Making a Double Loop with Specified Indices $(i,j)$

Question

I want to make a double loop with Do. Let me first make a simple example. Consider the following simple double loop

Do[f[i,j]=i*j,{i,{1,2}},{j,{1,2}}]

does the job for $(i,j)=(1,1),(1,2),(2,1),(2,2)$ respectively. Now, suppose that I just want to do the same job but for $(i,j)=(1,1),(2,2)$. Clearly, one can use if statements inside Do but I don't like such a solution since as the range of $(i,j)$ becomes large the process becomes time consuming. Also, if there is not any pattern for the range of loop indices $(i,j)$ then making use of if statements becomes complicated.

Now, let us express the general problem. Suppose that we have a random list for index $i$ and another random list for index $j$. I want to do a sequence of operations for each $(i,j)$ inside a Do loop. What is the best way to do this?

Answer 1

Bastian already showed MapThread but that function builds output (and uses memory) that I suspect you do not want. Here is another approach using a single separate index:

a = {1,2};
b = {1,2};

Do[f[i,i] = a[[i]]*b[[i]], {i, 2}]

Update: looking again I think I misunderstood and instead you would want something like:

Do[(f[##] = #*#2) &[a[[i]], b[[i]]], {i, 2}]

In either case the index approach should be applicable with adjustment.

Consider also using an Association as an alternative to DownValues function definitions.

fn = <| MapThread[{#, #2} -> #*#2 &, {a, b}] |>

<|{1, 1} -> 1, {2, 2} -> 4|>

You can then use fn like fn[{1, 1}] to get your value.

Also be aware of Scan, which could be applied in this manner:

Scan[Apply[(f[##] = #*#2) &], {a, b}\[Transpose]]

Answer 2

If you always have (1,1), (2,2), (3,3) etc. just use f[i,i] = i*i.

In general: Note that in Mathematica it is nicer to use functional coding style instead of a procedural one. For your case it would be most easy to use

MapThread[f[#1,#2] = #1*#2 &, {{1,2},{1,2}}]]