Showing papers in "Bit Numerical Mathematics in 1972"
TL;DR: The sin ϑ theorem for Hermitian linear operators in Davis and Kahan as discussed by the authors is applicable to computational solution of overdetermined systems of linear equations and especially cover the rank deficient case when the matrix is replaced by one of lower rank.
Abstract: LetA be anm ×n-matrix which is slightly perturbed. In this paper we will derive an estimate of how much the invariant subspaces ofA
H
A andAA
H
will then be affected. These bounds have the sin ϑ theorem for Hermitian linear operators in Davis and Kahan [1] as a special case. They are applicable to computational solution of overdetermined systems of linear equations and especially cover the rank deficient case when the matrix is replaced by one of lower rank.
553 citations
TL;DR: A fast algorithm is given to produce a small set of short sentences from a context free grammar such that each production of the grammar is used at least once.
Abstract: A fast algorithm is given to produce a small set of short sentences from a context free grammar such that each production of the grammar is used at least once. The sentences are useful for testing parsing programs and for debugging grammars (finding errors in a grammar which causes it to specify some language other than the one intended). Some experimental results from using the sentences to test some automatically generated simpleLR(1) parsers are also given.
210 citations
TL;DR: In this article, a modified Regula Falsi method is described which is appropriate for use when an interval bracketing of the root is known, and the algorithm appears to exhibit superior asymptotic convergence properties to other modified linear methods.
Abstract: A modified Regula Falsi method is described which is appropriate for use when an interval bracketing of the root is known. The algorithm appears to exhibit superior asymptotic convergence properties to other modified linear methods.
140 citations
CERN1
TL;DR: In this paper, the SSOR method with one relaxation parameter ω has been applied, yielding a spectral condition number approximately equal to the square root of that of A. The necessary number of iterations is thus only O(h��−1/2).
Abstract: To solve large sparse systems of linear equations with symmetric positive definite matrixA=D+L+L*,D=diag(A), with iteration, the SSOR method with one relaxation parameter ω has been applied, yielding a spectral condition number approximately equal to the square root of that ofA, if the condition
$$S(\tilde L\tilde L*) \leqq \tfrac{1}{4}$$
, where
$$\tilde L = D^{ - \tfrac{1}{2}} LD^{ - \tfrac{1}{2}} $$
, is satisfied and if 0 0 likeO([ζ
−1 +ζ/λ
1]h
−1),h → 0, whereλ
1
h
2 is the smallest eigenvalue ofD
−1
A, carries over for variable smooth coefficients and even for certain kinds of discontinuities among the coefficients, if the mesh-width is adjusted properly in accordance with the discontinuity. Since the resulting matrix of iteration has positive eigenvalues, a semi-iterative technique can be used. The necessary number of iterations is thus onlyO(h
−1/2).
135 citations
TL;DR: In this article, a class of methods for solving the initial value problem for ordinary differential equations is studied and a sub-class of formulas is derived which is related to Newton-Cotes quadrature and it is shown that for block sizes r = 1, 2, 8 these methods are stable while those for r = 9, 10 are not.
Abstract: A class of methods for solving the initial value problem for ordinary differential equations is studied We developr-block implicit one-step methods which compute a block ofr new values simultaneously with each step of application These methods are examined for the property ofA-stability A sub-class of formulas is derived which is related to Newton-Cotes quadrature and it is shown that for block sizesr=1,2,, 8 these methods areA-stable while those forr=9,10 are not We constructA-stable formulas having arbitrarily high orders of accuracy, even stiffly (strongly)A-stable formulas
92 citations
TL;DR: The construction of a program to solve a simple problem, written using a topdown structural approach, is described and an independent analysis of this program is provided commenting on the possible problems that arise from the use of such a technique.
Abstract: The construction of a program to solve a simple problem, written using a topdown structural approach, is described. An independent analysis of this program is provided commenting on the possible problems that arise from the use of such a technique.
59 citations
TL;DR: An algorithm for assigning the members of two disjoint equal sets to each other under the criterion of Stable Marriage is analysed and the results compared to the experimental results obtained on a computer.
Abstract: An algorithm for assigning the members of two disjoint equal sets to each other under the criterion of Stable Marriage is analysed and the results compared to the experimental results obtained on a computer The number of comparisons is shown to be of ordern logn (wheren is the size of the sets) which is better than that achieved by algorithms for solving the Classical Assignment Problem
56 citations
TL;DR: This note discusses methods of defining the for statement in high level languages and suggests a proof rule intended to reflect the proper role of a for Statement in computer programming and a suggestion for possible generalisation.
Abstract: This note discusses methods of defining the for statement in high level languages and suggests a proof rule intended to reflect the proper role of a for statement in computer programming. It concludes with a suggestion for possible generalisation.
53 citations
TL;DR: In this article, it was shown that the backward difference multistep method is stable in the sense of Dahlquist if f 1≦q≦6, where f is the number of nodes in the graph.
Abstract: It is shown that the backward difference multistep method
$$\mathop \Sigma \limits_{m = 1}^q \frac{1}{m}
abla ^m y_p = hf_p$$
for the numerical integration ofy′(x)=f(x, y) is stable in the sense of Dahlquist iff 1≦q≦6.
38 citations
TL;DR: This paper contains a detailed, step-by-step account of the considerations leading to a program for solving the 8-queens problem.
Abstract: As a contribution to programming methodology, the paper contains a detailed, step-by-step account of the considerations leading to a program for solving the 8-queens problem. The experience is related to the method of stepwise refinement and to general problem solving techniques.
36 citations
TL;DR: An algorithm is devised that improves an eigenvector approximation corresponding to the largest (or smallest) eigenvalue of a large and sparse symmetric matrix and solves the linear systems that arise in inverse iteration by means of the c-g algorithm.
Abstract: An algorithm is devised that improves an eigenvector approximation corresponding to the largest (or smallest) eigenvalue of a large and sparse symmetric matrix. It solves the linear systems that arise in inverse iteration by means of the c-g algorithm. Stopping criteria are developed which ensure an accurate result, and in many cases give convergence after a small numer of c-g steps.
TL;DR: In this article, the results of computations with coarse grids are used for the construction of an initial approximation on a finer grid, after a rather small number of iterations high accuracy is obtained.
Abstract: By “Richardson extrapolation from 2h andh toh/2” the results of computations with coarse grids are used for the construction of an initial approximation on finer grid. After a rather small number of iterations high accuracy is obtained. Numerical results are given for the two-dimensional Laplace equation.
CERN1
TL;DR: In this article, a class of A-stable quadrature methods, or equivalently, a certain implicit Runge-Kutta scheme, is presented, and a short proof of the strong A-stability is presented.
Abstract: Formulae for a class ofA-stable quadrature methods, or equivalently a certain implicit Runge-Kutta scheme, are given. A short proof of the strongA-stability is presented.
TL;DR: In this paper, the authors investigated the effect of dependence on the error of one operation in an interval arithmetic procedure and proved the quadratic convergence of the "centered form" and of a method of Hansen and Smith for solving linear algebraic systems.
Abstract: The size of the error incurred by one operation in an interval arithmetic procedure depends on the extent to which the operands are dependent, i.e., depend on the same initial variables. In this part we will investigate the effect of such dependence. Our results are applied to prove the quadratic convergence of the “centered form” and of a method of Hansen and Smith for solving linear algebraic systems.
TL;DR: In this paper, the numerical calculation of the distribution function when the characteristic function is known is discussed and two different methods are presented for computing the distribution functions with respect to the characteristic functions.
Abstract: In this paper characteristic functions of probability distributions are considered. The numerical calculation of the distribution function when the characteristic function is known is discussed and two different methods are presented.
TL;DR: In this article, product type quadrature formulas are applied to obtain approximate solutions of Fredholm integral equations, and several numerical examples which demonstrate the efficacy of the technique, are presented.
Abstract: Product type quadrature formulas are applied to obtain approximate solutions of Fredholm integral equations. A convergence theorem, and several numerical examples which demonstrate the efficacy of the technique, are presented.
TL;DR: Miller's recurrence algorithm for tabulating the subdominant solution of a second-order difference equation is modified so as to take the asymptotic behaviour of the solution into account as discussed by the authors.
Abstract: Miller's recurrence algorithm for tabulating the subdominant solution of a second-order difference equation is modified so as to take the asymptotic behaviour of the solution into account. The asymptotic solutions of various types of equations are listed, and a method is given for estimating the error in the tabulated solution.
TL;DR: An algorithmic language, GRAAL, is defined, as an extension of ALGOL 60 (Revised), for describing and implementing graph algorithms of the type arising in applications based on a set algebraic model of graph theory which defines the graph structure in terms of user specified morphisms between certain setgebraic structures over the node and arc set.
Abstract: An algorithmic language, GRAAL, is defined, as an extension of ALGOL 60 (Revised), for describing and implementing graph algorithms of the type arising in applications. It is based on a set algebraic model of graph theory which defines the graph structure in terms of user specified morphisms between certain set algebraic structures over the node and arc set. Several examples of graph algorithms written in GRAAL are included.
TL;DR: It is proved that the algorithm converges and that the solution found is the ε-optimal solution of the chance-constrained programming problem.
Abstract: This paper describes a new algorithm solving the deterministic equivalents of chance-constrained problems where the random variables are normally distributed and independent of each other. In this method nonlinear chance-constraints are first replaced by uniformly tighter linear constraints. The resulting linear programming problem is solved by a standard simplex method. The linear programming problem is then revised using the solution data and solved again until the stopping rule of the algorithm terminates the process. It is proved that the algorithm converges and that the solution found is the e-optimal solution of the chance-constrained programming problem. The computational experience of the algorithm is reported. The algorithm is efficient if the random variables are distributed independently of each other and if they number less than two hundred. The computing system is called “CHAPS”, i.e. “Chance-ConstrainedProgrammingSystem”.
TL;DR: In this paper, a procedure for inversion of a symmetric band matrix with all elements in a certain diagonal equal is given, starting from the main diagonal, where the elementsk,k−1, k−2,k −2,... 2,1 with zeros in the remaining diagonals.
Abstract: A procedure is given for inversion of a symmetric band matrix with all elements in a certain diagonal equal. Starting from the main diagonal we have the elementsk,k−1,k−2, ... 2,1 with zeros in the remaining diagonals. Letting the second order difference operator δ2 operate on the rows of the matrix we obtain a new matrix which can be inverted by a partition method after certain permutations of the elements.
TL;DR: The run time storage administration of a planned Simula system for PDP-10 is described with emphasis on record formats, record relationships and optimizations for the virtual memory organization.
Abstract: The run time storage administration of a planned Simula system for PDP-10 is described with emphasis on record formats, record relationships and optimizations for the virtual memory organization. A parametrized garbage collector for variable size records in virtual memory is outlined.
TL;DR: The method of parallel shooting for the solution of two-point boundary value problems is investigated in this article, where bounds for the norms of the fundamental matrices of the differential equations and their inverses are obtained for estimation of the condition number and for determining the shooting intervals.
Abstract: The method of parallel shooting for the solution of two-point boundary value problems is investigated. Bounds are obtained for the norms of the fundamental matrices of the differential equations and their inverses. These bounds are used for estimation of the condition number and for determining the shooting intervals.
TL;DR: Implicit and explicit Adams-like multistep formulas are derived for equations of the typeP(d/dt)y=f(t,y) whereP is a polynomial with constant coefficients and where ∣∂f/∂y∣ is considered small compared with the roots ofP.
Abstract: Implicit and explicit Adams-like multistep formulas are derived for equations of the typeP(d/dt)y=f(t,y) whereP is a polynomial with constant coefficients and where ∣∂f/∂y∣ is considered small compared with the roots ofP. Such equations appear for instance in control theory. An analysis of the local truncation error is performed and some examples are discussed where a considerable gain of computation time is obtained compared with classical methods. Finally some extensions of this method are mentioned in order to treat more general systems of differential equations.
TL;DR: It is shown that the functional dependence defined by anR-automaton is a continuous rational function in a neighborhood of any data point except in a point set, the Lebesgue measure of which is zero.
Abstract: A result quantity in a numerical algorithm is considered as a function of the input data, roundoff and truncation errors. In order to investigate this functional relationship using the methods of mathematical analysis a structural model of the numerical algorithm calledR-automaton is introduced. It is shown that the functional dependence defined by anR-automaton is a continuous rational function in a neighborhood of any data point except in a point set, the Lebesgue measure of which is zero. An effective general-purpose algorithm is presented to compute the derivative of any result quantity with respect to the individual roundoff and truncation errors. Some ways of generalizing theR-automation model without losing the results achieved are finally suggested.
TL;DR: In this paper, the Neville-Romberg algorithm is used to construct the conventional T-table, where the error functional satisfies an asymptotic expansion of a quadrature rule.
Abstract: WhenR(m)f is anm copy version of a quadrature ruleRf, the error functional satisfies an asymptotic expansion
$$R^{(m)} f - If \simeq d_2 h^2 + d_4 h^4 + ...,m = 1/h.$$
In the conventional form of Romberg Integration,Rf is the trapezoidal rule and early terms of this expansion are “eliminated.” For this purpose the Neville-Romberg algorithm is used to construct the conventionalT-table.
TL;DR: This paper shows that the error of Hansen and Smith's method for solving linear algebraic systems with interval coefficients isO(W2), whereW is the width of the set of coefficients.
Abstract: Hansen and Smith have proposed a method for solving linear algebraic systems with interval coefficients which produces good results if the intervals are narrow. In this paper we show that the error of their method isO(W
2), whereW is the width of the set of coefficients. The effects of round-off errors are not considered.
TL;DR: In this article, the error term related to a Romberg type extrapolation scheme based on the use of an arbitrary quadrature formula is derived and a complete discussion utilizing the known properties of the Bernoulli polynomials and their related periodic functions is presented in the case of a repeated halving of the integration interval.
Abstract: The error term related to a Romberg type extrapolation scheme based on the use of an arbitrary quadrature formula is derived. A complete discussion, utilizing the known properties of the Bernoulli polynomials and their related periodic functions, is presented in the case of a repeated halving of the integration interval. The general expression for the error term is derived in the case of an arbitrary subdivision of the integration interval.
TL;DR: In this paper, the authors estimate the distribution of the eigenvalues of the discrete Laplacian on a bounded set in Rn and prove stability in the maximum norm for the Crank-Nicolson method for the heat equation.
Abstract: We estimate the distribution of the eigenvalues of the discrete Laplacian on a bounded set inRn. The proof is based on a variational technique similar to that used by Weyl for the Laplacian. As an application of our estimates we prove stability in the maximum norm for the Crank-Nicolson method for the heat equation on a bounded set.