Concept explainers
Employ the following methods to find the maximum of
(a) Golden-section search
(b) Parabolic interpolation
(c) Newton's method
(a)
To calculate: The maximum of the function
Answer to Problem 6P
Solution:
The maximum of the function
Explanation of Solution
Given information:
The function
Formula used:
In the golden–section search algorithm two interior points are chosen which satisfies the golden ratio.
The method starts with two initial guesses,
Here d is the difference and
The value of the function is evaluated at these two interior points. Two results can occur:
(1). If
(2). If
And the process is repeated for a number of times till the value reaches close to a particular number.
Calculation:
Consider function
With
Now,
Let’s continue with the iterations to reach to
Iteration 1: First golden ratio is used to create two interior points as,
The two interior points are,
Now, comparing the value of function at these interior points as shown below:
For
For
As
Therefore, the domain of x to the left of
Iteration 2: Here
The two new interior points are,
Now, comparing the value of function at these interior points as shown below:
For
For
As
Therefore, the domain of x to the left of
Iteration 3: Here
The two new interior points are,
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:
(b)
To calculate: The maximum of the function
Answer to Problem 6P
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
Substituting these values in equation (1) to get value of
And value of function at
Therefore,
Iteration 2: Now the initial guesses are
Function values at these three initial points is,
For
For
For
Substituting these values in equation (1) to get value of
And value of function at
Therefore,
Iteration 3: Now the initial guesses are
Function values at these three initial points is,
For
The function for
And for
Substituting these values in equation (1) to get 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
Substituting these values in equation (1) to get value of
And value of function at
Therefore,
And the process continues with a summary shown below in a table:
(c)
To calculate: The maximum of the function
Answer to Problem 6P
Solution:
The maximum of the function
Explanation of Solution
Given information:
The function
Formula used:
Newton Method is open method 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,
And
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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