Multi-Variable Second Order Taylor Series Expansion: Ignoring Terms

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Discussion Overview

The discussion revolves around the accuracy of a second order Taylor series expansion for a multi-variable function, particularly whether including certain second order terms improves the approximation compared to a first order expansion. The scope includes theoretical considerations and mathematical reasoning related to Taylor series in the context of functions with multiple variables.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that including some second order terms will always provide a more accurate approximation than just the first order terms, as long as those terms are not zero.
  • Others argue that it is not a general rule that a partial second-order expansion will always improve accuracy, suggesting that second-order terms could theoretically cancel each other out, making the first-order approximation more accurate in certain cases.
  • A participant mentions the importance of including mixed partial derivatives (e.g., Ixy(dx dy)) in the expansion for a more accurate representation.
  • There is a suggestion that practical knowledge about the function's behavior in spatial dimensions versus temporal dimensions may influence the decision to include second order terms.
  • One participant expresses uncertainty about the function's behavior and questions whether adding second order terms would, on average, improve the approximation.
  • Another participant acknowledges that, without knowledge of the function's behavior, additional second order terms might improve the approximation "on average."
  • A participant critiques the vagueness of the original question, indicating that the purpose of the Taylor series expansion could affect its effectiveness.

Areas of Agreement / Disagreement

Participants express differing views on whether including certain second order terms is beneficial. Some assert that it will always improve accuracy, while others caution that this is not universally true and that specific conditions could lead to different outcomes. The discussion remains unresolved regarding the general applicability of these claims.

Contextual Notes

Participants note that the effectiveness of including second order terms may depend on the specific context of the Taylor series expansion, such as the intended application (e.g., curve sketching, asymptotic expansion) and the nature of the function being approximated.

cvanloon
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So I'm computing a second order Taylor series expansion on a function that has multiple variables. So far I have this

I(x,y,t)=dI/dx(change in x)+dI/dy(change in y)+dI/dt(change in t)+2nd order terms

Would it still be a better approximation than just he first order if I included some second order terms and not others or no? To be more clear I would use something like this :

I(x,y,t)=First Order Terms+Ixx(dx^2)+Iyy(dy^2)

If this is better than just the first order terms, do you have an explanation as to why it is theoretically? Thanks,

Chris
 
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No. Including some second order terms is never less accurate than the linear approximation and, as long as those included second order terms are not 0, will be more accurate. That can be seen by looking at the "error" terms for linear and second order Taylor's series.

Since this question says nothing about differential equations, why was it posted in this section? Did it arise in an attempted series solution to a differential equation?
 


Sure, it might be better, (but you probably also want an Ixy(dx dy) in there).

I don't think one can state as a general rule that this partial second-order expansion will be an improvement over the first-order. It is possible, at least in theory, that the second-order terms all nearly cancel, so that the first-order expression is accurate but any partial second-order expansion is worse. (Of course, these statements will only be true if, in approximating I, the differentials are replaced with specific finite numbers, such that the cancellation happens.) However, as a practical matter, one may know that there is much more action in the spatial dimensions than the temporal on scales of interest. Or maybe you have just chosen dt to be much smaller relative to Itt specifically to allow the neglect of the dt^2 terms.
 


HallsofIvy said:
No. Including some second order terms is never less accurate than the linear approximation and, as long as those included second order terms are not 0, will be more accurate. That can be seen by looking at the "error" terms for linear and second order Taylor's series.

Since this question says nothing about differential equations, why was it posted in this section? Did it arise in an attempted series solution to a differential equation?

I wasn't sure of a place to put it. Taylor series involve taking derivatives? ;) thanks for your answer though. Still looking for a solid mathematical reason why though. Thanks,

Chris
 


pmsrw3 said:
Sure, it might be better, (but you probably also want an Ixy(dx dy) in there).

I don't think one can state as a general rule that this partial second-order expansion will be an improvement over the first-order. It is possible, at least in theory, that the second-order terms all nearly cancel, so that the first-order expression is accurate but any partial second-order expansion is worse. (Of course, these statements will only be true if, in approximating I, the differentials are replaced with specific finite numbers, such that the cancellation happens.) However, as a practical matter, one may know that there is much more action in the spatial dimensions than the temporal on scales of interest. Or maybe you have just chosen dt to be much smaller relative to Itt specifically to allow the neglect of the dt^2 terms.

Thank you for your reply. Let's just say that you have no knowledge of what the function does and that you have second derivative information for I in x and y, but not xy t xt and yt. Would you rather use just the first order approximation or does "on average" or with greater probability, the few second order terms that you add improve your answer?

Thanks,

Chris
 


cvanloon said:
Thank you for your reply. Let's just say that you have no knowledge of what the function does and that you have second derivative information for I in x and y, but not xy t xt and yt. Would you rather use just the first order approximation or does "on average" or with greater probability, the few second order terms that you add improve your answer?
I think, in that case, that I'd agree with HallsofIvy: "on average", additional terms will improve the approximation.
 


cvanloon said:
Would it still be a better approximation than just he first order if I included some second order terms and not others or no?

Your question is vague, to say the least. It depends upon what you intend doing with your Taylor series expansion. Curve sketching, asymptotic expansion,...(?)
 

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