Multi-Variable Second Order Taylor Series Expansion: Ignoring Terms

[cvanloon]

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

 

[HallsofIvy]

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?

 

[pmsrw3]

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.

 

[cvanloon]

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

 

[cvanloon]

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

 

[pmsrw3]

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.

 

[matphysik]

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,...(?)