Using Array and Table Functions in Mathematica. Which is best when

Question

I have been mostly a Table functions user in mathematica. However I have noticed that in several examples where I used Array instead of Table to express the same result, it ran markedly faster, especially as the dimension of table grew larger.

So my question is this: When speed of execution is the primary concern, when is it most appropriate to use table?

What explains this difference?

My guess is that because Arrays assume a functional relationship between the items in the list, it stores them more efficiently, therefore use less memory, thus facilitating storage and subsequent processing?

Is it what is going on?

Answer 1

Array has no performance advantages over Table. There are differences between them which make one preferred over another.

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EDIT It was noted by several persons that Table is slower on multi-dimensional arrays. All of them used variable to hold the table size. Table has HoldAll attributes and only auto-evaluates outer-most interation bound. Because internal iterators remain unevaluated, the element of the table fails to compile. Using explicit numbers or With with result in auto-compilation:

In[2]:= With[{b = 10^4, c = 10^4},
 {Timing@(#[[1, 1]] &[ar = Array[(# + #2) &, {b, c}]]) , 
  Timing@(#[[1, 1]] &[ta = Table[(i + j), {i, b}, {j, c}]])}
 ]

Out[2]= {{4.93, 2}, {4.742, 2}}

In[3]:= Attributes[Table]

Out[3]= {HoldAll, Protected}

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The Array allows you to build an array of function values just as much as the Table. They take different arguments. Array takes a function:

In[34]:= Array[Function[{i, j}, a[i, j]], {3, 3}]

Out[34]= {{a[1, 1], a[1, 2], a[1, 3]}, {a[2, 1], a[2, 2], 
  a[2, 3]}, {a[3, 1], a[3, 2], a[3, 3]}}

while table takes an explicit form:

In[35]:= Table[a[i, j], {i, 3}, {j, 3}]

Out[35]= {{a[1, 1], a[1, 2], a[1, 3]}, {a[2, 1], a[2, 2], 
  a[2, 3]}, {a[3, 1], a[3, 2], a[3, 3]}}

Array can only go over regular arrays, while Table can do arbitrary iterating over list:

In[36]:= Table[a[i, j], {i, {2, 3, 5, 7, 11}}, {j, {13, 17, 19}}]

Out[36]= {{a[2, 13], a[2, 17], a[2, 19]}, {a[3, 13], a[3, 17], 
  a[3, 19]}, {a[5, 13], a[5, 17], a[5, 19]}, {a[7, 13], a[7, 17], 
  a[7, 19]}, {a[11, 13], a[11, 17], a[11, 19]}}

Sometimes Array can be more succinct. Compare multiplication table:

In[37]:= Array[Times, {5, 5}]

Out[37]= {{1, 2, 3, 4, 5}, {2, 4, 6, 8, 10}, {3, 6, 9, 12, 15}, {4, 8,
   12, 16, 20}, {5, 10, 15, 20, 25}}

versus

In[38]:= Table[i j, {i, 5}, {j, 5}]

Out[38]= {{1, 2, 3, 4, 5}, {2, 4, 6, 8, 10}, {3, 6, 9, 12, 15}, {4, 8,
   12, 16, 20}, {5, 10, 15, 20, 25}}

Array allows one to build expression with any head, not just list:

In[39]:= Array[a, {3, 3}, {1, 1}, h]

Out[39]= h[h[a[1, 1], a[1, 2], a[1, 3]], h[a[2, 1], a[2, 2], a[2, 3]],
  h[a[3, 1], a[3, 2], a[3, 3]]]

By default the head h is chosen to be List resulting in creation of the regular array. Table does not have this flexibility.