多参数MRV算法的算法设计与数值实验

MRV迭代法是求非线性方程组的数值解的一种Newton型迭代法. 它通过修改右端向量, 使得迭代过程中各步的线性方程组具有相同的系数矩阵. 在每步迭代过程中,利用一个参数的选择,来优化步长修正量. MRV迭代法的收敛速度较快, 界于定点Newton法和Newton迭代法之间. 借助于LU分解, 可使其计算成本降低, 低于定点Newton法. 这是一种非常实用的算法. 然而,其收敛速度仍需提高. 为此, 文献9]利用多个参数, 得到一种新的迭代法--多参数MRV迭代法, 并对其收敛性进行了严格的证明. 通过对该算法进行进一步的研究,特别是对那些仅含少量非线性方程的非线性方程组,设计出一些比较好的算法, 既克服了Newton法每个迭代步都要计算Jacobi矩阵的缺点, 又保持了和Newton型迭代法相同的收敛速度. 并通过数值实验, 对这些算法的优点进行了验证.

Design of Multi-parameter MRV Algorithm and Its Value Experiments

MRV iteration method is a Newton-like method for solving the numerical solution of non-linear system.By modifying the right-hand-side vector,the linear systems in each iterate have the same coefficient matrix.It optimizes the step correction by selecting one appropriate parameter at each iteration.It converges faster than fixed Newton method and slower than Newton method.When LU decomposition method is used,its cost is less than that of Newton method.This is a kind of practical algorithm.For accelerating its convergence,a new iteration method,multi-parameter MVR algorithm is proposed by utilizing multiple parameters on MVR algorithm in reference9].It is proved that the new algorithm converges faster than MVR algorithm.In this paper,the new algorithm is further researched.When the non-linear system has only a few non-linear equations,better multi-parameter MVR algorithm,which convergences the same fast as Newton method can be designed easily.In the meantime,it needn't calculus Jacobi matrix repeatedly at each iteration.Relevant numerical examples are calculated to demonstrate the advantage of the proposed method.