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

Laguerre polynomials are orthogonal polynomials defined on positive half line with respect to weight $e^{-x}$. They have wide applications in scientific and engineering computations. However, the exponential growth of Laguerre polynomials of high degree makes it hard to apply them to complicated systems that need to use large numbers of Laguerre bases. In this paper, we introduce modified three-term recurrence formula to reduce the round-off error and to avoid overflow and underflow issues in generating generalized Laguerre polynomials and Laguerre functions. We apply the improved Laguerre methods to solve an elliptic equation defined on the half line. More than one thousand Laguerre bases are used in this application and meanwhile accuracy close to machine precision is achieved. The optimal scaling factor of Laguerre methods are studied and found to be independent of number of quadrature points in two cases that Laguerre methods have better convergence speeds than mapped Jacobi methods.