Direct and indirect methods

1
 Gauss – Jacobi Iteration Method
 Gauss - Seidal Iteration Method

Iterative Method
 Simultaneous linear algebraic equation occur in various fields of Science
and Engineering.
 We know that a given system of linear equation can be solved by
applying Gauss Elimination Method and Gauss – Jordon Method.
 But these method is sensitive to round off error.
 In certain cases iterative method is used.
 Iterative methods are those in which the solution is got by successive
approximation.
 Thus in an indirect method or iterative method, the amount of
computation depends on the degree of accuracy required.
2
Introduction:

Iterative Method
 Iterative methods such as the Gauss – Seidal method give the user
control of the round off.
 But this method of iteration is not applicable to all systems of equation.
 In order that the iteration may succeed, each equation of the system
must contain one large co-efficient.
 The large co-efficient must be attached to a different unknown in that
equation.
 This requirement will be got when the large coefficients are along the
leading diagonal of the coefficient matrix.
 When the equation are in this form, they are solvable by the method of
successive approximation.
 Two iterative method - i) Gauss - Jacobi iteration method
ii) Gauss - Seidal iteration method
3
Introduction (continued..)

Gauss – Jacobi Iteration Method:
The first iterative technique is called the Jacobi method named after
Carl Gustav Jacob Jacobi(1804- 1851).
Two assumption made on Jacobi method:
1)The system given by
4
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa







2211
22222121
11212111 - - - - - - - - (1)
- - - - - - - - (2)
- - - - - - - - (3)
has a unique solution.

Gauss – Jacobi Iteration Method
5
Second assumption:
761373 321  xxx
2835 321  xxx
10312 321  xxx
10312 321  xxx
2835 321  xxx
761373 321  xxx

Gauss– Jacobi Iteration Method
6




n
j
1j
ijaa
i
ii

To begin the Jacobi method ,solve

7
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa







2211
22222121
11212111


8
(7)


9
8


10
9

11

12
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa







2211
22222121
11212111 - - - -(1)
- - - -(2)
- - - -(3)

13

14

15

Solution:
In the given equation , the largest co-efficient is attached to a
different unknown.
Checking the system is diagonally dominant .
Here
Then system of equation is diagonally dominant .so iteration method
can be applied.
16
7162727 131211  aaa

From the given equation we have
17
27
685 32
1
xx
x


15
2672 31
2
xx
x


54
110 21
3
xx
x


(1)

18
27
)0()0(685)1(
1

x
15
)0(2)0(672
2
)1( 
x
54
)0()0(110
3
)1( 
x
=3.14815
=4.8
=2.03704
First approximation

19

The results are tabulated
20
S.No Approximation
(or) iteration
1 0 0 0 0
2 1 3.14815 4.8 2.03704
3 2 2.15693 3.26913 1.88985
4 3 2.49167 3.68525 1.93655
5 4 2.40093 3.54513 1.92265
6 5 2.43155 3.58327 1.92692
7 6 2.42323 3.57046 1.92565
8 7 2.42603 3.57395 1.92604
9 8 2.42527 3.57278 1.92593
10 9 2.42552 3.57310 1.92596
11 10 2.42546 3.57300 1.92595

21

Gauss –Seidal Iteration Method
 Modification of Gauss- Jacobi method,
named after Carl Friedrich Gauss and Philipp Ludwig Von Seidal.
 This method requires fewer iteration to produce the same degree
of accuracy.
 This method is almost identical with Gauss –Jacobi method except
in considering the iteration equations.
 The sufficient condition for convergence in the Gauss –Seidal
method is that the system of equation must be strictly diagonally
dominant
22

Consider a system of strictly diagonally dominant equation as
23
nnnnnn
nn
nn
bxaxaxa
bxaxaxa
bxaxaxa







2211
22222121
11212111
- - - - -(1)
- - - - - (2)
- - - - - (3)

24


25

26
The successive iteration are generated by the scheme called
iteration formulae of Gauss –Seidal method are as
The number of iterations k required depends upon the desired
degree of accuracy

Soln: From the given equation ,we have
- - - - - - - (1)
- - - - - - -(2)
- - - - - - - (3)
27
27
685 32
1
xx
x


15
2672 31
2
xx
x


54
110 21
3
xx
x



28
91317.1
54
)54074.5()14815.3(110
3
)1(


x
14815.3
27
)0()0(685
1
)2(


x
54074.3
15
)0(2)14815.3(672
2
)1(


x

1st Iteration:
29
91317.1
3
)1(
54074.3
2
)1(
14815.3
1
)1(



x
x
x

30
For the second iteration,
91317.1
3
)1(
54074.3
2
)1(
14815.3
1
)1(



x
x
x
43218.2
27
685 3
)1(
2
)1(
1
)2(



xx
x
57204.3
15
2672 3
)1(
1
)2(
2
)2(



xx
x
92585.1
54
110 2
)2(
1
)2(
3
)2(



xx
x

Thus the iteration is continued .The results are tabulated.
S.No Iteration or
approximation
1 0 0 0 0
2 1 3.14815 3.54074 1.91317
3 2 2.43218 3.57204 1.92585
4 3 2.42569 3.57294 1.92595
5 4 2.42549 3.57301 1.92595
6 5 2.42548 3.57301 1.92595
31
..)3,2,,( ii
1,
i
x  ..)3,2,,(
2
ii
i
x  ..)3,2,,(
3
ii
i
x 
4th and 5th iteration are practically the same to four places.
So we stop iteration process.
Ans: 9260.1
3
;57301.3
2
;4255.2
1
 xxx

Comparison of Gauss elimination and Gauss- Seidal Iteration methods:
 Gauss- Seidal iteration method converges only for special systems of
equations. For some systems, elimination is the only course
available.
 The round off error is smaller in iteration methods.
 Iteration is a self correcting method
32

Direct and indirect methods

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