Concept explainers
Perform the same calculations as in (a) Example 11.1, and (b) Example 11.3, but for the tridiagonal system,
(a)
To calculate: The solution of following tridiagonal system with Thomas Algorithm:
Answer to Problem 1P
Solution:
The solution is
Explanation of Solution
Given:
A tridiagonal system:
Formula used:
(a) The decomposition is implemented as:
(2) Forward substitution is implemented as:
(3) Back substitution is implemented as:
For
Calculation:
Consider the system of equation:
Where,
First the decomposition is implemented as:
Now,
And,
And,
Thus, the matrix [A] is transformed to,
Now, the LU decomposition of the above matrix is,
The forward substitution is implemented as:
And,
Thus, the right-hand-side vector is modified to,
The right-hand-side vector can be used in conjunction with the [U] matrix to perform back substitution and obtain the solution as:
Also,
And,
Thus, the solution of the given system of equation is
(b)
To calculate: The solution of following tridiagonal system of equation with Gauss-Seidel method:
Answer to Problem 1P
Solution:
Using Gauss-Seidel method four iteration are performed to get
Explanation of Solution
Given:
A system of equation:
Formula used:
(1) The values of
(2) True percent relative error is given by,
(3) Convergence can be checked using the criterion
For all i, where j and j- 1 are the present and previous iterations.
Calculation:
The true solution is
Consider the system of equation:
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 true percent relative error is
Thus,
The value of
Thus,
Now, substitute the calculated values of
Thus,
For the third iteration, the same process is repeated with
The true percent relative error is
Thus,
The value of
Thus,
Now, substitute the calculated values of
Thus,
For the fourth iteration, the same process is repeated with
The true percent relative error is
Thus,
The value of
Thus,
Now, substitute the calculated values of
Thus,
The method is thus converging on the true solution.
Now, estimate the error:
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Chapter 11 Solutions
Numerical Methods for Engineers
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