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evaluation - How to make a function that evaluates an expression?


I'm trying to make a function where the input is an expression, but somehow it just won't be evaluated. For example, this little function doesn't work:


test[exp_] := Module[{x}, NestList[Function[x, exp], 2, 3]]
test[x^2]
(* {2, x^2, x^2, x^2} *)


And I would really like this output:


NestList[Function[x, x^2], 2, 3]
(* {2, 4, 16, 256} *)

Can anybody help?



Answer



Good question (and one that has bitten me in the past:) The reason for your troubles is the ways scoping of Function works. That's easiest to demonstrate with module however.


Module[{x},x]
(*x$642*)


You see that instead of x x$nnn was returned. For more info I recommend reading this.


Block has the expected behavior by the way:


Block[{x},x]
(*x*)

How about function? The file states: help file


A small test


test[exp_] := Function[x, exp]
test[x^2]

(*Function[x$, x^2]*)

So we see that the formal parameters of function have been renamed to x$ and no longer match the x you have put in. Here are some solutions:


Pass the function to your method


As has been suggested in the comments, you can pass the function as the argument.


ClearAll[test];
test[f_] := Module[{x}, NestList[f, 2, 3]]
test[#^2 &]
test[Function[x, x^2]]


(* ==> {2, 4, 16, 256} *)    
(* ==> {2, 4, 16, 256} *)

Pass the variables along with the expression


ClearAll@test
test[exp_, vars_] := Function[Evaluate@vars, exp]
test[x^2, x]
(*Function[x, x^2]*)

Or inserted into your original example



test[exp_, vars_] := NestList[Function[Evaluate@vars, exp], 2, 3]
test[x^2, x]
(*{2, 4, 16, 256}*)

Oh and if you give your function the HoldAll attribute, then it will not be bothered by existing assignments to x


SetAttributes[test, HoldAll]
test[exp_, vars_] := NestList[Function[vars, exp], 2, 3]
x = 5
test[x^2, x]
(*{2, 4, 16, 256}*)

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