nnz

Create an identity matrix and determine the number of nonzeros it contains.

X = eye(4)

X = 4×4

     1     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1

N = nnz(X)

N = 
4

Use nnz in conjunction with a relational operator to determine how many matrix elements meet a condition. Since relational operators produce logical matrices of 1s and 0s, the nnz function counts the 1s where the condition is true.

Create a matrix and determine how many elements are greater than 10.

X = magic(5)

X = 5×5

    17    24     1     8    15
    23     5     7    14    16
     4     6    13    20    22
    10    12    19    21     3
    11    18    25     2     9

nnz(X>10)

ans = 
15

The density of a matrix is the ratio of nonzeros to the total number of elements, nnz(X)/numel(X).

Create a sparse matrix representing the finite difference Laplacian on an L-shaped domain and calculate its density.

X = delsq(numgrid('L',20));
spy(X)

d = nnz(X)/numel(X)

d = 
0.0194

The result indicates that only about 2% of the elements in the matrix are nonzero.