How can I write a Do loop that iterates over N iterators when N is not known a-priori?

Question

For any positive integer $n$ which is not specified a priori, I can make $n$-tuple vector $v$ by v = Array[a, 1, n]. Now, given that $n$ is a positive integer which is not specified a priori, I want to write an iteration sentence which performs over $1\leq a_1,\ldots, a_n\leq 3$. ($a_i$ are integers and hence I will get $3^n$ iterations).

For example, if $n=2$, my sentence plays the same role as the following loop:

Do[EXPERIMENT, {i, 3}, {j, 3}]

If $n=3$, my sentence plays the same role as the following:

Do[EXPERIMENT, {i, 3}, {j, 3}, {k, 3}].

I can do it for any given SPECIFIED integer $n$, but I want to write the universal looping sentence which iterates for $1\leq a_1,\ldots, a_n\leq 3$ ($a_i$ are integers.)

Answer 1

Following belisarius' suggestion you might use something like:

SetAttributes[f, HoldFirst]

f[body_, n_Integer?Positive, m__:3] := Block[{a},
  Do[body, ##] & @@ Thread[{Array[a, n], m}]
]

Use:

f[Print[a[1], a[2], a[3]], 3]

f[Print[a[1], a[2]], 2, 4]

f[Print[a[1], a[2]], 2, 5, 7]

How are parameters evaluated for a Plot in Manipulate

The second parameter is your n value. The third and following parameters are the range specification for the loops.

I Blocked a to prevent a global assignment to that symbol from breaking the function.

Answer 2

Say for $n=1$, you have defined f[{i1}] as your first experiment. For $n=2$, your experiments are defined by f[{i1,i2}], for $n=3$ the experiments are defined by f[{i1,i2,i3}] and etc, with the idea that you wish to allow $n$ such experiments. If you structure your experiments this way, then you can create a list of all experiments by

m = 3; n = 3; c = Tuples[Range[m], n]

Here $m$ is the number of values each variable is to take on, and $n$ is how many variables you have. Now you can run all your experiments at once using

Map[f, c]

or use the shortcut

f /@ c

For $m=n=3$, this gives

{f[{1, 1, 1}], f[{1, 1, 2}], f[{1, 1, 3}], f[{1, 2, 1}], f[{1, 2, 2}],
 f[{1, 2, 3}], f[{1, 3, 1}], f[{1, 3, 2}], f[{1, 3, 3}], 
 f[{2, 1, 1}], f[{2, 1, 2}], f[{2, 1, 3}], f[{2, 2, 1}], f[{2, 2, 2}],
 f[{2, 2, 3}], f[{2, 3, 1}], f[{2, 3, 2}], f[{2, 3, 3}], 
 f[{3, 1, 1}], f[{3, 1, 2}], f[{3, 1, 3}], f[{3, 2, 1}], f[{3, 2, 2}],
 f[{3, 2, 3}], f[{3, 3, 1}], f[{3, 3, 2}], f[{3, 3, 3}]}

so you can see all 3^3 experiments will run. In general you will have $n^m$ experiments. As Mr. Wizard suggests, an alternative to Map would be to use Scan:

Scan[Print[f[#]] &, c]

This might be preferred if your experiment function f has "side effects", such as making assignments, or generating output, since Scan will suppress these.