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RE: How to operate on strictly numerical functions ?

My answer see below

> -----Original Message-----
> From:	Mark Harder [SMTP:harderm at]
To: mathgroup at
> Sent:	Tuesday, June 27, 2000 6:52 AM
> To:	mathgroup at
> Subject:	[mg24102] How to operate on strictly numerical functions ?
>     Some of the Mathematica functions made for numerical procedures 
> appear to actually evaluate by substitution of symbolic variables with 
> replacement rules. An example of this, which has blocked me for more 
> than a week now, is NonlinearRegress, which seems to evaluate the 
> user-supplied model function symbolically, then evaluate the resulting 
> expression through replacement of the independent variables and the 
> current set of adjustable parameters through replacement rules that it 
> constructs.  My model function requires evaluation of the SVD (with 
> SingularValues[] ) of a matrix computed from the independent variables 
> and the parameters of the model, and so NonlinearRegress fails, since 
> SingularValues[] can't accept a non-numeric matrix.  A simpler example I 
> have encountered is the numeric derivative function, which I'll use to 
> illustrate my problem.
>     First, construct a simple test function which, through an If[] test, 
> won't evaluate for non-numeric arguments:
> In[804]:=
> ClearAll[x, tstFn1]
> tstFn1[x_] := If[NumericQ[x  ], Return[x^2];, Print["NonNumeric x 
> and/or y."];]
> In[808]:=tstFn1[2]
> Out[808]=4
> In[809]:=tstFn1[u]
> & try to use ND[] on this function:
> <<NumericalMath`NLimit`
> In[806]:=ClearAll[u, v]
>             ND[tstFn1[u ], u, 1.]
> Out[807]= 0
>     So, I'm looking for some means of modifying testFn1 to cause ND to 
> evaluate it only after numeric substitution for u.  Is there some way of 
> Hold-ing evaluation until numeric values are assigned to arguments of a 
> function?  Is this sort of thing impossible in Mathematica?  Do I have 
> to write my own numeric routines for finding derivatives, gradients, 
> Jacobians, least-squares fits, etc. by procedural routines ala FORTRAN?  
> Thanks for any help you can offer.
> mark e. harder
> harderm at
[Wolf Hartmut]  Dear Mark,

perhaps this observation might help you:

In[20]:= << NumericalMath`NLimit`

In[23]:= Attributes[ND]
Out[23]= {Protected, ReadProtected}

so there are no Hold Attributes here, this will evaluate tstFn1[u] in first
place. However there might be more layers of evaluation lurking inside. So
let's test that:

In[44]:= uu = NestList[Unevaluated, Hold[tstFn1[u]], 5]

This contains up to five 'protection layers' agains evaluation.

In[51]:= Print[Block[{u = 1.}, ReleaseHold[#]]] & /@ uu
>From In[51]:= 1.
>From In[51]:= 1.
>From In[51]:= 1.
>From In[51]:= Unevaluated[tstFn1[u]]
>From In[51]:= Unevaluated[Unevaluated[tstFn1[u]]]
>From In[51]:= Unevaluated[Unevaluated[Unevaluated[tstFn1[u]]]]
Out[51]= {Null, Null, Null, Null, Null, Null}

This was for getting familiar with the test, now apply it:

In[47]:= (Print["---"]; ND[ReleaseHold[#], u, 1.]) & /@ uu
>From In[47]:= "---"
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "---"
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "---"
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "---"
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "---"
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "NonNumeric x and/or y."
>From In[47]:= "---"
Out[47]= {0, 0, 0, 0, 0, 2.}

The last result looks very pleasing, not so pleasing would be the next layer
(I avoided here).

So if you need to have your function unevaluated up to a certain --
generally unknown -- point in computation with supplied packages you
certainly live on the knife's edge.

However perhaps this might be a 'cheating device' against unwanted

In[59]:= tstFn2[x_] := If[NumericQ[x], x^2,

In[60]:= tstFn2[u]
Out[60]= Unevaluated[tstFn2[u]]

In[61]:= ND[tstFn2[u], u, 1.]
Out[61]= 2.

Kind regards,  Hartmut

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