"Mathematica employs the copy-on-write paradigm."

Ja, I understand that. I wrote several compilers and interpreters myself.

Hence myArray["NonzeroValues"] does not create a copy of that list, since it has that list already in the internal structure of a sparse array. But ArrayRules[myArray] are not directly available and have to be constructed.

Ja, that sounds understandable.

Thank you for your comments. I learned a lot from them.

There are two separate issues here:

• How to get the number of explicitly stored elements efficiently? Use Length[myArray["NonzeroValues"]]. This is undocumented syntax, but it is widely used publicly and—in my opinion—very unlikely to go away in the future. See here. I expect this method to be the most efficient way (in speed and memory usage) to accomplish your task.

  Caveat: Despite its name, "NonzeroValues" returns non-default values even in the case when the default value is something else than zero.

• How to get the number of elements that are different from the default value? This is a separate issue because some of the explicitly stored elements may be equal to the default value. To get rid of these, we must re-build the sparse array: Length[ SparseArray[myArray]["NonzeroValues"] ]

If you are curious to see a sparse array that has explicitly stored elements that are equal to the background value, here is an example:

In[25]:= sa = SparseArray[{0, 1, 2, 3}];

In[26]:= sa2 = If[# > 2, 0, #] & /@ sa;

In[27]:= sa2["NonzeroValues"]

Out[27]= {1, 2, 0}

In[28]:= ArrayRules[sa2]
Out[28]= {{2} -> 1, {3} -> 2, {4} -> 0, {_} -> 0}

In[29]:= sa3 = SparseArray[sa2]; (* rebuild the array *)

In[30]:= ArrayRules[sa3]
Out[30]= {{2} -> 1, {3} -> 2, {_} -> 0}