一种LU分解与迭代法的结合策略及算法实现

在矩阵求解算法中，直接法或迭代法都不能有效地求解大规模稀疏或病态矩阵,因此提出一种LU分解与迭代法结合的策略。采用LU分解对矩阵进行预处理，以提高迭代法的收敛性,并采用一种判断策略使矩阵的LU分解结果可最大限度地重复利用。此结合策略应用于两种共轭梯度（CG）法，得到CLUCG和CLUTCG两种算法。它们已应用于模拟和混合信号电路模拟器ZeniVDE中。大量实验结果表明此结合策略是很有效的，得到的两种算法具有较快的速度和较好的收敛性。

A combination strategy of LU factorization and iterative method and its implementation

In matrix solution algorithms,direct methods and iterative methods cannot efficiently solve large-scale sparse or ill matrix. This paper presents a strategy that combines LU factorization and iterative methods. In the strategy,LU factorization is used to precondition on matrix before iterative methods are used for matrix solution,which is to accelerate the convergence of the iterative methods. Furthermore,an effective judgment method is presented for re-factorization of matrix in preconditioning,by which the LU factorization result can be used furthest. The strategy has been used on two conjugate gradient(CG) methods,which generate two algorithms: CLUCG and CLUTCG. They have been used in an analog and mixed-signal circuit simulator ZeniVDE. Large numbers of experiments show that the combination strategy is effective,and the two algorithms have better performance and convergence than LU factorization and conjugate gradient method.