基于三角方法的Cauchy主值积分数值计算

Numerical computation of principal value integrals with Cauchy kernel based on trigonometric method

用三角变量替换的方法把含Cauchy核的主值积分变换到[0,π)上含三角函数核的主值积分,用非等距结点的π(反)周期三角插值多项式作为工具去逼近新的主值积分的被积函数,构造出含Cauchy核主值积分的一个新的内插型求积公式,根据求积公式视结点个数的奇偶性不同给出了求积公式的不同表达式,推导出求积公式中求积系数的循环关系式.最后以一个实例在计算机上用Matlab编程实现,用得到的数值结果和图像来说明所得求积公式的误差渐进性.

Using the method of changing trigonometric variable, a principal value integral with Cauchy kernel was transformed to a principal value integral with trigonometric functions kernel. The new interpolatory-type quadrature quadrature formulae were constructed for the principal value integral with Cauchy kernel, in which the inteture forns of the coefficients were derived. Finally, the asymptotic error of the quadrature formulae was illustrated,using the numerical result and images from a case realized by Matlab.