精度保証付き反復法による連立一次方程式解法

概要

論文の詳細を見る
Let P and Q be a regular n×n matrix and a constant vector of n-dimensions, respectively, and consider solving a set of simultaneous linear equations Px=Q. When we apply iterative methods for solving a set of simultaneous linear equations, the above equation becomes [numerical formula], where A and B depend only on P and Q. For a given small number ε>0,the inequalities [numerical formula] are generally used as the criterion for convergence. The first two inequalities may be used to test convergence, but they can not give any information about the computation error (i.e. the difference between the numerical and the true solutions of the set of simultaneous linear equations). On the other hand, the last two inequalities can give the information about the computation error, but the additional computation to obtain residual norms is necessary at every stage of iterations. From the view point of numerical computation, these are troublesome problems. So, in this paper we propose a new iterative method that gives the iteration number m which guarantees [numerical formula], (i=1,2,…, n), where a numerical solution is x^<(m)>=(x_1^<(m)>, x_2^<(m)>, x_n^<(m)>) and the true solution is x=(x_1,x_2,…, x_n). Finally, we evaluate the new iterative method that we proposed by the numerical computation of a simple Dirichlet problem.
日本シミュレーション学会の論文
2001-09-15