sparse_test, a MATLAB code which calls sparse(), which is a built-on MATLAB function for creating a sparse matrix.
MATLAB provides a sparse function of the form
matrix = sparse ( i, j, s, m, n, nz_max )This function can be used to create a sparse matrix. The input arguments have the following meaning:
Although this is not the usual way to use the sparse command, the following example should help to understand what is going on. We mean to define the following matrix:
11 12 0 0 15
0 22 23 0 0
31 0 33 34 35
by the following MATLAB commands:
i = [ 1, 1, 1, 2, 2, 3, 3, 3, 3 ];
j = [ 1, 2, 5, 2, 3, 1, 3, 4, 5 ];
s = [ 11, 12, 15, 22, 23, 31, 33, 34, 35 ];
m = 3;
n = 5;
nz_max = 9;
a = sparse ( i, j, s, m, n, nz_max );
Of course, in many applications, the matrix data is only available one piece at a time, or has to be modified as you go. This is easy to do, as well. You may start by defining the matrix to be an "empty" sparse matrix of a particular size, as, for example:
i = [];
j = [];
s = [];
m = 100;
n = 100;
a = sparse ( i, j, s, m, n );
The matrix will be empty, and entries of the matrix are
by default equal to zero. Then you can simply reference
entries of the matrix as you need them. For instance,
a(1,1) = 3
a(3,8) = a(3,8) + 7
a(4,7) = a(9,5) + 2 * a(8,12)
a(4,7) = a(4,7) + 1
If you reference, on the right hand side of these equations,
a matrix entry that doesn't exist, it is by default zero.
If you assign a value on the left hand side to a matrix entry
that doesn't exist, a space is created for it, and it is given
this value. If the entry already existed, it is overwritten.
In some cases, Matlab's sparse matrix structure allows you to ignore the fact that you are using a sparse matrix. We have already seen that you can reference the (i,j) element of the matrix in the same way you would do for a full matrix, and this is true whether you are simply asking to "read" the current value of the element, or to "write" a new value for the element.
A particular example of this is the fact that you can solve a sparse linear system using the same "backslash" formula that you would use if the matrix were full:
x = A \ b;
Matlab includes many commands specifically for dealing with a sparse matrix. For our examples, we will be considering
Note that, in a sense, MATLAB actually uses two formats for sparse matrices. The user communicates with MATLAB by specifying what is known as a sparse triplet, that is, a set of row indices, column indices, and values. But internally, MATLAB uses what is known as the sparse compressed column format. This format allows MATLAB to access matrix entries rapidly.
To copy the nonzero entries from a sparse matrix, creating the sparse triplet structure:
[i,j,s] = sparse ( A );
[m,n] = size ( A );
Correspondingly, to create the sparse matrix from this data:
A = sparse ( i, j, s, m, n );
When using MATLAB's sparse matrix format, it is possible to refer to matrix entries directly, using the usual index notation like A(i,j). However, accessing specific entries in this way, whether to initialize, extract, increment, or zero them, is an expensive process. The MathWorks suggests extracting the sparse triplet information, working on it in the natural way, and then "repacking" it with the sparse() command.
sparse_test is available in a MATLAB version and a Python version.
fem2d_heat_sparse, a MATLAB code which solves the time dependent heat equation in an arbitrary triangulated 2D region, using MATLAB's sparse matrix storage format and solver.
hb_io, a MATLAB code which reads and writes sparse linear systems stored in the Harwell-Boeing (HB) Sparse Matrix format.
hb_to_msm, a MATLAB code which converts a Harwell Boeing (HB) sparse matrix file to a MATLAB sparse matrix.
hb_to_st, a MATLAB code which converts the sparse matrix information stored in a Harwell Boeing (HB) file into a Sparse Triplet (ST) file.
mm_io, a MATLAB code which reads and writes sparse linear systems stored in the Matrix Market format.
msm_to_hb, a MATLAB code which converts a MATLAB sparse matrix into a Harwell Boeing (HB) file.
sparse_ccs, a data directory which contains examples of compressed column storage (CCS), equivalent to MATLAB's sparse format, and a file format suitable for storing such information.
sparse_crs, a data directory which contains a description and examples of the compressed row storage (CRS) format, for storing a sparse matrix, including a way to write the matrix as a set of three files.
sparse_display, a MATLAB code which can read information defining a matrix of numbers and display the sparsity pattern or location of the nonzero elements using gnuplot. This operation is already available in the built-in MATLAB "spy" command.
st, a data directory which contains examples of the "sparse triplet" format for storing sparse matrices. This format is equivalent to the form in which sparse matrix data is passed into MATLAB's sparse command (although the sparse compressed column format is used internally).
wathen, a MATLAB code which compares storage schemes (full, banded, sparse triplet, sparse) and solution strategies (A\x, Linpack, conjugate gradient) for linear systems involving the Wathen matrix, which can arise when solving a problem using the finite element method (FEM).