On a direct method for the solution of nearly uncoupled Markov chains

Summary This note is concerned with the accuracy of the solution of nearly uncoupled Markov chains by a direct method based on the LU decomposition. It is shown that plain Gaussian elimination may fail in the presence of rounding errors. A modification of Gaussian elimination with diagonal pivoting and correction of small pivots is proposed and analyzed. It is shown that the accuracy of the solution is affected by two condition numbers associated with aggregation and the coupling respectively.This work was supported in part by the Air Force Office of Sponsored Research under Contract AFOSR-87-0188     

Almost strictly totally positive matrices

A determinantal identity, frequently used in the study of totally positive matrices, is extended, and then used to re-prove the well-known univariate knot insertion formula for B-splines. Also we introduce a class of matrices, intermediate between totally positive and strictly totally positive matrices. The determinantal identity is used to show any minor of such matrices is positive if and only if its diagonal entries are positive. Among others, this class of matrices includes B-splines collocation matrices and Hurwitz matrices.This author acknowledges a sabbatical stay at IBM T.J. Watson Research Center in 1990, which was supported by a DGICYT grant from Spain.     

Bivariate Hermite-Birkhoff interpolation and Vandermonde determinants

The concepts of Vandermonde determinant and confluent Vandermonde determinant are extended to the multidimensional setting by relating them to multivariate interpolation problems. With an approach different from that of other recent papers on this subject, the values of these determinants are computed, recovering and extending the results of those papers.Partially supported by Research Grant PS900121 DGICYT.     

Error analysis of a pairwise summation algorithm to compute the sample variance

Summary We give an error analysis of an algorithm for computing the sample variance due to Chan, Golub, and LeVeque [The American Statistician 7 (1983), pp. 242–247]. It is shown that this algorithm is numerically stable. The algorithm computes the sample variance (and the sample mean) using just one pass through the sample data. It is amenable to pairwise summation and thus requires onlyO(logn) parallel steps.Research supported by the Air Force Office of Scientific Research under grant no. AFOSR-88-0161 and by the Office of Naval Research under grant no. N00024-85-C-6041     

Analyse numerique des ecoulements quasi-Newtoniens dont la viscosite obeit a la loi puissance ou la loi de carreau

Summary We prove abstract error estimates for the approximation of the velocity and the pressure by a mixed FEM of quasi-Newtonian flows whose viscosity obeys the power law or the Carreau law. These estimates are optimal in some cases. They can be applied to most finite elements used for the solution of Stokes's problem.

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Analyse numerique des ecoulements quasi-Newtoniens dont la viscosite obeit a la loi puissance ou la loi de carreau

Résumé On prouve des estimations d'erreur abstraites pour l'approximation de la vitesse et la pression par une MEF mixtes d'écoulements quasi-Newtoniens dont la viscosité obéit à la loi puissance ou la loi de Carreau. Ces estimations sont optimales dans certains cas. Elles peuvent être appliquées à la plupart des éléments finis utilisés pour la résolution du problème de Stokes.

     

Steplength optimization and linear multigrid methods

Summary Recently steplength parameters have been used in linear multigrid methods. In this paper we give a theoretical analysis of the effects of steplength optimization in a rather general framework which covers two different implementations of steplength optimization in standard multigrid methods.     

A continuous selection of the metric projection in matrix spaces,addendum

Summary In an earlier paper [1] the authors proved that the metric projection of the space ofm byn matrices onto the subset of matrices of rank less than or equal tok has a continuous selection. In this note we sharpen the result to show that the selection is in fact globally Lipschitz-continuous.Dedicated to R. S. Varga on the occasion to his sixtieth birthday     

A dual algorithm for convex-concave data smoothing by cubicC 2-splines

Summary In this paper the problem of smoothing a given data set by cubicC ²-splines is discussed. The spline may required to be convex in some parts of the domain and concave in other parts. Application of splines has the advantage that the smoothing problem is easily discretized. Moreover, the special structure of the arising finite dimensional convex program allows a dualization such that the resulting concave dual program is unconstrained. Therefore the latter program is treated numerically much more easier than the original program. Further, the validity of a return-formula is of importance by which a minimizer of the orginal program is obtained from a maximizer of the dual program.The theoretical background of this general approach is discussed and, above all, the details for applying the strategy to the present smoothing problem are elaborated. Also some numerical tests are presented.     

Numerical integration by means of adapted Euler summation formulas

Summary The purpose of the paper is the study of formulas and methods for numerical integration based on Euler summation formulas. Cubature formulas are developed from multidimensional generalizations. Estimates of the truncation error are given in adaptation to specific properties of the integrand.