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# sprand

sparse random matrix

### Calling Sequence

sp=sprand(nrows,ncols,density [,typ])

### Arguments

- nrows
integer (number of rows)

- ncols
integer (number of columns)

- density
filling coefficient (density)

- typ
character string,

`"uniform"`

(default) or`"normal"`

- sp
sparse matrix

### Description

`sp=sprand(nrows,ncols,density)`

returns a sparse
matrix `sp`

with `nrows`

rows,
`ncols`

columns and approximately
`density*nrows*ncols`

non-zero entries.

The `density`

parameter is expected to be in the
`[0,1]`

interval. If not, it is automatically
projected into this interval. Therefore, using a density which is
lower than 0 or greater than 1 will generate neither an error,
nor a warning: the formula `density=max(min(density,1),0)`

is used.

If `typ="uniform"`

uniformly distributed values on
[0,1] are generated. If `typ="normal"`

normally
distributed values are generated (mean=0 and standard deviation=1).

The entries of the output matrix are computed from the given
distribution function `typ`

.
The indices of the non-zeros entries are computed
randomly, so that the average number of nonzeros is equal to
`density`

.
The actual indices values are computed from the exponential distribution
function, where the parameter of the distribution function is
computed accordingly.

As a side effect, the states of the random number generators `rand`

and `grand`

are modified by this
function.
The indices of the nonzeros entries are computed from an
exponential distribution function and the grand function. The values of
the matrix are computed from the distribution function given by the user
(i.e. uniform or normal) and the rand function.

### Examples

In the following example, we generate a 100x1000 sparse matrix with approximate density 0.001, i.e. with approximately 100*1000*0.001=100 nonzero entries.

// The entries of the matrix are uniform. W=sprand(100,1000,0.001); // The entries of the matrix are normal. W=sprand(100,1000,0.001,"normal");

In the following script, we check that the entries of the matrix have the expected distribution. We use the spget function in order to get the nonzero entries. Then we compute the min, mean and max of the entries and compare them with the limit values.

typ = "normal"; // typ = "uniform"; nrows = 1000; ncols = 2000; density = 1/100; s=sprand(nrows,ncols,density,typ); nnzs=nnz(s); [ij,v]=spget(s); [%inf -%inf 0 %inf 1] // Limit values for "normal" [nnzs min(v) mean(v) max(v) variance(v)] [%inf 0 0.5 1 1/12] // Limit values for "uniform"

In the following script, we check that the entry indices, which
are also chosen at random, have the correct distribution.
We generate `kmax`

sparse random matrices with
uniform distribution.
For each matrix, we consider the indices of the nonzero
entries which were generated, i.e. we see if the event
`Aij = {the entry (i,j) is nonzero}`

occured for
each `i`

and `j`

, for `i=1,2,...,nrows`

and `j=1,2,...,ncols`

.
The matrix `C(i,j)`

stores the number of times that the event `Aij`

occured.
The matrix `R(k)`

stores the actual density of the try number `k`

,
where `k=1,2,...,kmax`

.

kmax = 1000; ncols=10; nrows=15; density=0.01; typ="uniform"; C=zeros(nrows,ncols); R=[]; for k=1:kmax M=sprand(nrows,ncols,density,typ); NZ=find(M<>0); NZratio = size(NZ,"*")/(nrows*ncols); R=[R NZratio]; C(NZ)=C(NZ)+1; end

Now that this algorithm has been performed (which may require some time), we can compute elementary statistics to check that the algorithm performed well.

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