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Test your knowledge of Gauss-Seidal Method
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Why do we need another method to solve a set of
simultaneous linear equations?
In certain cases, such as when a
system of equations is large, iterative methods of solving equations
such as Gauss-Seidal method are more advantageous. Elimination methods
such as Gaussian elimination are prone to round off errors for a large
set of equations. Iterative methods, such as Gauss-Seidal method, allow
the user the control of the round-off error. Also if the physics of the
problem are well known for faster convergence, initial guesses needed in
iterative methods can be made more judiciously.
You convinced me, so what is the algorithm for Gauss-Seidel method?
Given a general set of n equations and n unknowns, we have
. .
. .
. .
If the diagonal elements are non-zero, each equation is rewritten for
the corresponding unknown, that is, the first equation is rewritten with
 on
the left hand side and the second equation is rewritten with
 on
the left hand side and so on as follows
These equations can be rewritten in the summation form as
.
.
.
Hence for any row i,
Now to find xi’s, one assumes an initial guess for the
xi’s and then use the rewritten equations to calculate
the new guesses. Remember, one always uses the most recent guesses to
calculate xi. At the end of each iteration, one
calculates the absolute relative approximate error for each xi
as
where  is
the recently obtained value of xi, and
 is
the previous value of xi.
When the absolute relative approximate error for each xi
is less than the pre-specified tolerance, the iterations are stopped.
Simulations on Gauss-Seidal Method
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Example 1
The upward velocity of a rocket is given at
three different times in the following table
|
Time, t |
Velocity, v |
|
s |
m/s |
|
5 |
106.8 |
|
8 |
177.2 |
|
12 |
279.2 |
The velocity data is approximated by a
polynomial as
The coefficients a1, a2,
a3 for the above expression were found in Chapter 5 to be
given by
Find the values of a1, a2,
a3 using Gauss-Seidal Method. Assume an initial guess of
the solution as  .
Rewriting the equations gives
Iteration #1:
Given the initial guess of the solution
vector as
we get
= 3.6720
=
=
The absolute relative approximate error for
each xi then is
= 72.76%
= 125.47%
= 103.22%
At the end of the first iteration, the guess
of the solution vector is
and the maximum absolute relative approximate error is 125.47%.
Iteration #2:
The estimate of the solution vector at the
end of iteration #1 is
Now we get
= 12.056
=
=
The absolute relative approximate error for
each  then
is
= 69.542%
= 85.695%
= 80.54%.
At the end of second iteration the estimate
of the solution is
and the maximum absolute relative approximate
error is 85.695%.
Conducting more iterations gives the
following values for the solution vector and the corresponding absolute
relative approximate errors.
|
Iteration |
a1 |
 |
a2 |
 |
a3 |
 |
|
1
2
3
4
5
6 |
3.672
12.056
47.182
193.33
800.53
3322.6 |
72.767
67.542
74.448
75.595
75.850
75.907 |
-7.8510
-54.882
-255.51
-1093.4
-4577.2
-19049 |
125.47
85.695
78.521
76.632
76.112
75.971 |
-155.36
-798.34
-3448.9
-14440
-60072
-249580 |
103.22
80.540
76.852
76.116
75.962
75.931 |
As seen in the above table, the solution is
not converging to the true solution of
a1
= 0.29048
a2
= 19.690
a3
= 1.0858
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The above system of equations does not seem to
converge? Why?
Well, a pitfall of most iterative methods is that they may or may not
converge. However, certain classes of systems of simultaneous equations
do always converge to a solution using Gauss-Seidel method. This class
of system of equations is where the coefficient matrix [A] in [A][X]
= [C] is diagonally dominant, that is
for all i
and  for
at least one i.
If a system of equations has a coefficient matrix that is not diagonally
dominant, it may or may not converge. Fortunately, many physical
systems that result in simultaneous linear equations have diagonally
dominant coefficient matrix, which then assures convergence for
iterative methods such as Gauss-Seidel method of solving simultaneous
linear equations.
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Example 2
Given the system of equations.
find the solution using Gauss-Seidal method.
Given
as the initial guess.
The coefficient matrix
is diagonally dominant as
and the inequality is strictly greater than
for at least one row. Hence, the solution should converge using
Gauss-Seidel method.
Rewriting the equations, we get
Assuming an initial guess of
Iteration #1:
= 0.50000
= 4.9000
= 3.0923
The absolute relative approximate error at
the end of first iteration is
= 67.662%
= 100.000%
= 67.662%
The maximum absolute relative approximate
error is 100.000%
Iteration #2:
= 0.14679
= 3.7153
= 3.8118
At the end of second iteration, the absolute
relative approximate error is
= 240.62%
= 31.887%
= 18.876%.
The maximum absolute relative approximate
error is 240.62%. This is greater than the value of 67.612% we obtained
in the first iteration. Is the solution diverging? No, as you conduct
more iterations, the solution converges as follows.
|
Iteration |
a1 |
 |
a2 |
 |
a3 |
 |
|
1
2
3
4
5
6 |
0.50000
0.14679
0.74275
0.94675
0.99177
0.99919 |
67.662
240.62
80.23
21.547
4.5394
0.74260 |
4.900
3.7153
3.1644
3.0281
3.0034
3.0001 |
100.00
31.887
17.409
4.5012
0.82240
0.11000 |
3.0923
3.8118
3.9708
3.9971
4.0001
4.0001 |
67.662
18.876
4.0042
0.65798
0.07499
0.00000 |
This is close to the exact solution vector of
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Example 3
Given the system of equations
Find the solution using Gauss-Seidel method.
Use  as
the initial guess.
Rewriting the equations, we get
Assuming an initial guess of
the next six iterative values are given in
the table below
|
Iteration |
a1 |
 |
a2 |
 |
| | 