-This paper illustrates comparison of two stationary Some convergence result for the block Gauss-Seidel method iteration methods for solving systems of linear equations. The for problems where the feasible set is the Cartesian product of aim of the research is to analyze, which method is faster solve m closed convex sets, under the assumption that the sequence tshoilsvienqgu.aTthioinsspaanpderhporwesmenatnsyo miteerwataioyns troeqdueiarledwietahcthhims eptrhoobdlefmor. generated by the method has limit points presented in.[5]. In [9] presented improving the modiefied Gauss-Seidel method Index Terms-comparison, analisys, stationary methods, for Z-matrices. In [11] presented the convergence Gauss-Seidel Jacobi method, Gauss-Seidel method iterative method, because matrix of linear system of equations is positive definite, but it does not afford a conclusion on the I. INTRODUCTION convergence of the Jacobi method. However, it also proved that Jacobi method also converges. Also these methods were present in [12], where described each method. This publication present comparison of Jacobi and GaussSeidel methods. These methods are used for solving systems of linear equations. We analyze, how many iterations required each method and which method is faster.
The term ”iterative method” refers to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system at each step. In this publication we will cover two types of iterative methods. Stationary methods are older, simpler to understand and implement, but usually not as effective. Nonstationary methods are a relatively recent development; their analysis is usually harder to understand, but they can be highly effective. The nonstationary methods we present are based on the idea of sequences of orthogonal vectors. (An exception is the Chebyshev iteration method, which is based on orthogonal polynomials.)
The rate at which an iterative method converges depends
greatly on the spectrum of the coeffcient matrix. Hence,
iterative methods usually involve a second matrix that
transforms the coeffcient matrix into one with a more favorable
spectrum. The transformation matrix is called a
preconditioner. A good preconditioner improves the convergence of
the iterative method, suffciently to overcome the extra cost of
constructing and applying the preconditioner. Indeed, without
a preconditioner the iterative method may even fail to converge
[
There are a lot of publications with stationary iterative
methods. As is well known, a real symmetric matrix can be
transformed iteratively into diagonal form through a sequence
of appropriately chosen elementary orthogonal transformations
(in the following called Jacobi rotations): Ak ! Ak+1 =
UkT AkUk (A0 = given matrix), this transformations
presented in [
Copyright held by the author(s). then
We consider the system of linear equations where A 2 Mn n i detA 6= 0.
We write the matrix in form of a sum of three matricies: where:
L - the strictly lower-triangular matrix D - the diagonal matrix
U - the strictly upper-triangular matrix If the matrix has next form:
Ax = B
A = L + D + U 2 a11 a21 . .
. an1 2 a12 a22 . . .
: : : a1n 3 : : : . . .
a2n 7 ... 775 an2 : : : ann 0 a21 . .
. an1 0 0 . .
. an2 : : : 0 : : : 0 3 : : : 0 7 . . . ... 775 L + U = 4
2 6 D = 6 6 4
If the matrix A is nonsingular (detA 6= 0), then we can rearrange its rows, so that on the diagonal there are no zeros, so aii 6= 0, 8i2(1;2;:::;n)
If the diagonal matrix D is nonsingular matrix (so aii 6= 0, i 2 1; : : : ; n), then inverse matrix has following form: 1 2 a11 4 1 0 = r 5 = 4
2 3 5 3 3 2 03 5 = 2 1 4 1 x = 0 3 third step (i=2) + 1 = +
T x3 =
1; 1; 1 Ax3 b
Here we will present differences between these tests. We carry out the series of research, where we analyze how much time require each test and also we compare number of iteration of each test.
First, we analize speed of each test. After research, we get data with time in ms and CPU ticks, which presented in Table.I. Also, we prepared graph of time comparison in ms. So, we can conclude that Gauss-Seidel method is faster than Jacobi method.
Next, we analize the efficiency of these tests. We do research for different number of n, where n is size of matrix A. When we compare the numbers of each tests, we can see that Gauss-Seidel method required less number of iterations than Jacobi method. So, we can conclude, that Gauss-Seidel method has better efficiency.
We have compared two method implemented to solve systems of linear equations. This paper can helps us to decide, which method is better. After all research, we can conclude that Gauss-Seidel method is better method than Jacobi method, because it faster and required less number of iterations. So, if we want to get better results and do it faster, we should use Gauss-Seidel method. Advantages of our proposition is ease and clarity of description of each method with shown algorithms. Also in our paper presented examples of each method, which show how these methods could be used.