# How to approximate $\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})\;$ as a linear function?

Suppose we have a finite set $$I \subseteq \mathbb{N}$$, and $$\alpha_i\in [0, 1]$$ are fixed numbers for all $$i\in I$$.

Is there a way to approximate $$\prod\limits_{(i, j)\in I^2} (1 - \alpha_i x_{i, j})$$, where $$0\leq x_{i, j}\leq 1 \; \forall (i, j)\in I^2$$, as a linear function in terms of the $$x_{i, j}$$? If so, how accurate is the approximation to the actual product?

• For any differentiable function (not just this one), the first order approximation $f(\vec x_0) + \nabla f(\vec x_0)\cdot (\vec x-\vec x_0)$ is the best linear approximation near the point $\vec x_0$. For example, near $\vec 0$ the best linear approximation to your function is $1-\sum_{i=1}^k x_i$. Aug 5 at 20:26
• @GregMartin Thanks for the response. I realized my question was too reduced though. I updated the question to capture the full nuance of my problem. Aug 5 at 20:39
• OR sorry, I think your answer would still work for this updated problem Aug 5 at 20:40
• Is there a norm you have in mind? A uniform approximation will differ from an $L^2$ approximation, and both from a Taylor series. Aug 5 at 22:23
• OP, best to state your application. Aug 6 at 0:37