A Chebyshev or Fourier series may be evaluated on the standard collocation grid by the fast Fourier transform (FFT). Unfortunately, the FFT does not apply when one needs to sum a spectral series at N points which are spaced irregularly. The cost becomes O(N’) operations instead of the FFTs O(N log N). This sort of “off-grid” interpolation is needed by codes which dynamically readjust the grid e...

A new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented. When the interpolation nodes are roots of unity or Chebyshev points, the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which...

This paper introduces a new set of exchange rules for a Remez-like algorithm for the Chebyshev design of IIR digital lters. It is explained that the essential di culty, in applying the Remez algorithm to rational functions, is that on some iteration, there may be no solution to the interpolation problem for which the denominator is strictly non-zero in the interval of approximation. Then the us...

This article describes a fast direct solver (i.e., not iterative) for partial hierarchically semiseparable systems. This solver requires a storage of O(N logN) and has a computational complexity of O(N logN) arithmetic operations. The numerical benchmarks presented illustrate the method in the context of interpolation using radial basis functions. The key ingredients behind this fast solver are...

Boundary integral formulations of the heat equation involve time convolutions in addition to surface potentials. If M is the number of time steps and N is the number of degrees of freedom of the spatial discretization then the direct computation of a heat potential involves order N M operations. This article describes a fast method to compute three dimensional heat potentials which is based on ...

We consider the Hamiltonian first-order coupled-mode system that occurs in nonlinear optics, photonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the linearized coupled-mode system, which is equivalent to a four-by-four Dirac system with sign-indefinite metric. In the special class of symmetric nonlinear potentials, we construct a block-diagonal repr...