By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast, stable, and simple algorithm for computing the NUDFT that costs O(N log N log(1/ǫ)/ loglog(1/ǫ)) operations based on the fast Fourier transform, where N is the size of the transform and 0 < ǫ < 1 is a working precision. Our key observation is that a NUDFT and DFT matrix divided entry-by-entry is often well-approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally-scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.

The bounds for the ratios of first and second kind modified Bessel functions of consecutive orders are important quantities appearing in a large number of scientific applications. We obtain new bounds which are accurate in a large region of parameters and which are shaper than previous bounds. The new bounds are obtained by a qualitative analysis of the Riccati equation satisfied by these ratios. A procedure is considered in which the bounds obtained from the analysis of the Riccati equation are used to define a new function satisfying a new Riccati equation which yields sharper bounds. Similar ideas can be applied to other functions.

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An algorithm for computing the incomplete gamma function γ * (a, z) for real values of the parameter a and negative real values of the argument z is presented. The algorithm combines the use of series expansions, Poincaré-type expansions, uniform asymptotic expansions and recurrence relations, depending on the parameter region. A relative accuracy ∼ 10 −13 in the parameter region (a, z) ∈ [−500, 500]×[−500, 0) can be obtained when computing the function γ * (a, z) with the Fortran 90 module IncgamNEG implementing the algorithm.

It is well known that one of the most relevant applications of the reverse Bessel polynomials 𝜃 n (z) is filter design. In particular, the poles of the transfer function of a Bessel filter are basically the zeros of 𝜃 n (z). In this article we discuss an algorithm to compute the zeros of reverse generalized Bessel polynomials 𝜃 n (z; a). A key ingredient in the algorithm will be a method to compute the polynomials. For this purpose, we analyze the use of recurrence relations and asymptotic expansions in terms of elementary functions to obtain accurate approximations to the polynomials. The performance of all the numerical approximations will be illustrated with examples.