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COMPUTATIONAL METHODS
&
STATISTICS
MTH 2212 -2011/2012 SEM 1
LECTURER
DR ZAHARAH WAHID
EXT 4514, ROOM E1-4-8-13
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Gauss-Siedel Method
Commonly used iterative method
Employs initial guesses & then iterates
to obtain estimate of the solutions
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Gauss-Seidel Method
Advantages of the Gauss-Seidel Method
Algorithm for the Gauss-Seidel Method
Pitfalls of the Gauss-Seidel Method
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Gauss-Seidel Method
An iterative method.
Basic Procedure:
Algebraically solve each linear equation for xi
Assume an initial guess solution array
Solve for each xi and repeat
Use absolute relative approximate error after each iteration to
check if error is within a pre-specified tolerance.
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Gauss-Seidel Method
Why?
The Gauss-Seidel Method allows the user to control round-off
error.
Elimination methods such as Gaussian Elimination and LU
Decomposition are prone to prone to round-off error.
Also: If the physics of the problem are understood, a close
initial guess can be made, decreasing the number of iterations
needed.
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Gauss-Seidel Method
Algorithm
A set of n equations and n unknowns:
. .
. .
. .
If: the diagonal elements are non-zero
Rewrite each equation solving for the corresponding unknown
ex:
First equation, solve for x1
Second equation, solve for x2
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Gauss-Seidel Method
Algorithm
Rewriting each equation
From Equation 1
From equation 2
From equation n-1
From equation n
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Gauss-Seidel Method
Algorithm
General Form of each equation
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If the diagonal elements are all non-zero then:
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With an initial guess of
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Gauss-Seidel Method
Algorithm
General Form for any row i
How or where can this equation be used?
Particularly well suited for large numbers of equations
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Gauss-Seidel Method
Solve for the unknowns
Assume an initial guess for [X]
Use rewritten equations to solve for each value of xi.
Important: Remember to use the most recent value of xi. Which
means to apply values calculated to the calculations remaining in
the current iteration.
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Gauss-Seidel Method
Calculate the Absolute Relative Approximate Error
So when has the answer been found?
The iterations are stopped when the absolute relative
approximate error is less than a prespecified tolerance for all
unknowns.
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Example: Surface Shape Detection
To infer the surface shape of an object from images taken of a
surface from three different directions, one needs to solve the
following set of equations
The right hand side values are the light intensities from the
middle of the images, while the coefficient matrix is dependent on
the light source directions with respect to the camera. The
unknowns are the incident intensities that will determine the shape
of the object.
Find the values of x1, x2, and x3 use the Gauss-Seidel
method.
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Example: Surface Shape Detection
The system of equations is:
Initial Guess:
Assume an initial guess of
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Example: Surface Shape Detection
Rewriting each equation
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Example: Surface Shape Detection
Iteration 1
Substituting initial guesses into the equations
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Example: Surface Shape Detection
At the end of the first iteration
The maximum absolute relative approximate error is 101.28%
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Example: Surface Shape Detection
Iteration 2
Using
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Example: Surface Shape Detection
Finding the absolute relative approximate error for the second
iteration
At the end of the first iteration
The maximum absolute relative approximate error is 197.49%
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ERROR
Perfect accuracy in most computational
processes is impossible
Error = true value- approximate value
Absolute error = |true value- approximate value|
The relative error is a measure of the error in relation to the
size of
true being sought
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ERROR
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Example: Surface Shape Detection
Repeating more iterations, the following values are obtained
Notice: The absolute relative approximate errors are not
decreasing.
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Iterationx1x2x3123456 1058.62117.0 4234.88470.1
169423388899.055150.00149.99150.00149.99150.00 1062.72112.9
4238.98466.0 169463388499.059150.30149.85150.07149.96150.01783.81
803.982371.9 3980.58725.7
16689101.28197.49133.90159.59145.62152.28
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Gauss-Seidel Method: Pitfall
What went wrong?
Even though done correctly, the answer is not converging to the
correct answer
This example illustrates a pitfall of the Gauss-Siedel method:
not all systems of equations will converge.
Is there a fix?
One class of system of equations always converges: One with a
diagonally dominant coefficient matrix.
Diagonally dominant: [A] in [A] [X] = [C] is diagonally dominant
if:
for all i and
for at least one i
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Gauss-Seidel Method: Pitfall
Diagonally Dominant: In other words.
For every row: the element on the diagonal needs to be equal
than or greater than the sum of the other elements of the
coefficient matrix
For at least one row: The element on the diagonal needs to be
greater than the sum of the elements.
What can be done? If the coefficient matrix is not originally
diagonally dominant, the rows can be rearranged to make it
diagonally dominant.
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Example: Surface Shape Detection
Examination of the coefficient matrix reveals that it is not
diagonally dominant and cannot be rearranged to become diagonally
dominant
This particular problem is an example of a system of linear
equations that cannot be solved using the Gauss-Seidel method.
Other methods that would work:
1. Gaussian elimination2. LU Decomposition
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Gauss-Seidel Method
The Gauss-Seidel Method can still be used
The coefficient matrix is not diagonally dominant
But this is the same set of equations used in example #2, which
did converge.
If a system of linear equations is not diagonally dominant,
check to see if rearranging the equations can form a diagonally
dominant matrix.
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Gauss-Seidel Method
Not every system of equations can be rearranged to have a
diagonally dominant coefficient matrix.
Observe the set of equations
Which equation(s) prevents this set of equation from having a
diagonally dominant coefficient matrix?
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Gauss-Seidel Method: Example 2
Given the system of equations
With an initial guess of
The coefficient matrix is:
Will the solution converge using the Gauss-Siedel method?
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Gauss-Seidel Method: Example 2
Checking if the coefficient matrix is diagonally dominant
The inequalities are all true and at least one row is strictly
greater than:
Therefore: The solution should converge using the Gauss-Siedel
Method
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Gauss-Seidel Method: Example 2
Rewriting each equation
With an initial guess of
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Gauss-Seidel Method: Example 2
The absolute relative approximate error
The maximum absolute relative error after the first iteration is
100%
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Gauss-Seidel Method: Example 2
After Iteration #1
Substituting the x values into the equations
After Iteration #2
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Gauss-Seidel Method: Example 2
Iteration #2 absolute relative approximate error
The maximum absolute relative error after the first iteration is
240.62%
This is much larger than the maximum absolute relative error
obtained in iteration #1. Is this a problem?
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Gauss-Seidel Method: Example 2
Repeating more iterations, the following values are obtained
The solution obtained
is close to the exact solution of
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Iterationa1a2a31234560.500000.146790.742750.946750.991770.9991967.662240.6280.2321.5474.53940.742604.9003.71533.16443.02813.00343.0001100.0031.88717.4094.50120.822400.110003.09233.81183.97083.99714.00014.000167.66218.8764.00420.657980.074990.00000
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Gauss-Seidel Method: Example 3
Given the system of equations
With an initial guess of
Rewriting the equations
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Gauss-Seidel Method: Example 3
Conducting six iterations, the following values are obtained
The values are not converging.
Does this mean that the Gauss-Seidel method cannot be used?
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Iterationa1A2a312345621.000196.151995.0201492.03641052.057910595.238110.71109.83109.90109.89109.890.8000014.421116.021204.6121401.2272105100.0094.453112.43109.63109.92109.8950.680462.304718.1476364.81441054.865310698.027110.96109.80109.90109.89109.89
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Gauss-Seidel Method: Example 3
Given the system of equations
With an initial guess of
Rearrange the equations
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Gauss-Seidel Method:example2
Iteration #1
With an initial guess of
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Gauss-Seidel Method: Example 4
Rewriting the equations
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Relaxation
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Gauss-Seidel Method: Example 4
Rewriting the equations
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Gauss-Seidel Method: cont..
Iteration #1
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Relaxation
Designed to Enhance converge nce.
After each new value of x is computed, that value is
modified using:
Where
20
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The choice of
is problem-specific and is often
determined empirically.
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