Showing papers in "Bit Numerical Mathematics in 1969"

Some efficient fourth order multipoint methods for solving equations

Abstract: In this paper we derive a simple and efficient fourth order multipoint iterative method for solving equations. Comparisons of computational efficiency are made with other well known techniques and a numerical example is given. The paper is concluded with an ALGOL procedure for implementing the process.

A class ofA-stable methods

Abstract: A class of methods for the numerical solution of systems of ordinary differential equations is given which—for linear systems—gives solutions which conserve the stability property of the differential equation. The methods are of a quadrature type $$y_{i,r} = y_{n,r - 1} + h\sum\limits_{k = 1}^n {a_{ik} f(y_{k,r} ), n = 1,2, \ldots ,n, r = 1,2, \ldots ,} y_{n,0} given$$ wherea ik are quadrature coefficients over the zeros ofP n −P n−1 (v=1) orP n −P n−2 (v=2), whereP n is the Legendre polynomial orthogonal on [0,1] and normalized such thatP m (1)=1. It is shown that $$\left| {y_{n,r} - y(rh) = 0(h^{2n - _v } )} \right|$$ wherey is the solution of $$\frac{{dy}}{{dt}} = f(y), t \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 0, y(0) given.$$

Programming by action clusters

Abstract: The paper describes a programming discipline, aiming at the systematic construction of programs from given global requirements. The crucial step in the approach is the conversion of the global requirements into sets of action clusters (sequences of program statements), which are then used as building blocks for the final program. The relation of the approach to proof techniques and to programming languages is discussed briefly.

An algorithm for solving linear algebraic equations using residue arithmetic I

Abstract: This paper is in two parts. Part I contains a survey of the elementary theory of residue arithmetic and a description of an algorithm for solvingAx=b using single-modulus residue arithmetic. In Part II there is a description of the algorithm using more than one modulus. Part II also contains a discussion of how to select the moduli along with some numerical results. There is a sketch of this procedure in Newman [1]. However, the treatment here is complete and uses the notation of Szabo and Tanaka [2]. It lays the foundation for a subsequent paper which will extend these results.

Notes on top-down languages

Abstract: Properties of context-free languages and grammars permitting deterministic top-down recognition with bounded lookahead are discussed. In particular, it is shown that for eachk>1 there are such languages requiring a lookahead of at leastk characters.

New quadrature formulas for the numerical inversion of the Laplace transform

Abstract: New quadrature formulas for the evaluation of the Bromwich integral, arising in the inversion of the Laplace transform are discussed. They are obtained by optimal addition of abscissas to Gaussian quadrature formulas. A table of abscissas and weights is given.

Smooth curve interpolation

Abstract: In a recent paper [1] a method was presented for smooth curve interpolation by means of minimising the energy of an idealised elastic beam constrained to pass through given data points. This method of obtaining the curve required an enormous amount of computation, since it involved the repeated solution of a fourth order differential equation and the minimisation of a function of many variables, the number of variables being equal to the number of data points. In this paper a much simpler method is presented for finding the minimum energy curve in which the minimisation procedure is eliminated. An ALGOL procedure to perform the curve fit is supplied.

A criterion forA(α)-stability of linear multistep methods

Abstract: Some easy-to-check conditions are given which together are sufficient forA(α)-stability. As an application the greatest α's are determined which giveA(α)-stability for the differentiation formulae withk=3, 4, 5 and 6.

Weight diagrams for lie group representations: A computer implementation of Freudenthal's algorithm in ALGOL and FORTRAN

Abstract: Implementations in FORTRAN and ALGOL of the Dynkin and Freudenthal algorithms for computing weight systems and for determining the multiplicities of the weights for irreducible representations of simple Lie algebras are described. Reasonable computing times are found for algebras of rank less than or equal to 8 and for representations of dimension less than 1000.

Derivation of explicit expressions for the error terms in the ordinary and the modified romberg algorithms

Abstract: It is proved that the error terms for both the ordinary and the modified Romberg algorithms may be expressed in terms of Bernoulli polynomials and their related periodic functions. The properties of the error terms may thus be described using the known properties of the Bernoulli polynomials and functions.

Estimation of a non-negative function

Abstract: It is pointed out that the problem of estimating a function from the values of a finite number of functionals is often posed, at least partially, as an aesthetic problem rather than as a mathematical one. A particular problem of interpretation of experimental results, arising in the study of radiation source distribution in a plasma, is formulated mathematically in two different ways. Although the two formulations are different in character, they both rely on the idea of imposing a relative likelihood distribution on an appropriate function space; this is particularly fruitful when observational errors are taken into account.

On a modification of the Clenshaw-Curtis quadrature formula

Abstract: In this paper we discuss a modification of the Clenshaw-Curtis quadrature formula. It is shown that for integrals, where the integrand may be expanded in a sufficiently rapid convergent Chebyshev series, we may split the sequence of calculated approximations into two sequences, one which approximates the integral from above and one which approximates it from below. Thus, at any step during the calculation we obtain both upper and lower bounds for the true value of the integral.

Some comments on the solution of linear equations

Abstract: Computational methods of homogeneous ill- conditioned and singular simultaneous linear equations

Computer cartography range map

Abstract: The location of every real estate property in Sweden will be given in the official cadastre by means of coordinates. The coordinates of this property register can also be transferred to all the other Swedish registers which give the names of the properties, such as the population and tax registers, and of course also to the censuses of population, housing, agriculture, etc. These coordinates, among other things, can be used for the purposes of cartography. It becomes possible to have a computer turn out maps cheaply and quickly. In this paper we will give some computer mapping programs which use the cadastre coordinates.

A note on top-down deterministic languages

Abstract: In recent years attempts have been made to characterize those languages to which deterministic top-down analysis is applicable. This note examines the relationships between the various classes of grammars that are known to this author.

The norm of a matrix after a similarity transformation

Abstract: The norm of a similarity transformation of a matrix is studied considered as a function of the transformation matrix, and conditions for stationary points of this function are given in terms of properties of the transformed matrix.

Algorithms over partially ordered sets

Abstract: We here study some problems concerned with the computational analysis of finite partially ordered sets. We begin (in § 1) by showing that the matrix representation of a binary relationR may always be taken in triangular form ifR is a partial ordering. We consider (in § 2) the chain structure in partially ordered sets, answer the combinatorial question of how many maximal chains might exist in a partially ordered set withn elements, and we give an algorithm for enumerating all maximal chains. We give (in § 3) algorithms which decide whether a partially ordered set is a (lower or upper) semi-lattice, and whether a lattice has distributive, modular, and Boolean properties. Finally (in § 4) we give Algol realizations of the various algorithms.

The rate of convergence of parabolic difference schemes with constant coefficients

Abstract: The paper describes how estimates can be obtained for the rate of convergence of certain parabolic difference schemes. These schemes approximate a Cauchy problem with constant coefficients. Interpolation technique is used for many of the proofs. The results are compared with previous work by several authors.

A new explicit three-level difference scheme for the solution of the heat flow equation

Abstract: A new explicit three-level difference scheme for the numerical solution of the heat flow equation is proposed. The main features of the new scheme are: i) it is unconditionally stable, ii) it is very highly accurate from the point of view of the truncation error, and iii) its solution converges to the solution of the heat equation even if the time and the distance increments tend to zero independently.

Common prime factors of the numbers $$A_n = a^{2^n } + 1$$

Abstract: A method to find certain common factors of numbers of the form $$A_n = a^{2^n } + 1$$ , when theA n are lacking algebraic factors, is discussed. Using known factors of the Fermat numbers $$F_n = 2^{2^n } + 1$$ , otherA n , containing these factors, are deduced.

A proof of Hamblin's algorithm for translation of arithmetic expressions from infix to postfix form

Abstract: We show that a particular algorithm for the translation of arithmetic expressions in infix form into postfix form is correct. The method used is ad hoc, in contrast to other work in this field, and it is hoped that this in itself is of interest.

Data structure and storage allocation

Abstract: The paper orders the many kinds of data encountered in current programming languages into a small number of classes and points out which are the properties of these classes which impose particular schemes of storage allocation inside compilers. The problems of dynamic storage allocation and garbage collection are then analyzed.

A method for acceleration of the convergence of infinite series

Abstract: l f(t)dt (1) lim Zco-Sn lim ~ = lira f(n) n_~cog(~) -g(n) ~_+~ -g(n) n~g'(n) must exist and be different from zero. Thus we should choose g(n) such tha t g'(n)=Ic.f(n), k~cons tan t , and g(~)=O. We now easily see when the choice g(n) = 1In works, namely when the terms of the series decrease as 1/n ~. However, if for other series we choose g(n) according to the formula above, we get a very pronounced increase in effectiveness. A very useful method for polynomial extrapolations of this kind is Aitken's method for iterated inter/extrapolation since it yields successive results which can be used to estimate the error of the extrapolation. A further advantage is tha t it can easily be programmed. We now exemplify the effectiveness of a correct choice of g(n). The table below also exemplifies the interesting fact tha t an incorrect choice of g(n) yields a monotonic series of extrapolation results while the errors vary in sign irregulary for the correct choice of g(n). This

A note on correctors with an arbitrary number of nonstep points

Abstract: In this note maximal order,k step correctors with one nonstep point for the solution ofy′=f(x,y),y(x0)=y0, introduced by Gragg and Stetter [1] are extended to an arbitrary numbers of nonstep points. These correctors have order 2k + 2s, are proved stable fork≦8,s≧2, and unstable for largek.

A branch-bound algorithm for a wiring problem

Abstract: A simple computational method for solving a specific wiring problem related to the construction of the RC 4000 computer is described. The intimate relationship between the wiring problem and the traveling salesman problem is established, and the algorithm is based upon the “branch and bound” technique as employed by J.D.C. Little et al. [1] for solving the latter problem.

A basic course on compiler principles

Abstract: An attempt to devise a methodology of compiler design is described, and an outline is given for a possible course on this subject. The theoretical basis is formed by the concepts of phrase-structure language, finite-state- and stack-acceptor, and transducer. As their extension capable of processing context-dependent elements of languages, a so-called Table-Transducer is postulated, and it serves as the core-algorithm upon which compilers are based. The developed theory and method of compiler construction is applied to an example of a simple programming language.

Unique representation of primitive factors of 2 n −1,n ODD, in certain quadratic forms

Abstract: Primitive factorsq of 2 n −1,n odd, are either of the form 8m−1 or 8m+1. In the first case there exists a unique representation ofq in the quadratic forma 2−2b 2, (a, b)=1,a andb odd,b<[√q/2], and in the latter a unique representation ofq in the quadratic formc 2+2 j d2, (c, d)=1,c andd odd,j even and ≧6. Thus the uniqueness ofq=a 2−2b 2 orc 2+2 j d2 exhibits a proof of the primality ofq. A program that determinesa, b, c, andd for anyq not exceeding 16 decimal digits is described, and as an example the 13-digit prime 4432676798593 (a primitive factor of 249−1) is uniquely represented by 13742732+214·124612.