CHEBINT: A MATLAB/Octave Toolbox for Fast Multivariate Integration and Interpolation Based on Chebyshev Approximations over Hypercubes

摘要

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.