How to use Contains functions on matrices?

Question

I need a quick test to check if a large matrix contains any non-zeros. The Contains functions work on Lists not matrices.

d1 = ConstantArray[0, 100];
d2 = ConstantArray[0, {100, 100}];
ContainsOnly[d1, {0}]
ContainsOnly[d2, {0}]
ContainsAny[d1, {0}]
ContainsAny[d2, {0}]

(* True False True False *)

I know that I could use Flatten[], but Flatten appears to create another entire list in memory, which takes time and (worse) consumes memory:

n = 20000;
d2 = ConstantArray[0, {n, n}];
Print[Timing[ContainsOnly[Flatten@d2, {0}]]];
Print[Timing[d3 = Flatten[d2]][[1]]];
Print[Timing[ContainsOnly[d3, {0}]]];

(* {2.46, True} 1.375 {0.75, True} *)

Is there a way to just search the matrix instead of copying it into a Flatten[]'d list to search?

I ultimately just need to know if the matrix contains any non-zeros. Hopefully, there's a method that will stop search once it finds a non-zero.

Answer 1

The function FirstCase[matrix,0,"No zero",2] should do the trick quickly. It gives 0 if there is at least one 0 in the matrix and "No zero" otherwise. The good thing is (a) you don't have to flatten your matrix first and (b) it stops as soon as the first zero is found, so as long as the only zero is not hidden at the very last position, it doesn't even go through the whole matrix to answer your question.

matrix=Table[RandomReal[],{2000},{2000}];

Here Timing[ContainsAny[Flatten[matrix],{0}]] gives on my laptop 4.4 seconds, while

Timing[FirstCase[matrix,0,"Non zero",2]]

gives 0.3 seconds i.e. about 10 times faster. If you have a zero early on in the matrix 'ContainsAny' is as slow as before, but FirstCase speeds up:

matrix[[223,445]]==0
Timing[FirstCase[matrix,0,"Non zero",2]

gives now 0.01 seconds.

Answer 2

Given your statement:

I know I'll only have positive integers.

the answer I gave for How do I check if any element in a list is positive? applies:

Positive @ Max @ d2                     (* negate with Not as needed *)

Early exit behaviors are also discussed in that answer. In the case of a matrix a block-based approach naturally presents itself: test each row separately and exit if a positive value is found:

AllTrue[d2, Not @* Positive @* Max]

In the case of an all-zero array this will be significantly slower than the first form but it can exit after only one row.