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I am trying to make a finite difference operator such that it would behave similar to the partial derivative $\partial_{x}f[x]$.

$\delta_n[f[n],m] := \frac{f[n+m]-f[n]}{m}\ .$

I couldn't find a way to define it where I can just pass the function with its argument explicitly shown to the operator. Note that I don't want to give the operator just the name of the function, that's because it cannot act on the function twice if I define it that way, e.g.

$\delta_n[\delta_n[f[n],m],m]\ .$

The finite difference operator is just an example. My goal is to learn how to do it so I could apply this to my work in general.

Edit:

I found a partial solution with the following lines of code

SetAttributes[$\delta$, HoldAll]
$\delta[f\_[n\_],m\_]:=\frac{f[n+m]-f[n]}{m}\ .$

There is still a problem when I do the operation twice, or directly pass the expression of the function into the operator, i.e.

$\delta[n^2,m]$

or if the function has multiple variables,

$\delta[f[n,l],m]\ .$

It would be nice if it could just work like the Derivative operator.

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    $\begingroup$ Before anything else, have you already seen DifferenceQuotient[]? $\endgroup$ Aug 6, 2022 at 1:20
  • $\begingroup$ No, I have not, but I would like to learn how to implement it so that I can define any arbitrary operator on a function. It was just a simple example that I came up with. $\endgroup$ Aug 6, 2022 at 3:07
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    $\begingroup$ One problem with wanting something like $\delta[n^2, m]$ to work is that you'd actually need to specify what the variables were somewhere. Note that when taking the derivative, you need to write D[n^2, n] so that Mathematica knows what to take the derivative with respect to. (Otherwise it might be Power, or even 2.) So you need to find a way to give Mathematica this info—either by having extra arguments/options, feeding in functions/bound variables instead (e.g. d[n |-> n^2, m |-> m]), marking them somehow, or declaring specific variables to be arguments globally (which might be odd). $\endgroup$
    – thorimur
    Aug 6, 2022 at 5:36
  • $\begingroup$ You might also be interested in how I implemented the product derivative in this answer, as a guide on how to implement operators. $\endgroup$ Aug 6, 2022 at 7:44
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    $\begingroup$ And thank you, @thorimur, for the great suggestion! $\endgroup$ Aug 7, 2022 at 5:33

1 Answer 1

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Here is the solution:

FiniteD[f_,{n_,m_}] := (Function[n,f][n+m] - Function[n,f][n])/m
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