase the degree of exactness of the formulas. To be more precise, let ai, a2, ■ ■ • ,ap be real numbers, which are assumed to be fixed, such that the polynomial

On the computation of Euler’s constant

Abstract: The computation of Eider’s constant, γ, to 3566 decimal places by a procedure not previously used is described. As a part of this computation, the natural logarithm of 2 has been evaluated to 3683 decimal places. A different procedure was used in computations of γ performed by J. C. Adams in 1878 [1] and J. W. Wrench, Jr. in 1952 [2], and recently by D. E. Knuth [3]. This latter procedure is critically compared with that used in the present calculation. The new approximations to γ and ln 2 are reproduced in extenso at the end of this paper.