论文标题
改进的Laguerre光谱方法具有较小的圆形错误和更好的稳定性
Improved Laguerre Spectral Methods with Less Round-off Errors and Better Stability
论文作者
论文摘要
Laguerre多项式是正交多项式，相对于重量$ e^{ - x} $定义了正交的多项式。它们在科学和工程计算中具有广泛的应用。但是，高度的Laguerre多项式的指数增长使得它们很难应用于需要使用大量laguerre碱基的复杂系统。在本文中，我们介绍了经过修改的三个复发公式，以减少圆形误差，并避免生成广义的laguerre多项式和laguerre功能，以避免溢出和下流问题。我们应用改进的拉瓜方法来求解半线上定义的椭圆方程。在此应用程序中使用了一千多个Laguerre基础，同时获得了接近机器精确度的精度。在两种情况下，研究Laguerre方法的最佳缩放系数与正交点的数量无关，而Laguerre方法比映射的Jacobi方法具有更好的收敛速度。
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.
