Solve by Gauss Seidel Method 2x+y+6z=9; 8x+3y+2z=13; x+5y+z=7
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Answer:
Convergence of Jacobi and Gauss-Seidel Method and Error
Reduction Factor
1HarpinderKaur,
2KhushpreetKaur
1,2Department of Mathematics Baba FaridCollege,Bathinda
Abstract: In this paper, it is shown that neither of the iterative methods always converges. That is, it is possible
to apply the Jacobi method or the Gauss-Seidel method to a system of linear equations and obtain a divergent
sequence of approximations. In such cases, it is said that the method diverges.So for convergence, the Diagonal
Dominance of the matrix is necessary condition before applying any iterative methods. Moreover, also
discussed about the error reduction factor in each iteration in Jacobi and Gauss-Seidel method.
Keywords: Jacobi Method, Gauss-Seidel Method, Convergence and Divergence, Diagonal Dominance,
Reduction of Error.
I. Introduction:
An iterative technique to solve an n×n linear system A =b starts with an initial approximation0
to the
solution ∗, and generates a sequence of vectors{x
(k)
}=0
∞
that converges to ∗. Most of these iterative
techniques involve a process that converts the system A =b into an equivalent system of the form = S+
for some n×n matrix S and vector . After the initial vector 0
is selected, the sequence of approximate solution
vectors is generated by computing
(+1) =
() + , k=1,2,3,……The Jacobi method is based on solving for
every variable locally with respect to the other variables; one iteration corresponds to solving for every variable
once. It is easy to understand and implement, but convergence is slow.The Gauss-Seidel method is similar to the
Jacobimethodexceptthatit uses updated valuesassoonastheyareavailable. ItgenerallyconvergesfasterthantheJacobi
method, although still relatively slowly.
1.1 Jacobi Method:
The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its
main diagonal. Each diagonal element is solved for, and an approximate value put in. The process is then
iterated until it converges.The Jacobi method is easily derived by examining each of the n equations in the linear
system of equations A =b . If in the i−th equation
=1 =
solve for the value of while assuming the
other entries of remain fixed. This gives
+1 =
1
− ≠
()
, i = 1,2, ... ,n. (i)
which is the Jacobi method. Note that thecomputation of
(+1)
requires each component in
()
except
()
.In this method, the order in which the equations are examined is irrelevant, since the Jacobi
method treats them independently.
The equation A =b, which is (L+D+U) =b, can be written asD = −(L+U) +b.This reduces to+1=-
−1
( + )+
−1
Fig. 1 : Splitting of matrix for Jacobi Method
Where = -
−1
( + ) and =
−1
Note that if A is a strictly diagonally dominance matrix, then
∞
= −
−1
( + ) ∞< −
−1 ∞
( + ) ∞ = ∞
−1 ( + ) ∞=
(+) ∞
< 1
Hence for any 0∈IRn, the sequence {x
(k)
}=0
∞
defined by= −1 +
; (k = 1,2,···) , ≠
0converges to the unique solution of = +
. Therefore Jacobi’s iterative techn
Answer:
Step-by-step explanation: