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Re: What is the purpose of the Defer Command?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg81996] Re: What is the purpose of the Defer Command?
  • From: "David Park" <djmpark at comcast.net>
  • Date: Tue, 9 Oct 2007 05:33:15 -0400 (EDT)
  • References: <fe4uhp$155$1@smc.vnet.net>

Still, my question is: "What is the purpose of the Defer Command?"

The only potential purpose that I can see is for didactic purposes, to show 
preliminary expressions before they are automatically evaluated. But the 
only way to get Defer to evaluate is by copying and pasting and then 
evaluating,  by using Shift-Ctrl-L and then evaluating, or by evaluating in 
place. All of these operations leave no record of the steps taken. That goes 
contrary to any didactic purpose.

-- 
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/


"David Park" <djmpark at comcast.net> wrote in message 
news:fe4uhp$155$1 at smc.vnet.net...
>I do not understand the utility of the new Defer statement in Mathematica
> Version 6. Also, it seems to me to be similar to, but not as good as, the
> HoldTemporary command introduced by Ted Ersek on MathSource a few years 
> ago.
>
> The help for Defer says: "Defer[expr] yields an object that displays as 
> the
> unevaluated form of expr, but which is evaluated if it is explicitly given
> as Mathematica input." What does 'given as Mathematica input' mean? The
> examples seem to only involve copying and pasting, which I don't consider 
> a
> great method for doing mathematics, or evaluation in place.
>
> I would like to understand how Defer might be used in expository notebooks
> to clarify some piece of mathematics. The problem is that it requires an
> interactive action, which would be invisible to a reader of a notebook.  I
> think the idea of 'modification in place' is poor in technical 
> communication
> because it destroys the record of what was done.
>
> (In the examples below, whenever an output resulted in an expression that
> copied as a box structure, I converted to InputForm to simplify the
> posting.)
>
> Here is a simple example:
>
> y = Defer[1 + 1]
> 1 + y                                               giving
>
> 1 + 1
>
> 1 + (1 + 1)
>
> I would prefer that the Defer expression would have evaluated in the 
> second
> statement but I guess it is logical that it didn't. If I write:
>
> 1 + y
>
> then select the y and Evaluate In Place I obtain the following, which must
> then be further evaluated to obtain 3.
>
> 1 + 1 + 1
>
> 3
>
> A second example. I want to show an integral without evaluation and then 
> the
> evaluated result. I have to write the following expression, then select 
> the
> second line of output, evaluate in place, and then I obtain the result - 
> but
> as an Input cell. This is certainly a place where HoldForm would be 
> better.
>
> Defer[Integrate[x^2 Exp[-x], {x, 0, 1}]]
> %
> giving
>
> Integrate[x^2/E^x, {x, 0, 1}]
>
> 2 - 5/\[ExponentialE]                        (which is an Input cell)
>
> Here is third example. Defer does not evaluate and we obtain an error
> message.
>
> numb = Defer[2^67 - 1]
> FactorInteger[numb]                                       giving
>
> 2^67 - 1
>
> FactorInteger::"exact" :  "\"Argument \!\(\*SuperscriptBox[\"2\", \
> \"67\"]\) - 1 in FactorInteger[\!\(\*SuperscriptBox[\"2\", \"67\"]\) \
> - 1] is not an exact number\""
>
> FactorInteger[2^67 - 1]
>
> But it works if I copy and paste into FactorInteger.
>
> Now, look at the behavior of Ted's MathSource package.
>
> Needs["Enhancements`HoldTemporary`"]
>
> y = HoldTemporary[1 + 1]
> 1 + y                                                        giving
>
> 1 + 1
>
> 3
>
> The expression is evaluated if it is an argument of some function.
>
> HoldTemporary[Integrate[x^2 Exp[-x], {x, 0, 1}]]
> Identity[%]
> giving
>
> Integrate[x^2/E^x, {x, 0, 1}]
>
> 2 - 5/\[ExponentialE]       (which is an Output cell)
>
> numb = HoldTemporary[2^67 - 1]
> FactorInteger[numb]                                            giving
>
> 2^67 - 1
>
> {{193707721, 1}, {761838257287, 1}}
>
> Much better. I might be missing the point, but I don't think that Defer is
> at all well designed.
>
> There is another Hold that is very useful. This is one that holds an
> operation but evaluates the arguments. We have a HoldOp statement in the
> Tensorial package.
>
> Needs["TensorCalculus4V6`Tensorial`"]
>
> ?HoldOp
>
> HoldOp[operation][expr] will prevent the given operation from being
> evaluated in expr. Nevertheless, other operations within expr will be
> evaluated. Operation may be a pattern, including alternatives, that
> represents heads of expressions. The HoldOp can be removed with 
> ReleaseHold.
>
> One reason we want the arguments to evaluate is that the arguments often
> contain tensor shortcut expressions and we want them evaluated to show the
> full tensor expression inside some operation. However, there are many 
> other
> uses.
>
> f[x_] := Sin[x] \[ExponentialE]^x
>
> We would like f[x] to be evaluated inside the Integrate statement, but 
> hold
> the actual itegration.
>
> Integrate[f[x], {x, 0, \[Pi]}] // HoldOp[Integrate]
> % // ReleaseHold
> giving
>
> HoldForm[Integrate[E^x*Sin[x], {x, 0, Pi}]]
>
> 1/2 (1 + \[ExponentialE]^\[Pi])
>
> For exposition purposes we might want to keep the following expression in
> the input order.
>
> \[Pi]  Sin[x] \[ExponentialE]^x // HoldOp[Times]
> % // ReleaseHold
> giving
>
> HoldForm[Pi*Sin[x]*E^x]
>
> \[ExponentialE]^x \[Pi] Sin[x]
>
> Often we will have cases where some operation has automatic built-in 
> rules,
> such as linear and Leibnizian breakouts with differentiation. Again, for
> exposition purposes, we might want to show the expression before these 
> rules
> are applied.
>
> g[x_] := x^2
>
> D[a f[x] g[x], x] // HoldOp[D]
> % // ReleaseHold                                    giving
>
> HoldForm[D[a*E^x*x^2*Sin[x], x]]
>
> a \[ExponentialE]^x x^2 Cos[x] + 2 a \[ExponentialE]^x x Sin[x] +
> a \[ExponentialE]^x x^2 Sin[x]
>
>
> -- 
> David Park
> djmpark at comcast.net
> http://home.comcast.net/~djmpark/
>
>
>
> 



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