How to generate a multidimensional array of indexed variables in Maxima? - maxima

Question

I want to take a list of non-negative integers D=[d1,...,dm] and and generate a multidimensional array of indexed symbols A in the form of:
where 0<=i_j<=d_j. For example if D=[2,3] then A should be
[[a_[0,0],a_[0,1],a_[0,2]],
[a_[1,0],a_[1,1],a_[1,2]]]
For this case I could nest two for loops to generate the said array, however D does not necessarily have a length of 2 and I don't know how to nest an arbitrary number of for loops!
I would appreciate if you could help me know how I can generate A from D.
P.S. What I want to finally achieve is to create a multivariate polynomial as explained here.

Answer 1

Here's one way to do it. The essential part is that I called cartesian_product to construct the list of all combinations of indices, and then arrayapply to create the subscripted expressions.
(%i11) ii:setify(makelist(i, i, 0, n)), n=2;
(%o11) {0, 1, 2}
(%i12) apply (cartesian_product, makelist (ii, m)), m=3;
(%o12) {[0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, 0], [0, 1, 1],
[0, 1, 2], [0, 2, 0], [0, 2, 1], [0, 2, 2], [1, 0, 0],
[1, 0, 1], [1, 0, 2], [1, 1, 0], [1, 1, 1], [1, 1, 2],
[1, 2, 0], [1, 2, 1], [1, 2, 2], [2, 0, 0], [2, 0, 1],
[2, 0, 2], [2, 1, 0], [2, 1, 1], [2, 1, 2], [2, 2, 0],
[2, 2, 1], [2, 2, 2]}
(%i13) map (lambda ([l], arrayapply (_a, l)), %);
(%o13) {_a , _a , _a , _a , _a ,
0, 0, 0 0, 0, 1 0, 0, 2 0, 1, 0 0, 1, 1
_a , _a , _a , _a , _a ,
0, 1, 2 0, 2, 0 0, 2, 1 0, 2, 2 1, 0, 0
_a , _a , _a , _a , _a ,
1, 0, 1 1, 0, 2 1, 1, 0 1, 1, 1 1, 1, 2
_a , _a , _a , _a , _a ,
1, 2, 0 1, 2, 1 1, 2, 2 2, 0, 0 2, 0, 1
_a , _a , _a , _a , _a ,
2, 0, 2 2, 1, 0 2, 1, 1 2, 1, 2 2, 2, 0
_a , _a }
2, 2, 1 2, 2, 2
(%i14) grind (%);
{_a[0,0,0],_a[0,0,1],_a[0,0,2],_a[0,1,0],_a[0,1,1],_a[0,1,2],
_a[0,2,0],_a[0,2,1],_a[0,2,2],_a[1,0,0],_a[1,0,1],_a[1,0,2],
_a[1,1,0],_a[1,1,1],_a[1,1,2],_a[1,2,0],_a[1,2,1],_a[1,2,2],
_a[2,0,0],_a[2,0,1],_a[2,0,2],_a[2,1,0],_a[2,1,1],_a[2,1,2],
_a[2,2,0],_a[2,2,1],_a[2,2,2]}$
(%o14) done
This is just working at the top-level interactive prompt; if you need to construct a function, I think you'll see how to do it.
EDIT: Here's a way to create the polynomial.
(%i16) S : {[0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, 0], [0, 1, 1],
[0, 1, 2], [0, 2, 0], [0, 2, 1], [0, 2, 2], [1, 0, 0],
[1, 0, 1], [1, 0, 2], [1, 1, 0], [1, 1, 1], [1, 1, 2],
[1, 2, 0], [1, 2, 1], [1, 2, 2], [2, 0, 0], [2, 0, 1],
[2, 0, 2], [2, 1, 0], [2, 1, 1], [2, 1, 2], [2, 2, 0],
[2, 2, 1], [2, 2, 2]} $
(%i17) L : listify (S) $
(%i18) A : map (lambda ([l], arrayapply (_a, l)), L);
(%o18) [_a , _a , _a , _a , _a ,
0, 0, 0 0, 0, 1 0, 0, 2 0, 1, 0 0, 1, 1
_a , _a , _a , _a , _a ,
0, 1, 2 0, 2, 0 0, 2, 1 0, 2, 2 1, 0, 0
_a , _a , _a , _a , _a ,
1, 0, 1 1, 0, 2 1, 1, 0 1, 1, 1 1, 1, 2
_a , _a , _a , _a , _a ,
1, 2, 0 1, 2, 1 1, 2, 2 2, 0, 0 2, 0, 1
_a , _a , _a , _a , _a ,
2, 0, 2 2, 1, 0 2, 1, 1 2, 1, 2 2, 2, 0
_a , _a ]
2, 2, 1 2, 2, 2
(%i19) U : map (lambda ([l], product (u[i]^l[i], i, 1, length(l))), L);
2 2 2 2 2 2
(%o19) [1, u , u , u , u u , u u , u , u u , u u , u ,
3 3 2 2 3 2 3 2 2 3 2 3 1
2 2 2 2
u u , u u , u u , u u u , u u u , u u , u u u ,
1 3 1 3 1 2 1 2 3 1 2 3 1 2 1 2 3
2 2 2 2 2 2 2 2 2 2 2 2
u u u , u , u u , u u , u u , u u u , u u u , u u ,
1 2 3 1 1 3 1 3 1 2 1 2 3 1 2 3 1 2
2 2 2 2 2
u u u , u u u ]
1 2 3 1 2 3
(%i20) A.U;
2 2 2 2 2 2 2
(%o20) u u _a u + u u _a u + u _a u
1 2 2, 2, 2 3 1 2 2, 1, 2 3 1 2, 0, 2 3
2 2 2 2 2
+ u _a u u + _a u u + u _a u u
1 1, 2, 2 2 3 0, 2, 2 2 3 1 1, 1, 2 2 3
2 2 2
+ _a u u + u _a u + _a u
0, 1, 2 2 3 1 1, 0, 2 3 0, 0, 2 3
2 2 2 2
+ u u _a u + u u _a u + u _a u
1 2 2, 2, 1 3 1 2 2, 1, 1 3 1 2, 0, 1 3
2 2
+ u _a u u + _a u u + u _a u u
1 1, 2, 1 2 3 0, 2, 1 2 3 1 1, 1, 1 2 3
+ _a u u + u _a u + _a u
0, 1, 1 2 3 1 1, 0, 1 3 0, 0, 1 3
2 2 2 2
+ u u _a + u u _a + u _a
1 2 2, 2, 0 1 2 2, 1, 0 1 2, 0, 0
2 2
+ u _a u + _a u + u _a u
1 1, 2, 0 2 0, 2, 0 2 1 1, 1, 0 2
+ _a u + u _a + _a
0, 1, 0 2 1 1, 0, 0 0, 0, 0
Note that the ordering of terms within each product doesn't conform to what humans would consider the usual convention, e.g. [1]^2*u[2]^2*_a[2,2,2]*u[3]^2 is the first term. Maxima is ordering the terms according to the subscripts, therefore _a[2,2,2] comes after u[1] and before u[3]. In some contexts this coincides with what humans expect, but here it doesn't; in any event, Maxima is consistent in hope of making programmatic manipulation work better.
(%i21) grind (%);
u[1]^2*u[2]^2*_a[2,2,2]*u[3]^2+u[1]^2*u[2]*_a[2,1,2]*u[3]^2
+u[1]^2*_a[2,0,2]*u[3]^2
+u[1]*_a[1,2,2]*u[2]^2*u[3]^2
+_a[0,2,2]*u[2]^2*u[3]^2
+u[1]*_a[1,1,2]*u[2]*u[3]^2
+_a[0,1,2]*u[2]*u[3]^2
+u[1]*_a[1,0,2]*u[3]^2
+_a[0,0,2]*u[3]^2
+u[1]^2*u[2]^2*_a[2,2,1]*u[3]
+u[1]^2*u[2]*_a[2,1,1]*u[3]
+u[1]^2*_a[2,0,1]*u[3]
+u[1]*_a[1,2,1]*u[2]^2*u[3]
+_a[0,2,1]*u[2]^2*u[3]
+u[1]*_a[1,1,1]*u[2]*u[3]
+_a[0,1,1]*u[2]*u[3]
+u[1]*_a[1,0,1]*u[3]
+_a[0,0,1]*u[3]
+u[1]^2*u[2]^2*_a[2,2,0]
+u[1]^2*u[2]*_a[2,1,0]
+u[1]^2*_a[2,0,0]
+u[1]*_a[1,2,0]*u[2]^2
+_a[0,2,0]*u[2]^2
+u[1]*_a[1,1,0]*u[2]
+_a[0,1,0]*u[2]+u[1]*_a[1,0,0]
+_a[0,0,0]$
(%o21) done