The Starting Points Are Given As

Homework #8

ChE 356

due 3/23/07

1. 5.6.

Our objective is to minimize the function using quadratic interpolation. We start with three points and use the quadratic interpolation formula to calculate a fourth point. We then discard the point that is not needed in order to bracket the minimum, and perform the calculations again. The quadratic interpolation formula is given as

The starting points are given as

The following table summarizes the results of the iterations:

Iteration / Points used / / / Point to be discarded
1 / / / /
2 / / / /
3 / / / /
4 / / / /
5 / / / /
6 / / / /
7 / / / /
8 / /

2. 5.7.

Our objective is to minimize the function using cubic interpolation. We start with two points and use the cubic interpolation formula to calculate a third point. We then discard the point that is not needed in order to bracket the minimum, and perform the calculations again. The cubic interpolation formula is given as

Where

The starting points are given as

The following table summarizes the results of the iterations:

Iteration / Points used / / / Point to be discarded
1 / / / /
2 / / / /
3 / / / /
4 / / / /
5 / / / /
6 / / / /
7 / / / /
8 / / / /
9 / / / /
10 / /

3.  Repeat calculations in Example 5.5 from two starting points: (0, 0) and (0, 2). Use the gradient for the search direction. Select three equally-spaced points along the search direction (one is the starting point) and perform quadratic interpolation to find the next search point. Use a as the single variable for interpolation.

Using the first starting point (0, 0), we calculate the search direction using the negative gradient

We start to bracket the minimum by taking

The step calculated above yields . At , , an improvement.

For the next step, we want equally spaced points so we let and take another step in the same direction

At , , so we have bracketed the minimum of the function in the search direction. We find the minimum using quadratic interpolation of the three points:

This corresponds to point . We would then calculate a new search direction and repeat the above calculations.

Using the second starting point (0, 2), we calculate the search direction using the negative gradient

We start to bracket the minimum by taking

The step calculated above yields . At , , an improvement over the starting point .

For the next step, we want equally spaced points so we let and take another step in the same direction

At , , so we have bracketed the minimum of the function in the search direction. We find the minimum using quadratic interpolation of the three points:

This corresponds to point . We would then calculate a new search direction and repeat the above calculations.