Math Question: Finding Best Approximation of Function

[mrchauncey]

Hi everybody.

I am in this computer software class and I have a quiz coming up. My professor gave us a list of what to expect on the quiz and this question came up:

Given a family of functions f1, f2, f3, .., fn and given a function f, find the best approximation of f by the family. Best here means you have to minimize an error.

I am thinking it his something to do with least squares or some sort of curve fitting, but I just want to get an idea of what I need to review. This class is hard to study for because it is like a review of every math class I ever taken (and some I have not). Then I have to write certain commands and codes on the software to get the right answers, so its pretty hard.

Anybody got any ideas of what my professor is asking.

Thanks for looking.

 

[LCKurtz]

Hi everybody.

I am in this computer software class and I have a quiz coming up. My professor gave us a list of what to expect on the quiz and this question came up:

Given a family of functions f1, f2, f3, .., fn and given a function f, find the best approximation of f by the family. Best here means you have to minimize an error.

I am thinking it his something to do with least squares or some sort of curve fitting, but I just want to get an idea of what I need to review. This class is hard to study for because it is like a review of every math class I ever taken (and some I have not). Then I have to write certain commands and codes on the software to get the right answers, so its pretty hard.

Anybody got any ideas of what my professor is asking.

Thanks for looking.

Well, you do have to know how the error is to be measured. If it is least squares, the problem would be to find the $c_i$ so that$$
\int_a^b \left( \sum_{i=1}^n c_i f_i - f\right)^2$$is minimized. Or if you want a uniform approximation it might be that$$
\max_{x\in [a,b]}\left|\sum_{i=1}^n c_i f_i(x) - f(x)\right|$$is minimized.

 

[mrchauncey]

Yea I know how to use least squares. I just wanted some more insight on what he could possibly be asking. Right now in maple I am taking families of different functions, plotting all the points (for a certain range) of each function in the family. Then I found the best fit curve or line of all the points from the family of functions. Then I am taking a function (which is f in the question I stated in my original post) and plotting its points along with the points I found from the curve that fits through the families that I plotted before. Then I am taking the least squares of all the points and finding the least squares solution. I don't know if you will read this but what do you think of that? On track or way off?

Thanks for your time. I'm just confused at the question, but I seem to be getting somewhere so its a start.

EDIT: Now that I think about it I don't really need to fit a curve through the family of functions. I don't think that is necessary.