# How can I compare two approximants to a bivariate function?

An application I'm working on has required me to find simple approximations to a rather complicated bivariate function $g(x,y)$ that also takes a long time to evaluate on the computer. Through sheer algebra and much sweat, I finally managed to derive two simpler approximations, which I shall call $g_1(x,y)$ and $g_2(x,y)$.

How do I (quantitatively) determine which approximation is a "better" approximation to the original function?

The original complicated bivariate function is physically sensible on a rectangular domain, and I thus thought to compare $g_1(x,y)$ and $g_2(x,y)$ by plotting, respectively, $\left|1-\frac{g_1(x,y)}{g(x,y)}\right|$ and $\left|1-\frac{g_2(x,y)}{g(x,y)}\right|$ (i.e. the relative error) over that domain. Two approaches I've thought of are 1. integrate the relative error function over the domain of definition (i.e. a double integral); and 2. find the maximum value of the relative error; after which I compare the results for the two relative error functions.

Are there other, better/more sensible ways of performing a quantitative comparison?

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You might want to consider scicomp.stackexchange.com. However, cross posting is typically frowned upon on stackexchange network.. – user2468 Jul 10 '12 at 2:50
I'm sure it is. :) I thought I'd ask here first, and then go there if I don't get any answers within the week. – Renata Porfirio Jul 10 '12 at 2:53
It is quite possible that $g_1$ is better for one purpose and $g_2$ is better for another. Sometimes we want to minimize the average error and sometimes we worry about the worst case scenario. Without context it's impossible to give any advice on what error measurement is the "best". For one thing, you somehow know that the relative error makes sense for your functions and I have no clue why. – user31373 Jul 11 '12 at 3:20