Recurrence relations and fast algorithms
Mark Tygert
TL;DR
Addresses fast transforms for function families governed by recurrence relations. It reduces forward and inverse transforms to eigen-decompositions of tridiagonal real self-adjoint matrices and exploits divide-and-conquer spectral methods, handling weighted orthonormal polynomials and Bessel-function families. The work provides concrete algorithms for quadrature nodes/weights, zeros of Bessel functions, and analysis/synthesis of polynomial weights, plus evaluations of Bessel-function combinations, with cost bounds $C n (ln n) (ln(1/epsilon))^3$. This extends fast spectral transforms beyond FFT-like cases, enabling efficient, high-precision pseudospectral computations and potential use in spherical harmonics and related transforms.
Abstract
We construct fast algorithms for evaluating transforms associated with families of functions which satisfy recurrence relations. These include algorithms both for computing the coefficients in linear combinations of the functions, given the values of these linear combinations at certain points, and, vice versa, for evaluating such linear combinations at those points, given the coefficients in the linear combinations; such procedures are also known as analysis and synthesis of series of certain special functions. The algorithms of the present paper are efficient in the sense that their computational costs are proportional to n (ln n) (ln(1/epsilon))^3, where n is the amount of input and output data, and epsilon is the precision of computations. Stated somewhat more precisely, we find a positive real number C such that, for any positive integer n > 10, the algorithms require at most C n (ln n) (ln(1/epsilon))^3 floating-point operations and words of memory to evaluate at n appropriately chosen points any linear combination of n special functions, given the coefficients in the linear combination, where epsilon is the precision of computations.