中文摘要：

Petrov—Galerkin方法是研究Cauchy型奇异积分方程的最基本的数值方法．用此方法离散积分方程可得一系数矩阵是稠密的线性方程组．如果方程组的阶比较大，则求解此方程组所需要的计算复杂度则会变得很大．因此，发展此类方程的快速数值算法就变成了必然．该文将就对带常系数的Cauchy型奇异积分方程给出一种快速数值方法．首先用一稀疏矩阵来代替稠密系数矩阵，其次用数值积分公式离散上述方程组得到其完全离散的形式，然后用多层扩充方法求解此完全离散的线性方程组．证明经过上述过程得到方程组的逼进解仍然保持了最优阶，并且整个过程所需要的计算复杂度是拟线性的．最后通过数值实验证明结论．

英文摘要：

The Petrov-Galerkin method based on Jacobi polynomials is the conventional and standard numerical method for solving the Cauchy singular integral equations with constant coefficients. This conventional numerical method leads to a linear system with a full coefficient matrix. When the order of the linear system is large, the computational cost for obtaining and then solving the fully discrete linear system is huge. So in this paper the author develops a fast fully discrete Petrov-Galerkin method for solving this kind of integral equations. First compress this full coefficient matrix into a sparse matrix. Then apply the numerical integration scheme to obtain the fully discrete truncated linear system with a nearly linear computational cost. At last, the fully discrete truncated linear system is solved. It is established that the optimal convergence order of the approximation solution remains optimal.