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

Efficiency-power curve

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

Hamed
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  • What's your criterion for "proper approximation"? It might heavily depends on purposes that what is the proper approximation. – Orat Aug 18 '14 at 09:13
  • I need this curve for efficiency curve of an inverter. I already have the curve but I cannot use all the points in my program. I do not want it to be very accurate but to have a curve which is close to the real curve. – Hamed Aug 18 '14 at 09:25
  • In the formula for $f$, replace $x$ with $P_1$, and $f$ with $\eta_1$; this gives you a linear equation relating the four unknowns, $a_1,\dots,a_4$. Then use $P_2$ and $\eta_2$ 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 Aug 18 '14 at 09:25
  • As long as you just interpolate, it seems that a polynomial would be a good idea. But use all the data points and perform a polynomial fit. Do not forget to check that the generated polynomial is really continuous (no extremum in the range). – Claude Leibovici Aug 18 '14 at 09:29
  • Thank you all very much and my special thank to Gerry Myerson :) – Hamed Aug 18 '14 at 09:45

1 Answers1

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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

Hamed
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Orat
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