Re: partials of Sum[x[i]^2, {i,1,n}] (e.g.)

*To*: mathgroup at smc.vnet.net*Subject*: [mg69554] Re: partials of Sum[x[i]^2, {i,1,n}] (e.g.)*From*: dimmechan at yahoo.com*Date*: Fri, 15 Sep 2006 06:45:56 -0400 (EDT)*References*: <eebfke$4d8$1@smc.vnet.net>

***Mathematica as far as I know can work with limits in symbolic form. ***You can define the function you are interested as follows: In[63]:= f[lst:{x_,y___}]:=Sum[(lst[[i]])^2,{i,1,Length[lst]}] ***where In[70]:= ?___ Out[70]= ___ (three _ characters) or BlankNullSequence[ ] is a pattern object that can \ stand for any sequence of zero or more Mathematica expressions. ___h or \ BlankNullSequence[h] can stand for any sequence of expressions, all of which \ have head h. More... ***Then e.g. In[65]:= D[f[{Cos[x],Sin[x],Tan[x],Exp[y]}],Sin[x]] Out[65]= 2 Sin[x] In[66]:= D[f[{Cos[x],Sin[x],Tan[x],Exp[y]}],y] Out[66]= 2*E^(2*y) ***In fact you do not need here the limits. Indeed In[82]:= f[lst:{x_,y___}]:=Apply[Plus,lst^2] ***where In[80]:= ?Apply Out[80]= Apply[f, expr] or f @@ expr replaces the head of expr by f. Apply[f, expr, \ levelspec] replaces heads in parts of expr specified by levelspec. More... ***Then e.g. In[84]:= D[f[{Cos[x],Sin[x],Tan[x]}],Tan[x]] Out[84]= 2*Tan[x] Dimitris Anagnostou Î?/Î? kj ÎÎ³Ï?Î±Ï?Îµ: > In symbolic manipulations, one often needs to leave some of the > limits of an expression in symbolic form. E.g. the n in: > > f[x[1],...,x[n]] = Sum[x[i]^2, {i, 1, n}], > > (where I've used Mathematica notation loosely). > > Then one often finds derivations like > > D[f[x[1],...,x[n]], x[k]] = 2 x[k], for all k in { 1,..., n } > > Is it possible to do something like this in Mathematica? > > More generally, can Mathematica fully understand expressions with > symbolic limits? > > Basically, I have a slightly hairier expression that I want to take > the partials of, set them all equal to zero to produce a system of > n equations. If that weren't enough, I'd like to solve this system > of n equations using Mathematica. This kind of manipulation is > far more difficult, as far as symbolic math goes, than anything > I've seen Mathematica do yet, because it requires Mathematica to > understand the notion of an array with a "symbolic cardinality", > but I thought I'd ask. > > Thanks! > > kj > -- > NOTE: In my address everything before the first period is backwards; > and the last period, and everything after it, should be discarded.