Understanding Taylor Series Error & Degrees of Variables

[brad sue]

Hi,

can someone explain me the relation between the degree of a taylor series (TS) and the error. It is for my class of numerical method, and I do not find a response to my question in my textbook.

I mean when we have a function Q with two variables x and y,and we use a version of TS to calculate the error of Q by doing:

∆ (Q(x,y))= (∂Q/∂x )*∆x + (∂Q/∂y )*∆y (1st order)

We want to compare the error of each term to know which is greater (the one in x or that in y.) or which one I need to measure with more precision.

I don't know if I am clear enough.

For example, if I have for the x term a degree of -2 and for y term a degree of -.5 after finding ∆ (Q(x,y)), considering the error which error is greater?

Thank you

Brad

 

[Crosson]

The best thing would be to calculate the exact value, and compare that to the approximation.

As far as an analytical solution, just calculate the next largest terms:

$$\Delta Q = \frac{\partial Q}{\partial x} \Delta x + \frac{\partial Q}{\partial y} \Delta Y + \frac{\partial^2 Q}{\partial x^2} (\Delta x )^2 + \frac{\partial^2 Q}{\partial y^2} (\Delta y)^2 +\frac{\partial^2 Q}{\partial x \partial y} \Delta x \Delta y$$

Compare the delta x squared term to the corresponding y term. Use the mixed term to calculate the whole thing to second order, then compare that to your first order approx.

 

[tacman]

Hi Brad,

The degree of a Taylor series refers to the highest exponent of the variable in the polynomial approximation. For example, a Taylor series with degree 3 would have terms up to x^3, while a Taylor series with degree 5 would have terms up to x^5. The higher the degree, the more accurate the approximation will be.

In terms of error, the degree of a Taylor series can give us an idea of how quickly the error decreases as we increase the number of terms in the series. Generally, the higher the degree, the faster the error decreases. So in your example, the error in the x term (degree -2) would decrease faster than the error in the y term (degree -0.5).

However, it's important to note that the degree of a Taylor series is not the only factor that affects the error. The size of the interval and the smoothness of the function also play a role. So while the degree can give us some information about the error, it's not the only factor to consider when determining which term to measure with more precision.

I hope this helps clarify the relationship between the degree of a Taylor series and the error. Let me know if you have any other questions. Good luck with your class!

Best regards,