一类弱非线性方程组的Picard-MHSS迭代方法

修正的Hermite/反Hermite分裂(MHSS)迭代方法是一类求解大型稀疏复对称线性代数方程组的无条件收敛的迭代算法.基于非线性代数方程组的特殊结构和性质,我们选取Picard迭代为外迭代方法,MHSS迭代作为内迭代方法,构造了求解大型稀疏弱非线性代数方程组的Picard-MHSS和非线性MHSS-like方法.这两类方法的优点是不需要在每次迭代时均精确计算和存储Jacobi矩阵,仅需要在迭代过程中求解两个常系数实对称正定子线性方程组.除此之外,在一定条件下,给出了两类方法的局部收敛性定理.数值结果证明了这两类方法是可行、有效和稳健的.

ON PICARD-MHSS METHODS FOR WEAKLY NONLINEAR SYSTEMS

Modified Hermitian and skew-Hermitian splitting(MHSS) iteration method is an unconditionally convergent method for solving large sparse complex symmetric linear systems. Based on the special structure and properties of the nonlinear systems, choosing Picard iteration as outer iteration and MHSS iteration as the inner solver of Picard iteration, we present the Picard-MHSS and nonlinear MHSS-like iteration methods for solving large scale systems of weakly nonlinear equations. The advantage of these methods is that they do not require explicit construction and accurate computation of the Jacobian matrix, and only need to solve linear sub-systems of constant coefficient real symmetric positive definite matrices. Moreover, Under suitable conditions, we establish local convergence theorems for both Picar-MHSS and nonlinear MHSS-like iteration methods. Numerical results show that these iteration methods are feasible, effective and robust.