Matakuliah
Tahun
: S0262-Analisis Numerik
: 2010
Linear Algebraic Equation System
Pertemuan 6
Material Outline
• Linear Algebraic Equations System
– Gauss Seidel method
– Jacobi method
•
Sys. of Linear Algebraic equation
a11x1  a12 x2    a1n xn  b1 
a21x1  a22 x2    a2 n x n  b2 

  Ax  b




an1 x1  an 2 x2    ann xn  bn 

In which:
 a11
a
A   21
 

an1
a12
a 22
 a1n
 a2 n

an 2

ann


 b1 
 x1 

b 
x 
 ; B   2 ; x   2 




 
 
b

 n
 xn 
4
• Gauss-Seidel Method
Gauss Seidel Meth Is an iterative or approximate method.
Ax  b
1. Step-1: write the system of linear equations to the following form
(you may need to interchange the rows if necessary)
x1 
b1  a12 x2  a13 x3    a1n xn
a11
x2 
b2  a21 x1  a 23 x3    a2 n xn
a22


xn 
bn  an1 x1  a n 2 x2    ann1 xn 1
ann



5
• Gauss-Seidel Method
2. Step-2: Choose initial for all x’s
3. Step-3: Use the previous set of equations to refine guess of the
values of x’s start with x1. Then the refine value of x1 together
with all other x’s will be used to refine the value of x2 etc. This
process will be done iteratively until the predetermined
convergence criteria is achieved.
 Convergence Criteria:
 a ,i
xij  xij 1

100%   s
j
xi
Note: xj = x at jth iteration
s= tolerance
6
• Gauss-Seidel Method
Then the iteration can be written as:
b1  a12 x2j 1  a13 x3j 1    a1n xnj 1
x 
a11
j
1
j 1
23 3
j 1
2n n
b2  a x  a x    a x
x 
a22
j
21 1
j
2
Initial values :
x10 ; x20 ; x30  xn0 are given
j
j
j 1
b

a
x

a
x



a
x
32 2
3n n
x3j  3 31 1
a33





bn  an1 x1j  an 2 x2j    ann1 xnj1
x 
ann
j
n
Note: xij = the values of xi at iteration j
7
• Gauss Seidel Method
Example: Solve the following linear system
using Gauss Seidel Method. Use your first
guess for x’s equal to zero.
 3  0.1  0.2  x1   7.85 
  

0.1

7

0
.
3
x


19
.
3




2


0.3  0.2 10   x3   71.4 
28-Jun-16
Paston Sidauruk
8
• Jacobi method
Al most the same procedure as Gauss-Siedel
method. In Jacobi method, however, the
new values of x is not immediately updated
until the current iteration completed.
9
• Jacobi Method
Then the iteration for Jacobi method can be written as:
j 1
j 1
j 1
b

a
x

a
x



a
x
13 3
1n n
x1j  1 12 2
a11
x2j 
j 1
21 1
b2  a x
j 1
23 3
j 1
2n n
 a x  a x
a22
Initial values :
x10 ; x20 ; x30  xn0 are given
j 1
j 1
j 1
b

a
x

a
x



a
x
32 2
3n n
x3j  3 31 1
a33





bn  an1 x1j 1  an 2 x2j 1    ann1 xnj1
x 
ann
j
n
Note: xij = the values of xi at iteration j
10
• Jacobi Method
Jacobi stopping criteria will be the same as GaussSeidel method

Convergence Criteria:
 a ,i
xij  xij 1

100%   s
j
xi
Note: xj = x at jth iteration
s= tolerance
11
• Jacobi Method
Example: Solve the following linear system
using Jacobi Method. Use your first guess
for x’s equal to zero.
 3  0.1  0.2  x1   7.85 
  

0.1

7

0
.
3
x


19
.
3




2


0.3  0.2 10   x3   71.4 
28-Jun-16
Paston Sidauruk
12
• Comment on Convergence
To guarantee the convergence of Gauss-Seidel
and jacobi methods, the diagonal element
of the matrix at each row must be greater
than the sum off-diagonal elements. It is
recommended than to re arrange the rows
as much as possible to fulfill the following
criteria
aii 
28-Jun-16
n
a
j , j i
Paston Sidauruk
i, j
13