Re: Re: Product

*To*: mathgroup at smc.vnet.net*Subject*: [mg87681] Re: [mg87592] Re: Product*From*: "Szabolcs HorvÃt" <szhorvat at gmail.com>*Date*: Tue, 15 Apr 2008 05:55:03 -0400 (EDT)*References*: <200804110542.BAA04653@smc.vnet.net>

> > Well, maybe I conceded too much too quickly ;-) > When I wrote that "it would be quite hard or impossible ", I had in mind > dealing with general tensors. Unfortunately I did not devote to this many > any detailed thought but instinctively felt that, in the case of "general > tensors", it would not be as easy to do the same things with Apply and Plus > as with Total. When I first read your reply I again reacted too quickly and > conceded the entire point. However, after cosnidering this issue a little > more carefully, I now think the truth is more subtle. It was certainly very > rash of me to write ""it would be quite hard or impossible", as it is > clearly possible and in fact not at all hard. But now, I think, in the case > of tensors there can be significant inconveniences in using the Apply, Plus > combination instead of Total. Let me illustrate this. Consider this tensor: > > A = Array[a, {2, 3, 4}]; > > Now, with Total we can add all elements in A with: > > Total[A, Infinity] > > a[1, 1, 1] + a[1, 1, 2] + a[1, 1, 3] + a[1, 1, 4] + a[1, 2, 1] + a[1, 2, > 2] + > a[1, 2, 3] + a[1, 2, 4] + > a[1, 3, 1] + a[1, 3, 2] + a[1, 3, 3] + a[1, 3, 4] + a[2, 1, 1] + > a[2, 1, 2] + a[2, 1, 3] + a[2, 1, 4] + > a[2, 2, 1] + a[2, 2, 2] + a[2, 2, 3] + a[2, 2, 4] + a[2, 3, 1] + > a[2, 3, 2] + a[2, 3, 3] + a[2, 3, 4] > > or we can do the same with Total[A,-1]. > > Note, however, that this will give a completely different answer: > > Apply[Plus, A, {0, Infinity}] > 144 > > Of course what happened is that all the indexes got added. If we want to > add all the tensor entries you > need to correctly specify the highest level: > > Apply[Plus, A, {0, 2}] > a[1, 1, 1] + a[1, 1, 2] + a[1, 1, 3] + a[1, 1, 4] + a[1, 2, 1] + a[1, 2, 2] > + > a[1, 2, 3] + a[1, 2, 4] + > a[1, 3, 1] + a[1, 3, 2] + a[1, 3, 3] + a[1, 3, 4] + a[2, 1, 1] + > a[2, 1, 2] + a[2, 1, 3] + a[2, 1, 4] + > a[2, 2, 1] + a[2, 2, 2] + a[2, 2, 3] + a[2, 2, 4] + a[2, 3, 1] + > a[2, 3, 2] + a[2, 3, 3] + a[2, 3, 4] > > Note also that now your rule: > > > > > > > > Total[expression, {n}] is equivalent to Apply[Plus, expression, {n-1}] > > > > > > > does not work for negative indexes. Total[A,{-1}] is Apply[Plus,A,{-3}] > and not Apply[Plus,A,-2}]. > The reason is again the same: Apply counts negative levels beginning with > the atoms, which are in this case just the indexes. > > OK, so now comes my main point. Suppose that now we define > > a[x_, y_, z_] := Times[x, y, z] > > With this definition the array A turns into an array of integers. Now, note > that > > Total[A, Infinity] > 180 > Total[A, -1] > 180 > > have the same meaning as before, they are both the sum of all the entries > of A. But the meaning of > > Apply[Plus, A, {0, Infinity}] > 180 > > has now changed. Recall that earlier the answer was 140, which was the sum > of the indexes, and now it has become 180, which is the sum of entries. The > same applies to some of the other cases of using the Apply, Plus > combination. This "change of meaning" can be a serious inconvenience in > certain situations. > > So while you can certainly do the same things with Apply and Plus and level > specifications as with Total and level specifications, they seem to be > intended for different purposes. Apply and Plus are more suitable for > handling general expressions while Total is much more convenient for dealing > with tensors. While you can always achieve the same purpose with Apply and > Plus as with Total, doing so is not completely automatic but has to be based > on individual cases. > > So I conclude that the reason for the existence of Total is not simply that > of syntax but more importantly that of purpose: the Apply, Plus combination > is more suitable for handling general expressions but Total is more suitable > for dealing with tensors. I think it is a fairly significant difference. Dear Andrzej, Thanks for pointing this out! Now I see that Apply[Plus, ...] and Total[] have a more important essential difference than the Method option. To sum up the main difference: Apply treats expressions with different heads in the same way. But Total only touches those subexpressions whose head matches the head of the complete expression. For example, Total[ f[...], Inifnity ] will only sum the terms of expressions with head f, and Total[ {...}, Infinity ] will only sum lists (but it treats everything else as atoms). Szabolcs

**References**:**Product***From:*"Steven" <steven_504@telenet.be>

**Re: Labelling contours on contour plots**

**Re: A Problem with Simplify**

**Re: Re: Product**

**RE: Product**