#### 22.1.2 Creating Sparse Matrices ¶

There are several means to create sparse matrix.

Returned from a function

  There are many functions that directly return sparse matrices. These include speye, sprand, diag, etc.

Constructed from matrices or vectors

  The function sparse allows a sparse matrix to be constructed from three vectors representing the row, column and data. Alternatively, the function spconvert uses a three column matrix format to allow easy importation of data from elsewhere.

Created and then filled

  The function sparse or spalloc can be used to create an empty matrix that is then filled by the user

From a user binary program

  The user can directly create the sparse matrix within an oct-file.

There are several basic functions to return specific sparse matrices. For example the sparse identity matrix, is a matrix that is often needed. It therefore has its own function to create it as `speye (n)` or `speye (r, c)`, which creates an n-by-n or r-by-c sparse identity matrix.

Another typical sparse matrix that is often needed is a random distribution of random elements. The functions sprand and sprandn perform this for uniform and normal random distributions of elements. They have exactly the same calling convention, where `sprand (r, c, d)`, creates an r-by-c sparse matrix with a density of filled elements of d.

Other functions of interest that directly create sparse matrices, are diag or its generalization spdiags, that can take the definition of the diagonals of the matrix and create the sparse matrix that corresponds to this. For example,

    s = diag (sparse (randn (1,n)), -1);

creates a sparse (n+1)-by-(n+1) sparse matrix with a single diagonal defined.

`B =` spdiags `(A)` ¶
`[B, d] =` spdiags `(A)` ¶
`B =` spdiags `(A, d)` ¶
`A =` spdiags `(v, d, A)` ¶
`A =` spdiags `(v, d, m, n)` ¶

  A generalization of the function `diag`.

  Called with a single input argument, the nonzero diagonals d of A are extracted.

  With two arguments the diagonals to extract are given by the vector d.

  The other two forms of `spdiags` modify the input matrix by replacing the diagonals. They use the columns of v to replace the diagonals represented by the vector d. If the sparse matrix A is defined then the diagonals of this matrix are replaced. Otherwise a matrix of m by n is created with the diagonals given by the columns of v.

  Negative values of d represent diagonals below the main diagonal, and positive values of d diagonals above the main diagonal.

  For example:

      spdiags (reshape (1:12, 4, 3), [-1 0 1], 5, 4)
         ⇒  5 10  0  0
            1  6 11  0
            0  2  7 12
            0  0  3  8
            0  0  0  4

  See also: diag.

`s =` speye `()` ¶
`s =` speye `(n)` ¶
`s =` speye `(m, n)` ¶
`s =` speye `([m, n])` ¶

  Return a sparse identity matrix of size mxn.

  If called with no arguments, return the sparse scalar value `1`.

  If invoked with a single scalar argument n, return a sparse square NxN identity matrix.

  If supplied two scalar arguments (m, n), or a 2-element vector `[m, n]`, return a sparse MxN identity matrix with m rows and n columns.

  Programming Note: The implementation is significantly more efficient than `sparse (eye (…))` as the full matrix is not constructed.

  See also: sparse, spdiags, eye.

`r =` spones `(S)` ¶

  Replace the nonzero entries of S with ones.

  This creates a sparse matrix with the same structure as S.

  See also: sparse, sprand, sprandn, sprandsym, spfun, spy.

`s =` sprand `(m, n, d)` ¶
`s =` sprand `(m, n, d, rc)` ¶
`s =` sprand `(s)` ¶

  Generate a sparse matrix with uniformly distributed random values.

  The size of the matrix is mxn with a density of values d. d must be between 0 and 1. Values will be uniformly distributed on the interval (0, 1).

  If called with a single matrix argument, a sparse matrix is generated with random values wherever the matrix s is nonzero.

  If called with a scalar fourth argument rc, a random sparse matrix with reciprocal condition number rc is generated. If rc is a vector, then it specifies the first singular values of the generated matrix (`length (rc) <= min (m, n)`).

  See also: sprandn, sprandsym, rand.

`s =` sprandn `(m, n, d)` ¶
`s =` sprandn `(m, n, d, rc)` ¶
`s =` sprandn `(s)` ¶

  Generate a sparse matrix with normally distributed random values.

  The size of the matrix is mxn with a density of values d. d must be between 0 and 1. Values will be normally distributed with a mean of 0 and a variance of 1.

  If called with a single matrix argument, a sparse matrix is generated with random values wherever the matrix s is nonzero.

  If called with a scalar fourth argument rc, a random sparse matrix with reciprocal condition number rc is generated. If rc is a vector, then it specifies the first singular values of the generated matrix (`length (rc) <= min (m, n)`).

  See also: sprand, sprandsym, randn.

`S =` sprandsym `(n, d)` ¶
`S =` sprandsym `(s)` ¶

  Generate a symmetric random sparse matrix.

  The size of the matrix will be nxn, with a density of values given by d. d must be between 0 and 1 inclusive. Values will be normally distributed with a mean of zero and a variance of 1.

  If called with a single matrix argument, a random sparse matrix is generated wherever the matrix s is nonzero in its lower triangular part.

  See also: sprand, sprandn, spones, sparse.

The recommended way for the user to create a sparse matrix, is to create two vectors containing the row and column index of the data and a third vector of the same size containing the data to be stored. For example,

      ri = ci = d = [];
      for j = 1:c
        ri = [ri; randperm(r,n)'];
        ci = [ci; j*ones(n,1)];
        d = [d; rand(n,1)];
      endfor
      s = sparse (ri, ci, d, r, c);

creates an r-by-c sparse matrix with a random distribution of n (<r) elements per column. The elements of the vectors do not need to be sorted in any particular order as Octave will sort them prior to storing the data. However, pre-sorting the data will make the creation of the sparse matrix faster.

The function spconvert takes a three or four column real matrix. The first two columns represent the row and column index respectively and the third and four columns, the real and imaginary parts of the sparse matrix. The matrix can contain zero elements and the elements can be sorted in any order. Adding zero elements is a convenient way to define the size of the sparse matrix. For example:

    s = spconvert ([1 2 3 4; 1 3 4 4; 1 2 3 0]')
    ⇒  Compressed Column Sparse (rows=4, cols=4, nnz=3)
          (1 , 1) -> 1
          (2 , 3) -> 2
          (3 , 4) -> 3

An example of creating and filling a matrix might be

    k = 5;
    nz = r * k;
    s = spalloc (r, c, nz)
    for j = 1:c
      idx = randperm (r);
      s (:, j) = [zeros(r - k, 1); ...
            rand(k, 1)] (idx);
    endfor

It should be noted, that due to the way that the Octave assignment functions are written that the assignment will reallocate the memory used by the sparse matrix at each iteration of the above loop. Therefore the spalloc function ignores the nz argument and does not pre-assign the memory for the matrix. Therefore, it is vitally important that code using to above structure should be vectorized as much as possible to minimize the number of assignments and reduce the number of memory allocations.

`FM =` full `(SM)` ¶

  Return a full storage matrix from a sparse, diagonal, or permutation matrix, or from a range.

  See also: sparse, issparse.

`s =` spalloc `(m, n, nz)` ¶

  Create an m-by-n sparse matrix with pre-allocated space for at most nz nonzero elements.

  This is useful for building a matrix incrementally by a sequence of indexed assignments. Subsequent indexed assignments after `spalloc` will reuse the pre-allocated memory, provided they are of one of the simple forms

  - `s(I:J) = x`
  - `s(:,I:J) = x`
  - `s(K:L,I:J) = x`

  and that the following conditions are met:

  - the assignment does not decrease `nnz (S)`.
  - after the assignment, `nnz (S)` does not exceed nz.
  - no index is out of bounds.

  Partial movement of data may still occur, but in general the assignment will be more memory and time efficient under these circumstances. In particular, it is possible to efficiently build a pre-allocated sparse matrix from a contiguous block of columns.

  The amount of pre-allocated memory for a given matrix may be queried using the function `nzmax`.

  Programming Note: Octave always reserves memory for at least one value, even if nz is 0.

  See also: nzmax, sparse.

`S =` sparse `(A)` ¶
`S =` sparse `(m, n)` ¶
`S =` sparse `(i, j, sv)` ¶
`S =` sparse `(i, j, sv, m, n)` ¶
`S =` sparse `(i, j, sv, m, n, "unique")` ¶
`S =` sparse `(i, j, sv, m, n, nzmax)` ¶

  Create a sparse matrix from a full matrix A or row, column, value triplets.

  If A is a full matrix, convert it to a sparse matrix representation, removing all zero values in the process. The matrix A should be of type logical or double.

  If two inputs m (rows) and n (columns) are specified then create an empty sparse matrix with the specified dimensions.

  Given the integer index vectors i and j, and a 1-by-`nnz` vector of real or complex values sv, construct the sparse matrix `S(i(k),j(k)) = sv(k)` with overall dimensions m and n. If any of i, j, or sv are scalars, they are expanded to have a common size.

  If m or n are not specified then their values are derived from the maximum index in the vectors i and j as given by `m = max (i)`, `n = max (j)`.

  Note: If multiple values are specified with the same i, j indices, the corresponding value in S will be the sum of the values at the repeated location. See accumarray, for an example of how to produce different behavior such as taking the minimum instead.

  If the option `"unique"` is given, and more than one value is specified at the same i, j indices, then only the last specified value will be used. For completeness, the option `"sum"` can be given and will be ignored as the default behavior is to sum values at repeated locations.

  `sparse (m, n)` will create an empty mxn sparse matrix and is equivalent to `sparse ([], [], [], m, n)`

  The optional final argument reserves space for nzmax values in the sparse array and is useful if the eventual number of nonzero values will be greater than the number of values in sv used during the initial construction of the array. See spalloc, for more information and usage instructions.

  Example 1 (convert full matrix to sparse to save memory):

      x = full (diag (1:1000));
      sizeof (x)
      ⇒   8000000
      s = sparse (x);
      sizeof (xs)
      ⇒   24008

  Example 2 (sum at repeated indices):

      i = [1 1 2]; j = [1 1 2]; sv = [3 4 5];
      sparse (i, j, sv, 3, 4)
      ⇒
         Compressed Column Sparse (rows = 3, cols = 4, nnz = 2 [17%])

           (1, 1) ->  7
           (2, 2) ->  5

  Example 3 ("unique" option):

      i = [1 1 2]; j = [1 1 2]; sv = [3 4 5];
      sparse (i, j, sv, 3, 4, "unique")
      ⇒
         Compressed Column Sparse (rows = 3, cols = 4, nnz = 2 [17%])

           (1, 1) ->  4
           (2, 2) ->  5

  See also: full, accumarray, spalloc, spdiags, speye, spones, sprand, sprandn, sprandsym, spconvert, spfun.

`x =` spconvert `(m)` ¶

  Convert a simple sparse matrix format easily generated by other programs into Octave’s internal sparse format.

  The input m is either a 3 or 4 column real matrix, containing the row, column, real, and imaginary parts of the elements of the sparse matrix. An element with a zero real and imaginary part can be used to force a particular matrix size.

  See also: sparse.

The above problem of memory reallocation can be avoided in oct-files. However, the construction of a sparse matrix from an oct-file is more complex than can be discussed here. See External Code Interface, for a full description of the techniques involved.