There are quite a few different concepts inside your code. Let's start with the most basic ones. Python lists and numpy arrays have different methodologies for indexation. Also you can build a numpy array by providing it a list:
S_list = [[1,2,3], [4,5,6], [7,8,9]]
S_array = np.array(S_list)
print(S_list)
print(S_array)
print(S_list[0][2]) # indexing element 2 from list 0
print(S_array[0,2]) # indexing element at position 0,2 of 2-dimensional array
This results in:
[[1, 2, 3], [4, 5, 6], [7, 8, 9]]
[[1 2 3]
[4 5 6]
[7 8 9]]
3
3
So for your first line of code:
np.array([[[S[i,j]] for i in range(order+1)] for j in range(order+1)])
You are building a numpy array by providing it a list. This list is being built with the concept of list comprehension. So the code inside the np.array(...) method:
[[[S[i,j]] for i in range(order+1)] for j in range(order+1)]
... is equivalent to:
order = 2
full_list = []
for j in range(order+1):
local_list = []
for i in range(order+1):
local_list.append(S_array[i, j])
full_list.append(local_list)
print(full_list)
This results in:
[[1, 4, 7], [2, 5, 8], [3, 6, 9]]
As for your second snippet its important to notice that although typically numpy arrays have very specific and constant (for all the array) cell types you can actually give the data type object to a numpy array. So creating a 2-dimensional array of lists is possible. It is also possible to create a 3-dimensional array. Both are compatible with the indexation rm[i,j][k]. You can check this in the following example:
rm = np.array(["A", 3, [1,2,3]], dtype="object")
print(rm, rm[2][0]) # Acessing element 0 of list at position 2 of the array
rm2 = np.zeros((3, 3, 3))
print(rm2[0, 1][2]) # This is also valid
The following code:
[rm[i,j][k]*someFuction(name,u[i],v[j])[k] for k in range(len(rm[i,j])) if rm[i,j][k]]
... could be written as such:
some_list = []
for k in range(len(rm[i,j])):
if rm[i, j][k]: # Expecting a boolean value (or comparable)
a_list = rm[i,j][k]*someFuction(name,u[i],v[j])
some_list.append(a_list[k])
The final detail is the tmp+=some_list. When you sum two list they'll be concatenated as can been seen in this simple example:
tmp = []
tmp += [1, 2, 3]
print(tmp)
tmp += [4, 5, 6]
print(tmp)
Which results in this:
[1, 2, 3]
[1, 2, 3, 4, 5, 6]
Also notice that multiplying a list by a number will effectively be the same as summing the list several times. So 2*[1,2] will result in [1,2,1,2].
