Determine whether matrix is sparse? - algorithm

Question

I have a matrix. I want to know it whether sparse or not. Is there any function in matlab to evaluate that property? I tried to used issparse function, but it always returns 0(not sparse). For example, my matrix (27 by 27) is
A=
[ 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
1 1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0
0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0
0 0 1 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0
0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0
0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0
250 243 247 245 244 244 244 122 61 144 72 36 18 9 4 2 1 1 0 0 0 0 0 0 0 0 0
151 197 236 118 181 212 106 53 26 13 136 68 34 17 8 4 2 0 1 0 0 0 0 0 0 0 0
24 12 6 3 143 201 234 117 180 90 45 152 76 38 19 9 4 0 0 1 0 0 0 0 0 0 0
18 9 138 69 172 86 165 220 224 112 56 28 128 64 32 16 8 0 0 0 1 0 0 0 0 0 0
27 131 207 103 189 94 47 153 194 239 119 59 29 128 64 32 16 0 0 0 0 1 0 0 0 0 0
44 22 133 204 232 116 58 147 199 237 248 124 62 31 129 64 32 0 0 0 0 0 1 0 0 0 0
238 119 181 90 45 152 76 38 19 135 205 232 116 58 29 128 64 0 0 0 0 0 0 1 0 0 0
48 24 12 6 3 143 201 100 50 25 130 207 233 116 58 29 128 0 0 0 0 0 0 0 1 0 0
168 84 42 21 132 66 33 158 79 39 19 135 205 232 116 58 29 0 0 0 0 0 0 0 0 1 0
235 117 58 29 128 64 32 16 8 4 2 1 142 201 234 117 58 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0
1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0
0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1
0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0
0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0]
This is figure for above matrix

Answer 1

This seemingly easy question is quite difficult to answer. There is actually no known standard that determines whether a matrix is sparse or full.
However, the most common measure I know is to measure a matrix's sparsity. This is simply the fraction of the total number of zeroes over the total number of elements. If this exceeds some sensible threshold, then you could say that the matrix is sparse.
If you're given the matrix A, perhaps something like this:
sparsity = (numel(A) - nnz(A)) / numel(A);
numel determines the total number of elements in the matrix A and nnz determines the total number of non-zero elements. Therefore, numel(A) - nnz(A) should give you the total number of zero elements.
So, going with the threshold idea, this is what I was talking about:
is_sparse = sparsity > tol;
tol would be a fraction from [0,1], so something like 0.75 could work. This would mean that if 75% of your matrix consisted of zeroes, this could be a sparse matrix. It's all heuristic though. Choose a threshold that you think makes the most sense.