Develop an M-file to implement parabolic interpolation to locate a minimum. The function s...

Develop an M-file to implement parabolic interpolation to locate a minimum. The function 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 function 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(x).

Test your program with the same problem as Example 7.3.

Example 7.3:

Parabolic Interpolation

Problem Statement. Use parabolic interpolation to approximate the minimum of

with initial guesses of x₁ = 0, x₂ = 1, and x₃ = 4.

Solution. The function values at the three guesses can be evaluated:

and substituted into Eq. (7.10) to give

which has a function value of f(1.5055) = −1.7691.

Next, a strategy similar to the golden-section search can be employed to determine which point should be discarded. Because the function value for the new point is lower than for the intermediate point (x₂) and the new x value is to the right of the intermediate point, the lower guess (x₁) is discarded. Therefore, for the next iteration:

which can be substituted into Eq. (7.10) to give

which has a function value of f(1.4903) = −1.7714. The process can be repeated, with the results tabulated here:

Thus, within five iterations, the result is converging rapidly on the true value of −1.7757 at x = 1.4276.

Equation (7.10):