Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, strictly diagonally dominant $M$-matrix. So I establish a stationary iteration scheme as follows,
$$Sx^{(k+1)} = Tx^{(k)} + b.$$
According to numerical results, it seems that this iterative scheme is always convergent. Is it rational from theoretical analysis? So, can we prove the convergence of this iterative scheme? i.e.,
show $\rho(S^{-1}T) < 1$, where $\rho(\cdot)$ is the spectral radius.
Please also refer to the URL for details.
http://scicomp.stackexchange.com/questions/24326/convergence-conditions-of-a-stationary-iteration-method-for-linear-systems
A spiltting form as you described is referred to as "regular splitting". It was a a hot topic in the 1950s to 1960s. It belongs to the classical iterative methods. It has been well studied by Varga, Householder, Kahan ,Fiedler and many other famous names. You'd better refer to Varga's classical book "Matrix Iterative Analysis" Section 3.6.
A spiltting form as you described is referred to as "regular splitting". It was a a hot topic in the 1950s to 1960s. It belongs to the classical iterative methods. It has been well studied by Varga, Householder, Kahan ,Fiedler and many other famous names. You'd better refer to Varga's classical book "Matrix Iterative Analysis" Section 3.6.
Zhu is right: the most classical reference is the book by Varga.
Under the conditions you described the method is NOT always convergent: take S=\eps A for constructing a counterexample.
However you can find mild conditions for convergence..
In this context there are many beautiful results based on 'monotonicity' arguments
@Zhu, @Stefano, Yeah, many thanks, I have found the details from Varga's book. But after check it, I find that my question cannot be proved if no other conditions are imposed. In fact, this problem is from the FDM discretization of FDEs. Initially, I feel that if the stationary iteration scheme is convergent, then I can set up a new smoother for multigrid method to solve FDEs. Surely, the stationary iterative method is also directly available for solving the linear systems from FDEs.
On the other hand, I also note that if I make the number of grid nodes larger, then I can prove that \| S^{-1} T\|_{\infty} < 1. Maybe, the stationary iteration method still can be tried.
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