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This is an old revision of this page, as edited by 155.97.201.118 (talk) at 10:33, 3 December 2007 (10.) The Method of Undetermined Coefficients). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

1.) Spline vs. Cubic Hermite

a.) Data:




The coefficients of the piecewise cubic function on the first interval (0-1) are found by solving:

And on the second interval (1-2):

b.) This becomes a cubic spline if the second derivative is continuous. Since we only have two intervals, this will hold if the second derivative for both equations is equal at 1. That is, if:

2.) More on Cubic Splines

Data:

a.) I'm not sure I should even bother solving the system of equations to find the coefficients - it's pretty obvious that is what we're looking for.

b.)

which yields the polynomial:

c.)

3.) Linear Programming

4.) Polynomial Interpolation

5.) Runge Interpolation and 6.) Judicious Interpolation

Uploading the 40 images for these two questions would be a headache and a chore. Therefore I provide 4 images - the interpolations at n=10 and n=20. If you'd like to generate more, then you can use these Maple worksheets I created:


Runge Interpolation

Judicious Interpolation


10th degree Runge

File:Runge10.jpg


20th degree Runge

File:Runge20.jpg


10th degree Judicious

File:Judicious10.jpg


20th degree Judicious

File:Judicious20.jpg

7.) Interpolation of Symmetric Data is Symmetric

8.) Linear Independence of Bernstein-Bezier Basis Functions

9.) Uniqueness of Interpolating Polynomial

a.) Power Form

This yield , , , and . So .


b.) Lagrange Form

So . Substituting and simplifying everything down leaves us with:


c.) Newton Form







So, . If we expand all the terms and then simplify, we arrive at:

10.) The Method of Undetermined Coefficients

We expect that since we have four points, we'll at least have accuracy up to cubic functions. We then want to check if we can find values for such that the we have exactness for quartic polynomials.

I claim that there are no that fulfill the conditions. To show this, consider this counter-example:

Let . In order to find , let's throw four specific functions at the equality, and see what turn up.


. Plugging in the four functions above yields:


Or, in matrix form:


However, if we consider a slightly different set of functions

And try to find , we get:

Which is different! So we can't find a solution for that works for all quartic functions. And if we can't find working weights for quartic function we aren't going to find them for higher order polynomials.

So! We now want to find the for cubic functions!

To do so, we use the procedure described in the 10/25 lecture.

11.) Error Analysis

12.) The Method of Undetermined Coefficients, Continued