Interpolation with knowledge of derivatives

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I want to interpolate a function between the points A, B, C. At A and C I only know the value of the function, but at B (lying between them) I also know the function's first and second derivatives. How would you interpolate between these points?
 

mathman

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Since you have 5 parameters, you can fit a function of the form y=P(x) where P(x) is a fourth degree polynomial and evaluate P(x) for points in the interval.
 
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mathman, could you clarify a bit? How are the two derivatives parameters? I don't get that. Sorry for my ignorance; where could I read further into this? I do understand why a fourth degree polynomial would be the starting point.

Edit: Sorry I get it now. What would be a more accurate way to do this?
 
Last edited:
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There are many different types of interpolations, all based on "what you know" at different points. Most are based on knowing the value of a function at different points. I am not aware of any that use derivatives at some points, but not others. You may just have to get creative and invent your own.

Start by making assumptions about what the function looks like across its domain, assumptions that use a first and second derivative.
 
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As mathman said, plug the numbers into a fourth degree polynomial. The coefficients are your unknowns. Now take the first and second derivatives of the fourth degree polynomial. Now you have 3 equations, and that's all you're gonna get. But the fourth degree polynomial has 5 unknowns. The leftovers could of course be anything; they represent the coordinates of where the point that you took the derivative of was. This point could of course be anywhere. So you can't interpolate.
 

mathman

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As mathman said, plug the numbers into a fourth degree polynomial. The coefficients are your unknowns. Now take the first and second derivatives of the fourth degree polynomial. Now you have 3 equations, and that's all you're gonna get. But the fourth degree polynomial has 5 unknowns. The leftovers could of course be anything; they represent the coordinates of where the point that you took the derivative of was. This point could of course be anywhere. So you can't interpolate.
The other two equations come from knowing the values of the function at the end points (A and C) of the interval.
 
32
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Many thanks for all your replies.
 

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