Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.

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Barycentric Lagrange Interpolation

A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

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A MATLAB differentiation matrix suite

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

In 1945, W. Taylor discovered the barycentric formula for evaluating the interpolating polynomial. In 1984, W. Werner has given first consequences of the fact that the formula usually is a rational interpolant. We review some advances since the latter paper in the use of the formula for rational interpolation.

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Recent Developments in Barycentric Rational Interpolation

In 1988 the second author presented experimentally well-conditioned linear rational functions for global interpolation. We give here arrays of nodes for which one of these interpolants converges exponentially for analytic functions.

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Exponential convergence of a linear rational interpolant between transformed Chebyshev points

We discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for Cebysev-Gauss-Lobatto points. We propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices.

The errors in calculating the pseudospectral differentiation matrices for C̆ebys̆ev-Gauss-Lobatto points

Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions Following [1], we suggest to determine for each set of n + 1 nodes another denominator of degree n , and then to interpolate every continuous function by a rational function with that same denominator, so that the resulting interpolation process remains a linear projection The optimal denominator is chosen so as to minimize the Lebesgue constant for the given nodes It has to be computed numerically For that purpose, the barycentric representation of rational interpolants, which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights, is used The optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in [1] We show here how to do so, and we present numerical results

Jean-Paul Berrut

Papers

Barycentric Lagrange Interpolation

Recent Developments in Barycentric Rational Interpolation

Exponential convergence of a linear rational interpolant between transformed Chebyshev points

The errors in calculating the pseudospectral differentiation matrices for C̆ebys̆ev-Gauss-Lobatto points

Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval