Preconditioned Iterative Method for Regular Splitting

Several preconditioners are proposed for improving the convergence rate of the iterative method derived from splitting. In this paper, the comparison theorem of preconditioned iterative method for regular splitting is proved. And the convergence and comparison theorem for any preconditioner are indicated. This comparison theorem indicates the possibility of finding new preconditioner and splitting. The purpose of this paper is to show that the preconditioned iterative method yields a new splitting satisfying the regular or weak regular splitting. And new combination preconditioners are proposed. In order to denote the validity of the comparison theorem, some numerical examples are shown.


Introduction
There are many iterative methods for solving a linear system of equations, Here, A is a n n × nonsingular M-matrix; x and b are n-dimensional vectors.Matrix A which arises from various problems is usually large and sparse matrix.Then large amount of computation times and memory are needed in order to solve efficiently the problems.Therefore, various preconditioners and iterative methods have been proposed.In this paper, Gauss-Seidel iterative method is treated as classical iterative method.Basically, the classical iterative method can be defined by splitting the coefficient matrix.It is assumed that the splitting for original linear equation satisfies the regular splitting.When Gauss-Seidel iterative method for preconditioned linear system, its splitting is Gauss-Seidel method.However, for the original coefficient matrix A it means to de-

T. Kohno
fine a new splitting.The new splitting also fulfils the condition of the regular or weak regular splitting.We propose new preconditioners by combining preconditioners satisfying the regular splitting.
The outline of the paper is as follows: In Section 2, we review the preconditioned iterative method and some known results.And the iterative algorithm based on the splitting is shown.Section 3 consists of a comparison theorem and some numerical examples.Finally, in Section 4, we make some concluding remarks.

Preconditioned Iterative Method and Some Results
We review some known results [1] [2].We write and the vector n x R ∈ positive ( writing 0 x > ) if all its elements are positive.
Let n n Z × denote that set of all real n n × matrices which have non-positive off-diagonal elements.A nonsingular matrix In addition, the splitting is  , E and F are strictly lower and strictly upper triangular n n × matrices, respectively.For using Diagonal preconditioner { } In this article, suppose the diagonal part of a coefficient matrix is unit diagonal element.So, we consider the matrix sum of a coefficient matrix as follows, When setting M I = , we have the point Jacobiiterative method.And if M I L = − , then we have the Gauss-Seidel iterative method.
Definition 2. We define M I L = − the Gauss-Seidel regular splitting of For some preconditioner P , we call the following equation the preconditioned iterative system, Many researchers proposed some preconditioner P .The preconditioner using the first column has been proposed [3] as follows, c P works to eliminate the first column of A .Then ( ) where and c D , c E and c F are the diagonal, strictly lower and strictly upper trian- gular parts of CU , respectively.If c M is nonsingular, then the iterative matrix of the Gauss-Seidel method is defined by .
In 1991, Gunawardena et al. proposed the preconditioner S P I S = + [4] to eliminates the elements of the first upper co-diagonal of A , In 1997, Kohno et al. proposed the preconditioner ( ) ( ) with parameter α to accelerate its convergence for the preconditioned iterative me- thod [5].Moreover, Kotakemori et al. proposed the preconditioner by using the upper triangular matrix [6], Parameters of each preconditioner are changed for each row.
The preconditioner max max P I S = + using the maximum absolute value of the element of the upper diagonal part was proposed [7], where { } min , : is maximal for 1

Comparison Theorem
We now consider the comparison theorem for the two regular splitting of normal and preconditioned linear system in Equation ( 1) and (2).By using some preconditioner P , we have preconditioned splitting Conversely, if ( ) We solve the comparison theorem for any preconditioner P .Theorem 7. Let  and ( ) . Proof.Clearly, From the assump- tion ( ) and Theorem 4, we have the following relation It follows that 0.
Because the iterative matrix 1 M N − is nonnegative, there exists a positive vector x satisfied the following equation ( ) .
Example 1.We test the following matrix, This matrix was shown in [10] as a counterexample to the condition of the parameter of preconditioner .We check whether or not the condition of Theorem 7 is satisfied.This matrix has two regular splitting ( ) are Gauss-Seidel regular splitting, respectively.The assumption of Theorem 7 is satisfied as following inequality, ( ) 1 0 1 0 0 0 0 1 0 .0.5 0.5 0 1 0 0.5 0 0 Using the preconditioner S P I S = + is equivalent to using the following splitting, This splitting satisfies the regular splitting.And the following inequality is satisfied, ( ) ( ) Therefore, we have the spectral radius of each iterative matrix, 0.500 0.707 0.794 1.
For display, eigenvalues are given in approximate values.When using with parameter α , the regular splitting is not satisfied for However, it is well-known that the spectral radius may be smaller than the one of S P I S = + in the range of 1 α > .For example, by using We show spectral radii of some preconditioners in Table 1 for examples 1 and 2.
iterative matrix.If the spectral radius of the iterative matrix is less than one, the sequence the solution of the linear system(1).We can express the matrix A as the matrix sum are nonsingular.Rewrite two splitting like following relation, matrix of PA transformed as follows, and theorems [8] [9] [10] are shown below.Lemma 3. Let A M N = − be a regular splitting of A .If

1
Then each of the following conditions is equivalent to the statement: A is a nonsingular M-matrix.Let T be a nonnegative matrix.If Tx x α ≥ for some positive vector x , then ( ) not nonnegative.And more, compari- son condition between S P and C P is not indicated.Because elements used in each preconditioner are different, comparison of matrices is not satisfied.Therefore, we show following corollary., the condition of Theorem 7 satisfies, we have the spectral radius of preconditioned Gauss-Seidel iterative matrix is 0.156.And more, by setting the combination preconditioner U C P I U C + = + + , weak regular splitting is sa- tisfied, the spectral radius is 0.078.