Showing papers in "Numerische Mathematik in 1969"
TL;DR: In this paper, Cook et al. gave an algorithm which computes the coefficients of the product of two square matrices A and B of order n with less than 4. 7 n l°g 7 arithmetical operations (all logarithms in this paper are for base 2).
Abstract: t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical operations (all logarithms in this paper are for base 2, thus tog 7 ~ 2.8; the usual method requires approximately 2n 3 arithmetical operations). The algorithm induces algorithms for invert ing a matr ix of order n, solving a system of n linear equations in n unknowns, comput ing a determinant of order n etc. all requiring less than const n l°g 7 arithmetical operations. This fact should be compared with the result of KLYUYEV and KOKOVKINSHCHERBAK [1 ] tha t Gaussian elimination for solving a system of l inearequations is optimal if one restricts oneself to operations upon rows and columns as a whole. We also note tha t WlNOGRAD [21 modifies the usual algorithms for matr ix multiplication and inversion and for solving systems of linear equations, trading roughly half of the multiplications for additions and subtractions. I t is a pleasure to thank D. BRILLINGER for inspiring discussions about the present subject and ST. COOK and B. PARLETT for encouraging me to write this paper. 2. We define algorithms e~, ~ which mult iply matrices of order m2 ~, by induction on k: ~ , 0 is the usual algorithm, for matr ix multiplication (requiring m a multiplications and m 2 ( m t) additions), e~,k already being known, define ~ , ~ +t as follows: If A, B are matrices of order m 2 k ~ to be multiplied, write
2,978 citations
TL;DR: In this paper, the authors specify a class of condition numbers for which those scalings can be given explicitly, and show how far at most for a certain scaling the quanti ty under consideration may be away from its minimum.
Abstract: Introduction In numerical linear algebra one meets condition numbers [[AII []A-11[ and similar quantities such as (max [a,i[)[[A-1H and [[Ai[[ [[A-l[], where A = (a,i) and A i is the /'-th column of A. The norms are very diverse. The problem then is to determine a rowand/or column-scaling of A which minimizes the quanti ty under consideration. I t is the purpose of this paper to specify a class of such quantities for which those scalings can be given explicitly. The results will be extensions of some results in [2]. They will also hold for non-square matrices. All proofs will be completely elementary. Also, in some cases where the minimizing scaling cannot be given explicitly, it can be said how far at most for a certain scaling the quanti ty under consideration may be away from its minimum.
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TL;DR: In this article, an integral equation method for computing the conformal mapping of a finite doubly-connected domain onto R <|w|<1, where R is uniquely determined, is presented.
Abstract: This paper describes an integral equation method for computing the conformal mapping of a finite doubly-connected domain ontoR <|w|<1, whereR is uniquely determined. The method is illustrated by numerical examples.
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TL;DR: The systematic relaxation method is analysed for consistently ordered matrices as defined by Broyden (1964) and has a better asymptotic rate of convergence than S.O.R. and requires less calculations and computer store.
Abstract: A systematic relaxation method is analysed for consistently ordered matrices as defined by Broyden (1964). The method is a generalisation of successive over-relaxation (S.O.R.). A relation is derived between the eigenvalues of the iteration matrix of the method and the eigenvalues of the Jacobi iteration matrix. Forp-cyclic matrices, the method corresponds to using a special type of diagonal matrix instead of a single relaxation factor. For certain choices of this diagonal matrix, the method has a better asymptotic rate of convergence than S.O.R. and requires less calculations and computer store.
TL;DR: In this paper, an alternation theorem is developed for approximating with functions having restricted ranges where equality is allowed to occur between the restraining curves, and an alternate theorem is also developed for functions with restricted ranges.
Abstract: In this paper an alternation theorem is developed for approximating with functions having restricted ranges where equality is allowed to occur between the restraining curves.
TL;DR: In this paper, the Fourier coefficients of a smooth function are computed using approximate values of the integrals of the function and some of its derivatives, denoted as the derivatives of / and its derivatives.
Abstract: J. N. LYNESS* and C. B. MOLER '~* Received September t9, 1967 1. Introduction Certain algorithms for calculating the Fourier coefficients of a smooth func- tion /(x) (Stetter [7], Lyness [4J) require approximate values of the integrals of / and some of its derivatives. We denote these by
TL;DR: In this article, an iterative method for factoring a polynomial that bears the same relation to Bairstow's method as the secant method in a single variable bears to Newton's method is described.
Abstract: This paper describes an iterative method for factoring a polynomial that bears the same relation to Bairstow's method as the secant method in a single variable bears to Newton's method. Like the secant method, the generalized secant method requires only one function evaluation for each iteration, and like the secant method it converges to a simple factor with order (1+75)/2.
Abstract: This paper deals with questions of nonlinear Tschebyscheff-approximation theory, the approximations being constrained by nonlinear relations We assume the approximating functions depending Frechet-differentiable on a parameter and the constraints satisfying certain regularity and differentiability properties Under these hypotheses in the main theorem we give necessary conditions to characterisize best approximations Using these results, some problems in approximating functions, the best approximations being regarded to satisfy interpolatory conditions, are discussed We deduce, that in this case best approximations admit a characterisation by generalized alternants
TL;DR: In this paper, the convergence of difference approximations to initial-value problems with first-order systems of quasilinear hyperbolic differential equations in two independent variables was studied.
Abstract: Using an idea due toCourant, Isaacson andRees, Tornig andZiegler showed convergence of difference approximations to initial-value problems with first-order systems of quasilinear hyperbolic differential equations in two independent variables by constructing an ordinary difference or differential equation, which has a solution being a majorant of the error and decreasing for decreasing step-sizes. Here this method is generalized to a wider class of quasilinear problems (not necessary hyperbolic, not necessary of first order, not necessary in only two independent variables).
TL;DR: In this paper, it was proved that any consistent one-step method for solving the initial value problem for a first-order ODE is convergent; no stability condition is required.
Abstract: It is proved that any consistent one-step method for solving the initial value problem for a first-order ordinary differential equation is convergent; no stability condition is required. An application is made to a similarly stated result, allowing part of the hypothesis in that case to be dropped.
TL;DR: In this paper, bounds for the eigenvalues of singular Sturm-Liouville problems from a finite difference method are obtained. But the accuracy of the method is not known.
Abstract: Generalizing the method of Wendroff [9] and using an estimate for the square integral of a normed eigenfunction outside a compact set, bounds are obtained for the eigenvalues of singular Sturm-Liouville problems from a finite difference method. The number of mesh points necessary to obtain. the accuracy ? behaves like ??½ ln? if? tends to zero. Some numerical examples are given.
TL;DR: Two well known high accuracy Alternating Direction Implicit difference schemes for solving Laplace's equation and the Biharmonic equation are considered and the set of iteration parameters of Douglas is used.
Abstract: Two well known high accuracy Alternating Direction Implicit difference schemes for solving Laplace's equation and the Biharmonic equation are considered. The set of iteration parameters of Douglas is used in both problems. More complete optimum values of the parameters involved are given.
TL;DR: In this article, the question of constructing stable numerical representations for the solutions of initial-boundary value problems for parabolic differential equations is examined, and a stable numerical representation for the solution of this problem is given.
Abstract: The question of constructing stable numerical representations for the solutions of initial-boundary value problems for parabolic differential equations is examined.
TL;DR: In this paper, a uniform approximation of vector-valued functions is defined, together with analogs of extreme points and H-sets, and characterizations of best approximations are given in terms of these.
Abstract: Uniform approximation of vector-valued functions is defined, together with analogs of extreme points and H-sets. Characterizations of best approximations are given in terms of these, and some applications are presented.
TL;DR: In this article, the authors generalize this idea and construct iterative processes of any orderk (k > 1) in a recursive manner, by adding higher derivatives of the operator.
Abstract: The Newton process for operator equations in say a linear normed complete space converges under certain hypothesis about the Frechet-derivatives of the operator with at least the order two. There are different ways to improve this Newton process. For instance you obtain a process of order three if you add a correction element containing the second Frechet-derivative of the operator [1]. In the following note we will generalize this idea. In a recursive manner -- by adding higher derivatives -- we will construct iterative processes of any orderk (k > 1). A general theorem due toCollatz provides us error estimates for this processes. Last we will illustrate the processes by several examples.