Trying to find the equation of a cubic

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To find a cubic equation that best fits the provided data, the general form y = ax^3 + bx^2 + cx + d can be used, where coefficients a, b, c, and d are determined to minimize the error in the fit. A regression analysis using least squares has yielded approximate values for these coefficients: a = -0.00407, b = 0.15356, c = -1.2757, and d = 18.27288. While matrix methods can sometimes lead to incorrect quartic equations, using finite differences or computational tools like MATLAB is recommended for accuracy. Manual calculations for cubic fitting can be tedious and are generally not advised. Proper use of regression techniques can yield a near-perfect cubic fit for the data.
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1. Find an equation that best fits the data?
Data
2 16.40
3 15.70
4 15.30
5 15.20
6 15.21
7 15.40
8 15.80
9 16.30
10 16.80
11 17.50
12 18.18
13 18.70
14 19.36
15 19.88
16 20.40
17 20.85
18 21.22
19 21.60
20 21.65

I have narrowed it down that it is most likely a cubic, because it seems to make sense.

I tried to use a matrix to solve this equation but found that it gave me a quartic, that did not match the data at all any help would be appreciated, is there a formula on how to solve cubics?
 
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If you mean "find the least squares best fit cubic", you would want to write, in general, y= ax^3+ bx^2+ cx+ d and then find a, b, c, d to minimize \sum (y_i- (ax_i^3+ bx_i^2+ cx_i+ d
 
Have you tried simply using Finite differences?
 
not really sure how i would use it, but ill try to
 
Last edited:
I did a regression by least squares and it seems to almost perfectly fit a cubic.
a = -0.00407
b= 0.15356
c=-1.2757
d=18.27288
(approximate values)
where it is in the form y = ax^3 + bx^2 + cx + d
normally you would use a computer package like MatLab or a programmable calculator to do these. I don't recommend doing this by hand, it's very long.
 

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