Showing papers in "Mathematics of Computation in 1969"

Perturbation Methods in Applied Mathematics

Abstract: 1 Introduction.- 2 Limit Process Expansions Applied to Ordinary Differential Equations.- 3 Multiple-Variable Expansion Procedures.- 4 Applications to Partial Differential Equations.- 5 Examples from Fluid Mechanics.- Author Index.

Rational Chebyshev approximations for the error function

Abstract: This note presents nearly-best rational approximations for the functions erf (x) and erfc (x), with maximal relative errors ranging down to between 6 X 10-19 and 3 X 10-20. for small x because of subtraction error, but they do not provide any alternative. Hastings' (2) approximations for erf (x) are no better, since they explicitly use the constant 1 as an additive term and are chosen to nearly minimize the maximum absolute error rather than the relative error. Clenshaw's (3) Chebyshev series ex- pansions for erf (x)/x come close to minimizing relative error, but his approximations are somewhat inefficient because of his choice of interval and his restriction to polynomials. For a computer subroutine with entries for both erf (x) and erfc (x), cancellation error can be avoided by evaluating erf (x) directly and erfc (x) indirectly (as 1 - erf (x)) when erf (x) is smaller in magnitude than erfc (x), and erf (x) indirectly and erfe (x) directly, otherwise. The changeover point occurs for IxI .47. In this note we present nearly-best rational approximations for the functions erf (x) and erfc (x) with maximal relative errors ranging down to between 6 X 10-19 and 3 X 10-20. The approximation forms and intervals used are erf (x) x 1.m (X2 ) , lxI 4 where the Rim(z) are rational functions of degree 1 in the numerator and m in the denominator. The relations erf (-x) = -erf (x) and erfc (-x) = 2 - erfc (x) canl be used to evaluate the functions for negative arguments.

Algebraic theory of machines, languages, and semigroups,

Abstract: : The book is an integrated exposition of the algebraic, and especially semigroup-theoretic, approach to machines and languages. It is designed to carry the reader from the elementary theory all the way to hitherto unpublished research results.

On the approximate minimization of functionals

Abstract: This paper considers in general the problem of finding the minimum of a given functional f(u) over a set B by approximately minimizing a sequence of functionals fn(un) over a "discretized" set Bn; theorems are given proving the convergence of the approximating points un in Bn to the desired point u in B. Applications are given to the Rayleigh-Ritz method, regularization, Chebyshev solution of differential equations, and the calculus of variations. 1. Introduction. Many theoretical and computational problems either arise or can be formulated as one of locating a minimizing point of some real-valued (non- linear) functional over a certain set; such variational settings often lead to existence theorems as well as to computational methods for solving the problems in question. Computationally, however, one is generally forced to deal with discrete data in place of the original functional; it is therefore necessary to analyze the relationships between variational problems and their discretized analogues. In (7, Section 4), we first studied under certain equicontinuity assumptions the question of approximately minimizing one functional by minimizing a sequence of nearby functionals. In this present note we state the problem generally, give some convergence theorems, and describe some particular examples.

Block implicit one-step methods

Abstract: A class of one-step methods which obtain a block of r new values at each step are studied. The asymptotic behavior of both implicit and predictor-corrector procedures is examined. 1. Introduction. We shall consider a class of implicit one-step methods for solving ordinary differential equations which generalize the trapezoidal rule. The idea is to determine a block of r new values at each stage, the trapezoidal rule being a case with r = 1. Implicit one-step methods have been studied by Stoller and Morrison (1), Ceschino and Kuntzmann (2) and Butcher (3). For linear problems these methods are quite useful but with the exception of the trapezoidal rule they have not found favor for nonlinear problems because of the relatively great amount of work in- volved in advancing one step. Rosser (4) has suggested obtaining a block of new values simultaneously which makes the implicit methods more competitive. He discusses in detail a procedure which calculates four new values at each stage. In addition to his references to earlier work let us note the procedure of Clippinger and Dimsdale (5)-formula (3) cf. Section 2 below-which obtains two new values at each stage. The methods we study can be described theoretically as block one-step methods. This situation prevails in practice for indefinite integrals and linear problems and also for general problems when we iterate to a fixed accuracy. In Section 2 we show convergence of these methods and study stability for a particular method. As Rosser indicates, one always expects good stability properties and indeed our example is a fourth order procedure which is A-stable. For theoretical purposes the trapezoidal rule can be conveniently regarded as a one-step method but its practicality depends on computing with it as a predictor- corrector procedure. This is what we shall do in the general case. In Section 3 we shall show that a suitable predictor-corrector approach leads to the same asymptotic behavior as iterating to completion. Again we discuss the stability of an example. Some comparative numerical examples are presented in Section 4. 2. Implicit Methods. We wish to approximate the solution of

Extensions and applications of the Householder algorithm for solving linear least squares problems

Abstract: Mathematical and numerical least squares solution of linear equations, using Householder algorithm

Generalized finite-difference schemes

Abstract: Finite-difference schemes for initial boundary-value problems for partial differential equations lead to systems of equations which must be solved at each time step. Other methods also lead to systems of equations. We call a method a generalized finite-difference scheme if the matrix of coefficients of the system is sparse. Galerkin's method, using a local basis, provides unconditionally stable, implicit generalized finite-difference schemes for a large class of linear and nonlinear problems. The equations can be generated by computer program. The schemes will, in general, be not more efficient than standard finite-difference schemes when such standard stable schemes exist. We exhibit a generalized finite-difference scheme for Burgers' equation and solve it with a step function for initial data. U

Chebyshev approximations for the exponential integral ()

Abstract: *** ************************************************************** .***. ************************************************** ************ * * * * * * * * * * * * * * * * * * * *

Perfectly symmetric two-dimensional integration formulas with minimal numbers of points

Abstract: Perfectly symmetric integration formula of degrees 9-15 with a minimal number of points are computed for the square, the circle and the entire plane with weight functions exp (-(x2 + y 2)) and exp (-(x2 + y 2)1 /2). These rules were computed by solving a large system of nonlinear algebraic equations having a special structure. In most cases where the minimal formula has a point exterior to the region or where some of the weights are negative, 'good' formulas, which consist only of interior points and have only positive weights, are given which contain more than the minimal number of points.

Gaussian Quadratures for the Integrals ∞ 0 exp(-x 2 )f(x)dx and b 0 exp(-x 2 )f(x)dx

Abstract: Gaussian quadratures are developed for the evaluation of the integrals given in the title. The weights and abscissae for the semi-infinite integral are given for two through fifteen points with fifteen places. For b = 1, the weights and abscissae are given for two through ten points with fifteen places.

Convergence estimates for essentially positive type discrete Dirichlet problems

Abstract: In this paper we consider a class of difference approximations to the Dirichlet problem for second-order elliptic operators with smooth coefficients. The main result is that if the order of accuracy of the approximate problem is v, and F (the right-hand side) andf (the boundary values) both belong to O' for X < v, then the rate of convergence is O(hx).

Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I

Abstract: with A a diagonal matrix. Appropriate boundary and initial conditions are given. The amplification matrix Q(t) need not be diagonal. However, he required that IQ() I < 1. We use certain results in matrix theory and Wiener-Hopf factorization to replace this restrictive assumption by certain reasonable assumptions on accuracy of W(t) and smoothness of an associated positive-definite symmetric matrix. This technique will be important in half-space problems in many space variables since for such problems the amplification matrix will certainly not be diagonal. M

A nonlinear alternating direction method

Abstract: An alternating direction iteration method is formulated, and con- vergence is proved, for the solution of certain systems of nonlinear equations. The method is applied to a heat conduction problem with a nonlinear boundary con- dition. e 1. Alternating direction methods are often used for solving the sets of linear equations arising from the discretization of elliptic boundary value problems (11, (2), (3). In this paper, an alternating direction method is formulated for a certain nonlinear system of equations. Convergence of the method is established in the case of a single iteration parameter. Finally, the method is applied to a set of equa- tions arising from a steady-state heat conduction problem with nonlinear boundary conditions. Such boundary conditions occur when energy is transmitted from the boundary of the region by means of radiation or by means of natural convection

Simultaneous approximation of a set of bounded real functions

Abstract: The problem of simultaneous Chebyshev approximation of a set F of uniformly bounded, real-valued functions on a compact interval I by a set P of continuous functions is equivalent to the problem of simultaneous approximation of two real-valued functions F+ (x), F(x), with F(x) _ F+ (x), for all x in I, where Fis lower semicontinuous and F+ is upper semicontinuous. 1. Formulation of the Approximation Problem. In this introductory section, which consists of nine "points," the "general problem of the simultaneous approximation of a family of functions" is formulated (see, in particular, point 4). Besides, a "heuristic derivation" of the basic equation (equation (T2) of point 8) is given. 1. Let g be a (finite) real-valued function defined for all real numbers x on the finite-closed real number interval [a, b] = {x a _ x o O o O_ Ix_Y 1 0 O Ix-y 1 0 O0Ix-yl

A method for solving nonlinear Volterra integral equations of the second kind

Abstract: The approach given in this paper leads to numerical methods for Volterra integral equations which avoid the need for special starting procedures. Formulae for a typical fourth-order method are derived and some numerical examples presented. A convergence theorem is given for the method described. 1. Introduction. In this paper we consider the numerical solution of the equation

The double points of Mathieu’s differential equation

Abstract: Mathieu's differential equation, y" + (a - 2q cos 2x)y 0, admits of solutions of period xr or 27r for four countable sets of characteristic values, a(q), which can be ordered as aT(q), r = O, 1, *. The power series expansions for the ar(q) converge up to the first double point for that order in the complex plane. (At a double point, ar(q) = ar+2(q).) The present work furnishes the double points for orders r up to and including 15. These double points are singular points, and the usual methods of determining the characteristic values break down at a singular point. However, it was possible to determine two smooth functions in which one could interpolate for both q and ar(q) at the singular point. The method is quite general and can be used in other problems as well. U 1. Introduction. Mathieu's differential equation (1.0) y" + (a-2q cos 2x)y = 0 admits of four countable sets of characteristic values, ar(q), corresponding to which the solutions y(x) are periodic, and of period wr or 27r. These four sets are associated with solutions defined below. 00

On Gauss’s class number problems

Abstract: Let h be the class number of binary quadratic forms (in Gauss's formulation). All negative determinants having some h = 6n i: 1 can be deter- mined constructively: for h = 5 there are four such determinants; for h = 7, six; for h = 11, four; and for h = 13, six. The distinction between class numbers for determinants and for discriminants is discussed and some data are given. The question of one class/genus for negative determinants is imbedded in the larger question of the existence of a determinant having a specific Abelian group as its composition group. All Abelian groups of order <25 so exist, but the noncyclic groups of order 25, 49, and 121 do not occur. Positive determinants are treated by the same composition method. Although most positive primes of the form n2 - 8 have h = 1, an interesting subset does not. A positive determinant of an odd exponent of irregularity also appears in the investigation. Gauss indicated that he could not find one. U

An iterative finite-difference method for hyperbolic systems

Abstract: Abstract. An iterative finite-difference scheme for initial value problems is presented. It is applied to the quasi-linear hyperbolic system representing the one-dimensional time dependent flow of a compressible polytropic gas. The emphasis in this research was on the handling of discontinuities, such as shock waves, and overcoming the post-shock oscillations resulting from nonlinear instabilities. The linear stability is investigated as well. The success of the method is indicated by the monotonie profiles which were obtained for almost all the cases tested.

On a Method to Subtract off a Singularity at a Corner for the Dirichlet or Neumann Problem

Abstract: Let D be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle 7ra > 0. Let U(x, y) be a solution in D of Poisson's equation such that either U or a U/an (the normal derivative) takes prescribed values on the boundary segments. Let U(x, y) be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer N there exists a function VN(X, y) which satisfies a related Poisson equation and which satisfies related boundary conditions such that U - VN is N-times con- tinuously differentiable at the corner. If 1/a is an integer VN may be found ex- plicitly in terms of the data of the problem for U. a In solving an elliptic partial differential equation by numerical methods the results proved about convergence of the numerical approximation to the actual solution frequently depend on differentiability properties of the (unknown) solu- tion. In the work of Gerschgorin (2) and other papers written since, it is assumed that the solution of the partial differential equation has derivatives of order four which are continuous up to the boundary. If the boundary and all the data are sufficiently smooth there is, of course, no problem. In many cases, however, the boundary pos- sesses a finite number of singularities, usually (in the two-dimensional case) in the. form of corners; occasionally too, the boundary data may have jumps. Laasonean (3) has proved that convergence of the discrete solution to the actual solution holds for the Dirichlet problem, but that the convergence is slow in a neighborhood of the corner. In this paper we will consider a method to subtract off the singularity. The method is quite old (see Fox (1)), but includes results on the asymptotic behavior of solutions near a corner. In this light see the works of Lewy (4), Lehman (5), Wasow (6), and the author (7). We consider a problem for which the solution is not known too be smooth. We then find, explicitly in terms of the boundary data, a solution to a related problem; then the difference between these two solutions is a solution to a. third problem, and is sufficiently well-behaved to insure convergence of difference schemes. Finally, the sought solution can be found by adding the explicitly given one to the numerically-solved one. Let D be a plane domain partly bounded by two open line segments ri and r2, which share the origin as a common endpoint and form there an interior angle 7ra > 0. We assume that ri is a subset of the positive x-axis and r2 makes an angle 7ra > 0 with the positive x-axis. Let F(x, y) be given in D and ib(x, y) (respectively

On the Solid-Packing Constant for Circles

Abstract: A solid packing of a circular disk U is a sequence of disjoint open circular subdisks Ul, U2, . whose total area equals that of U. The Mergelyan- Wesler theorem asserts that the sum of radii diverges; here numerical evidence is presented that the sum of ath powers of the radii diverges for every a < 1.306951. This is based on inscribing a particular sequence of 19660 disks, fitting a power law for the radii, and relating the exponent of the power law to the above constant. U 1. We shall be concerned here with solid packings of a closed circular disk U. Such a packing P consists of a sequence of open pairwise disjoint circular disks U1, U2, which are subsets of U; P is called solid if the areas of U and U n= Un are the same. Let r be the radius of U and rn that of Un so that the condition for a

Stochastic quadrature formulas

Abstract: A class of formulas for the numerical evaluation of multiple integrals is described, which combines features of the Monte-Carlo and the classical methods. For certain classes of functions-defined by smoothness conditions-these formulas provide the fastest possible rate of convergence to the integral. Asymptotic error estimates are derived, and a method is described for obtaining good a posteriori error bounds when using these formulas. Equal-coefficients formulas of this class, of degrees up to 3, are constructed.

Factorization of polynomials over finite fields

Abstract: If f(x) is a polynomial over GF(q), we observe (as has Berlekamp) that if h(x)2 =_h(x) (modf(x)), thenf(x) = IHa eGF(q) gcd (f(x), h(x) a). The object of this paper is to give an explicit construction of enough such h's so that the repeated application of this result will succeed in separating all irreducible factors of f. The h's chosen are loosely defined by hi(x) = xi + xiq + xiq2 + * * * (mod f(x)). A detailed example over GF(2) is given, and a table of the factors of the cyclotomic polynomials 4b(x) (mod p) for p = 2, n < 250; p = 3, n < 100; p = 5, 7, n < 50, is included.