Showing papers in "Numerische Mathematik in 1970"

Singular value decomposition and least squares solutions

Abstract: Let A be a real m×n matrix with m≧n. It is well known (cf. [4]) that $$A = U\sum {V^T}$$ (1) where $${U^T}U = {V^T}V = V{V^T} = {I_n}{\text{ and }}\sum {\text{ = diag(}}{\sigma _{\text{1}}}{\text{,}} \ldots {\text{,}}{\sigma _n}{\text{)}}{\text{.}}$$ The matrix U consists of n orthonormalized eigenvectors associated with the n largest eigenvalues of AA T , and the matrix V consists of the orthonormalized eigenvectors of A T A. The diagonal elements of ∑ are the non-negative square roots of the eigenvalues of A T A; they are called singular values. We shall assume that $${\sigma _1} \geqq {\sigma _2} \geqq \cdots \geqq {\sigma _n} \geqq 0.$$ Thus if rank(A)=r, σ r+1 = σ r+2=⋯=σ n = 0. The decomposition (1) is called the singular value decomposition (SVD).

A three-stage variable-shift iteration for polynomial zeros and its relation to generalized rayleigh iteration

Abstract: We introduce a new three-stage process for calculating the zeros of a polynomial with complex coefficients. The algorithm is similar in spirit to the two stage algorithms studied by Traub in a series of papers. We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order. To obtain additional insight we recast the problem and algorithm into matrix form. The third stage is inverse iteration with the companion matrix, followed by generalized Rayleigh iteration.

Simultaneous iteration method for symmetric matrices

Abstract: The “ordinary” iteration method with one single iteration vector (sometimes called v. Mises-Geiringer iteration) can often yield an eigenvector and its eigenput value in very short time. But since this cannot be guaranteed, not even with improvements such as shifts of origin, Aitken-Wynn acceleration or Richardson’s purification, the method cannot be recommended for general use. In order to prevent possible poor convergence, the computation is carried in parallel with several iteration vectors, between which an orthogonality relation is maintained.

On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems

Abstract: Consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin; according to the nature of this singularity we must consider either the two-point boundary-value problem or the one-point boundary value problem. Finite-difference schemes are studied; results are given concerning error analysis and monotone convergence.

TheQ R algorithm for real hessenberg matrices

Abstract: The QR algorithm of Francis [1] and Kublanovskaya [4] with shifts of origin is described by the relations $$ \matrix{ {{Q_s}({A_s} - {k_s}I) = {R_s},} & {{A_{s + 1}} = {R_s}Q_s^T + {k_s}I,} & {giving} \cr } \matrix{ {{A_{s + 1}} = } \hfill \cr } {Q_s}{A_s}Q_s^T, $$ (1) where Q s is orthogonal, R s is upper triangular and k s is the shift of origin. When the initial matrix A 1 is of upper Hessenberg form then it is easy to show that this is true of all A s . The volume of work involved in a QR step is far less if the matrix is of Hessenberg form, and since there are several stable ways of reducing a general matrix to this form [3,5, 8], the QR algorithm is invariably used after such a reduction.

Numerical integration of products of fourier and ordinary polynomials

Abstract: Sets of coefficients for four finite difference methods of numerical integration are presented that will integrate without truncation error products of fourier and ordinary polynomials. These sets are formulated such that they are free from computational difficulties.

Eigenvectors of real and complex matrices byLR andQR triangularizations

Abstract: In a recent paper [4] the triangularization of complex Hessenberg matrices using the LR algorithm was described. Denoting the Hessenberg matrix by H and the final triangular matrix by T we have $${P^{ - 1}}HP = T$$ (1) , where P is the product of all the transformation matrices used in the execution of the LR algorithm. In practice H will almost invariably have been derived from a general complex matrix A using the procedure comhes [3] and hence for some nonsingular S we have $${P^{ - 1}}{S^{ - 1}}ASP = T$$ (2) .

Numerical methods of high-order accuracy for singular nonlinear boundary value problems

Abstract: is a 2n-th order self-adjoint linear differential operator, and it was shown that the Rayleigh-Ritz-Galerkin method is an efficient scheme, both theoretically and numerically, for solving such problems. Our aim here is to extend the results of [2] and [3] to the case of nonsel[adjoint linear differential operators whose coefficients have a singularity at one or both end points of the interval [0, t ]. For ease of exposition, we shall restrict ourselves here to second order operators, as in the particular case of

Properties of a matrix with a very ill-conditioned eigenproblem

Abstract: It is proved that a matrix that has a very ill-conditioned eigenvector matrix is close to one that has multiple eigenvalues, and an estimate of this distance is given, measured in the Euclidean matrix norm

Residuenabschätzung für Polynom-Nullstellen mittels Lagrange-Interpolation

Abstract: If, for each zero of a polynomial, an approximation is known, estimates for the errors of these approximations are given, based on the evaluation of the polynomial at these points. The procedure can be carried over to the case of multiple roots and root clusters using derivatives up to the orderk - 1, wherek is the multiplicity of the cluster.

Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben

Abstract: In this paper, sufficient and necessary maximum conditions are established for a class of mathematical programming problems with an infinite set of restrictions, which is described by a finite number of inequalities. The criteria may be applied to nonlinear approximation problems and to the numerical solution of boundary value problems.

Solution to the complex eigenproblem by a norm reducing Jacobi type method

Abstract: Let C = (c ij ) = A+iZ be a complex n×n matrix having real part A =(a ij ) and imaginary part Z = (z ij ). We construct a complex matrix W = T +iU = W 1 W 2…W k as a product of non-singular two dimensional transformations W j such that the off diagonal elements of W -1 CW =C are arbitrarily small1. The diagonal elements of C are now approximations to the eigenvalues and the columns of W are approximations to the corresponding right eigenvectors.

Die Konvergenz des Rand- und Eigenwertproblems linearer gewöhnlicher Differenzengleichungen

Abstract: In this paper the convergence of finite difference approximations for the general eigenvalue and boundary value problem of ordinary differential equations is proved under the condition of consistency and stability. The eigenvalues are shown to converge preserving multiplicity. Estimates are given for the rate of convergence of difference quotients and eigenvalues.

A finite element procedure of the second order of accuracy

Abstract: A finite element procedure of the second order of accuracy for solving second order boundary value problems is presented and justified and numerical results are given.

Error analysis of algorithms for matrix multiplication and triangular decomposition using Winograd's identity

Abstract: The number of multiplications required for matrix multiplication, for the triangular decomposition of a matrix with partial pivoting, and for the Cholesky decomposition of a positive definite symmetric matrix, can be roughly halved if Winograd's identity is used to compute the inner products involved. Floating-point error bounds for these algorithms are shown to be comparable to those for the normal methods provided that care is taken with scaling.

Regularities in congruential random number generators

Abstract: This paper suggests, as did an earlier one, [1] that points inn-space produced by congruential random number generators are too regular for general Monte Carlo use. Regularity was established in [1] for multiplicative congruential generators by showing that all the points fall in sets of relatively few parallel hyperplanes. The existence of many containing sets of parallel hyperplanes was easily established, but proof that the number of hyperplanes was small required a result of Minkowski from the geometry of numbers--a symmetric, convex set of volume 2 n must contain at least two points with integral coordinates. The present paper takes a different approach to establishing the coarse lattice structure of congruential generators. It gives a simple, self-contained proof that points inn-space produced by the general congruential generatorr i+1 ?ar i +b modm must fall on a lattice with unit-cell volume at leastm n?1 There is no restriction ona orb; this means thatall congruential random number generators must be considered unsatisfactory in terms of lattices containing the points they produce, for a good generator of random integers should have ann-lattice with unit-cell volume 1.

Discrete variational Green's function

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On a free interface problem for linear ordinary differential equations and the one-phase Stefan problem

Abstract: A numerical method for the solution of the one-phase Stefan problem is discussed. By discretizing the time variable the Stefan problem is reduced to a sequence of free boundary value problems for ordinary differential equations which are solved by conversion to initial value problems. The numerical solution is shown to converge to the solution of the Stefan problem with decreasing time increments. Sample calculations indicate that the method is stable provided the proper algorithm is chosen for integrating the initial value problems.

An implicit one-step method of high-order accuracy for the numerical integration of ordinary differential equations

Abstract: For a differential equationdx/dt=f(t, x) withf t (t, x),f x (t, x) computable, the author presents a new one-step method of high-order accuracy. A rule of controlling the mesh size is given and the method is compared with the Runge-Kutta method in two numerical examples.

Global approximation of a compact set by elements of a convex set in a normed space

Abstract: : The problem of the simultaneous approximation of a compact set of functions by an element of a convex set is considered in an arbitrary normed space and then in three particular function spaces. (Author)

Note on an inverse characteristic value problem

Abstract: Given an arbitraryn ×n real matrixA andn real numberss 1, ...,s n , we study the problem of the existence of a diagonal matrixM such that the characteristic values ofA +M bes 1, ...,s n .

On the remainder term in theN-dimensional Euler Maclaurin expansion

Abstract: The form of the remainder term in theN-dimensional Euler Maclaurin expansion is investigated. A concise formalism is developed for handling expressions which are lengthy to state using conventional notation. Conditions under which an integral representation involving only derivatives of the same total order with conventional kernel functions exists for the remainder term are derived.