1
$\begingroup$

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?

$\endgroup$
6
  • $\begingroup$ 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 $1-\sum_{i=1}^k x_i$. $\endgroup$
    – Greg Martin
    Aug 5 at 20:26
  • $\begingroup$ @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. $\endgroup$
    – graphtheory123
    Aug 5 at 20:39
  • $\begingroup$ OR sorry, I think your answer would still work for this updated problem $\endgroup$
    – graphtheory123
    Aug 5 at 20:40
  • $\begingroup$ Is there a norm you have in mind? A uniform approximation will differ from an $L^2$ approximation, and both from a Taylor series. $\endgroup$
    – A rural reader
    Aug 5 at 22:23
  • 1
    $\begingroup$ OP, best to state your application. $\endgroup$
    – A rural reader
    Aug 6 at 0:37

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.