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Re: Symbolic vector handling
*To*: mathgroup at smc.vnet.net
*Subject*: [mg83468] Re: Symbolic vector handling
*From*: David Bailey <dave at Remove_Thisdbailey.co.uk>
*Date*: Wed, 21 Nov 2007 02:50:19 -0500 (EST)
*References*: <200711180946.EAA01450@smc.vnet.net> <fhrshf$66f$1@smc.vnet.net>
danl at wolfram.com wrote:
>> Hello group,
>>
>> this is most probably a very simple question but still ...
>>
>> Consider a set of vectors vA, vB, vC, ... and scalars a, b, c, ...
>>
>> What I want to do is a symbolic handling of expressions like vA.(b vB +
>> vC)
>> which should be expanded to
>>
>> (1) expr = vA.(b vB + vC) -> b vA.vB + vA.vC
>>
>> i.e. I want Mathematica to apply the distributive law and the
>> extraction of scalars.
>> I don't want to use any coordinate representation.
>>
>> The first thing that comes into mind would be to define a type "vector"
>> and another type "scalar" and then define the usual rules.
>>
>> I'd like to execute a comand like
>>
>> FunctionExpand[ expr , Assumptions->{{vA, vB, vC} "elem" vectors,
>> {a,b,c} "elem" scalars}]
>>
>> Here's an example of a simple problem in which my question arises:
>>
>> Let the vectors vR1 and vR2 be defined in terms of four other vectors
>> vA1, vA2, vN1, vN2 and two scalars a and b as follows:
>>
>> vR1 = vA1 + a vN1
>> vR2 = vA2 + b vN2
>>
>> and consider the function
>>
>> U = (vR1 - vR2).(vR1 - vR2)
>>
>> The task is to find the minimum of U in terms of a combination of
>> scalar product of the vectors vA1, vA2, vN1, vN2 and of the optimizing
>> parameters a and b.
>>
>> Any help is greatly appreciated.
>>
>> Regards,
>> Wolfgang
>
> There are probably a few reasonable ways to go about this. One is to use a
> special vector "head" and define rules for handling distribution involving
> Plus, Times, and whatever else might be of relevance. I encapsulate your
> vectors with vec[].
>
> I find it convenient to move to a new function, myDot, for purposes of
> intermediate computation. This avoids placing new behavior on Dot itself.
>
> myDot[a_] /; Head[a] =!= vec := a
> myDot[x___, y_Plus, z___] := Map[myDot[x, #, z] &, y]
> myDot[x___, y_Dot, z___] := myDot[x, Apply[Sequence, y], z]
> myDot[x___, y_Times, z___] := myDot[x, Apply[Sequence, y], z]
> myDot[x___, y_myDot, z___] := myDot[x, Apply[Sequence, y], z]
> myDot[] = 1;
> myDot[x___, y_, z___] /; Head[y] =!= vec := y*myDot[x, z]
>
> DotExpand[expr_] := (expr /. Dot -> myDot) /. myDot -> Dot
>
> In[54]:= DotExpand[vec[vA].(b*vec[vB] + vec[vC])]
>
> Out[54]= b vec[vA].vec[vB] + vec[vA].vec[vC]
>
> The above myDot is an alteration of some code I cribbed from the section
> "Some noncommutative algebraic manipulation" in the 1998 conference
> notebook at:
>
> http://library.wolfram.com/infocenter/Conferences/325/
>
>
> Daniel Lichtblau
> Wolfram Research
>
>
>
>
Or why not use one of the undefined operators - such as CircleTimes -
instead of Dot? Done that way, you can avoid the need for the DotExpand
function because you can define the properties of CircleTimes directly.
David Bailey
http://www.dbaileyconsultancy.co.uk
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