# Least Squares Fit

1. Dec 3, 2009

### rey242

1. The problem statement, all variables and given/known data
For the following data, find the least squares fit of the given form
x=1,2,3,4,6
y=14,10,8,6,5
h(x)=ae^x+be^(-x)

2. Relevant equations

3. The attempt at a solution
So I tried to linearize the equation by taking the natural log of everything
ln(h)=ln(ae^-x)+ln(be^ex)
ln(h)=ln(a)+ln(b)+x-x

that when I run into a problem, I eliminate the x's.
My question is, is there another way to linearize the equation or should I continue though?

2. Dec 4, 2009

### clamtrox

That's not right. In particular,
$$\ln (ae^{-x} + be^x) \neq \ln (ae^{-x}) + \ln (be^x)$$

3. Dec 4, 2009

### HallsofIvy

There is no reason to linearize anything. Nor are you trying to fit the curve to those points- just the "least squares" fit.

If $h(x)= ae^{x}+ be^{-x}$ then $h(1)= ae+ be^{-1}$. Since you are told that y= 14 when x= 1, the "error" is $ae+ be^{-1}- 14$ and the "error squared" is $(ae+ be^{-1})^2$. Similarly, for x= 2, y= 10, the "error squared" is $(ae^2+ be^{-2}- 10)^2$.

For all 5 x,y pairs, the sum of errors squared is $(ae+ be^{-1}- 14)^2$$+ (ae^2+ be^{-2}- 10)^2$$+ (ae^3+ be^{-3}- 8)^2$$+ (ae^4+ be^{-4}- 6)^2+ (ae^6+ be^{-6}-5)^2$. Find a and b to minimize that.

4. Dec 4, 2009

### rey242

What procedure would I use to find a and b? I know about the error...would I convert the sum of the errors squared into normal equations?