Some results on a generalized alternating iterative method

Abstract

Climent and Perea [J.-J. Climent, C. Perea, Convergence and comparison theorems for a generalized alternating iterative method, Appl. Math. Comput. 143 (2003) 1–14] introduced a generalized alternating iterative method. In this paper, we establish convergence results for a nonsingular H-matrix, upper bound to the spectral radius of iterative matrix and comparison theorem for a monotone matrix. Moreover, we give some numerical examples to show our results.

Introduction

Firstly, let us introduce some alternating iterative methods which are used for the numerical solution of the real linear system

$$\mathit{Ax}=b\text{,}$$

where $A\in R^{n\times n}$ is a given nonsingular matrix, $x\in R^n$ is unknown vector and $b\in R^n$ is given vector.

In paper [1], Benzi and Szyld introduced the following stationary alternating iterative method:

For $x^{(0)}$, $k=0\text{,}1\text{,}2\text{,}\dots \text{,}$

$$\begin{array}{cc}\hfill & x^{(k+\frac{1}{2})}=M^{-1}{\mathit{Nx}}^{(k)}+M^{-1}b\text{,}\hfill \\ \hfill & x^{(k+1)}=P^{-1}{\mathit{Qx}}^{(k+\frac{1}{2})}+P^{-1}b\text{,}\hfill \end{array}$$

where $A=M-N=P-Q$ are two splittings of A. Thus

$$x^{(k+1)}={\mathit{Rx}}^{(k)}+P^{-1}({\mathit{QM}}^{-1}+I)b\text{,}$$

where

$$R=P^{-1}{\mathit{QM}}^{-1}N\text{.}$$

In paper [2], Climent and Perea introduced the following nonstationary alternating iterative method:

For $x^{(0)}$, $k=0\text{,}1\text{,}2\text{,}\dots \text{,}$

$$\begin{array}{cc}\hfill & x^{(k+\frac{1}{2})}=(M^{-1}N{)}^{\mu (k)}{x}^{(k)}+\sum _{j=0}^{\mu (k)-1}(M^{-1}N{)}^j{M}^{-1}b\text{,}\hfill \\ \hfill & x^{(k+1)}=(P^{-1}Q{)}^{\nu (k)}{x}^{(k+\frac{1}{2})}+\sum _{j=0}^{\nu (k)-1}(P^{-1}Q{)}^j{P}^{-1}b\text{,}\hfill \end{array}$$

where $A=M-N=P-Q$ are two splittings of A. Thus

$$x^{(k+1)}={\mathit{Sx}}^{(k)}+\left[(P^{-1}Q{)}^{\nu (k)}\sum _{j=0}^{\mu (k)-1}(M^{-1}N{)}^j{M}^{-1}+\sum _{j=0}^{\nu (k)-1}(P^{-1}Q{)}^j{P}^{-1}\right]b\text{,}$$

where

$$S=(P^{-1}Q{)}^{\nu (k)}(M^{-1}N{)}^{\mu (k)}\text{.}$$

Let $A=M_l-N_l(l=1\text{,}2\text{,}\dots \text{,}p)\text{,}$ where $M_l$ is nonsingular matrices.

In paper [2], Climent and Perea also introduced a generalized alternating iterative method:

For $x^{(0)}$, $k=0\text{,}1\text{,}2\text{,}\dots \text{,}$

$$x^{(k+\frac{l}{p})}=(M_l^{-1}{N}_l{)}^{\mu (l\text{,}k)}{x}^{(k+\frac{l-1}{p})}+\sum _{j=0}^{\mu (l\text{,}k)-1}(M_l^{-1}{N}_l{)}^j{M}_l^{-1}b\text{.}$$

Thus

$$x^{(k+1)}={\mathit{Tx}}^{(k)}+\sum _{i=1}^p\left(\prod _{l=1}^{p-i}(M_{p-l+1}^{-1}{N}_{p-l+1}{)}^{\mu (p-l+1\text{,}k)}\sum _{j=0}^{\mu (i\text{,}k)-1}(M_i^{-1}{N}_i{)}^j{M}_i^{-1}b\right)\text{,}$$

where

$$T=\prod _{l=1}^p(M_{p-l+1}^{-1}{N}_{p-l+1}{)}^{\mu (p-l+1\text{,}k)}\text{,}\prod _{l=1}^0(M_{p-l+1}^{-1}{N}_{p-l+1}{)}^{\mu (p-l+1\text{,}k)}=I\text{.}$$

In Section 2, we introduce some results about the convergence of a generalized alternating iterative method when matrix A is a nonsingular H-matrix using H-splittings. In Section 3, we introduce upper bound to the spectral radius of the iterative matrix when matrix A is a nonsingular H-matrix using H-splittings. In Section 4, we introduce comparison result when A is monotone and the splittings are nonnegative.