向量范数、矩阵范数的应用——线性方程组解的误差估计的推广

Applicatioa of Vector norm and matrix norm——A gene ralization of error estimate of solution of linear equations

本文对[1]中线性方程组解的误差估计的定理作了推广,即证明了下面的定理: 定理设1)矩阵A=(a_(ij))∈C^(n×n)的摄动矩阵为δ=(δ_(ij))∈C^(n×n),向量B=(b_1,b_2,…,b_n)~T∈C^n的摄动向量为δ~*=(δ_1,δ_2,…,δ_n)~T∈C^n; 2)||A||_a是与某向量范数||X||_a相容的算子范数; 3)A可逆,B≠(O,O,…,O)~T; 4)||A^(-1)δ||_a<1,如果X,X~*分别满足 AX=B(x+δ)X~*=B+δ~*=B~*

In this paper, the theorem of estimation of error about the so lution of linear eguations is extended, i. e., the following result is obtained. Theorem Let 1)δ= (δ_(ij))∈C^nx^n bedisturbance matrix of A=(a_(ij)) ∈C^nx^n, δ=(δ_1, δ_2, …δ_n)T~∈C^n be disturbance Vector of B= (b_1, b_2, …, b_n) ∈C^n; 2) ||A||. be an operator norm compatible with a vector norm ||X||_a; 3) A be an invertibie matrix, B≠(o, …, o)~T; 4)||A-1δ||_a<1, If there exist X, X such that AX=B (A+δ) X= B+δ=B tben 1~O ||X-X||_a/||X||_a≤||A||_a||A||A^(-1)||_a (||B-B||_a/||B||_a+||A^(-1)δ||_a/1-||A||^(-1)δ||_a ||B||_a/||B||_a 2~O ||X-X||_a/||X||_a≤||A||_a||A||A^(-1)||_a||B-B||_a/||B||_a+||A^(-1)δ||a/1-||A^(-1)δ||_a× (1+||A||_a||A^(-1)δ||_a/||B||_a)