中文摘要：

研究奇异两点边值问题的高精度数值方法.首先,将奇异两点边值问题转化为奇异积分的计算问题.其次,利用改进的复合梯形公式离散奇异积分,针对几种不同情形给出了误差渐近展开式.再次,由误差估计式设计了一种改进的龙贝格算法,利用该算法可以得到问题的高精度数值解.最后,通过数值算例说明了算法的有效性.

英文摘要：

High precision numerical method for singular two-point boundary value problems （SBVPs） is studied. Firstly, the SBVPs are transformed into the computation of singular integrals. Secondly, the singular integrals are discretized by modi- fied composite trapezoidal rule and the error asymptotic expansions are obtained for some kind of cases. Thirdly, a modified Romberg extrapolation algorithm is designed to obtain high precision numerical results based on the error expansions. Fi- nally, numerical examples show the effectiveness of the algorithm.