求解病态线性方程的一种精细格式及迭代终止准则

A precise form for solving ill-conditioned algebraic equations and its iteration stopping criterion

研究了求解病态线性方程组的一种简化精细迭代格式和相应的迭代终止准则。首先将线性病态方程组系数矩阵的逆,归结为一矩阵指数的无穷积分形式;然后选择一个固定步长t,建立前述矩阵指数积分在区间[0,2τ]与[0,τ]上的递推关系,并通过区间倍增的方式逼近无穷积分。算法以2~n指数收敛,经过数十次迭代即可获得高精度解,因此具有极高的效率。在迭代过程中解的精度随着积分区间的增加而迅速提高,但当积分区间达到一定程度后,矩阵自乘过程中的误差积累以及矩阵的病态性,反而会导致精度随着区间的增加迅速下降。故一个可行的迭代终止准则,才使得算法具有实际意义。本文以迭代残差为指标,如果该指标连续n次出现增加,则计算停止。n与问题的病态程度及矩阵规模有关,一般情况下n取2即可,最大不超过10。在算例中,n取为5进行计算,都能使得迭代在解较为精确的次数时停止,证明了准则是有效的。

In this paper, a simplified iteration form of precise integration method for ill-conditioned algebraic equations and its iteration stopping criterion are studied. Firstly, the inverse of positive definite real coefficient matrix of ill-conditioned algebraic equation is taken as an infinite integral with matrix exponential. Secondly, a fixed step t is chosen to establish the recursive relation in integrating range of [0, 2τ] and [0, τ ], and through the way of multiplying interval to approximate the infinite integral algorithm. After several iterations, high precision solutions can be obtained. Solutions precision are rapidly increased then decreased with increase of the integral interval. So the iteration stopping criterion makes the algorithm has practical significance. In this paper, the iterative residual is used as the index. Calculation stops if the index is successive addition in n times. n is related to the degree of ill and the size of the matrix. In general, n taken as 2 is alright and less than 10. For example, n is taken as 5, then high precision solutions can be got when iteration stops. This proves that the criterion is effective.