为提高大型线性方程组的求解效率,在VRP-GMRES(m)算法基础上,利用截断技术,即在构造Krylov子空间的基向量和Hessenberg矩阵时采用不完全正交的Arnoldi过程,提出截断型变参数广义极小残余算法(VRP-IGMRES(m)),并利用连续2次迭代残余向量的夹角余弦与模的关系给出算法的收敛性证明. 最后通过数值算例分析了截断指标对计算精度和计算效率的影响,表明VRP-IGMRES(m)算法在保证计算精度的前提下,可以有效地提高计算效率,并得到了最优截断比的取值大约为0.1,为实际工程问题的求解提供了新的方法.
In order to solve large linear equations efficiently, based on the Generalized Minimal Residual with Variable Restart Parameter algorithm(VRP-GMRES(m)), a truncation-pattern Incomplete Generalized Minimal Residual with Variable Restart Parameter algorithm(VRP-IGMRES(m))is proposed using the truncation technology,namely, using incomplete orthogonal Arnoldi process to constructe the base vector of the Krylov subspace and the Hessenberg matrix. The convergence of the algorithm is proved by the relationship between the angle cosine and the modulus of the two successive iterated residual vectors. Finally, through numerical examples to analyze the influence of truncation index on computational accuracy and efficiency, it is found that VRP-IGMRES(m)algorithm can effectively improve the computation efficiency under the premise of guaranteeing the accuracy. The obtained optimal truncation ratio is about 0.1. Our work provides a new method for solving practical engineering problems.
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