  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.
