MATLAB®

Accessing Sparse Matrices

Nonzero Elements

There are several commands that provide high-level information about the nonzero elements of a sparse matrix:

• nnz returns the number of nonzero elements in a sparse matrix.

• nonzeros returns a column vector containing all the nonzero elements of a sparse matrix.

• nzmax returns the amount of storage space allocated for the nonzero entries of a sparse matrix.

To try some of these, load the supplied sparse matrix west0479, one of the Harwell-Boeing collection.

load west0479
whos

  Name            Size             Bytes  Class     Attributes

  M_full       1100x1100         9680000  double              
  M_sparse     1100x1100            5004  double    sparse    
  west0479      479x479            24564  double    sparse    

This matrix models an eight-stage chemical distillation column.

Try these commands.

nnz(west0479)

ans =
1887

format short e
west0479
   
west0479 =
    
  (25,1)      1.0000e+00
  (31,1)     -3.7648e-02
  (87,1)     -3.4424e-01
  (26,2)      1.0000e+00
  (31,2)     -2.4523e-02
  (88,2)     -3.7371e-01
  (27,3)      1.0000e+00
  (31,3)     -3.6613e-02
  (89,3)     -8.3694e-01
  (28,4)      1.3000e+02
    .
    .
    .
   

nonzeros(west0479);
    ans =
    
   1.0000e+00
  -3.7648e-02
  -3.4424e-01
   1.0000e+00
  -2.4523e-02
  -3.7371e-01
   1.0000e+00
  -3.6613e-02
  -8.3694e-01
   1.3000e+02
    .
    .
    .

Note   Use Ctrl+C to stop the nonzeros listing at any time.

Note that initially nnz has the same value as nzmax by default. That is, the number of nonzero elements is equivalent to the number of storage locations allocated for nonzeros. However, MATLAB^® software does not dynamically release memory if you zero out additional array elements. Changing the value of some matrix elements to zero changes the value of nnz, but not that of nzmax.

However, you can add as many nonzero elements to the matrix as desired. You are not constrained by the original value of nzmax.

Back to Top

Indices and Values

For any matrix, full or sparse, the find function returns the indices and values of nonzero elements. Its syntax is

[i,j,s] = find(S)

find returns the row indices of nonzero values in vector i, the column indices in vector j, and the nonzero values themselves in the vector s. The example below uses find to locate the indices and values of the nonzeros in a sparse matrix. The sparse function uses the find output, together with the size of the matrix, to recreate the matrix.

[i,j,s] = find(S)
[m,n] = size(S)
S = sparse(i,j,s,m,n)

Back to Top

Visualizing Sparse Matrices

It is often useful to use a graphical format to view the distribution of the nonzero elements within a sparse matrix. The MATLAB spy function produces a template view of the sparsity structure, where each point on the graph represents the location of a nonzero array element.

For example:

spy(west0479)

Back to Top

Constructing Sparse Matrices

Sparse Matrix Operations