Second Degree Refinement Jacobi Iteration Method for Solving System of Linear Equation

Several iterative techniques for the solution of linear system of equations have been proposed in different literature in the past.In this paper, we present a Second degree of refinement Jacobi Iteration method for solving system of linear equation, Ax = b and we consider few numerical examples and spectral radius to show that the effective of the Second degree of refinement Jacobi Iteration Method (SDRJ) in comparison with other methods of First degree Jacobi (FDJ), First degree Refinement Jacobi (FDRJ) and Second degree Jacobi (SDJ) method.


Second Degree Refinement Jacobi Iteration Method for Solving System of Linear Equation Tesfaye Kebede
Abstract-Several iterative techniques for the solution of linear system of equations have been proposed in different literature in the past.In this paper, we present a Second degree of refinement Jacobi Iteration method for solving system of linear equation, Ax = b and we consider few numerical examples and spectral radius to show that the effective of the Second degree of refinement Jacobi Iteration Method (SDRJ) in comparison with other methods of First degree Jacobi (FDJ), First degree Refinement Jacobi (FDRJ) and Second degree Jacobi (SDJ) method.
Index Terms-Jacobi iteration, second degree refinement, system of linear equations.

I. INTRODUCTION
I N many scientific and engineering applications, one often comes across with a problem of finding the solution of a system of linear equations written as the following equation in matrix form: where A is a nonsingular matrix of size n × n, x and b are n-dimensional vectors.Splitting the matrix A [1] as : where D is a diagonal matrix and −L and −U are strictly lower and upper triangular part of A respectively.A general first degree linear stationary iterative method for the solution of the system of equation ( 1) may be defined in the form: where x (n+1) and x (n) are the approximation for x at the (n + 1) th and n th iterations respectively, H is called the iterative matrix depending on matrix A and C is a column vector.The iteration system x (n+1) = Hx (n) +C is converge if and only if the spectral radius of H are less than unity, i.e. σ (H) < 1.
The first degree iterative method of Jacobi (FDJ) method for the solution of ( 1) is defined as: and the first degree refinement Jacobi (FDRJ) method can be obtained in the form of : where The linear stationary any second degree method is given by [2] x Here x (n+1) appearing in the right hand side as given in equation ( 3) is completely consistent for any constant a and b 1 such that b 1 = 0. where The linear stationary any second degree method is given by [2] can be written in number (6) or number (7) with (8), ( 9) and (10) conditions.On the other way equation ( 1) can be solved using the second degree Jacobi(SDJ) stationary iterative method using For optimal values of a and b 1 Where If A is a row strictly diagonal dominant (SDD) matrix, then the Jacobi method converges for any arbitrary choice of the initial approximation [3].
In this paper, we construct a new method of solving a linear system of the form Ax = b that arise in any engineering and applied science.
The outline of this paper is as follows: we introduce second degree refinement Jacobi (SDRJ) iterative method in accordance this we will see the relationship between spectral radius of first degree Jacobi (FDJ), first degree refinement Jacobi (FDRJ), Second degree Jacobi (SDJ) methods and Second degree refinement Jacobi iteration (SDRJ) methods are given.Based on the methods and results, few numerical examples are considered to show that the efficiency of the new method in comparison with the existing FDJ, FDRJ and SDJ methods.Finally discussion and conclusion made at Section V.

II. SECOND DEGREE REFINEMENT JACOBI (SDRJ) ITERATIVE METHOD
Theorem 1: If matrix A is non singular PD and SDD matrix with A = D − L − U, then the Second degree of Refinement Jacobi iterative method is: for any initial guess and the optimal values for a and b 1 .
Given: A is non singular PD and SDD matrix and A = D-L-U.Required: the second degree of refinement Jacobi iterative method is: 1) .Proof: now consider equation ( 5)and (6), so one can get : This also can be written as follows after some computation: where where The above equation converges to the exact solution if σ ( G) < 1, i.e. the spectra radius of G is less than one.In order to solve the spectra radius of G, first we have to solve the eigenvalues λ of G.
Thus, the eigenvalues λ RJ of G are related to the eigenvalues µ RJ of H RJ with H J is As the image of the circle, Let the eigenvalue λ RJ = ve iθ = v(cos θ +i sin θ ) is the ellipse, then substituting this in equation ( 18), we obtain We know that cos 2 θ + sin 2 θ = 1 Length of semi-minor axis Before we prove the theorem let us prove the following Lemmas.
Proof: We know that µ RJ is a real number.We require that α ≤ µ RJ ≤ β < 1, for any foci α and β which are real.The proof is as follows: since µ RJ is a real number.In this equation θ varies.
Because α and β from equation ( 26) and (27) and to be convergent ρ(H RJ ) < 1 so all the eigenvalues must be less than 1.Lemma 3: If the eigenvalues µ RJ of H RJ < 1 are real and lie in the interval.
Proof: α ≤ µ RJ ≤ µ J ≤ β < 1, then the optimal choices of a and b 1 must satisfy the following conditions: Given: the eigenvalues µ RJ of H RJ < 1 are real and lie in the interval α ≤ µ RJ ≤ µ J ≤ β < 1. Required: proof of a) until d).
b) From the (26) and ( 27) and from Lemma 2(a) ,we get: , then one can get Divide equation ( 31) by ( 33),we get Lemma 4: If µ RJ is the spectral radius of H RJ , then Proof: Given: µ RJ is the spectral radius of H RJ .Required: µ RJ = β −α 2−(α+β ) .Proof: we know that . By definition of derivative of functions in calculus To calculate the maximum and minimum value, the above equation equates to zero.
From the above the maximum value occurs at θ = 0 and 2π The minimum value occurs at by equation ( 31) and (33).Therefore µ RJ = β .
From the previous two results, we obtain 2µ RJ = β − α and 2 = 2 − (α + β ).Then divide the previous two equations, we get µ RJ = β −α 2−(α+β ) .Now let us determine the values of a and b 1 .First, let us find a from Lemma 3d equation ( 32) by lemma 3d, we have This is the equation of quadratic whose graph is a parabola and the minimum value occurs at p=( 1 ) since µ RJ > 0. One can solve by quadratic formula of the above equation: Secondly, let us find b 1 ⇒ b 1 = 4v β −α by using equation ( 27) .
Lemma 5: If matrix A is positive definite matrix and if H RJ is Jacobi iterative matrix, then β = −α = µ RJ = σ (H RJ ).
Proof: Given: matrix A is positive definite matrix and if H RJ is Jacobi iterative matrix.Required: β = −α = µ RJ = σ H RJ .Proof: In order to prove this Lemma, we have to use Lemma 3 Now we can find the optimal value of a and b 1 i.e. a =

III. RELATIONSHIP BETWEEN SPECTRAL RADIUS
As we have seen above the spectral radius of • First degree Jacobi method(FDJ) is µ.
That is one can see IV. NUMERICAL EXAMPLES a. Solve the following SDD matrix using FDJ ,FDGJ ,SDJ and SDRJ iterative methods.
Solution: all results are based on the given data and we get the spectral radius as follows Method FDJ FDRJ SDJ SDRJ Spectral radius 0.4295 0.1845 0.2257 0.0931 b.Solve the following PD matrix using FDJ, FDGJ, SDJ and SDRJ iterative methods.
Solution: all results are based on the given data we get the spectral radius as follows Method FDJ FDRJ SDJ SDRJ Spectral radius 0.3333 0.1111 0.1716 0.0557 The detailed experimental results are written in the appendix.Table I shows that FDRJ method converges faster than FDJ method for SDD matrix.Table II shows that SDRJ method converges faster than the SDJ method for SDD matrix.Table III shows that FDRJ method converges faster than FDJ method for PD matrix.Table IV shows that SDRJ method converges faster than the SDJ method for PD matrix.

V. CONCLUSIONS
As we have seen in this report for SDD and PD matrix, we can notice that FDJ, FDRJ and SDJ are reasonable to approximate the exact solution of system of linear equations at a certain given condition.But they are relatively slow to converge to the exact solution.However, the Second degree of refinement Jacobi iterative method for solving system of linear equations are reasonable and efficient way of approximating the exact solution of system of linear equations.Numerical results of spectral radius show that, SDRJ methods converge with a small number of iteration steps for solving systems of linear equations.
In general, the results of numerical examples considered clearly demonstrate the accuracy of the methods developed in this paper.It is conjectured that the rate of convergence of the method that developed in this paper can be further enhanced by using extrapolating techniques.

TABLE IV ALL
NUMERICAL RESULT OF POSITIVE DEFINITE (PD) MATRIX FOR SDJ AND SDRJ OF EXAMPLE 2