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.
翻译:Laguerre 多元二次曲线是正半径对正数的圆形多面体,与美元相比,它们具有广泛的科学和工程计算应用。然而,由于Laguerre 高度多面体的指数增长,因此很难将其应用于需要使用大量Laguerre基体的复杂系统。在本文中,我们引入了经过修改的三期复发公式,以减少交错,避免溢出和下流问题,从而产生普遍的Laguerre 多面体和Laguerre函数。我们采用改进的拉盖尔方法来解决在半线上定义的椭圆方程。在这项应用中使用了一千多个Laguerre基体,同时达到了接近机器精确度的精确度。对Laguerre方法的最佳缩放系数进行了研究,发现在两个案例中,Laguerre方法的趋同速度优于绘制的雅各布方法。