Richard S. Varga

On recurring theorems on diagonal dominance

Abstract The elementary, but very useful, concept of a strictly diagonally dominant n x n complex matrix has seen various generalizations over the course of years. The primary purpose of this note is to give a number of equivalent conditions which reproduce, and in some cases strengthen, many consequences of recent generalizations of the property of diagonal dominance.

L-Splines

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∞ andL2 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 readers use is desirable, and such a list is found in the References at the end of this paper.

Error Bounds for Spline and L-Spline Interpolation

One of our basic aims here is to obtain improved error bounds for spline and L-spline interpolation at knots, and to obtain certain stability (or perturbation) results for such forms of interpolation. To give a concrete example to illustrate our aim, consider for simplicity the interpolation of a given function f defined on [a, b] by a smooth cubic spline s over a uniform partition A, of [a, b]. Normally, if f E Cl[u, b], then its unique cubic spline interpolant s is defined by

ALTERNATING DIRECTION IMPLICIT METHODS

Publisher Summary Alternating direction implicit methods, or ADI methods as they are called for short, constitute powerful techniques for solving elliptic and parabolic partial difference equations. However, in contrast with systematic overrelaxation methods, their effectiveness is hard to explain rigorously with any generality. Indeed, to provide a rational explanation for their effectiveness must be regarded as a major unsolved problem of linear numerical analysis. The current status of this problem with regard to the elliptic partial difference equation in the plane is discussed in the chapter. It is divided into four chapters and four appendices. Part I deals with ADI methods that iterate a single cycle of alternating directions. In this case, the theory of convergence is reasonably satisfactory. Part II studies the rate of convergence of AD1 methods using m > 1 iteration parameters, in the special case that the basic linear operators H, V, Σ in question are all permutable. In this case, the theory of convergence and of the selection of good iteration parameters is now also satisfactory. Part III surveys that that is known about the comparative effectiveness of ADI methods and methods of systematic overrelaxation, from a theoretical standpoint. Part IV analyzes the results of some systematic numerical experiments that were performed to test comparative convergence rates of different methods. The four appendices deal with various technical questions and generalizations.

Error Bounds for Gaussian Quadrature of Analytic Functions

For Gaussian quadrature rules over a finite interval, applied to analytic or meromorphic functions, we develop error bounds from contour integral representations of the remainder term. As in previous work on the subject, we consider both circular and elliptic contours. In contrast with earlier work, however, we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus. We succeed in answering this question for a large class of weight distributions (including all Jacobi weights) when the contour is a circle. In the more difficult case of elliptic contours, we can settle the question for certain special Jacobi weight distributions with parameters

Numerical Methods of Higher-Order Accuracy for Diffusion- Convection Equations

\pm \frac{1} {2}

On the zeros and poles of Padé approximants toez

, and we provide empirical results for more general Jacobi weights. We further point out that the kernel of the error functional, at any complex point outside the interval of integration, can be evaluated accurately and efficiently by a recursive procedure. The same procedure is useful also t...

The analysis ofk-step iterative methods for linear systems from summability theory

A numerical formulation of high-order accuracy, based on variational methods, is proposed for the solution of multi-dimensional diffusion-convection type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that very accurate solutions of a one-domensional problem can be obtained in the neighborhood of a sharp front without doing a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in 2 dimensions. (14 refs.)

Reactor Criticality and Nonnegative Matrices

In this paper, we study the location of the zeros and poles of general Pade approximats toe z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.

A study of semiiterative methods for nonsymmetric systems of linear equations

SummaryUsing the theory of Euler methods from summability theory, we investigate general iterative methods for solving linear systems of equations. In particular, for a given Euler method, a regionS of the complex plane is determined such that ak-step iterative method converges if the eigenvalues of an iteration operatorT are contained inS. For a givenS, optimal methods are described, and upper and lower bounds are derived for the associated asymptotic rate of convergence. Special attention is given to two-step methods with complex parameters.