# Summation question

## Homework Statement

I have a set of data (i, yi). A polynomial fit of 1st degree would be y=ai+b, right?
If I have c=Σ(i2*yi) is it correct to substitute y=ai+b inside the summation?

## The Attempt at a Solution

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Ray Vickson
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## Homework Statement

I have a set of data (i, yi). A polynomial fit of 1st degree would be y=ai+b, right?
If I have c=Σ(i2*yi) is it correct to substitute y=ai+b inside the summation?

## The Attempt at a Solution

You are being careless with notation, and it is landing you in trouble. You have data ##\{ (i, y_i)
\}## and fit a formula of the form ##Y(x) = ax + b## to the data; that is, you are approximating ##y_i## by the value ##Y(i) = ai + b##. Hopefully, the approximation is good in some sense, but that is another, separate issue. Anyway, you have a quantity ##c = \sum i^2 y_i##. When you substitute ##Y(i)## instead of ##y_i## you are computing an approximation ##C = \sum i^2 Y(i)## instead of the exact value of ##c##.

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I don't care about the exact approximation because I test various polynomials to figure out for which c approximates a specific known value. But my problem is what to do about the coefficients of the polynomials. Is there a way to calculate them in order to calculate c?

Ray Vickson
Homework Helper
Dearly Missed
I don't care about the exact approximation because I test various polynomials to figure out for which c approximates a specific known value. But my problem is what to do about the coefficients of the polynomials. Is there a way to calculate them in order to calculate c?
How do you perform the fit to the data? If you use the least-squares method there are formulas for the coefficients. If you use some other method, there may not be formulas---only algorithms. For example, if you do a least average absolute-deviation fit, you can set up the problem as a linear program and solve it using a standard package (such as the EXCEL Solver). The solution of the linear program will include values of the coefficients.

1 person
Okay I understand that, thank you. But what if I want to test with polynomials of second degree or higher?
My y data can be manually changed and c takes a specific value. What I'm trying to do is figure out which ys to change to make c take that value and/or by how much. Is that possible?

Ray Vickson
$$\sum_{i=1}^n i^2 (ai+b) = c$$