Concept explainers
Use the Gauss-Seidel method(a) without relaxation and (b) with relaxation
(a)
To calculate: The solution of following system of equation with Gauss-Seidel method without relaxation to a tolerance of
Answer to Problem 13P
Solution:
Using Gauss-Seidel method three iterations are performed to get the values
Explanation of Solution
Given:
A system of equation:
With
Formula used:
(1) The values of
(2) Convergence can be checked using the criterion
For all i, where j and j- 1 are the present and previous iterations.
Calculation:
Consider the system of equation:
The sufficient condition for convergence is:
The diagonal coefficient in each of the equations must be larger than the sum of the absolute values of the other coefficients in the equation. The systems where this condition holds are called diagonally dominant.
Thus, the equations should first be arranged so that they are diagonally dominant.
It can be written in the form:
Where,
First, solve each of the equations for its unknown on the diagonal
For initial guess, assume
Thus, equation (1) becomes,
This value, along with the assumed value of
Now, substitute the calculated values of
For the second iteration, the same process is repeated with
The value of
Now, substitute the calculated values of
Now, the error can be computed as:
For the third iteration, the same process is repeated with
The value of
Now, substitute the calculated values of
Now, the error can be computed as:
Thus, after three iterations the maximum error is 2.92% which is less than
(b)
To calculate: The solution of following system of equation with Gauss-Seidel method with relaxation
Answer to Problem 13P
Solution:
Using Gauss-Seidel method six iterations are performed with relaxation
Explanation of Solution
Given:
A system of equation:
With relaxation
Formula used:
(1) The values of
(2) Relaxation
Where
(3) Convergence can be checked using the criterion
For all i, where j and j- 1 are the present and previous iterations.
Calculation:
Consider the system of equation:
The sufficient condition for convergence is:
The diagonal coefficient in each of the equations must be larger than the sum of the absolute values of the other coefficients in the equation. The systems where this condition holds are called diagonally dominant.
Thus, the equations should first be arranged so that they are diagonally dominant.
It can be written in the form:
Where,
First, solve each of the equations for its unknown on the diagonal
For initial guess, assume
Thus, equation (1) becomes,
Relaxation yields:
This value, along with the assumed value of
Relaxation yields:
Now, substitute the calculated values of
Relaxation yields:
For the second iteration, the same process is repeated with
Relaxation yields:
The value of
Relaxation yields:
Now, substitute the calculated values of
Relaxation yields:
Now, the error can be computed as:
For the third iteration, the same process is repeated with
Relaxation yields:
The value of
Relaxation yields:
Now, substitute the calculated values of
Relaxation yields:
Now, the error can be computed as:
The table showing values calculated for further iterations is given below
Therefore, after six iterations, the maximum error is 3.60% which is less than 5%. Thus, the results are
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