摘要
§1.算法的建立为简单计,本文讨论的问题为 Ax=b, (1)其中,A是n阶非奇异实方阵,b是已知的n维向量。定理1.设(1)中的b为非零向量,n阶非异方阵H使得Hb=se_n,其中s为一非零常数,e_n=(0,…,0,1)~T。设HA=LQ,L为下三角阵,Q为直交阵,则Q^T的第n列平行于解向量x。证。记Q^T=(q_1,q_2,…,q_n),L阵的第n个对角元为l_(nn)。
In this paper a new algorithm for solving an ill-conditioned system of linear equations is proposed and a better practical estimate of the solution is given. The main results are the following: Theorem 1. Let x. be the solution of the system Ax=b, in which A is an n×n nonsingular matrix, and b≠0, an n-vector. If H is an n×n nonsingular matrix such that HA=LQ and Hb=se, where Q is orthogonal, L is lower trian-gular, s is a scalar, and e, =(0,…,0, 1)~T, then the last column of Q^T is parallel to x_g. AlgorithmⅠ. (i) Construct a Householder matrix H such that Hb=se_m; (ii) Determine the QR-decomposition of(HA)~T; (iii) Compute in which b_(i_0) is the i_o-th component of b and satisfies |b_(i_o)|=max|b_i|,α_(i_o)~T is the row of A corresponding to b_(i_o), and q_n is the last column of Q;(iv) Compute x=αq_n. Corollary 3. If x is the solution of the system Ax=b from Algorithm I, then we have the estimate of the solution. where r_(mm) is the n-th diagonal element of R, can be obtained by using the method of backward error analysis introduced by J.H. Wilkinson. From Corollary 3 we can easily get the absolute error bound of the solution in the computing process.
出处
《计算数学》
CSCD
北大核心
1990年第4期434-439,共6页
Mathematica Numerica Sinica
