Slightly modifying the Gauss Jacobi method, the Gauss Seidal method was introduced. If the linear equations are strictly diagonally dominant then this method is also convergent. At each step of the iteration, the improved value from the previous step is used.
DERIVATION
The iteration successively will be followed as:
Where,
Now, Gauss Seidal iteration can be written as,
And the stopping criteria of this method will be:
GRAPHICAL REPRESENTATION:
INPUT:
A matrix
OUTPUT:
The values after solving the linear equations.
PROCESS:
Step 1: [Taking the inputs from the user]
Read n [the order of matrix]
for i = 0 to n - 1 repeat
for j = 0 to n repeat
Read a[i][j]
[End of ‘for’ loop]
[End of ‘for’ loop]
Step 2: [Gauss Seidal Method]
Step 2.1: for i = 0 to n - 1 repeat
Set x[i] ← 0.0
Set y[i] ← 0.0
[End of ‘for’ loop]
Step 2.2: Set iteration ← iteration + 1
Print iteration
for i = 0 to n - 1 repeat
Set x[i] ← a[i][n]
for j = 0 to n - 1 repeat
If i = j then
continue
[End of ‘if’]
Set x[i] ← x[i] - a[i][j] * x[j]
[End of ‘for’ loop]
Set x[i]x[i]/a[i][i]
[End of ‘for’ loop]
for i = 0 to n - 1 repeat
Set flag ← 0
if absolute value of(x[i] - y[i]) > 0.0005 ) then
Set flag ←1
for j = 0 to n - 1 repeat
Set y[j] ← x[j]
[End of ‘for’ loop]
break
[End of ‘if’]
[End of ‘for’ loop]
for j = 0 to n - 1 repeat
Print x[j]
While flag = 1 repeat 2.2
[end of ‘gauss seidal’]
ADVANTAGES
1. The calculation in this method is simple.
2. Programming for this method is easy.
3. The memory spaces required are less.
4. This method is useful for the small system of linear equations.
DISADVANTAGES
1. It requires a large number of iterations to reach convergence.
2. It is not efficient for a large system
3. If the size of the system increases, the convergence time also increases.
APPLICATIONS
1. It is used to solve linear equations with unknown variables.
2. It is used in digital computers for computing.
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