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Concept explainers
(a)
To calculate: The positive real root of
(a)
![Check Mark](/static/check-mark.png)
Answer to Problem 3P
Solution:
The positive real root of the given equation
Explanation of Solution
Given Information:
The given equation is,
Use Müller’s method.
Formula used:
The expression for the new roots is,
The expression for the error is,
Calculation:
Recall the equation mentioned in the problem,
Draw the plot of the equation.
From the above plot it is clear that one root is at about
Consider the initial guess be
, and
Thus, the values of
Now calculate the
Substituting the value of
, and
Now calculate the
Substitute the above calculated values.
And,
Calculate the value of constants
For b,
For c,
Thus, the new root is calculated as follows.
Substitute the all values.
Calculate the error estimate.
Because the error is large, so required new guesses for
,
, and
Therefore, for the new iteration,
,
,
Thus, the values of
Now calculate the
Substituting the value of
, and
Now calculate the
Substitute the above calculated values.
And,
Calculate the value of constants
For b,
For c,
Thus, the new root is calculated as follows.
Substitute the all values.
Calculate the error estimate.
Because the error is near about 1, so required new guesses for
,
, and
. And repeat the same process will get the root is
error | ||
0 | 1.979384 | 26.03 |
1 | 1.8 | |
2 | 2 | 0.00241 |
Similarly repeat for all other roots.
To find the exact positive real root use MATLAB.
Write the following code in command window.
Therefore, the only positive real root of the given equation
(b)
To calculate: The positive real root of
(b)
![Check Mark](/static/check-mark.png)
Answer to Problem 3P
Solution:
The only positive real root of the given equation
Explanation of Solution
Given Information:
The given equation is,
Use Müller’s method.
Formula used:
The expression for the new roots is,
The expression for the error is,
Calculation:
Recall the equation mentioned in the problem,
Draw the plot of the equation.
From the above plot it is clear that one root is at about
Consider the initial guess be
, and
Thus, the values of
Now calculate the
Substituting the value of
, and
Now calculate the
Substitute the above calculated values.
And,
Calculate the value of constants
For b,
For c,
Thus, the new root is calculated as follows.
Substitute the all values.
Calculate the error estimate.
Because the error is large, so required new guesses for
,
, and
Therefore, for the new iteration,
,
,
Thus, the values of
Now calculate the
Substituting the value of
,
,
Now calculate the
Substitute the above calculated values.
And,
Calculate the value of constants
For b,
For c,
Thus, the new root is calculated as follows.
Substitute the all values.
Calculate the error estimate.
Because the error is zero, the root is
error | ||
0 | 200% | |
1 | 0% |
Similarly repeat for all other roots.
To find the exact positive real root use MATLAB.
Write the following code in command window.
Therefore, the only positive real root of the given equation
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Chapter 7 Solutions
Numerical Methods for Engineers
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