Multidimensional MATLAB conversion

Question

I try to convert this MATLAB code: From:

(useful to compute some prox of some functions):

case 3
    %% 3D field %%
    V          = cat(4, ...
                U.M{1}(1:end-1,:,:) + U.M{1}(2:end,:,:), ...
                U.M{2}(:,1:end-1,:) + U.M{2}(:,2:end,:), ...
                U.M{3}(:,:,1:end-1) + U.M{3}(:,:,2:end));
case 4
    %% 4D field %%
    V          = cat(5, ...
                U.M{1}(1:end-1,:,:,:) + U.M{1}(2:end,:,:,:), ...
                U.M{2}(:,1:end-1,:,:) + U.M{2}(:,2:end,:,:), ...
                U.M{3}(:,:,1:end-1,:) + U.M{3}(:,:,2:end,:),...
                U.M{4}(:,:,:,1:end-1) + U.M{4}(:,:,:,2:end));

"The dimension of U.M{1} is (N+1,N) while that of U.M{2} is (N,N+1) for 2D field". I would like to have the same behavior on Mathematica, without a switch case, on arbitrary dimension. I try to do some mix with Take, Drop, ... Without any success.

rank = 4;
size = 30;
baseDim = ConstantArray[size, rank];
U = Table[Array[Subscript[m, ##] &, ReplacePart[baseDim, k -> size + 1]], {k, 1, rank}];

The idea of the MATLAB code is to concatenate N sub-tensor by taking the begining of and the end of each dimension and sum them up.

Here the '4' stand for the dimension where I would like to catenate, that do the same job as "Join". This is the intuition behind the Mathematica code above. In pure Mathematica for a Tensor with Rank 4 above the special case code can look like:

Join[U[[1]][[2 ;; ,   ;; ,   ;; ,   ;;]] + U[[1]][[;; -2, ;;   , ;;   , ;;   ]],
     U[[2]][[  ;; , 2 ;; ,   ;; ,   ;;]] + U[[2]][[;;   , ;; -2, ;;   , ;;   ]],
     U[[3]][[  ;; ,   ;; , 2 ;; ,   ;;]] + U[[3]][[;;   , ;;   , ;; -2, ;;   ]],
     U[[4]][[  ;; ,   ;; ,   ;; , 2 ;;]] + U[[4]][[;;   , ;;   , ;;   , ;; -2]], 5]

[Edit] The solution based on the answer of @Henrik-Schumacher:

With[{all = ConstantArray[All, rank]},
    Table[U[[k]][[Sequence @@ ReplacePart[all, k -> 2 ;;]]] + U[[k]][[Sequence @@ ReplacePart[all, k -> ;; -2]]], {k, 1, rank}]
]

Since most people here don't know Matlab, perhaps you could explain what the constructions do? For example, what are U, U.M, and what does "cat" do? — bill s, Oct 18 '18 at 00:47
you're right I update the message, remove the irrelevant U.M and add more explanation. — chkone, Oct 18 '18 at 01:05
The last Mathematica code is an example which is the working example, which I'd to generalize for arbitrary rank. — chkone, Oct 18 '18 at 05:54
I am not convinced that the last code example returns what you want. Are you aware that you try to join arrays of different Dimensions? — Henrik Schumacher, Oct 18 '18 at 06:11
Since you're dealing with tensor, are you aware that the Mathematica and MATLAB represents tensor in different way?: https://mathematica.stackexchange.com/q/10582/1871 — xzczd, Oct 18 '18 at 07:11
Also, neither of your Mathematica code sample works, please double-check it. — xzczd, Oct 18 '18 at 07:40
@xzczd that is for memory storage, in my understanding for tensor indexing seem the same m[[a, b, c, d]] will point out the same number in Matlab m(a,b,c,d). This one work: tmp = Array[Subscript[m, ##] &, {3, 3, 3, 3}]; Join[tmp[[2 ;; , ;; , ;; , ;;]] + tmp[[;; -2, ;; , ;; , ;;]], tmp[[;; , 2 ;; , ;; , ;;]] + tmp[[;; , ;; -2, ;; , ;;]], tmp[[;; , ;; , 2 ;; , ;;]] + tmp[[;; , ;; , ;; -2, ;;]], tmp[[;; , ;; , ;; , 2 ;;]] + tmp[[;; , ;; , ;; , ;; -2]], 5] — chkone, Oct 19 '18 at 03:51
The dimession give me: Dimensions /@ {tmp[[2;;,;;,;;,;;]] + tmp[[;;-2,;;,;;,;;]], tmp[[;;,2;;,;;,;;]] + tmp[[;;,;;-2,;;,;;]], tmp[[;;,;;,2;;,;;]] + tmp[[;;,;;,;;-2,;;]], tmp[[;;,;;,;;,2;;]] + tmp[[;;,;;,;;,;;-2]]} {{2, 3, 3, 3}, {3, 2, 3, 3}, {3, 3, 2, 3}, {3, 3, 3, 2}} And the join: Join[tmp[[2 ;;, ;; , ;; , ;;]] + tmp[[;; -2, ;; , ;; , ;;]], tmp[[;; , 2 ;;, ;; , ;;]] + tmp[[;; , ;; -2, ;; , ;;]], tmp[[;; , ;; , 2 ;;, ;;]] + tmp[[;; , ;; , ;; -2, ;;]], tmp[[;; , ;; , ;; , 2 ;;]] + tmp[[;; , ;; , ;; , ;; -2]], 5] // Dimensions {3, 3, 3, 3} — chkone, Oct 19 '18 at 03:52
Then your understanding is wrong, and your new sample is not working, please look at the output carefully, it contains terms like 4 Subscript[m, 1, 1, 1, 1] + Subscript[m, 1, 1, 1, 2] + Subscript[m, 1, 1, 2, 1] + Subscript[m, 1, 2, 1, 1] + Subscript[m, 2, 1, 1, 1], which indicates the output is not the one you expect. This behavior can be simplified to the following: Join[{A[x]}, {A[x]}, 2], and your mistake will be even more obvious if you replace tmp with tmp = RandomReal[1, {3, 3, 3, 3}];. — xzczd, Oct 19 '18 at 07:25
BTW, in v9.0.1 the sample will cause Join::headsd: Expression {A[x]} at position 1 is expected to have head List for all subexpressions through level 2. warning, I'm not sure when the change happens though. I've posted a separate question about this here. And, your MATLAB sample doesn't seem to be correct, either. The following sample will cause error: cat: dimension mismatch warning and fail in Octave 4.2.0: U=[1 2 3;4 5 6;7 8 9];V= cat(3, U(1:end-1,:) + U(2:end,:), U(:,1:end-1,:) + U(:,2:end)). — xzczd, Oct 19 '18 at 11:15
Maybe you can edit your post to mention where you got the MATLAB code from? — J. M.'s missing motivation, Oct 19 '18 at 11:54
@J.M.iscomputer-less sure it's from here: https://github.com/gpeyre/2013-SIIMS-ot-splitting/blob/master/code/toolbox/%40staggered/interp.m https://github.com/gpeyre/2013-SIIMS-ot-splitting/blob/master/code/toolbox/interp_adj.m — chkone, Oct 19 '18 at 18:02
-1 for posting the wrong MATLAB sample and a wrong Mathematica sample (the first one). You cannot simplify the U.M{1}, U.M{2}, etc. in the original code to a single U, Notice e.g. the dimension of U.M{1} is $(N+1, N)$ while that of U.M{2} is $(N, N+1)$ for 2D field. Please test your code first before posting it in the question. — xzczd, Oct 20 '18 at 08:11
Your samples still involve multiple mistakes. Once again, please test your code before posting it in the question. — xzczd, Oct 22 '18 at 08:18
I fix few typo issue, the code were tested. If you find something else let me know cheers. — chkone, Oct 22 '18 at 13:56
Thanks fixed. I didn't test the Matlab code. I work mainly on Mathematica. — chkone, Oct 22 '18 at 15:15

Henrik Schumacher · Accepted Answer · 2018-10-18T18:12:31.410

3

Up to the final Join which I don't understand, the following might help:

r = 4;
tmp = Array[Subscript[m, ##] &, ConstantArray[3, r]];

With[{idx = ConstantArray[All, r]},
  Table[
   tmp[[Sequence @@ ReplacePart[idx, i -> 2 ;;]]] + 
    tmp[[Sequence @@ ReplacePart[idx, i -> ;; -2]]],
   {i, 1, r}]
  ]

or

Table[Map[2 MovingAverage[#, {1, 1}] &, tmp, {i - 1}], {i, 1, r}]

edited Oct 18 '18 at 18:12

answered Oct 18 '18 at 06:15

Henrik Schumacher

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Dimension of both method give me {4} and not a tensor. – chkone Oct 19 '18 at 03:56
thanks that is the solution edit the original message – chkone Oct 21 '18 at 22:50
You're welcome. – Henrik Schumacher Oct 21 '18 at 22:53

Multidimensional MATLAB conversion

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