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Least Squares Fit

  • Thread starter rey242
  • Start date
  • #1
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Homework Statement


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)

Homework Equations





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?
 

Answers and Replies

  • #2
938
9
That's not right. In particular,
[tex] \ln (ae^{-x} + be^x) \neq \ln (ae^{-x}) + \ln (be^x) [/tex]
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,770
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There is no reason to linearize anything. Nor are you trying to fit the curve to those points- just the "least squares" fit.

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

For all 5 x,y pairs, the sum of errors squared is [itex](ae+ be^{-1}- 14)^2[/itex][itex]+ (ae^2+ be^{-2}- 10)^2[/itex][itex]+ (ae^3+ be^{-3}- 8)^2[/itex][itex]+ (ae^4+ be^{-4}- 6)^2+ (ae^6+ be^{-6}-5)^2[/itex]. Find a and b to minimize that.
 
  • #4
41
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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?
 

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