Koen Poppe

CHEBINT: A MATLAB/Octave toolbox for fast multivariate integration and interpolation based on chebyshev approximations over hypercubes

We present the fast approximation of multivariate functions based on Chebyshev series for two types of Chebyshev lattices and show how a fast Fourier transform (FFT) based discrete cosine transform (DCT) can be used to reduce the complexity of this operation. Approximating multivariate functions using rank-1 Chebyshev lattices can be seen as a one-dimensional DCT while a full-rank Chebyshev lattice leads to a multivariate DCT. We also present a MATLAB/Octave toolbox which uses this fast algorithms to approximate functions on a axis aligned hyper-rectangle. Given a certain accuracy of this approximation, interpolation of the original function can be achieved by evaluating the approximation while the definite integral over the domain can be estimated based on this Chebyshev approximation. We conclude with an example for both operations and actual timings of the two methods presented.

In search for good Chebyshev lattices

Recently we introduced a new framework to describe some point sets used for multivariate integration and approximation (Cools and Poppe, BIT Numer Math 51:275–288, 2011), which we called Chebyshev lattices. The associated integration rules are equal weight rules, with corrections for the points on the boundary. In this text we detail the development of exhaustive search algorithms for good Chebyshev lattices where the cost of the rules, i.e., the number of points needed for a certain degree of exactness, is used as criterium. Almost loopless algorithms are considered to avoid dependencies on the rank of the Chebyshev lattice and the dimension. Also, several optimisations are applied: reduce the vast search space by exploiting symmetries, lower the cost of the point set creation and minimise the cost of the degree verification. The concluding summary of the search results indicates that higher rank rules in general are better and that the blending formulae due to Godzina lead to the best rules within the class of Chebyshev lattice rules: no better rules have been found in the searches conducted in up to five dimensions.

Error handling in Fortran 2003

Although the Fortran programming language is evolving steadily, it still lacks a framework for error handling-- not to be confused with floating point exceptions. Therefore, the commonly used techniques for handling errors did not change much since the early days and do not benefit from the new features of Fortran 2003. After discussing some historical approaches, a Fortran 2003 framework for error handling is presented. This framework also proved to be valuable in the context of unit testing and the design-by-contract (DBC) paradigm.