# spral_random_matrix - Pseudo-random Matrix Generator¶

## Purpose¶

This package generates a random sparse matrix of specified size and density in compressed sparse column format. Either the pattern or both the pattern and values can be generated. Both symmetric and unsymmetric matrices can be generated, and structural non-degeneracy can optionally be ensured, and the row indices can be sorted within columns.

### Version history¶

2016-09-08 Version 1.1.0
Add long support
2014-03-06 Version 1.0.0
Initial release

## Routines¶

function random_matrix_generate(state, matrix_type, m, n, nnz, ptr, row, flag[, stat, val, nonsingular, sort])

Generate an $$m\times n$$ random matrix with $$nnz$$ non-zero entries.

If matrix_type specifies a symmetric or skew symmetric matrix, only the lower half matrix will be returned to the user.

Parameters:
• state [random_state,inout] :: State of the pseudo-random number generator to use.
• matrix_type [integer,in] ::

Type of matrix to generate. One of:

 0 Undefined (matrix generated will be unsymmetric/retangular). 1 Rectangular matrix. 2 Unsymmetric. 3 Symmetric positive-definite.nnz entries in lower triangle returned, matrix made diagonally dominant if val is present, non-singularity assured regardless of argument nonsingular. 4 Symmetric indefinite.nnz entries in lower triangle returned. 5 Skew symmetric.nnz entries in lower triangle returned.
• m [integer,in] :: Number of rows in the matrix.
• n [integer,in] :: Number of columns in the matrix.
• nnz [integer(long),in] :: Number of non-zeroes in the matrix.
• ptr (n+1) [integer(long),out] :: Column pointers of the matrix (see CSC format).
• row (nnz) [integer,out] :: Row indices of the matrix (see CSC format).
• flag [integer,out] :: Exit status of the algorithm, 0 on success, otherwise an error occured, see table below.
Options:
• stat [integer,out] :: Stat parameter of last allocate() call.
• val (nnz) [integer,out] :: non-zero values of the matrix (see CSC format).
• nonsingular [logical,in] :: Ensure matrix is non-singular if present with value .true.. Such a matrix is guaranteed to have a transversal of size $$\min({\tt m}, {\tt n})$$. In the symmetric or skew symmetric case this will be the natural diagonal. In the unsymmetric and rectangular cases a random matching is used. Otherwise the matrix may or may not be structurally singular. Note that symmetric positive-definite matrices are always non-singular.
• sort [logical,in] :: Sort entries in each column into ascending order if present with value .true.. Otherwise entries may be in any order within a column.

Possible exit status values are:

flag Status
-1 An allocation error has occurred. If present, stat will contain the Fortran stat value returned by the failed allocate() call.
-2 An invalid value of matrix_type was supplied.
-3 At least one of m, n, or nnz was less than 1.
-4 The (in)equality of m and n was inconsistent with matrix_type.
-5 A non-singular matrix was requested, but $$nnz<\min(m,n)$$.

Note

A version is also provided in which nnz and ptr have type default integer, however users are encouraged to use 64-bit integers to ensure code can handle large matrices.

## Example¶

The following code generates a random $$4 \times 5$$ matrix with $$8$$ non-zeroes that is non-singular.

! examples/Fortran/random_matrix.f90 - Example code for SPRAL_RANDOM_MATRIX pkg
program random_matrix_example
use spral_matrix_util, only : print_matrix
use spral_random, only : random_state
use spral_random_matrix
implicit none

integer, parameter :: wp = kind(0d0)

integer, parameter :: m=4, n=5, nnz=8
integer :: ptr(n+1), row(nnz)
real(wp) :: val(nnz)
integer :: flag
type(random_state) :: state

! Generate matrix
write(*, "(a,i3,a,i3,a,i3,a)") &
"Generating a ", m, " x", n, " non-singular matrix with ", nnz, &
" non-zeroes"
call random_matrix_generate(state, 0, m, n, nnz, ptr, row, flag, val=val, &
nonsingular=.true.)

! Print matrix using utility routine from SPRAL_MATRIX_UTILS package
write(*, "(a)") "Generated matrix:"
call print_matrix(6, -1, 0, m, n, ptr, row, val=val)

end program random_matrix_example


This produces the following output:

Generating a   4 x  5 non-singular matrix with   8 non-zeroes
Generated matrix:
Matrix of undefined type, dimension 4x5 with 8 entries.
1:                                         -1.0744E-01   9.1000E-01
2:                             9.5364E-01                1.0912E-01
3:                                          1.1631E-01  -5.8957E-01
4:  -9.0631E-01                                          7.7313E-01


## Method¶

If structural non-singularity is requested, first $$\min(m, n)$$ entries are generated as follows:

Unsymmetric or Rectangular
Random permutations of the rows and columns are generated. The first $$\min(m, n)$$ entries of these permutations are used to specify the entries of a maximum transversal.
Symmetric
The diagonal is added to the matrix explicitly.

The remaining non-zero entries are then assigned to columns uniformally at random. In the symmetric case, a weighting is used in proportion to the number of entries below the diagonal. If the selected column for a given non-zero is already full, a new random sample is drawn.

Once the number of entries in each column has been determined, and any required maximum transversal inserted, row indices are determined uniformally at random. Should a non-zero in that row already be present in the column, a new random sample is drawn.

In all cases, values are drawn uniformally at random from the range $$(-1,1)$$. In the positive-definite case, a post-processing step sums the absolute values of all the entries in each column and replaces the diagonal with this value.