A convergence theorem of some iteration methods
Tian Qiu-ju · Journal of Shandong University · 2009
For solving a system of linear equations of the form Ax=b, Ais often split into A=M-N, Where M is nonsingular. It is known that x(k+1)=M-1Nx(k)+M-1b(k=0,1,2,…) converges to the solution x=A-1b for each x(0), if and only if spectral radius ρ(M-1N)1. The matrix M-1N is called an iterative matrix. It is easy to see that the estimations for the bounds of ρ(M-1N) are of interest.Some iteration methods for solving linear systems were studied, when coefficient matrix was doubly α-diagonal strictly dominance, and a convergence theorem and some corollaries were given. Results obtained were applicable for doubly α-diagonal strictly dominance matrix, and for generalized diagonal strictly dominance matrix. Finally, two numerical examples were given to show the advantage of this results.