RE: can Mathematica generate itself a function?

• To: mathgroup at smc.vnet.net
• Subject: [mg31800] RE: [mg31769] can Mathematica generate itself a function?
• From: "David Park" <djmp at earthlink.net>
• Date: Sun, 2 Dec 2001 04:25:32 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Alexandre,

You don't indicate how you actually generate your functions. Still, here is
an example of a function which has a variable number of parameters. The
function generates an (N-1)th degree polynomial with the parameters being
the coefficients. The pattern a__ stands for a sequence of 1 or more
arguments.

f[a__][x_] := Fold[x(#1 + #2) &, 0, Reverse[{a}]]/x

f[a0, a1, a2, a3][x] // Expand

a0 + a1*x + a2*x^2 + a3*x^3

David Park

> [mailto:alexandre.romoscanu at imes.mavt.ethz.ch]
>
> Hello
>
> I am trying to set Mathematica to create itself a function, and then
> consider it as a function definition.
>
> Suppose I want to create a function of the form
>
> function[w,a1,...aN]
>
> where w is a variable and a1...aN, a total of N parameters (such a
> function can for example be used as a fitting model with N parameters in
> NonlinearRegress, etc...)
>
> In some cases, 2 parameters will be requested, i.e. N=2.
>
> In some, one would like the function to have 5 parameters(N=5) to
> define, say a 1-variable, 5-parameter fitting model.
>
> I would like to program Mathematica to create the function itself on a
> given pattern, my only input being the number of parameters (here, N). I
> just do not want to retype a new function each time for the cases I want
> 2 paramaters, or 10...or N.
>
> Keeping in mind that a function is defined as f[x_,y_]=Sin[x+y] (for
> instance), if the total number of parameter pairs is defined by the
> positive integer
>
> NumberOfModules (=5 for example)
>
> then the following expressions obviously create the left part of a
> potential definition statement (like "f[x_,y_]=Sin[x+y]"):
>
> In[1]:= AddUnderscore2[varname_String] := varname <> "_";
>
> In[2]:= function[w_,
>   Sequence @@ (ToExpression[
>         AddUnderscore2 /@ (ToString /@ Array[a, NumberOfModules])])
>
>
> give
>
> Out[2]:= function[w_, a[1] _, a[2] _, a[3] _, a[4] _, a[5] _]
>
> because NumberOfModules was chosen to be 5.(+notice the problematic
> blank in front of the _.)
>
> The right part could look as
>
> Sum[(a[i]*w^2)/(1 + a[i]^2*w^2), {i, 1, NumberOfModules}]
>
> Hence,
>
> function[x_, y_,
>   Sequence @@ (AddUnderscore2 /@ (ToString /@ Array[a,
> NumberOfModules]))] =
>
> Sum[(a[i]*w^2)/(1 + a[i]^2*w^2), {i, 1, NumberOfModules}]
>
> Looks like the definition of a formula.
>
> My problem is the last step...Mathematica does not (without surprize)
> look at the expression right above as it looks to "f[x_,y_]=Sin[x+y]".
>
> There must be a trick with ToExpression, ToString, or some other
> low-level Mathematica command.
>
> Thank you for any hint-
>
> Alexandre Romoscanu