Re: Summing above diagonal in a tensor

• To: mathgroup at smc.vnet.net
• Subject: [mg112601] Re: Summing above diagonal in a tensor
• From: Leonid Shifrin <lshifr at gmail.com>
• Date: Wed, 22 Sep 2010 01:58:01 -0400 (EDT)

```Hi Alice,

Here is a test tensor:

In[111]:== Clear[a,tensor]
(tensor == Array[a,{3,3,3,3}])//Short[#,5]&
Out[112]//Short==
{{{{a[1,1,1,1],a[1,1,1,2],a[1,1,1,3]},{a[1,1,2,1],a[1,1,2,2],a[1,1,2,3]},{a[1,1,3,1],a[1,1,3,2],a[1,1,3,3]}},{{a[1,2,1,1],a[1,2,1,2],a[1,2,1,3]},{a[1,2,2,1],a[1,2,2,2],a[1,2,2,3]},{a[1,2,3,1],a[1,2,3,2],a[1,2,3,3]}},{{a[1,3,1,1],a[1,3,1,2],a[1,3,1,3]},{a[1,3,2,1],a[1,3,2,2],a[1,3,2,3]},{a[1,3,3,1],a[1,3,3,2],a[1,3,3,3]}}},{{{a[2,1,1,1],a[2,1,1,2],a[2,1,1,3]},{<<1>>},{<<1>>}},{<<1>>},{<<1>>}},{<<1>>}}

Here are a couple of brute force solutions:

In[113]:==
Total@Extract[tensor,
Select[Tuples[Range/@Dimensions[tensor]],#[[1]]<#[[2]]&&#[[3]]<#[[4]]&]]

Out[113]==
a[1,2,1,2]+a[1,2,1,3]+a[1,2,2,3]+a[1,3,1,2]+a[1,3,1,3]+a[1,3,2,3]+a[2,3,1,2]+a[2,3,1,3]+a[2,3,2,3]

In[114]:==
Total@Flatten@ MapIndexed[#1*Boole[#2[[1]] < #2[[2]] && #2[[3]] < #2[[4]]]
&, tensor, {4}]

Out[114]==
a[1,2,1,2]+a[1,2,1,3]+a[1,2,2,3]+a[1,3,1,2]+a[1,3,1,3]+a[1,3,2,3]+a[2,3,1,2]+a[2,3,1,3]+a[2,3,2,3]

The following one I consider somewhat more elegant, since it does not go to
the low-level position
details and explicitly shows your request to "double - UpperTriangularize":

In[115]:==
Clear[fourthRankTensorUpperTriangularSum];
fourthRankTensorUpperTriangularSum[tensor_,n_]:==
Module[{b,outerMatrix,rules},
outerMatrix== Array[b,Take[Dimensions[tensor],2]];
rules ==Flatten[
Total[Flatten[UpperTriangularize[outerMatrix,n]/.rules]]];

In[117]:== fourthRankTensorUpperTriangularSum[tensor,1]

Out[117]==
a[1,2,1,2]+a[1,2,1,3]+a[1,2,2,3]+a[1,3,1,2]+a[1,3,1,3]+a[1,3,2,3]+a[2,3,1,2]+a[2,3,1,3]+a[2,3,2,3]

Hope this helps.

Regards,
Leonid

On Tue, Sep 21, 2010 at 10:05 AM, Alice Lesser <alice.lesser at bwin.org>wrote:

> Hi group,
>
> Summing the elements of a matrix which are above the diagonal is easy:
>
> m == Table[Subscript[a, x, y], {x, 1, 3}, {y, 1, 3}]
> Total[UpperTriangularize[m, 1], 2]
>
> gives
> a_1,2 + a_1,3 + a_2,3
> of course.
>
> If I instead have a 4-dim tensor
>
> t == Table[ Subscript[a, x, y, z, q], {x, 1, 3}, {y, 1, 3}, {z, 1, 3}, {q,
> 1, 3}]
>
> is there a nifty way to get the sum of the elements which are above
> the diagonal in the outer matrix and above the diagonal in the submatrices?
> In other words I'm looking for the sum
>
> a_1,2,1,2 + a_1,2,1,3 + a_1,2,2,3 + a_1,3,1,2 + a_1,3,1,3 + a_1,3,2,3+ a_2,
> 3,1,2 + a_2,3,1,3 + a_2,3,2,3
>
> UpperTriangularize only seems to work on matrices, not tensors.
> I could of course solve this with a few ugly for-loops, but it would be
> nice to know if there is a better way.
>
> Thanks,
> Alice
>
>
>

```

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