三项递归公式的数值稳定性分析

Numerical Stability Analysis of the Three Recursion Relations

三项递归公式是计算正交多项式的高效方法,有些递归计算稳定,但有些递归计算具有数值不稳定性,可能影响正交多项式的实际应用效果。为判定三项递归计算的数值稳定性,将三项递归公式按阶数转化为离散状态方程,基于离散控制理论,利用李雅普诺夫稳定性原理分析三项递归计算的数值稳定性,提出了一个判定三项递归计算数值稳定的充分条件,并通过实例验证。

Three recursive relations are effective way to calculate the orthogonal polynomials.Some recursive calculations are stable,but some recursive calculations are not,which may affect the application effect of orthogonal polynomials.To determine the numerical stability of three recursive calculations,we transformed the three recursion formula into the discrete state equation of numerical errors according to the order number.Based on the discrete control theory,we analyzed numerical stability of three recursive calculations by using the lyapunov second method and put forward a set of sufficient conditions for the robust stability of the three recursion relations.An experiment was designed to verify the feasibility of our conditions.