Showing papers in "Numerische Mathematik in 1971"

A floating-point technique for extending the available precision

Abstract: A technique is described for expressing multilength floating-point arithmetic in terms of singlelength floating point arithmetic, i.e. the arithmetic for an available (say: single or double precision) floating-point number system. The basic algorithms are exact addition and multiplication of two singlelength floating-point numbers, delivering the result as a doublelength floating-point number. A straight-forward application of the technique yields a set of algorithms for doublelength arithmetic which are given as ALGOL 60 procedures.

Projection method for solving a singular system of linear equations and its applications

Abstract: The iterative method for solving system of linear equations, due to Kaczmarz [2], is investigated. It is shown that the method works well for both singular and non-singular systems and it determines the affine space formed by the solutions if they exist. The method also provides an iterative procedure for computing a generalized inverse of a matrix.

Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation

Abstract: In this paper we give two derivative-free computational algorithms for nonlinear least squares approximation. The algorithms are finite difference analogues of the Levenberg-Marquardt and Gauss methods. Local convergence theorems for the algorithms are proven. In the special case when the residuals are zero at the minimum, we show that certain computationally simple choices of the parameters lead to quadratic convergence. Numerical examples are included.

Circular arithmetic and the determination of polynomial zeros

Abstract: Suppose all zeros of a polynomialp but one are known to lie in specified circular regions, and the value of the logarithmic derivativep?p ?1 is known at a point. What can be said about the location of the remaining zero? This question is answered in the present paper, as well as its generalization where several zeros are missing and the values of some derivatives of the logarithmic derivative are known. A connection with a classical result due to Laguerre is established, and an application to the problem of locating zeros of certain transcendental functions is given. The results are used to construct (i) a version of Newton's method with error bounds, (ii) a cubically convergent algorithm for the simultaneous approximation of all zeros of a polynomial. The algorithms and their theoretical foundation make use of circular arithmetic, an extension, based on the theory of Moebius transformations, of interval arithmetic from the real line to the extended complex plane.

Bounds for a class of linear functionals with applications to Hermite interpolation

Abstract: A general estimation theorem is given for a class of linear functionals on Sobolev spaces. The functionals considered are those which annihilate certain classes of polynomials. An interpolation scheme of Hermite type is defined inN-dimensions and the accuracy in approximation is bounded by means of the above mentioned theorem. In one and two dimensions our schemes reduce to the usual ones, however our estimates in two dimensions are new in that they involve only the pure partial derivatives.

An algorithm for Gaussian quadrature given modified moments

Abstract: An algebraic algorithm, the long quotient- modified difference (LQMD) algorithm, is described for the Gaussian quadrature of the one-dimensional product integral ?f(x)w(x)dx when the weight function ?(x) is known through modified momentsv l =; theP l (x) are any polynomials of degreel satisfying 3-term recurrence relations with known coefficients. The algorithm serves to establish the co-diagonal matrix, the eigenvalues of which are the Gaussian abscissas. Applied to ordinary moments it requires far fewer divisions than the quotient-difference algorithm; if theP l (x) are themselves orthogonal with a kernelw 0 03F0;, there is no instability due to rounding errors. For smooth kernels ?(x) it is safe to use secondorder interpolation in determining the eigenvalues by Givens' method. The Christoffel weights can be expressed as ratios of two terms which are most easily calculated in a Sturm sequence beginning with the highest value ofl. A formula for the Christoffel weights applicable for rational versions of theQR algorithm is also derived. Convergence and the propagation of rounding errors are illustrated by several examples, and anAlgol procedure is given.

A stabilization of the simplex method

Abstract: This paper considers the effect of round-off errors on the computations carried out in the simplex method of linear programming. Standard implementations are shown to be subject to computational instabilities. An alternative implementation of the simplex method based upon L U decompositions of the basic matrices is presented, and its computational stability is indicated by a round-off error analysis. Some computational results are given.

Deflation techniques for the calculation of further solutions of a nonlinear system

Abstract: This paper defines several classes of methods which can be used to find additional solutions of a nonlinear system of equations. A theory which embraces these classes is presented and the theory is extended to the multiple root problem. The techniques developed can also be used in avoiding previously found extreme points when performing function minimization. Results of computer experiments are presented.

The QR and QL Algorithms for Symmetric Matrices

Abstract: The QR algorithm as developed by Francis [2] and Kublanovskaya [4] is conceptually related to the LR algorithm of Rutishauser [7]. It is based on the observation that if $$A = QR{\text{ and }}B{\text{ = }}RQ{\text{,}}$$ (1) where Q is unitary and R is upper-triangular then $$B = RQ = {Q^H}AQ,$$ (2) that is, B is unitarily similar to A. By repeated application of the above result a sequence of matrices which are unitarily similar to a given matrix A 1 may be derived from the relations $${A_s} = {Q_s}{R_s},{\rm{ }}{A_{s + 1}} = {R_s}{Q_s} = Q_s^H{A_s}{Q_s}$$ (3) and, in general, A s tends to upper-triangular form.

An axiomatic approach to rounded computations

Abstract: The present paper is intended to give an axiomatic approach to rounded computations. A rounding is defined as a monotone mapping of an ordered set into a subset, which in general is called a lower respectively an upper screen. The first chapter deals with roundings in ordered sets. In the second chapter further properties of roundings in linearly ordered sets are studied. The third chapter deals with the two most important applications, the approximation of the real arithmetic on a finite screen and the approximation of the real interval arithmetic on an upper screen. Beyond these examples various further applications are possible.

Newton's method under mild differentiability conditions with error analysis

Abstract: The concept of majorizing sequences introduced by Rheinboldt (SIAM J.N.A. 1968) is used to prove convergence for Newton's method for operator equations of the formT f=? when the operator satisfied the condition that the Frechet derivative is Holder continuous. A detailed analysis of computational errors is given for Newton's method applied to operators with Holder continuous derivatives. This analysis is shown to reduce the analysis of Lancaster (Num. Math. 1968) when the operator has a continuous second derivative. The above analysis is applied to an example of a second order differential equation.

Chebyshev approximation by spline functions with free knots

Abstract: This paper is concerned with Chebyshev approximation by spline functions with free knots. Necessary and sufficient conditions for the best approximations are derived. It is shown by examples that the gap between these conditions cannot be bridged. The situation is less complicated, if the given function satisfies a generalized convexity condition. In dieser Arbeit wird die Tschebyscheff-Approximation mit Splines bei freien Knoten behandelt. Notwendige und hinreichende Alternantenkriterien werden hergeleitet. Anhand von Beispielen zeigt sich, daβ diese nicht ohne weiteres verscharft werden konnen. Dies hat erhebliche Konsequenzen fur die Konstruktion bester Approximationen. Fur Funktionen, die in einem bestimmten Sinne konvex sind, ist die Lage wesentlich ubersichtlicher. An verschiedenen Stellen ergeben sich Parallelen zur Approximation mit ?-Polynomen, die in einer fruheren Arbeit untersucht wurden.

A product integration method for a class of singular first kind Volterra equations

Abstract: Convergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t?s) ?? , 0

Attenuation factors in practical Fourier analysis

Abstract: Given a 2 ?-periodic functionf, it is desired to approximate itsn-th Fourier coefficientc n (f) in terms of function valuesf μ atN equidistant abscisses $$x_\mu = \mu 2\pi /N, \mu = 0, 1, ..., N - 1.$$ A time-honored procedure consists in interpolatingf at these points by some 2 ?-periodic function ? and aproximatingc n (f) byc n (?). In a number of cases, where ? is piecewise polynomial, it has been known that $$c_n (\varphi ) = \tau _n \hat c_n (f)$$ where $$\hat c_n (f)$$ is the trapezoidal rule approximation ofc n (f) and? n is independent off. Our interest is in the factors? n , called attenuation factors. We first clarify the conditions on the approximation processP:f?? under which such attenuation factors arise. It turns out that a necessary and sufficient condition is linearity and translation invariance ofP. The latter means that shifting the periodic dataf={f μ } one place to the right has the effect of shifting ?=P f by the same amount. An explicit formula for? n is obtained for any processP which is linear and translation invariant. For interpolation processes it suffices to obtain a factorizationc n (?)=?(n)? f (n), where ? does not depend onf and? f (n) has periodN. This also implies existence of attenuation factors? n , which are expressible in terms of ?. The results can be extended in two directions: First, the processP may also approximate successive derivative valuesf μ (x) , ?=0,1,...,k?1, of the functionf, in which case formulas of the type $$c_n (\varphi ) = \sum\limits_{x = 0}^{k - 1} {\tau _{_{n_{,x} } } \hat c} (f^{(x)} )$$ emerge. Secondly,P may be translation invariant overr subintervals,r>1, in which case $$c_n (\varphi ) = \sum\limits_{\varrho = 0}^{r - 1} {\tau _{n,\varrho } \hat c_n + \varrho N/} r(f)$$ . All results are illustrated by a number of examples, in which ? are polynomial and nonpolynomial spline interpolants, including deficient splines, as well as other piecewise polynomial interpolants. These include approximants of low, and medium continuity classes permitting arbitrarily high degree of approximation.

An eigenvalue algorithm for skew-symmetric matrices

Abstract: A Jacobi-like algorithm is presented for the skew-symmetric eigenvalue problem. The process constructs iteratively, with elementary orthogonal transformations, a sequence of matrices which converges to the so-called Murnaghan form of the intial matrix.

The implicit QL algorithm

Abstract: In [1] an algorithm was described for carrying out the QL algorithm for a real symmetric matrix using shifts of origin. This algorithm is described by the relations $$\matrix{ {{Q_s}({A_s} - {k_s}I) = {L_s},} & {{A_{s + 1}} = {L_s}Q_s^T + {k_s}I,} & {{\rm{giving}}} & {{A_{s + 1}} = {Q_s}{A_s}Q_s^T,} \cr } $$ (1) where Q s is orthogonal, L s is lower triangular and k s is the shift of origin determined from the leading 2×2 matrix of A s .

Solving confluent Vandermonde systems of Hermite type

Abstract: In this paper the authors develop two algorithms to solve systems of linear equations where the coefficients form a confluent Vandermonde matrix of Hermite type, or its transpose. These algorithms reduce the given system to upper triangular form by means of elementary matrix transformations. Recursive formulas to obtain the upper triangular form in an economical way are derived. Applications and numerical results are included.Algol-60 programs are appended.

On the structure of error estimates for finite-difference methods

Abstract: In this paper we study in an abstract setting the structure of estimates for the global (accumulated) error in semilinear finite-difference methods. We derive error estimates, which are the most refined ones (in a sense specified precisely in this paper) that are possible for the difference methods considered. Applications and (numerical) examples are presented in the following fields: 1. Numerical solution of ordinary as well as partial differential equations with prescribed initial or boundary values. 2. Accumulation of local round-off error as well as of local discretization error. 3. The problem of fixing which methods out of a given class of finite-difference methods are "most stable". 4. The construction of finite-difference methods which are convergent but not consistent with respect to a given differential equation.

Sur le calcul des produits de matrices

Abstract: The purpose of this paper is to prove that Strassen's algorithm is a representation of a product of (2,2) matrices by a Hadamard product in a space of 7 dimensions. For matrices (n, n) it is possible to obtain this representation in spaces withn 3?n+1 dimensions.

Méthodes numériques pour la décomposition et la minimisation de fonctions non différentiables

Abstract: In this article we present numerical methods in Reflexive Banach spaces for the decomposition and the minimisation of non differentiable functions with contraints. In fact these methods are generalisation of the successive overrelaxation method. Jacobi and Southwell method.

Tangent methods for nonlinear equations

Abstract: Equation-solving methods that utilize alternate function and derivative values are developed. The procedures are similar to the secant or to Muller's method, and are especially competitive when the derivative is simpler to obtain than the function itself. Related hybrid methods are also found to be attractive.

Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix

Abstract: We prove the convergence of Wielandt's method in the computation of the maximal eigenvalue and eigenvector of a non-negative irreducible matrix.

An exact determination of serial correlations of pseudo-random numbers

Abstract: Exact expressions for serial correlations of sequences of pseudo-random numbers are derived. The reduction to generalized Dedekind sums is of optimum simplicity and covers all cases of the linear congruential method. The subsequent evaluation of the generalized Dedekind sums is based on a modified Euclidean algorithm whose quotients are recognized as the main contributors to the size of the serial correlations. This leads to the establishment of bounds as well as of fast computer programs. Moreover, some light is thrown upon the general question of quality in random number generation.

Inclusion domains for the eigenvalues of stochastic matrices

Abstract: Inclusion domains for the nontrivial eigenvalues of stochastic matrices are given which are closely related to a bound for the nontrivial eigenvalues given by Bauer, Deutsch and Stoer. The inclusion domains are constructed by adapting Bauer's concept of a field of values subordinate to norms to the more general case of seminorms.

On exponents of primitive matrices

Abstract: A theorem of Heap and Lynn is slightly strengthened and a number of new sharp bounds and some old ones for exponents of certain special cases of primitive matrices are presented. A new characterisation of primitive matrices in introduced.