Showing papers in "Numerische Mathematik in 1967"

Smoothing by spline functions. II

Abstract: In this paper we generalize the results of [4] and modify the algorithm presented there to obtain a better rate of convergence.

L-Splines

Abstract: In this paper, we study the problem of unique interpolation and approximation by a class of spline functions,L-splines, containing as special cases the deficient and generalized spline functions ofAhlberg, Nilson, andWalsh [3, 5, 6], the Chebyshevian spline functions ofKarlin andZiegler [27], and the piecewise Hermite polynomial functions, as considered in [17]. We first give sufficient conditions for unique interpolation byL-spline functions in Section 2. Then, we obtain newL ? andL 2 error estimates for interpolation byL-splines in Section 4, and show that these error estimates are, in a certain sense, sharp. In addition, we make a similar study for theg-splines ofSchoenberg, cf. [44, 3], in Section 5. In Section 6, an application of these new error estimates is made to the analysis of the error made in the use of finite dimensional subspaces ofL-splines andg-splines. in the Rayleigh-Ritz procedure for the class of nonlinear two-point boundary value problems studied in [17]. Because of the rapid growth of the number of papers devoted to or connected with the topic of splines, we believe that a compilation of papers on splines for the reader's use is desirable, and such a list is found in the References at the end of this paper.

Construction of fully symmetric numerical integration formulas of fully symmetric numerical integration formulas

Abstract: The paper develops a construction for finding fully symmetric integration formulas of arbitrary degree 2k+1 inn-space such that the number of evaluation points isO((2n)k)/k!),n ? ?. Formulas of degrees 3, 5, 7, 9, are relatively simple and are presented in detail. The method has been tested by obtaining some special formulas of degrees 7, 9 and 11 but these are not presented here.

An effective algorithm for minimization

Abstract: An algorithm is proposed for minimizing certain niceC 2 functionsf onE n assuming only a computational knowledge off and?f. It is shown that the algorithm provides global convergence at a rate which is eventually superlinear and possibly quadratic. The algorithm is purely algebraic and does not require the minimization of any functions of one variable. Numerical computation on specific problems with as many as six independent variables has shown that the method compares very favorably with the best of the other known methods. The method is compared with theFletcher andPowell method for a simple two dimensional test problem and for a six dimensional problem arising in control theory.

Iterated deferred corrections for nonlinear operator equations

Abstract: : It is shown that a Theorem by Stetter on asymptotic expansions can be obtained with fewer hypotheses, and that correction operators for improving the order of convergence of discretization algorithms can be constructed under fairly weak assumptions. (Author)

Solution of symmetric and unsymmetric band equations and the calculation of eigenvectors of band matrices

Abstract: In an earlier paper in this series [2] the triangular factorization of positive definite band matrices was discussed. With such matrices there is no need for pivoting, but with non-positive definite or unsymmetric matrices pivoting is necessary in general, otherwise severe numerical instability may result even when the matrix is well-conditioned.

High speed computation of group characters

Abstract: This paper describes a practical procedure for computing the ordinary irreducible characters of finite groups (of orders up to 1000 or so). The novelty of the method consists of transposing the problem from the field of complex numbers into the field of integers modulop for a suitable primep. It is much easier to compute the modular characters in the latter field, and from these characters we can calculate the ordinary irreducible characters in algebraic form.

Convergence of the conjugate gradient method with computationally convenient modifications

Abstract: The conjugate gradient (CG) method has been well analyzed [6] as a method of solving linear operator equations; its practical aspects in this regard have also been thoroughly reported for systems of linear algebraic equations in an excellent s tudy b y ENGELI, et al. [2]. For nonlinear equations, it is possible to implement the CG idea in several ways [3, 4, 7a, 7b, 8]; earlier papers [la, 1hi described the theoretical aspects of one particular implementation, while the present report continues with a discussion of the effect on the convergence of introducing certain computat ional ly convenient alterations. Numerical examples are given comparing different methods of solution and describing certain difficulties tha t m a y arise with CG methods for finite systems of (nonlinear) algebraic equations. Let 1 J(x) be a norm continuous operator in x from a real separable Hilbert space ~ with inner product ( . , • ) into ~ ; for each x in ~ let there exist the Frechet derivative J~ with range ~ and Gateaux derivative J~'. We assume tha t IIJ~lt and fIJ~'ll are uniformly bounded and tha t J~ is a self-adjoint uniformly positive definite operator, i.e. O < a I < ~ J ~ A I , lI/2'fl T~e CG iteration to solve J(x) = 0 is as follows:

The condition numbers of the matrix eigenvalue problem

Abstract: For ann ×n matrixA with distinct eigenvalues explicit expressions are obtained for certain condition numbers associated with the reduction ofA to its Jordan normal form. These condition numbers are also related by inequalities to (i) the departure from normality ofA, (ii) the discriminant of the eigenvalues ofA, (iii) the Gram determinant of the eigenvectors ofA.

Optimal approximation and error bounds in seminormed spaces

Abstract: The general problem of linear approximation is concerned with the characterization of the range of a given functional on the subset of a linear space which is defined by the data. In most applications that domain is the intersection of a sphere and a linear variety so that the conventional techniques based upon classical inequalities that are sharp on a sphere cannot yield optimal appraisals. In Hilbert spaces the required characterization is given by the well known hypercircle inequality which is based uponPythagoras' theorem and therefore cannot be extended to the general case where the norm is not expressible in terms of an inner product. The optimization theory in seminormed spaces and a practical method of solution are developed by resorting to duality theorems and more specially to the theory of convex bodies. A simple but nontrivial application is given by way of illustration.

Relative error propagation in the recursive solution of linear recurrence relations

Abstract: This paper is concerned with the numerical solution of the general initial value problem for linear recurrence relations. An error analysis of direct recursion is given, based on relative rather than absolute error, and a theory of relative stability developed.Miller's algorithm for second order homogeneous relations is extended to more general cases, and the propagation of errors analysed in a similar manner. The practical significance of the theoretical results is indicated by applying them to particular classes of problem.

Numerical mapping of exterior domains

Abstract: This paper describes an integral equation method for computing the conformal mapping of the exterior of a given simply-connected region onto the exterior of the unit circle. The method includes evaluation of the transfinite, diameter of the given region. Results are presented for a number of trial problems.

Some high accuracy difference schemes with a splitting operator for equations of parabolic and elliptic type

Abstract: High accuracy alternating direction implicit difference schemes for the heat equation, LAPLACE's equation and the biharmonic equation are considered. In addition to surveying the existing methods, several new methods are introduced. Sequences of iteration parameters are obtained for the elliptic problems and a numerical example is given.

A generalized inverse ∈-algorithm for constructing intersection projection matrices, with applications

Abstract: Givenk linear manifolds ?1, ..., ? k and corresponding perpendicular projection matricesP 1, ...,P k , a closed formula is derived for[Figure not available: see fulltext.] the perpendicular projection matrix with range[Figure not available: see fulltext.]. The derivation uses results taken from the theory of generalized inverses together with an application ofWynn's ?-Algorithm to a convergent sequence of matrices. A variant of this formula is then used in solving arbitrary complex linear systems by iteration and in computing generalized inverses; the latter application provides a solution to least squares linear regression problems.

Some fifth degree integration formulas for symmetric regions II

Abstract: Here Sn is a region in n-dimensional Euclidean space (n _ 2), the Ai are constants and the (vil, ***, vin) are points in the space. We discuss a general class of 5th degree formulas for a class of regions Sn which includes the n-dimensional cube and sphere; a precise description of the regions to which our discussion applies will be given in the next section. Each formula in the class contains no more than N = 2n(n + 1) points and has all positive coefficients Ai. For the n-sphere we give constants in four useful formulas which contain 2 n(n + 1), 2"n + 1, 2n+1 - 1 and 2n + 2n points. The 2n(n + 1) point formula has all coefficients equal. The 2n + 2n point formula has fewer points than any other known 5th degree formula in which all coefficients are positive (for the n-sphere, n > 4). The corresponding formulas for the n-cube are not as useful since, for most values of n, they have some points which lie outside the cube. Previously, Hammer and Stroud [3] have given formulas of 5th degree for the n-cube and n-sphere which use only 2n2 + 1 points. Those formulas, however, have the undesirable feature that, for the n-sphere (n > 5), some of the coefficients Ai are negative. The only other known general class of 5th degree formulas for the n-sphere, for arbitrary n, are the spherical product formulas; these contain 3n - 3n-1 + 1 points and have all positive coefficients. (Spherical product formulas were first described, for n = 2, 3, by Peirce [6, 7], and for arbitrary n independently by Hetherington [4] and lMlysovskih [5]; an alternative description is given by

Monte Carlo path-integral calculations for two-point boundary-value problems

Abstract: A computational technique based on the method of path integral is studied with a view to finding approximate solutions of a class of two-point boundary-value problems. These solutions are "rough" solutions by Monte Carlo sampling. From the computational point of view, however, once these rough solutions are obtained for any nonlinear cases, they serve as good starting approximations for improving the solutions to higher accuracy. Numerical results of a few examples are also shown.

Finite difference solution of the third boundary problem in elliptic and parabolic equations

Abstract: Finite difference methods (including the Peaceman-Rachford method) are considered for the solution of the third boundary value problem for parabolic and elliptic equations. Conditions on the coefficients involved in the boundary conditions are obtained from the stability requirements of the difference methods and shown to coincide with those necessary for asymptotic stability of the differential system.

Symmetry and self-duality in nonlinear programming

Abstract: This paper is concerned with a pair of naturally symmetric problems related by duality. Self-duality has been investigated for the class of non-differentiable convex programs.

Remarks on band matrices

Abstract: In this note we consider band- or tridiagonal-matrices of orderk whose elements above, on, and below the diagonal are denoted byb i ,a i,c i . In the periodic case, i.e.b i+m =b i etc., we derive fork=nm andk=nm?1 formulas for the characteristic polynomial and the eigenvectors under the assumption that $$\mathop \prod \limits_{i = 1}^m c_i b_i > 0$$ i=1 In the latter case it is shown that the characteristic polynomial is divisible by them?1-th minor, as was already observed byRosa. We also give estimations for the number of real roots and an application to Fibonacci numbers.

Transformed rational Chebyshev approximation

Abstract: The approximation problem is: Given f~ C(X), to find a coefficient vector A* minimizing e(A) = Ilf F(A, .)\I over A E 9’. Such a coefficient vector A* is called best and &A*, .) is called a best approximation tofon X. 1,~ is a transformation operator. The first study of transformations was that of the author [2], who studied transformations of ordinary rational functions on an interval. Kaufman and Belford [6] have studied transformations of alternating families. Williams [9] has studied some special cases of transformations of Haar subspaces on an interval.