Showing papers in "Mathematics of Computation in 1978"

Monte Carlo methods for index computation ()

Abstract: We describe some novel methods to compute the index of any integer relative to a given primitive root of a prime p. Our flrst method avoids the use of stored tables and apparently requires O(p 1/2) operations. Our second algorithm, which may be regarded as a method of catching kangaroos, is applicable when the index is known to lie in a certain interval; it requires O(w/2) operations for an interval of width w, but does not have complete certainty of success. It has several possible areas of application, including the f1actorization of integers. 1. A Rho Method for Index Computation. The concept of a random mapping of a finite set is used by Knuth [1, pp. 7-8] to explain the behavior of a type of random number generator. A sequence obtained by iterating such a function in a set of p elements is 'rho-shaped' with a tail and cycle which are random variables with expectation close to (1) /(irp/8) 0.6267 N/p, (as shown first in [2], [3]). Recently [4], we proposed that this theory be applied to recurrence relations such as (2) xi l x ? 1 (mod p), and showed how a very simple factorization method results, in which a prime factor p of a number can be found in only 0(pW2) operations. The method has been further discussed by Guy [5] and Devitt [6], who have found it suitable for use in programmable calculators. We now suggest that the same theory can be applied to sequences such as xo= 1, i1 qx1 0< x1 < jp 13~~~ (3)~SXi 2? 2i3 ()Xi+ 1 xi' (mod p) for 3' p

On computing the discrete Fourier transform

Abstract: New algorithms for computing the Discrete Fourier Transform of n points are described. For n in the range of a few tens to a few thousands these algorithms use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.

Analysis of some difference approximations for a singular perturbation problem without turning points

Abstract: Some three point difference schemes are considered for a singular perturbation problem without turning points Bounds for the discretization error are obtained which are uniformly valid for all h and e > 0 The degeneration of the order of the schemes at e = 0 is considered

Finding zeroes of maps: homotopy methods that are constructive with probability one

Abstract: We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is "constructive with probability one" and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.

Accelerated overrelaxation method

Abstract: This paper describes a method for the numerical solution of linear systems of equations. The method is a two-parameter generalization of the Successive Over- relaxation (SOR) method such that when the two parameters involved are equal it co- incides with the SOR method. Finally, a numerical example is given to show the su- periority of the new method. 1. Introduction. For the numerical solution of linear systems, numerous direct as well as indirect methods exist. Among the indirect or iterative methods the Succes- sive Overrelaxation (SOR) and related methods play a very important role and are the most popular ones. These methods are fully covered in the excellent books by Varga (1), by Wachspress (2) and in the most recent one by Young (3). The purpose of this paper is to present a two-parameter generalization of the SOR method and also the first basic results concerning this method which has been called Accelerated Overrelaxation (AOR) method. As will be seen, the well-known methods of Jacobi, of Gauss-Seidel, of Simultaneous Overrelaxation and of Successive Overrelaxation can be derived, as special cases, from the AOR method. Finally a characteristic numerical example, which we give in a special case, shows the superiority of the AOR method. 2. Derivation of the AOR Method. We consider a system of N linear equations with N unknowns written in matrix form

The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes

Abstract: This paper presents a new computational method for the calculation of discontinuous solutions of hyperbolic systems of conservation laws, which deal effectively with both shock and contact discontinuities. The method consists of two stages: in the first stage a standard finite-difference scheme is hybridized with a nonoscillatory first order accurate method to provide for the monotonic variation of the solution near discontinuities, and in the second stage artificial compression is applied to sharpen transitions at discontinuities. This modification of a standard finite-difference method results in a scheme which preserves the order of truncation error of the original method and yet yields a sharp and oscillation free transition for both shocks and contact discontinuities. The modification can be easily implemented in existing computer codes.

Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation

Abstract: Shepard developed a scheme for interpolation to arbitrarily spaced discrete bivariate data. This scheme provides an explicit global representation for an interpolant which satisfies a maximum principle and which reproduces constant functions. The interpolation method is basically an inverse distance formula which is generalized to any Euclidean metric. These techniques extend to include interpolation to partial derivative data at the interpolation points.

Maximum norm estimates in the finite element method on plane polygonal domains. I

Abstract: The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Rate of convergence estimates in the maximum norm, up to the boundary, are given locally. The rate of convergence may vary from point to point and is shown to depend on the local smoothness of the solution and on a possible pollution effect. In one of the applications given, a method is proposed for calculating the first few coefficients (stress intensity factors) in an expansion of the solution in singular functions at a corner from the finite element solution. In a second application the location of the maximum error is determined. A rather general class of non-quasi-uniform meshes is allowed in our present investigations. In a subsequent paper, Part 2 of this work, we shall consider meshes that are refined in a systematic fashion near a corner and derive sharper results for that case. 0. Introduction. Let Q2 be a bounded simply connected domain in the plane with boundary U2 consisting of a finite number of straight line segments meeting at vertices v;, j = 1, . . . , M, of interior angles 0 < oil S < aM < 27r (in a suitable ordering). We shall consider the Dirichlet problem -Au = f in Q2, (0.1) u= 0 on Q, where f is a given function, which for simplicity we assume to be smooth. To solve the problem (0.1) numerically, let Sh = Sh(2), 0 < h < 1, denote a one-parameter family of finite dimensional subspaces of H1 (2) n w2 (Q). We have in mind piecewise polynomials of a fixed degree on a sequence of partitions of Q2. In our considerations the partitions do not have to be quasi-uniform, not even locally (cf. examples in Section 9). Let uh E Sh be the approximate solution of (0.1) defined by the relation (0.2) A(uh, X) = (f X) for all X Esh. Here A(v, w) = fI Vv VW dx, and (v, w) = fvw dx. We wish to obtain local estimates up to the boundary in the maximum norm for the error u uh. Although our present assumptions allow meshes that are refined near a corner, in the subsequent paper, Part 2, we shall investigate the error in more detail in that case, and obtain sharper results. The general results derived in the present papei will be essential in those investigations. Received April 25, 1977. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. *This work was supported in part by the National Science Foundation. Copyright i 1978, American Mathematical Society

Superconvergence and reduced integration in the finite element method

Abstract: The finite elements considered in this paper are those of the Serendipity family of curved isoparametric elements. There is given a detailed analysis of a superconvergence phenomenon for the gradient of approximate solutions to second order elliptic boundary value problems. An approach is proposed how to use the superconvergence in practical computations.

Convergence of vortex methods for Euler’s equations

Abstract: A numerical method for approximating the flow of a two dimensional incompressible, inviscid fluid is examined. It is proved that for a short time interval Chorin's vortex method converges superlinearly toward the solution of Euler's equations, which govern the flow. The length of the time interval depends upon the smoothness of the flow and of the particular cutoff. The theory is supported by numerical experiments. These suggest that the vortex method may even be a second order method.

The Fourier method for nonsmooth initial data

Abstract: Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing. Introduction. In recent years the Fourier method for the numerical approximation of solutions to hyperbolic initial value problems has been used quite successfully. In fact, if the initial function is C°° and the coefficients of the equation are constant the method converges arbitrarily fast, i.e. is limited in practice only by the method of time discretization which is chosen. This is the reason that the Fourier method is caled "infinite order" accurate. However, the situation is drastically different when the initial function is not smooth. We take as a model the one space dimension scalar problem ut = ux to be solved for 2ir periodic u on the interval n < x < n with initial values

Approximants de Padé

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On the $‘‘3x+1”$ problem

Abstract: It is an open conjecture that for any positive odd integer m the function C(m) = (3m + 1)/2e(m) where e(m) is chosen so that C(m) is again an odd integer, satisfies Ce(m) = 1 for some h. Here we show that the number of m < x which satisfy the conjecture is at least x for a positive constant c. A connection between the validity of the conjecture and the diophantine equation 2X -3Y p is established. It is shown that if the conjecture fails due to an occurrence m = Ck(m), then k is greater than 17985. Finally, an analogous "qx + r" problem is settled for certain pairs (q, r) 4 (3, 1).

An improved multivariate polynomial factoring algorithm

Abstract: A new algorithm for factoring multivariate polynomials over the integers based on an algorithm by Wang and Rothschild is described. The new algorithm has improved strategies for dealing with the known problems of the original algorithm, namely, the leading coefficient problem, the bad-zero problem and the occurrence of extraneous factors. It has an algorithm for correctly predetermining leading coefficients of the factors. A new and efficient p-adic algorithm named EEZ is described. Bascially it is a linearly convergent variable-by-variable parallel construction. The improved algorithm is generally faster and requires less store then the original algorithm. Machine examples with comparative timing are included.

Numerical solution of an exterior Neumann problem using a double layer potential

Abstract: Introduction. Solving boundary value problems for partial differential operators by integral equation methods is not a new idea. However, the classical way to do it consists in representing the unknown solution as a potential of the type that will lead to an integral equation of the second kind. Then, Fredholm's theorems can be used. Thus, the Dirichlet problem is usually solved with the help of a double layer potential, and the Neumann problem with the use of a single layer potential. We shall have a different point of view. Our aim will be to obtain a variational formulation of the problem in order to obtain the existence and unicity of a solution and error estimates. This philosophy leads to opposite choices for the representation of the solution. Thus, J. C. Nedelec and J. Planchard, for the three-dimensional case, and M. N. Leroux for the two-dimensional case, have solved the Dirichlet problem by using a single layer potential. We propose here the solution of a Neumann problem by using a double layer potential. Let 2 be a bounded open set of R3. Let r be the boundary of 2 and Qc denote the complementary set of Q. We assume that r is sufficiently smooth, and we put the coordinates' origin in 2. We shall write nt, for the exterior normal to r, r, for the distance to the origin, [V] = vlint vie,xt, for the jump through r, of the function v defined in Ri3.

Computation of the bivariate normal integral

Abstract: This paper presents a simple and efficient computation for the bivariate normal integral based on direct computation of the double integral by the Gauss quadrature method.

The irregular primes to 125000

Abstract: We have determined the irregular primes below 125000 and tabulated their distribution Two primes of index five of irregularity were found, namely 78233 and 94693 Fermat's Last Theorem has been verified for all exponents up to 125000 We computed the cyclotomic invariants,Mp, Ap , and found that -O 0 for all p < 125000 The complete factorizations of the numerators of the Bernoulli numbers B2k for 2k < 60 and of the Euler numbers E2k for 2k < 42 are given

A note on the operator compact implicit method for the wave equation

Abstract: : In a previous paper a fourth order compact implicit scheme was presented for the second order wave equation. A very efficient factorization technique was developed when only second order terms were present. In this note we implement the operator compact implicit spatial discretization method for the second order wave equation when first order terms are present. The resulting algorithm is completely analogous to the compact implicit algorithm when lower order terms were not present. For this more general operator compact implicit spatial approximation the same factorization as in our previous paper is developed. (Author)

Interpolation by convex quadratic splines

Abstract: Algorithms are presented for computing a quadratic spline interpolant with variable knots which preserves the monotonicity and convexity of the data. It is also shown that such a spline may not exist for fixed knots.

Some numerical results using a sparse matrix updating formula in unconstrained optimization

Abstract: This paper presents a numerical comparison between algorithms for unconstrained optimization that take account of sparsity in the second derivative matrix of the objective function. Some of the methods included in the comparison use difference approximation schemes to evaluate the second derivative matrix and others use an approximation to it which is updated regularly using the changes in the gradient. These results show what method to use in what circumstances and also suggest interesting future developments.

Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II

Abstract: : Convenient stability criteria are obtained for difference approximations to hyperbolic initial-boundary value problems. The approximations consist of arbitrary basic schemes and a wide class of boundary conditions. The new criteria are given in terms of the outlfow part of the boundary conditions and are independent of the basic scheme. The results easily imply that a number of well known boundary treatments, when used in combination with arbitrary stable basic schemes, always maintain stability. Consequently, many special cases studied in recent literature are generalized. (Author)

Fast Poisson solvers for problems with sparsity

Abstract: Fast Poisson solvers, which provide the numerical solution of Poisson's equation on regions that permit the separation of variables, have proven very useful in many applications. In certain of these applications the data is sparse and the solution is only required at relatively few mesh points. For such problems this paper develops algorithms that allow considerable savings in computer storage as well as execution speed. Results of numerical experiments are given.

Pisot and Salem numbers in intervals of the real line

Abstract: Based on the work of Dufresnoy and Pisot, we develop an algorithm for determining all the Pisot numbers in an interval of the real line, provided this number is finite. We apply the algorithm to the problem of determining small Salem numbers by Salem's construction, and to the proof that certain Pisot sequences satisfy no linear re-

Discrete variable methods for a boundary value problem with engineering applications

Abstract: In this paper we develop numerical techniques of order 2, 4 and 6 for the solution of a fourth order linear equation. A priori error bound is obtained for the fourth order method to prove the convergence of the finite difference scheme. A suf- ficient condition guaranteeing the uniqueness of the solution of the boundary value problem is also given. Numerical illustrations are tabulated and results compared with the classical Runge-Kutta method. 1. Introduction. We consider the problem of bending a rectangular clamped beam of length / resting on an elastic foundation. The vertical deflection w of the beam satis- fies the system

The inverse Sturm-Liouville problem and the Rayleigh-Ritz method

Abstract: In this paper we present an algorithm for solving the inverse Sturm-Liouville problem with symmetric potential and Dirichlet boundary conditions. The algorithm is based on the Rayleigh-Ritz method for calculating the eigenvalues of a two point boundary value problem, and reduces the inverse problem for the differential equation to a nonstandard discrete inverse eigenvalue problem. It is proved that the solution of the discrete problem converges to the solution of the continuous problem. Finally, we establish the stability of the method and give numerical examples. Introduction. In this paper we will present a numerical method for solving the inverse Sturm-Liouville problem. We will prove that the algorithm converges, and will give numerical results. The inverse Sturm-Liouville problem is primarily a model problem. In essence it amounts to determining the density of a vibrating string from its fundamental tone and overtones, see Borg [4, p. 83] and Krein [19]. Similar, but more complicated, problems occur in geophysics and engineering. One of the fundamental problems in geophysics is to determine the variation of the density within the earth from the eigenfrequencies of the earth, see Backus and Gilbert [2]. These data can be obtained from seismograph recordings after major earthquakes. In mechanical engineering inverse problems arise in the design of driving shafts. Here it is important that the eigenfrequencies of the shaft do not coincide with the contemplated rotational frequency, see Niordson [25]. The basic paper on the mathematical aspects of the inverse Sturm-Liouville problem is that of Borg [4]. Later more elegant uniqueness proofs have been given by Marcenko [23], Levinson [21], and Hochstadt [151. Alternate constructive methods have been suggested by Krein [18], Gel'fand and Levitan [10], Levitan [22], Niordson [25], Barcilon [3], Friedland [8], Hochstadt [16], and Hald [13]. The algorithm presented here is based on the classical Rayleigh-Ritz method. The idea is to expand the potential and the eigenfunctions in Fourier series, truncate the series and reduce the problem to a nonstandard finite dimensional inverse eigenvalue problem. The existence of solutions to this problem is easily proved by the contraction mapping theorem. The difflcult step is to show that the solutions of the finite dimensional problems converge toward the correct solution of the continuous problem. This crucial point is missing in the discrete algorithms presented by Gantmacher and Krein [9], Anderson [1], Hald [12] and Morel [24]. Received November 8, 1976; revised September 22, 1977. AMS (MOS) subject classifications (1970). Primary 65L1 5. Copyright i) 1978, American Mathematical Society

Simultaneous approximation in scales of Banach spaces

Abstract: The problem of verifying optimal approximation simultaneously in different norms in a Banach scale is reduced to verification of optimal approximation in the highest order norm. The basic tool used is the Banach space interpolation method developed by Lions and Peetre. Applications are given to several problems arising in the theory of finite element methods. 1. Introduction. In many papers concerning the mathematical analysis of finite element methods, certain approximation properties are assumed. In particular, it is often supposed that a given function may be approximated by a function in another space and that this approximation is "optimal" simultaneously in different norms. More precisely, let I2 be a bounded domain in R^ and Hs = ws2(£l) the Sobolev space of order s with norm ||-||s (cf. Lions and Magenes (9)). Let k and r be positive integers with k < r, and let {Sh' 0