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.