Showing papers in "Mathematics of Computation in 1979"
TL;DR: Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed and the resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
Abstract: Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed. The aspects of this method considered include error estimation, adaptive mesh selection, B-spline basis function evaluation, linear system solution and nonlinear problem solution. The resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
514 citations
TL;DR: It is shown how a modification called selective orthogonalization stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step signifi'cantly.
Abstract: The simple Lanczos process is very effective for finding a few extreme eigenvalues of a large symmetric matrix along with the associated eigenvectors. Unfortunately, the process computes redundant copies of the outermost eigen- vectors and has to be used with some skill. In this paper it is shown how a modification called selective orthogonalization stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step signifi'cantly. The degree of linear independence among the Lanczos vectors is controlled without
416 citations
TL;DR: In this paper, it was shown that the Riemann zeta function has 75,000,000 zeros of the form σ+ it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the line σ = 1/2.
Abstract: We describe a computation which shows that the Riemann zeta function ζ(s) has exactly 75,000,000 zeros of the form σ+ it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the line σ = 1/2. (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of “Rosser’s rule” are given. Comments Only the Abstract is given here. The full paper appeared as [1]. For further work, see [2, 3].
212 citations
TL;DR: A class of linear implicit methods for numerical solution of stiff ODE's is presented that require only occasional calculation of the Jacobian matrix while maintaining stability, and an effective second order stable algorithm with automatic stepsize control is designed and tested.
Abstract: A class of linear implicit methods for numerical solution of stiff ODE's is presented. These require only occasional calculation of the Jacobian matrix while maintaining stability. Especially, an effective second order stable algorithm with automatic stepsize control is designed and tested. 1. Introduction. During the last decade there has been a considerable amount of research on the numerical integration of stiff systems of ODE's. This work indicates that all efficient integration methods for such problems are implicit in character. This is due to the fact that only such methods have the required stability properties. Thus, the practical problem is not the stability restrictions, but the implicitness the need to avoid these give rise to. The relevant question is now, what is the cheapest type of implicitness we have to require. Mainly, two different approaches to the implicitness can be found in the litera- ture. The first approach involves the numerical solution of nonlinear algebraic equations by the simplified Newton iteration. The simplification consists of treating the iteration matrix as piecewise constant (which means the use of an approximate Jacobian matrix). Examples of such an approach are semi-implicit Runge-Kutta formulas in Ne'rsett (4) and the formulas based on backward-differences in Gear (3). Among recent methods proposed for numerical solution of stiff ODE's are the class of modified Rosenbrock methods introduced in Wolfbrandt (6). When solving the system of equations
141 citations
TL;DR: For a two-point boundary-value problem the existence of a unique optimal mesh distribution is proved and its properties are analyzed, allowing for rather straightforward extensions to more general problems in one dimension as well as to higher-order elements.
Abstract: A theory of a posteriori estimates for the finite element method has been developed. On the basis of this theory, for a two-point boundary-value problem the existence of a unique optimal mesh distribution is proved and its properties are analyzed. This mesh is characterized in terms of certain, easily computable local error indicators which in turn allow for a simple adaptive construction of the mesh and also permit the computation of a very effective a posteriori error bound. While the error estimates are asymptotic in nature, numerical experiments show the results to be excellent already for 10% accuracy. The approaches are not restricted to the model problem considered here only for clarity; in fact, they allow for rather straightforward extensions to more general problems in one dimension as well as to higher-order elements. 11 tables.
137 citations
TL;DR: In this article, the Sinc-Galerkin method is applied to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximation solution of some linear elliptic and parabolic partial differential equations in the plane, based on approximating functions and their derivatives by use of the Whittaker cardinal function.
Abstract: : This paper illustrates the application of a Sinc-Galerkin method to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The method is based on approximating functions and their derivatives by use of the Whittaker cardinal function. The DE is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products, the evaluation of which does not require any numerical integration. Using n function evaluations the error in the final approximation to the solution of the DE is 0 exp (-c(n to the 1/2 d power)) where c is independent of n, and d denotes the dimension of the region on which the DE is defined. This rate of convergence is optimal in the class of n-point methods which assume that the solution is analytic in the interior of the interval, and which ignore possible singularities of the solution at the end-points of the interval. (Author)
130 citations
TL;DR: In this article, a band Lanczos algorithm for the iterative computation of eigenvalues and eigenvectors of a large sparse symmetric matrix is described and tested on numerical examples.
Abstract: A band Lanczos algorithm for the iterative computation of eigenvalues and eigenvectors of a large sparse symmetric matrix is described and tested on numerical examples. It starts with a p dimensional subspace, and computes an orthonormal basis for the Krylov spaces of A, generated from this starting subspace, in which A is represented by a 2p + 1 band matrix, whose eigenvalues can be computed. Special emphasis is given to devising an implementation that gives a satisfactory numerical orthogonality, with a simple program and few arithmetic operations.
129 citations
TL;DR: Schatz et al. as mentioned in this paper considered the finite element method when applied to a model Dirichlet problem on a plane polygonal domain and gave local error estimates for the case when the finite elements partitions are refined in a systematic fashion near corners.
Abstract: The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Local error estimates are given for the case when the finite element partitions are refined in a systematic fashion near corners. 0. Introduction. We assume that the reader is familiar with Part 1, [21, of this paper; some notation is briefly recollected in Section 1. General references to the literature were given in the Bibliography of Part 1. Of these references, the following are particularly relevant to our present situation: Babuska [1], Babuska and Aziz [21, Babuska and Rheinboldt [4], Babuska and Rosenzweig [5], Eisenstat and Schultz [1 1], Thatcher [36]. Let Q be a bounded simply connected plane polygonal domain with interior angles 0 2 denote the optimal order of the parameter h to which the spaces S" can approximate smooth functions in Lq norms. Furthermore, let Q2j, j = 1, . .. , M, be the intersection of Q with a disc of radius Rj centered at the jth vertex and such that Q2j contains no other vertex, and set Qo = 2\(UL, , 1). Also, put f3 = 7r/aj. In Part 1 we showed that with e > 0 arbitrarily small (see Part 1, Theorem 4.1 Received March 1, 1978. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. * This work was supported in part by the National Science Foundation. ? 1979 American Mathematical Society 0025-571 8/79/0000-0051 /$08.00 465 This content downloaded from 157.55.39.161 on Mon, 23 May 2016 06:01:22 UTC All use subject to http://about.jstor.org/terms 466 A. H. SCHATZ AND L. B. WAHLBIN for the precise hypotheses), IIU UhIIL (Q2i) r, we may take hM,k h (i.e., no refinement is necessary); whereas if OM r/2, no refinement is necessary at that vertex. If g3 2) the refinement process can be taken to start fairly close to the corner according to (0.10), and is less stringent than at the Mth vertex (even if f3 OM)The conditions (0.10)-(0.12) can also be motivated from simple approximation considerations, see Section 4. Let us remark that if an hr-E rate of convergence is desired only on the interior domain QO, then the weaker kind of refinement described in (0.10)-(0.12) suffices at each corner. To elucidate the above, let us give three examples. Example 0.1. A procedure for placing the nodes in the radial direction near VM. Consider the problem of how to place N + 1 nodes over [0, 1] so as to obtain an efficient approximation of the function xg (= gM) with piecewise polynomials of degree r 1. This problem was solved by Rice [1], who explicitly prescribed the location of the nodes so as to obtain a good approximation, asymptotically as N oo. Essentially, the N + 1 nodes xi, i =0, . . ., N, were taken as xi = (jIN)rl. In the two dimensional situation, one can, e.g., construct a triangular mesh near VM in the following fashion, Figure 1. Draw N + 1 radial lines (including the boundaries) from vM; along each of these mark down the N + 1 points xi. Then connect the ith points on the successive radial lines, thus obtaining a cobweb-like set of quadrilaterals. Now triangulate those by drawing one diagonal in each. The family of triangulations obtained in this simple way will, as N X oo, satisfy a maximum angle condition, but not a minimum angle one. In order to satisfy the latter, a more complicated construction would be necessary.
98 citations
TL;DR: Two quadrature rules for approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed in this paper.
Abstract: Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integral, are proved.
91 citations
TL;DR: In this article, the error to the discretization in time of a parabolic evolution equation by a single-step method or by a multistep method when the initial condition is not regular was studied.
Abstract: We study the error to the discretization in time of a parabolic evolution equation by a single-step method or by a multistep method when the initial condition is not regular. Introduction. The problem we are considering is the parabolic evolution equation 5 u'(t)+Au(t)=O, O 3 is documented in [8] and [2]. It is shown in [8] that for p > 3, rp is in fact strongly A(0p)-stable for some 0 < Op < ir/2. For small p, Op is close to ir/2 and in the special cases p = 3, 4, rp is A-stable. Examples of rational approximations to eZ which are strongly A(O)-stable with r(oo) = 0 are provided by the family r,,(z) developed in [2]. In the second part, we investigate error estimates when the discretization in time is carried out by means of a multistep method. Zlamal gives an error bound under the assumption that the operator A is selfadjoint and the method strongly A(O)-stable. Here, error estimates are obtained if the operator A is maximal sectorial and the method strongly A(0)-stable (O < 0 < ir/2). I. Semidiscretization in Time by a Single-Step Method.
79 citations
TL;DR: In this article, a new formulation of the generalized linear least squares problem is given, based on some ideas in estimation and allowing complete generality in that there are no restrictions on the matrices involved.
Abstract: A new formulation of the generalized linear least squares problem is given. This is based on some ideas in estimation and allows complete generality in that there are no restrictions on the matrices involved. The formulation leads directly to a numerical algorithm involving orthogonal decompositions for solving the problem. A perturbation analysis of the problem is obtained by using the new formulation and some of the decompositions used in the solution. A rounding error analysis is given to show that the algorithm is numerically stable.
TL;DR: A theorem is proved which shows that a set of independent transforms can be computed by performing a partial transformation on a single vector and this theorem also applies to nonvector machines.
Abstract: Two algorithms are presented for performing a Fast Fourier Transform on a vector computer and are compared on the Control Data Corporation STAR-100. The relative merits of the two algorithms are shown to depend upon whether only a few or many independent transforms are desired. A theorem is proved which shows that a set of independent transforms can be computed by performing a partial transformation on a single vector. The results of this theorem also apply to nonvector machines and have reduced the average time per transform by a factor of two on the CDC 6600 computer.
TL;DR: In this article, the nonlinear transformations to accelerate the convergence of sequences due to Levin are considered and bounds on the errors are derived, and convergence theorems for oscillatory and some monotone sequences are proved.
Abstract: The nonlinear transformations to accelerate the convergence of sequences due to Levin are considered and bounds on the errors are derived. Convergence theorems for oscillatory and some monotone sequences are proved.
TL;DR: In this article, a general setting for smooth interpolating splines depending on a parameter such that as this parameter approaches infinity the spline converges to the piecewise linear interpolant is given.
Abstract: A general setting is given for smooth interpolating splines depending on a parameter such that as this parameter approaches infinity the spline converges to the piecewise linear interpolant. The theory includes the standard exponential spline in tension, a rational spline, and several cubic splines. An algorithm is given for one of the cubics; the parameter for this example controls the spacing of new knots which are introduced.
TL;DR: Evaluation of Polynomials Iterative Processes Direct Methods for Solving Sets of Linear Equations The Fast Fourier Transform Fast Multiplications of Numbers Internal Sorting External Sorting Searching.
Abstract: Evaluation of Polynomials Iterative Processes Direct Methods for Solving Sets of Linear Equations The Fast Fourier Transform Fast Multiplications of Numbers Internal Sorting External Sorting Searching.
TL;DR: In this article, the authors give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k, of the least prime in the class, is usually about 0 (k) log k log 0(k), where 0 is Euler's phi-function.
Abstract: We give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k, of the least prime in the class, is usually about 0(k) log k log 0(k), where 0 is Euler's phi-function.
TL;DR: In this paper, it is shown that if A01 satisfies any of four certain properties, the rate of convergence of the mesh to zero is much faster than that predicted by the general case.
Abstract: Let AABC be a triangle with vertices A,, B and C. It is "bisected" as follows: choose a/the longest side (say AB) of AABC, let D be the midpoint of AB, then replace AABC by two triangles, AADC and ADBC. Let A01 be a given triangle. Bisect A01 into two triangles A 11' A 12. Next, bisect each Ali, i = 1, 2, forming four new triangles A2i, i = 1, 2, 3, 4. Continue thus, forming an infinite sequence Tj, j = 0, 1, 2, . . ., of sets of triangles, where Tj = {A1ji: 1 < i < 21 }. It is known that the mesh of T. tends to zero as j -e -. It is shown here that if A01 satisfies any of four certain properties, the rate of convergence of the mesh to zero is much faster than that predicted by the general case.
TL;DR: In this article, a graph is called transitive if its automorphism group acts transitively on the vertex set, i.e., it acts on the set of all the vertices of the graph.
Abstract: A graph is called transitive if its automorphism group acts transitively on the vertex set. We list the 1031 transitive graphs with fewer than 20 vertices, together with many of their properties.
TL;DR: In this paper, it was shown that the condition number IIMnj IjMn Mlloo increases at an exponential rate if the interval [a, b] is symmetric or on one side of the origin, the rate of growth being at least equal to 1 + y/T.
Abstract: A study is made of the numerical condition of the coordinate map Mn which associates to each polynomial of degree 6 n 1 on the compact interval [a, b I the n-vector of its coefficients with respect to the power basis. It is shown that the condition number IIMnj IjMn Mlloo increases at an exponential rate if the interval [a, b] is symmetric or on one side of the origin, the rate of growth being at least equal to 1 + y/T. In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.
TL;DR: In this paper, it was shown that the Rayleigh quotient can often give a good approximation to the dominant eigenvalue after a very few iterations, even when the order of the matrix is large.
Abstract: The power method for computing the dominant eigenvector of a positive definite matrix will converge slowly when the dominant eigenvalue is poorly separated from the next largest eigenvalue. In this note it is shown that in spite of this slow convergence, the Rayleigh quotient will often give a good approximation to the domi- nant eigenvalue after a very few iterations-even when the order of the matrix is large.
TL;DR: In this article, the root separation of an arbitrary polynomial P is defined as the minimum of the distances between distinct (real or complex) roots of P. The root separation is defined in terms of the distance between distinct real or complex roots.
Abstract: The minimum root separation of an arbitrary polynomial P is defined as the minimum of the distances between distinct (real or complex) roots of P. Some asymptotically good lower bounds for the root separation of P are given, where P may have multiple zeros. There are applications in the analysis of complexity of algorithms and in the theory of algebraic and transcendental numbers.
TL;DR: A description and explanation of a simple multigrid algorithm for solving finite element systems and shows the high efficiency, essentially independent of the grid spacing, predicted by the theory.
Abstract: A description and explanation of a simple multigrid algorithm for solving finite element systems is given. Numerical results for an implementation are reported for a number of elliptic equations, including cases with singular coefficients and indefinite equations. The method shows the high efficiency, essentially independent of the grid spacing, predicted by the theory.
TL;DR: In this paper, the Tikhonov regularizaron procedure was used to solve ill-posed problems of the form (1) g(t) = jQKit, s)fis)ds, 0 < t « 1, where g and K are given, and we must compute /.
Abstract: We consider ill-posed problems of the form (1) g(t) = jQKit, s)fis)ds, 0 < t « 1, where g and K are given, and we must compute /. The Tikhonov regularizaron procedure replaces (1) by a one-parameter family of minimization problems— 2 Minimize i\\\\Kf g\\\\ + aft(/))—where ft is a smoothing norm chosen by the user. We demonstrate by example that the choice of ft is not simply a matter of convenience. We then show how this choice affects the convergence rate, and the condition of the problems generated by the regularization. An appropriate choice for ft depends upon the character of the compactness of K and upon the smoothness of the desired solution.
TL;DR: A parallel algorithm for the solution of the general tridiagonal system is presented, based on an efficient implementation of Cramer's rule, in which the only divisions are by the determinant of the matrix.
Abstract: . A parallel algorithm for the solution of the general tridiagonal system is presented. The method is based on an efficient implementation of Cramer's rule, in which the only divisions are by the determinant of the matrix. Therefore, the algorithm is defined without pivoting for any nonsingular system. 0(n) storage is required for n equations and 0(log n) operations are required on a parallel computer with n processors. 0(n) operations are required on a sequential computer. Experimental results are presented from both the CDC 7600 and CRAY-1 computers.
TL;DR: In this paper, an expression for the surface area of an ellipsoid in the form of a convergent series was derived based on an n-point Gauss-Chebyshev numerical quadrature.
Abstract: An expression is derived for the surface area of an ellipsoid in the form of a convergent series. The derivation is based upon an n-point Gauss-Chebyshev numerical quadrature. The rate of convergence and accuracy of the formula are demonstrated by computing the surface area of several ellipsoids.
TL;DR: In this paper, it was shown that the Artin L-functions can be expressed as linear combinations of Epstein zeta functions of positive definite binary quadratic forms, and that these functions have rapidly convergent expansions in terms of incomplete gamma functions.
Abstract: This paper gives a method for computing values of certain nonabelian Artin L-functions in the complex plane. These Artin L-functions are attached to irreducible characters of degree 2 of Galois groups of certain normal extensions K of Q. These fields K are the ones for which G = Gal(K/Q) has an abelian subgroup A of index 2, whose fixed fileld Q(,fd) is complex, and such that there is a a G G A for which aaa 1 = a-1 for all a E A. The key property proved here is that these particular Artin L-functions are Hecke (abelian) L-functions attached to ring class characters of the imaginary quadratic field Q(-d) and, therefore, can be expressed as linear combinations of Epstein zeta functions of positive definite binary quadratic forms. Such Epstein zeta functions have rapidly convergent expansions in terms of incomplete gamma functions. In the special case K = Q(\/, a1/3), where a > 0 is cube-free, the Artin L-function attached to the unique irreducible character of degree 2 of Gal(K/Q) -S3 is the quotient of the Dedekind zeta function of the pure cubic field L = Q(a1/3) by the Riemann zeta function. For functions of this latter form, representations as linear combinations of Epstein zeta functions were worked out by Dedekind in 1879. For a = 2, 3, 6 and 12, such representations are used to show that all of the zeroes p = a + it of these L-functions with 0 < a < 1 and Iti < 15 are simple and lie on the critical line a = 1/2. These methods currently cannot be used to compute values of L-functions with Im(s) much larger than 15, but approaches to overcome these deficiencies are discussed in the final section.
TL;DR: In this paper, a user's guide to the dilogarithm function Li2(z) fZ log(1 z) dz 0 z of a real argument is given.
Abstract: This paper is a user's guide to the dilogarithm function Li2(z) fZ log(1 z) dz 0 z of a real argument. It is intended for those who are primarily interested in the values of the dilogarithm rather than in its functional relationships. The paper is deliberately written in the style of the book Computer Approximations by Hart, Cheney et al. a. Definition and Analytical Behavior. The dilogarithm function Li2 (z) is defined [1] by (1) Li2(z) = j z log(l Z) dz. The function is real-valued for real values of z S 1 and has a logarithmic branch point at z = 1. It is usual to assign a branch cut along the real line from 1 to oo and to assign the imaginary part -i7r log(x) to Li2(X) for real values of x > 1. In what follows, we deal only with the real part of the function Li2(x) for real arguments x. Li2 is asymptotic to 7r2/3 _ ?2 log2 (x) for large x and to -7r2/6 _log2(-x) for large negative x. Li2 has a maximum at x = 2 and the value there is 7r2 /4. Li2 has a zero at the origin and at x = 12.5951703698450161286398965... [4]. Li2 has infinite slope at x = 1. b. Fundamental Identities.
TL;DR: It is shown that the maximum order attainable is i when s 4; but when i = 6, an .4-stable method of order 5 is obtained, and this method has five nonzero diagonal elements, and these elements are equal.
Abstract: An s — 1 stage semiexplicit Runge-Kutta method is represented by an i X i real lower triangular matrix where the number of implicit stages is given by the number of nonzero diagonal elements. It is shown that the maximum order attainable is i when s 4; but when i = 6, an .4-stable method of order 5 is obtained. This method has five nonzero diagonal elements, and these elements are equal. Finally, a six stage bi- stable method of order six is given. Again, this method has five nonzero (and equal) di- agonal elements. 1. Introduction. Consider an initial value problem, for a system of ordinary dif- ferential equations, of the form x = f(x), x(t0) = x0. An s - 1 stage, semiexplicit, Runge-Kutta method computes a sequence of approximations