对称正定线性方程组具有任意权矩阵的多分裂迭代求解

MULTISPLITTING ITERATIVE METHODS WITH GENERAL WEIGHTING MATRICES FOR SOLVING SYMMETRIC POSITIVE DEFINITE LINEAR SYSTEMS

本文利用优化模型研究求解对称正定线性方程组Ax=6的多分裂并行算法的权矩阵.在我们的多分裂并行算法中,m个分裂仅要求其中之一为P-正则分裂而其余的则可以任意构造,这不仅大大降低了构造多分裂的难度,而且也放宽了对权矩阵的限制(不像标准的多分裂迭代方法中要求权矩阵为预先给定的非负数量矩阵).并且证明了新的多分裂迭代法是收敛的.最后,通过数值例子展示了新算法的有效性.

By making use of optimal models, we study the weighting matrices of the multisplitting parallel methods for solving the symmetric positive definite linear system Ax = b. In our multisplitting there is only one that is required to be P-regular splitting and all the others can be constructed arbitrarily, which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also relaxes the constraints to the weighting matrices （unlike the standard methods, they are not necessarily nonnegative diagonal scalar matrices or given in advance）. We then prove the convergence of this new method. Finally. numerical experiments show that the method is efficient.