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Gauss-Seidel
Iterative or approximate methods provide analternative to the elimination methods. The Gauss-Seidel method is the most commonly used iterative
method.
The system [A]{X}={B} is reshaped by solving thefirst equation for x1, the second equation for x2, andthe third for x3, and n
th equation for xn. For
conciseness, we will limit ourselves to a 3x3 set ofequations.
PPS-UB-PAT-TM-2010 Chapter 11 1
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33
2321313
1
22
32312122
11
31321211
a
xaxab
x
a
xaxabx
a
xaxabx
PPS-UB-PAT-TM-2010 Chapter 11 2
Now we can start the solution process by choosing
guesses for the xs. A simple way to obtain initialguesses is to assume that they are zero. These zeros
can be substituted into x1equation to calculate a new
x1=b1/a11.
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New x1 is substituted to calculatex2 and x3. The
procedure is repeated until the convergence criterion
is satisfied:
sj
i
j
i
j
iia
x
xx %100
1
,
PPS-UB-PAT-TM-2010 Chapter 11 3
For all i, where j and j-1 are the present and previous
iterations.
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Fig. 11.4
PPS-UB-PAT-TM-2010 Chapter 11 4
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Convergence Criterion for Gauss-Seidel
Method
The Gauss-Seidel method has two fundamental
problems as any iterative method:
It is sometimes nonconvergent, and
If it converges, converges very slowly. Recalling that sufficient conditions for convergence
of two linear equations, u(x,y) and v(x,y) are
1
1
y
v
x
v
y
u
x
u
PPS-UB-PAT-TM-2010 Chapter 11 5
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Similarly, in case of two simultaneous equations,the Gauss-Seidel algorithm can be expressed as
0
0
),(
),(
222
21
1
11
12
21
1
22
21
22
221
2
11
12
11
121
x
v
a
a
x
v
a
a
x
u
x
u
xaa
abxxv
xa
a
a
bxxu
PPS-UB-PAT-TM-2010 Chapter 11 6
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Substitution into convergence criterion of two linear
equations yield:
1122
21
11
12
a
a
a
a
n
ij
j
jiaaii
aa
aa
1
,
2122
1211
:equationsnFor
PPS-UB-PAT-TM-2010 Chapter 11 7
In other words, the absolute values of the slopes
must be less than unity for convergence:
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Figure 11.5
PPS-UB-PAT-TM-2010 Chapter 11 8