Chapter 13, Problem 14P

Develop a program using a programming or macro language to implement the parabolic interpolation algorithm. Design the program so that it is expressly designed to locate a maximum and selects new points as in Example 13.2. The subroutine should have the following features:

• Base it on two initial guesses, and have the program generate the third initial value at the midpoint of the interval.

• Check whether the guesses bracket a maximum. If not, the subroutine should not implement the algorithm, but should return an error message.

• Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations.

• Return both the optimal x and $f\left(x\right)$.

• Minimize the number of function evaluations.

Test your program with the same problem as Example 13.2.