       Re: A simple question?

• To: mathgroup at smc.vnet.net
• Subject: [mg3651] Re: A simple question?
• From: lou (Lou D'Andria)
• Date: Sun, 31 Mar 1996 00:52:59 -0500
• Organization: Wolfram Research, Inc.
• Sender: owner-wri-mathgroup at wolfram.com

```In article <4jd5ej\$pmr at dragonfly.wolfram.com>, nader at math.chalmers.se

>        I guess this must be a simple question, but I can't seem to
>find a good solution to it. For a function f[x,y] I want to create a
>2x2x2x2 matrix called fourthDs. The elements of fourthDs will be the
>fourth derivatives of f with respect to x and y in the following form:
>
>fourthDs[[1,1,1,1]]=D[f[x,y],{x,4}]
>fourthDs[[2,2,2,2]]=D[f[x,y],{y,4}]
>fourthDs[[1,2,1,1]]=D[f[x,y],{x,3},{y,1}]
>
>        This simply means that the number of 1's in the index will
>give the number of derivations with respect to x and the number of 2's
>in the index will give the number of derivations with respect to y. In
>this case there will be 5 different elements in fourthDs. For example,
>the last element in the above list will be the same for
>fourthDs[[2,1,1,1]], fourthDs[[1,1,2,1]] and so on.

As usual with Mathematica, there are many, many different ways to do what
you need.  The most straight-forward, given your description in the last
paragraph above, is this.

Array[D[f[x,y],{x,Count[{##},1]},{y,Count[{##},2]}]&,{2,2,2,2}]

There is at least one problem with this though.  It calculates most of the
derivatives more than once.  In fact, D[f[x,y],{x,2},{y,2}] is calculated
6 different times!  That's a bit wasteful.  Here's a solution that
calculates each derivative exactly once.

Array[Plus,{2,2,2,2},0] /. Table[i->D[f[x,y],{x,4-i},{y,i}],{i,0,4}]

The third argument to Array sets the "origin" of the array.  So that
instead of elements running from {1,1,1,1} through {2,2,2,2}, the elements
will run from {0,0,0,0} to {1,1,1,1}.

Here are some more methods which will work - almost one for every day of
the week.  ;-)  I believe that each one uses a different idea, and all of
them except the last one are wasteful in their calculation of
derivatives.  Enjoy!

Array[D[f[x,y],##]&@@{x,y}[[{##}]]&,{2,2,2,2}]

Array[D[f[x,y],{x,4-#},{y,#}]&[Plus[##]]&,{2,2,2,2},0]

Array[Unevaluated[D[f[x,y],##]]/.{1->x,2->y}&,{2,2,2,2}]

Table[D[f[x,y],{x,4-i-j-k-l},{y,i+j+k+l}],
{i,0,1},{j,0,1},{k,0,1},{l,0,1}]

Fold[
Map[Function[{z},Partition[z,#2,1]],#,{-3}]&,
Table[D[f[x,y],{x,4-i},{y,i}],{i,0,4}],
{4,3,2}
]

Lou D'Andria

==== [MESSAGE SEPARATOR] ====

```

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