Curve approximation by some known points on the curve

Question

I want to approximate a curve by some known points on the curve.
I can choose these point.
My curve is shown as below:

I have to use such a equation:

f(x) = a1x^1 + a2x^2 + a3x^3 + a4x^4   

1) Is it a proper equation for my approximation ?
2) In my case, I should consider P as a and Eta as x?

Thanks in advance

Answer 1

If rough approximation is enough for your use as you commented, then Lagrange polynomial will suit your purpose.

If the number of known points increases in future and you wanted to calculate a fitting curve as number of points increases, then Newton polynomial may suit your purpose better since it doesn't waste computations which was done before.

In the formula for f, replace x with P1, and f with η1; this gives you a linear equation relating the four unknowns, a1,…,a4. Then use P2 and η2 to get a second equation. Then get similarly a third and a fourth equation. Now you have 4 equations for the 4 unknwons; you can solve, and you're done. A quicker way is to look up "Lagrange interpolation" and use the formulas you will find. – Gerry Myerson