Concept explainers
Employ the following methods to find the maximum of the function from Prob. 13.7:
(a) Golden-section search
(b) Parabolic interpolation
(c) Newton's method
(a)

To calculate: The maximum of the function
Answer to Problem 8P
Solution:
The maximum of the function
Explanation of Solution
Given information:
The function
Formula used:
The golden-search method with two initial guesses,
Evaluated the function at the above two interior points. Two results can occur,
If
If
Calculation:
Consider function
With
Iteration 1: First golden ratio is used to create two interior points as,
The two interior points are as follows:
First point is,
Second point is,
Now, comparing the value of function at these interior points as shown below:
For
For
As
Therefore, the maximum is in the interval defined by
Where,
The error at this point can be computed as follow:
Therefore, the domain of x to the left of
For this case,
Iteration 2: Here,
The two new interior points are as follows:
First point is,
Second point is,
Now, comparing the value of function at these interior points as shown below:
For
For
As
For this case,
Now,
Iteration 3: Here,
The two new interior points are as follows:
First interior point is,
Second interior point is,
Now, comparing the value of function at these interior points as shown below:
For
For
As
Therefore, for this case,
Proceeding like this the iterations can be tabulated below as:
Thus, the result converges to true value
(b)

To calculate: The maximum of the function
Answer to Problem 8P
Solution:
The maximum of the function
Explanation of Solution
Given information:
The function
Formula used:
Consider three points jointly bracket an optimum, thus a unique parabola through these three points can be determined. On differentiating and setting it equal to zero estimate of optimal can be computed.
Consider
Calculation:
Consider function
With initial guesses
Iteration 1: Function values at these three initial points is,
For
For
For
Substitute the value of
The value of function at
Therefore,
Iteration 2: Now the initial guesses are
Function values at these three initial points are,
For
For
For
Substitute the value of
The value of the provided function at
Therefore,
Iteration 3: Now the initial guesses are
Function values at these three initial points is,
For
The function for
And for
Substitute the value of
And value of function at
Therefore,
Iteration 4: Now the initial guesses are
Function values at these three initial points is,
For
For
For
Substitute the value of
And value of function at
Therefore,
And the process continues with a summary shown below in a table:
Thus, after four iterations result is converging to true value
(c)

To calculate: The maximum of the function
Answer to Problem 8P
Solution:
The maximum of the function
Explanation of Solution
Given information:
The function
Formula used:
Newton Method is similar to Newton Raphson as it does not require initial guesses that bracket the optimum solution.
For any function
Calculation:
Consider function
With initial guesses
First and second derivatives of function that is,
Iteration 1:Initially for
For second derivative,
Therefore,
And
Iteration 2:Now for
For second derivative,
Therefore,
And
Iteration 3:Now for
For second derivative,
Therefore,
And
Iteration 4:Now for
For second derivative,
Therefore,
Maintaining the error percentage using equation (3) iterations can be summarized as shown in table below:
Thus, within four iterations, the result converges to true value
Therefore, the maximum of the function
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