一类积分的Gauss-Jacobi方法及其误差分析

Gauss-Jacobi rule of some integral and error estimate

针对某类积分,从正交多项式的性质和带权Gauss型数值积分的一些结论出发,利用Jacobi多项式推导出Gauss-Jacobi求积方法,估计了截断误差,并给出应用实例。Gauss-Jacobi求积方法在应用中可得到与广义单节点数值积分公式完全相同的近似结果及误差估计。最后将此方法进行了推广,指出对另外两类积分可完全类似地进行推导,有相应的Gauss-Jacobi求积方法。Gauss-Jacobi求积方法具有精度高、误差估计简单及应用范围广的优点。

The Gauss-Jacobi quadrature formula of some type of integral is derived by using Jacobi polynomials on the basis of the properties of orthogonal polynomials and known conclusions of Gauss quadrature with weight. The truncation error analysis is made,and an application example is given.The approximate result and error estimate totally identical to those with the generalized numerical integral method of single node can be got by means of Gauss-Jacobi quadrature. Lastly the application range of the quadrature is extended to the other two types of integral,and it is pointed out that corresponding Gauss-Jacobi quadrature formulas can be obtained similarly. The Gauss-Jacobi quadrature has the advantages of high precision,simple error estimate and extensive application range.