降元迭代法

ITERATIVE ALGORITHM OF REDUCING DEMINSIONS

本文提出了一种新的求解大型线性方程组AX=b(其中A∈R^(n×n)为对称正定矩阵)的降元迭代法,降元的手段是在n维空间中,取k个线性无关的向量,然后将n元线性方程的未知数X的取值范围限定在由这k个线性无关向量所组成的一个子空间内;同时将原来n元线性方程组降成k元线性方程组.求解这个k元线性方程组可得到X在此子空间内的一个最优解.从这个最优解出发,在n维空间中,取另外k个线性无关的向量,并作同样的降元处理.这样就形成了一个降元迭代过程.

The paper presents the method of solving huge linearly equations AX = b (where A∈Rn×n is symmetrical positive metrix) by iterative algorithm of reducing demin-sions. Means of reducing deminsions is that value range of unknown vector X in the huge linearly equations is limited to the subspace whith is defined by k linearly independent vectors which are selected from n deminsion space and that the linearly equations with n unknown numbers is reduced the linearly equations with k unknown unmbers. Solving the linearly equations with k unknown numbers, the optimal solution of X is obtained in the subspace. From the optimal solution, the other k linearly independent vectors are selected from n deminsion space and the linearly equations with n unknown numbers is reduced to linearly equations with k unknown numbers So the iterative algorithm of reducing deminsions is formed.