Sellmeier's equation & Least-squares Fitting

[yo56]

Homework Statement

Determine the values of Sellmeier's coefficients (S and λ₀) using the least-squares method and Sellmeier's dispersion equation:
n²=1+Sλ²/(λ²-λ₀²)

Homework Equations

n²=1+Sλ²/(λ²-λ₀²)

The Attempt at a Solution

I understand how to use the least-squares method with a simple equation like y=mx+b (see below), but when trying to do the same thing with Sellmeier's equation, I get confused with where to put the summation symbols. Also, I am not sure if I use the squared version of Sellmeier's equation or to take the square root of it.

With y=mx+b, I know to take a y' point on the line corresponding to one of the points. A general equation would be achieved, i.e. D_i=y_i-y'_i --> D_i=y_i-mx_i+b. You would then square the equation, take the summation of it, and then the derivatives with respect to m and b to minimize, setting both equations equal to 0:

(1) Ʃ(y_ix_i)-mƩ(x_i²)-bƩ(x_i)=0
(2) Ʃ(y_i)-mƩ(x_i)-bn=0, n is the number of data points

The solved linear equations yielded:
m=[Ʃ(x_i)][Ʃ(y_i)]-nƩ(y_ix_i)/[[Ʃ(x_i)]²-nƩ(x_i²)]
b=[Ʃ(y_ix_i)][Ʃ(x_i)]-[Ʃ(x_i²)][Ʃ(y_i)]/[[Ʃ(x_i)]²-nƩ(x_i²)]

With Sellmeier's however, I am achieving some complex starting equations. Is there some mathematical "trick" I don't know about? Thanks for any help!

 

[DrDu]

You could try to massage your equation somehow so that $1/\lambda^2=x$ and $y=1/(n^2-1)$.