Accessing Sparse Matrices
There are several commands that provide high-level information about the nonzero elements of a sparse matrix:
To try some of these, load the supplied sparse matrix
one of the Harwell-Boeing collection.
load west0479 whos
Name Size Bytes Class Attributes west0479 479x479 34032 double sparse
This matrix models an eight-stage chemical distillation column.
Try these commands.
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 . . .
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 . . .
Use Ctrl+C to stop the
at any time.
Note that initially
nnz has the same value
nzmax by default. That is, the number of nonzero
elements is equivalent to the number of storage locations allocated
for nonzeros. However, MATLAB® 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
but not that of
However, you can add as many nonzero elements to the matrix
as desired. You are not constrained by the original value of
Indices and Values
For any matrix, full or sparse, the
returns the indices and values of nonzero elements. Its syntax is
[i,j,s] = find(S);
find returns the row indices of nonzero values
i, the column indices in vector
and the nonzero values themselves in the vector
The example below uses
find to locate the indices
and values of the nonzeros in a sparse matrix. The
find output, together with the size of
the matrix, to recreate the matrix.
S1 = west0479; [i,j,s] = find(S1); [m,n] = size(S1); S2 = sparse(i,j,s,m,n);
Indexing in Sparse Matrix Operations
Because sparse matrices are stored in compressed sparse column format, there are different costs associated with indexing into a sparse matrix than there are with indexing into a full matrix. Such costs are negligible when you need to change only a few elements in a sparse matrix, so in those cases it’s normal to use regular array indexing to reassign values:
B = speye(4); [i,j,s] = find(B); [i,j,s]
ans = 1 1 1 2 2 1 3 3 1 4 4 1
B(3,1) = 42; [i,j,s] = find(B); [i,j,s]
ans = 1 1 1 3 1 42 2 2 1 3 3 1 4 4 1
(3,1), MATLAB inserts an additional row into the nonzero values vector and subscript vectors, then shifts all matrix values after
Using linear indexing to access or assign an element in a large
sparse matrix will fail if the linear index exceeds
which is the current upper bound for the number of elements allowed
in a matrix.
S = spalloc(2^30,2^30,2); S(end) = 1
Maximum variable size allowed by the program is exceeded.
To access an element whose linear index is greater than
use array indexing:
S(2^30,2^30) = 1
S = (1073741824,1073741824) 1
While the cost of indexing into a sparse matrix to change a single element is negligible, it is compounded in the context of a loop and can become quite slow for large matrices. For that reason, in cases where many sparse matrix elements need to be changed, it is best to vectorize the operation instead of using a loop. For example, consider a sparse identity matrix:
n = 10000; A = 4*speye(n);
Awithin a loop is slower than a similar vectorized operation:
tic A(1:n-1,n) = -1; A(n,1:n-1) = -1; toc
Elapsed time is 0.003344 seconds.
tic for k = 1:n-1 C(k,n) = -1; C(n,k) = -1; end toc
Elapsed time is 0.448069 seconds.
Aduring each pass through the loop.
Preallocating the memory for a sparse matrix and then filling it in an element-wise manner similarly causes a significant amount of overhead in indexing into the sparse array:
S1 = spalloc(1000,1000,100000); tic; for n = 1:100000 i = ceil(1000*rand(1,1)); j = ceil(1000*rand(1,1)); S1(i,j) = rand(1,1); end toc
Elapsed time is 2.577527 seconds.
Constructing the vectors of indices and values eliminates the need to index into the sparse array, and thus is significantly faster:
i = ceil(1000*rand(100000,1)); j = ceil(1000*rand(100000,1)); v = zeros(size(i)); for n = 1:100000 v(n) = rand(1,1); end tic; S2 = sparse(i,j,v,1000,1000); toc
Elapsed time is 0.017676 seconds.
For that reason, it’s best to construct sparse matrices
all at once using a construction function, like the
For example, suppose you wanted the sparse form of the coordinate
Construct the five-column matrix directly with the
sparse function using the triplet pairs for the row subscripts, column
i = [1 5 2 5 3 5 4 5 1 2 3 4 5]'; j = [1 1 2 2 3 3 4 4 5 5 5 5 5]'; s = [4 1 4 1 4 1 4 1 -1 -1 -1 -1 4]'; C = sparse(i,j,s)
C = (1,1) 4 (5,1) 1 (2,2) 4 (5,2) 1 (3,3) 4 (5,3) 1 (4,4) 4 (5,4) 1 (1,5) -1 (2,5) -1 (3,5) -1 (4,5) -1 (5,5) 4
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.
Load the supplied sparse matrix
west0479, one of the Harwell-Boeing collection.
View the sparsity structure.