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I need to populate a matrix $A_{kl}$, where

$$ k = (m-1)J+n$$ $$ l = (p-1)J+q$$

And

$$m,p = 1, 2, ..., I$$ $$n,q = 1, 2, ..., J$$

Its components are (mnpq). For populate it, I'm using an expensive 4 Do loop

Do[ Do[ Do[ Do[ Print[m, n, p, q]; k = (m - 1) nC + n; l = (p - 1) nC + q; If[k <= l, A[[k, l]] = cf[Nfunc, xi, yi, wix, wiy, m, n, p, q], 0]; , {q, 1, J, 1}] , {p, 1, I, 1}] , {n, 1, J, 1}] , {m, 1, I, 1}]

Knowing that $A_{kl}$ for a $I=J=2$, its components are (mnpq)

$$ \begin{bmatrix} (1111) & (1112) & (1121) & (1122)\\ & (1212) & (1221) & (1222)\\ symm. & & (2121) & (2122)\\ & & & (2222)\\ \end{bmatrix} $$

Does anyone know a more efficient way to populate it? Maybe using a bult-in function?

UPDATE

cf = Compile[{{Nfunc, _Real, 2}, {xi, _Real, 1}, {yi, _Real, 1}, {wix, _Real, 1}, {wiy, _Real, 1}, {m, _Integer}, {n, _Integer}, {p, _Integer}, {q, _Integer}}, Module[{sum}, sum = 0.0; For[i = 1, i <= Length@xi, i++, For[j = 1, j <= Length@yi, j++, sum = sum + (8 \[Pi]^2)/ a^2 m p Cos[(m \[Pi] xi[[i]])/((1/ 2) a)] Sin[(n \[Pi] yi[[j]])/((1/ 2) b)] Cos[(p \[Pi] xi[[i]])/((1/ 2) a)] Sin[(q \[Pi] yi[[j]])/((1/2) b)]*wix[[i]]* wiy[[j]]*Nfunc[[i, j]]; ] ]; sum], CompilationTarget -> "C", RuntimeAttributes -> {Listable}, Parallelization -> True, RuntimeOptions -> "Speed"];
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  • $\begingroup$ At least you can just use only one Do. $\endgroup$ Commented May 5, 2020 at 13:32
  • $\begingroup$ What is the nature of cf? Is it Listable in any of its arguments? I.e. would cf[nFunc, xi, yi, wix, wiy, m, n, p, {q1, q2, q3}] correctly give a list of three values? $\endgroup$ Commented May 5, 2020 at 13:35
  • $\begingroup$ @ΑλέξανδροςΖεγγ, that is the exact point. I'm not getting create a algorithm to only one Do $\endgroup$ Commented May 5, 2020 at 13:53
  • $\begingroup$ @MariusLadegårdMeyer, m,n,p,q are integers. All others are lists. cf = Compile[{{Nfunc, _Real, 2}, {xi, _Real, 1}, {yi, _Real, 1}, {wix, _Real, 1}, {wiy, _Real, 1}, {m, _Integer}, {n, _Integer}, {p, _Integer}, {q, _Integer}}. $\endgroup$ Commented May 5, 2020 at 13:55
  • $\begingroup$ Yes, but you want to evaluate this function not for only one value of these integers, but for many different ones. If we put these many different ones in a list, and feed that list to cf, will it work? For instance, if I want to find Sin[x] for x = 1, 2, 3, ..., 100 I don't need to loop, I just do Sin[Range[100]]. I'm asking whether this applies to your function cf. $\endgroup$ Commented May 5, 2020 at 13:58

2 Answers 2

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Your $m-1, n-1$ and $p-1, q-1$ are the two digits of $k-1$ and $l-1$, respectively, in base $J$. This should be pretty fast:

dim = i*j; A = ConstantArray[0, {dim, dim}]; Do[ A[[k, l]] = A[[l, k]] = cf[Nfunc, xi, yi, wix, wiy, Quotient[k - 1, j] + 1, Mod[k - 1, j] + 1, Quotient[l - 1, j] + 1, Mod[l - 1, j]+ 1] , {k, 1, dim}, {l, k, dim} ]

If $IJ$ is huge and your cf is very very fast, then you can probably shave off a little bit more time by making an outer Do over k where you compute the quotient and mod once for that k, followed by an inner Do over l.

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  • $\begingroup$ I will update my question adding cf. Could you analyse it? Thank you for answering! $\endgroup$ Commented May 5, 2020 at 14:29
  • $\begingroup$ It doing i = 2; j = 2; dim = i*j; A = ConstantArray[0, {dim, dim}]; Do[A[[k, l]] = A[[l, k]] = {Quotient[k, j] + 1, Mod[k, j], Quotient[l, j] + 1, Mod[l, j]}, {k, 1, dim}, {l, k, dim}], we do not get the @bill s answer, as should be. $\endgroup$ Commented May 5, 2020 at 14:58
  • $\begingroup$ I have fixed the off-by-one issue. $\endgroup$ Commented May 5, 2020 at 15:10
  • $\begingroup$ Perfect, @Marius. Thank you very much! $\endgroup$ Commented May 5, 2020 at 15:29
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One way to create your data structure is to realize that your list of integers is closely related to all the Tuples taken in groups. For the case I=J=2, the permutations are of 1 and 2 taken 4 at a time:

mat = Partition[Tuples[{1, 2}, 4], 4]; mask = UpperTriangularize[ConstantArray[1, {4, 4}]]; outMat = mat mask + Transpose[mat mask] - mat IdentityMatrix[4] outMat//MatrixForm

enter image description here

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  • $\begingroup$ Thank you answering. But I did not understand how I will access each component of the matrix A using your approach. $\endgroup$ Commented May 5, 2020 at 14:28
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    $\begingroup$ Use Part ([[ ]]). For example, outMat[[2, 3]] will give you the 2,3 element of the matrix. $\endgroup$ Commented May 5, 2020 at 15:07
  • $\begingroup$ What a dumb! Sorry for this dumb question! Thank you very much! I accepted the @Marius's answer because it was more straight for me. But, without doubt, your approach is really great too! $\endgroup$ Commented May 5, 2020 at 15:28
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    $\begingroup$ I should also note that generating tuples from a length-$k$ list is isomorphic to getting base-$k$ digits, so: mat = Partition[IntegerDigits[Range[0, 15], 2, 4] + 1, 4] $\endgroup$ Commented May 5, 2020 at 16:39

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