Improved Laguerre Spectral Methods with Less Round-off Errors and Better Stability

Laguerre polynomials are orthogonal polynomials with respect to exponentially decaying weights defined on the positive half-line. They are widely utilized in scientific and engineering computations. However, it is difficult to use them in challenging problems that require a large number of Laguerre basis functions due to the stability issue associated with the exponential growth of high-degree Laguerre polynomials. To solve this issue, we introduce in this paper an improved three-term recurrence formulae to avoid underflow (or overflow) and reduce round-off errors in the computation of generalized Laguerre polynomials and functions. To demonstrate the effectiveness of the improved methods, we apply them to solve a commonly used prototype elliptic equation, where one-dimensional bases consisting of more than one thousand Laguerre polynomials are used and an accuracy that converges to machine precision without observable deterioration is achieved. The optimal scaling factors of the Laguerre methods are investigated as well and found to be independent of the number of Gauss quadrature points in two typical cases where the Laguerre methods have faster convergence speeds than the mapped Jacobi methods. This finding is different to the existing results in the literature. It allows us to tune optimal scaling factor on coarse grids.