中文摘要：

球谐函数展开是一种广泛应用的数学方法.在研究Rokhlin-Tygert球谐函数展开快速算法（RT算法）的基础上,扩充并完善了RT算法,建立了所有阶连带Legendre多项式展开系数的计算过程,进而研究了该算法的MPI并行策略并给出了RT并行算法.数值实验表明,RT算法的计算量随三角截断波数的增大而急剧增长;MPI并行技术能够有效提高其运算速度,较好地提升算法的整体性能,但并行效率会受多核处理器cache大小和访存带宽的限制.

英文摘要：

Spherical harmonic expansions are a widely used tool of applied mathematics.Based on the research of Rokhlin-Tygert＇s fast algorithm for spherical harmonic expansions（RT algorithm）,the procedure for computing expansion coefficients of all orders of associated Legendre functions is established,while the RT algorithm is extended and consummated.Furthermore,the strategy of MPI parallelization of RT algorithm is developed,and its parallel algorithm is designed and implemented.Numerical results show that,the computation of RT algorithm grows violently with the lift of the highest Fourier wave number of triangular truncations;MPI parallelization can efficiently bring up the computing speed and improve the performance of the algorithm.However,the efficiency of parallelization could be limited by the cache size of multi-cores processors and the bandwidth of memory access.