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Use simple fixed-point iteration to locate the root of
Use an initial guess of
![Check Mark](/static/check-mark.png)
To calculate: The root of the function
Answer to Problem 1P
Solution:
The root of the function
Explanation of Solution
Given:
The function,
The initial condition,
Formula used:
The simple fixed-point iteration formula for the function
And, formula for approximate error is,
Calculation:
Consider the function,
The function can be formulated as fixed-point iteration as,
Use initial guess of
Therefore, the approximate error is,
Use
Therefore, the approximate error is,
Use
Therefore, the approximate error is,
Similarly, all the iteration can be summarized as below,
0 | 0.5 | |
1 | 0.6496 | 23.03% |
2 | 0.7215 | 9.965% |
3 | 0.7509 | 3.915% |
4 | 0.7621 | 1.47% |
5 | 0.7662 | 0.535% |
6 | 0.7678 | 0.208% |
7 | 0.7683 | 0.0651% |
8 | 0.76852 | 0.029% |
9 | 0.7686 | 0.01% |
Since, the approximate error in the ninth iteration is 0.01%. So, stop the iteration.
Hence, the root of the function is 0.7686.
Now, to verify that the process is linearly convergent, the condition to be satisfied is
The fixed-point iteration is,
Therefore,
Differentiate the above function with respect to x,
Therefore,
Since,
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Chapter 6 Solutions
Numerical Methods for Engineers
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