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Determine the roots of the simultaneous nonlinear equations
Use a graphical approach to obtain your initial guesses. Determine refined estimates with the two-equation Newton-Raphson method described in Sec. 6.6.2.
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To calculate: The root of the non-linear simultaneous equations,
By the two-equation Newton-Raphson method and find the initial guess by the graphical method.
Answer to Problem 23P
Solution:
The root of the simultaneous non-linear equations
For the initial root
For the initial root
Explanation of Solution
Given information:
The non-linear simultaneous equation,
Formula used:
The Newton-Raphson formula for two non-linear equation is,
Calculation:
Use MATLAB to draw the graph of the equations,
Code:
%x-coordinates spacing is defined.
%first function is defined.
%second function is defined.
%Plot command is used to draw the two function.
Output:
From the above graph, it is observed that there are two roots at
Consider theequations,
Rewrite the equation as below,
Partial differentiate the above functions with respect to x,
And,
Now, partial differentiate the above functions with respect to y,
And,
Use initial guesses
And,
Now, use
And,
Now, use
And,
Thus, all the iteration can be summarized as below,
0 | 1.8 | 3.6 |
1 | 1.8056 | 3.5694 |
2 | 1.80583 | 3.56917 |
3 | 1.80583 | 3.56917 |
Hence, the root of the simultaneous non-linear equations
Use initial guesses
And,
Now, use
And,
Now, use
And,
Thus, all the iteration can be summarized as below,
0 | 3.6 | 1.8 |
1 | 3.56944 | 1.80556 |
2 | 3.56917 | 1.80583 |
3 | 3.56917 | 1.80583 |
Hence, the root of the simultaneous non-linear equations
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