# Numbering indices / array element positions while creating an array (using 3 argument form)

I am new to Mathematica and trying to create a list that I can use within the InterpolatingPolynomial function. e.g. {{-1, 0},{ 0, 0},{1, 1}}.

I extended the "linearMesh" function from here: https://mathematica.stackexchange.com/a/32721 to insert n array elements

linearmesh[a_, b_, n_Integer] := Array[# &, n, {a, b}]


became:

linearmesh[a_, b_, n_Integer] :=
Array[# &,
n, {{a, KroneckerDelta[i, j]}, {b, KroneckerDelta[i, j]}}];


My aim is to produce, e.g. for a==-1 and b==1 and 3 elements:

{{-1, KroneckerDelta[i, j]}, {0, KroneckerDelta[i, j]}, {1,
KroneckerDelta[i, j]}}


where each of the KroneckerDeltas is calculated with respect to a variable i, and j is the index of the array element. How can I get the index, since the array is being defined at the same time?

What I tried:

linearmesh[a_, b_, n_Integer] :=
Array[# &,
n, {{a, KroneckerDelta[i, #1]}, {b, KroneckerDelta[i, #1]}}];


and then linearmesh[-1,1,3] gives me results:

{{-1, KroneckerDelta[i, #1]}, {0, KroneckerDelta[i, #1]}, {1, KroneckerDelta[i, #1]}}


rather than substituting the values of each of the # indices and obtaining

{{-1, KroneckerDelta[i, 1]}, {0, KroneckerDelta[i, 2]}, {1, KroneckerDelta[i, 3]}}


I also considered using Parts, but I can't do that since the array doesn't exist yet while linearmesh is being defined.

Thanks!

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– user9660
Mar 3, 2016 at 18:22
• Thanks Louis! I came over from StackOverflow so I am familiar with the system!! I appreciate it. Mar 3, 2016 at 18:27

One possibility:

linearmesh[a_, b_, n_Integer] := {a + (b - a)/(n - 1) #, KroneckerDelta[i, # + 1]} & /@ Range[0, n - 1]


The reason that

linearmesh[a_, b_, n_Integer] :=
Array[# &,
n, {{a, KroneckerDelta[i, #1]}, {b, KroneckerDelta[i, #1]}}];


doesn't work is that the third argument to Array can't be a function (and certainly the expressions put in place of # in the first argument don't get also fed to the third argument).

• Thanks! This seems perfect, but it doesn't work for other values of n (except 3). The range is not being properly calculated. Could you please help check? e.g. for 2 items I have {{0, KroneckerDelta[1, i]}, {2, KroneckerDelta[2, i]}} instead of {{-1, KroneckerDelta[1, i]}, {1, KroneckerDelta[2, i]}} Mar 3, 2016 at 18:25
• @Cogicero. Indexing problem. Edited. Does it work now? Mar 3, 2016 at 18:27
• Thanks for the edit! I further edited it to linearmesh[a_, b_, n_Integer] := {a + (b - a)/(n - 1) #, KroneckerDelta[i, # + 1]} & /@ Range[0, n - 1] nd it works exactly as intended. Accepting your answer! Thanks again Mar 3, 2016 at 18:30
• @Cogicero. Yes, missed that. I advise not accepting this just yet. Other people are likely to come up with more clever solutions, and the more answers the better. Mar 3, 2016 at 18:31
• An alternative: linearmesh[a_, b_, n_Integer] := Transpose[{Subdivide[a, b, n - 1], Thread[KroneckerDelta[i, Range[n]]]}] Mar 3, 2016 at 22:42

You can use MapIndexed.

linearmesh[a_, b_, n_Integer] :=
MapIndexed[{#1, KroneckerDelta[i, First@#2]} &]@Array[# &, n, {a, b}]


Then

linearmesh[-1, 1, 3]
(* {{-1, KroneckerDelta[1, i]}, {0, KroneckerDelta[2, i]}, {1, KroneckerDelta[3, i]}} *)


The order changes but KroneckerDelta is orderless. Actually, that is perhaps why it changes since it knows what the numbers are but not the i.

Hope this helps.

• Checking out MapIndexed. Thank you! Mar 3, 2016 at 19:08