Interpolation with knowledge of derivatives

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Discussion Overview

The discussion revolves around the problem of interpolating a function between three points A, B, and C, where only the values of the function at A and C are known, while both the first and second derivatives are known at point B. The scope includes theoretical aspects of interpolation methods and the implications of using derivatives in the interpolation process.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests using a fourth degree polynomial to interpolate, noting that it has five parameters which can be adjusted based on the known values and derivatives.
  • Another participant expresses confusion about how the derivatives serve as parameters in the interpolation process and seeks clarification and further reading.
  • A different participant points out that traditional interpolation methods typically rely on function values and questions the feasibility of using derivatives at some points but not others, suggesting a need for creative solutions.
  • One participant reiterates the polynomial approach, explaining that while there are three equations from the derivatives, there are five unknowns, leading to a conclusion that interpolation may not be possible under these conditions.
  • Another participant echoes the previous point about the polynomial approach and emphasizes the challenge of having more unknowns than equations, which complicates the interpolation process.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for interpolation given the specific conditions of the problem. There are competing views on the feasibility of using derivatives in this context and the implications of the polynomial approach.

Contextual Notes

Participants note limitations related to the number of equations versus unknowns in the polynomial interpolation approach, highlighting the unresolved nature of how to effectively incorporate derivatives into the interpolation process.

bruno67
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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?
 
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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.
 
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?
 
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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.
 
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 going to 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.
 
Nyxie said:
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 going to 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.
 
Many thanks for all your replies.
 

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