`sparse`

Creates a sparse matrix, or converts a full matrix to sparse form.

Syntax

S = sparse(A)

Converts the full matrix A to sparse storage.

S = sparse(m, n)

Creates an m-by-n sparse matrix of zeros.

S = sparse(i, j, v)

Creates a matrix with S(i(k), j(k)) = v(k); repeated positions are added together.

S = sparse(i, j, v, m, n)

Same, with an explicit size m-by-n.

S = sparse(i, j, v, m, n, nzmax)

Also reserves space for nzmax nonzeros.

Description

A sparse matrix keeps only its nonzero entries, which lets large matrices that are mostly zeros, such as finite-difference grids or graphs, be handled efficiently. Display lists each stored entry as (row,column) value.

Arithmetic follows simple rules: sparse times sparse is sparse, sparse plus a nonzero scalar becomes full, and most elementwise functions keep sparsity if f(0) = 0. Use full to convert back, issparse to test, and find to list the entries.

Building from triplets (i, j, v) is the fast way to assemble a matrix. Duplicate positions are summed, which is convenient for assembling finite-element matrices.

Input arguments

A: Numeric or logical matrix.
i, j: Row and column indices of the entries, as vectors of positive integers.
v: Values for the entries; a scalar is expanded to all positions.
m, n: Number of rows and columns.

Output arguments

S: Sparse matrix. Zero entries are not shown when it is displayed.

Example

Triplets, a sum and a conversion

S = sparse([1 2 3 3], [1 2 3 3], [4 5 6 1])
full(S)
T = sparse([1 0 2; 0 0 3])
size(sparse(2, 3))

Output

S = 

   (1,1)        4
   (2,2)        5
   (3,3)        7
ans = 3×3

     4     0     0
     0     5     0
     0     0     7
T = 

   (1,1)        1
   (1,3)        2
   (2,3)        3
ans = 1×2

     2     3

Notes

Complex values are not kept: the imaginary part is dropped when building a sparse matrix. MatLivre stores sparse matrices densely, so they save no memory, and sizes above 200 million elements are rejected. Results and display match MATLAB for ordinary sizes.

sparse