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### 21.1.2 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 (`

or `n`)`speye (`

, which
creates an `r`, `c`)`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 (`

,
creates an `r`, `c`, `d`)`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.

__Function File:__[`b`,`c`] =**spdiags***(*`a`)__Function File:__`b`=**spdiags***(*`a`,`c`)__Function File:__`b`=**spdiags***(*`v`,`c`,`a`)__Function File:__`b`=**spdiags***(*`v`,`c`,`m`,`n`)A generalization of the function

`diag`

. Called with a single input argument, the non-zero diagonals`c`of`A`are extracted. With two arguments the diagonals to extract are given by the vector`c`.The other two forms of

`spdiags`

modify the input matrix by replacing the diagonals. They use the columns of`v`to replace the columns represented by the vector`c`. 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`v`.Negative values of

`c`represent diagonals below the main diagonal, and positive values of`c`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

__Function File:__`y`=**speye***(*`m`)__Function File:__`y`=**speye***(*`m`,`n`)__Function File:__`y`=**speye***(*`sz`)Returns a sparse identity matrix. This is significantly more efficient than

`sparse (eye (`

as the full matrix is not constructed.`m`))Called with a single argument a square matrix of size

`m`by`m`is created. Otherwise a matrix of`m`by`n`is created. If called with a single vector argument, this argument is taken to be the size of the matrix to create.

__Function File:__`y`=**spfun***(*`f`,`x`)Compute

`f(`

for the non-zero values of`x`)`x`. This results in a sparse matrix with the same structure as`x`. The function`f`can be passed as a string, a function handle or an inline function.

__Mapping Function:__**spmax***(*`x`,`y`,`dim`)__Mapping Function:__[`w`,`iw`] =**spmax***(*`x`)This function has been deprecated. Use

`max`

instead.

__Mapping Function:__**spmin***(*`x`,`y`,`dim`)__Mapping Function:__[`w`,`iw`] =**spmin***(*`x`)This function has been deprecated. Use

`min`

instead.

__Function File:__`y`=**spones***(*`x`)Replace the non-zero entries of

`x`with ones. This creates a sparse matrix with the same structure as`x`.

__Function File:__**sprand***(*`m`,`n`,`d`)__Function File:__**sprand***(*`s`)Generate a random sparse matrix. The size of the matrix will be

`m`by`n`, with a density of values given by`d`.`d`should be between 0 and 1. Values will be uniformly distributed between 0 and 1.Note: sometimes the actual density may be a bit smaller than

`d`. This is unlikely to happen for large really sparse matrices.If called with a single matrix argument, a random sparse matrix is generated wherever the matrix

`S`is non-zero.**See also:**sprandn.

__Function File:__**sprandn***(*`m`,`n`,`d`)__Function File:__**sprandn***(*`s`)Generate a random sparse matrix. The size of the matrix will be

`m`by`n`, with a density of values given by`d`.`d`should be between 0 and 1. Values will be normally distributed with mean of zero and variance 1.Note: sometimes the actual density may be a bit smaller than

`d`. This is unlikely to happen for large really sparse matrices.If called with a single matrix argument, a random sparse matrix is generated wherever the matrix

`S`is non-zero.**See also:**sprand.

__Function File:__**sprandsym***(*`n`,`d`)__Function File:__**sprandsym***(*`s`)Generate a symmetric random sparse matrix. The size of the matrix will be

`n`by`n`, with a density of values given by`d`.`d`should be between 0 and 1. Values will be normally distributed with mean of zero and variance 1.Note: sometimes the actual density may be a bit smaller than

`d`. This is unlikely to happen for large really sparse matrices.If called with a single matrix argument, a random sparse matrix is generated wherever the matrix

`S`is non-zero in its lower triangular part.

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)(1: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 preassign 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.

__Loadable Function:__`FM`=**full***(*`SM`)returns a full storage matrix from a sparse, diagonal, permutation matrix or a range.

**See also:**sparse.

__Function File:__`s`=**spalloc***(*`r`,`c`,`nz`)Returns an empty sparse matrix of size

`r`-by-`c`. As Octave resizes sparse matrices at the first opportunity, so that no additional space is needed, the argument`nz`is ignored. This function is provided only for compatibility reasons.It should be noted that this means that code like

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

will reallocate memory at each step. It is therefore vitally important that code like this is vectorized as much as possible.

__Loadable Function:__`s`=**sparse***(*`a`)__Loadable Function:__`s`=**sparse***(*`i`,`j`,`sv`,`m`,`n`,`nzmax`)__Loadable Function:__`s`=**sparse***(*`i`,`j`,`sv`)__Loadable Function:__`s`=**sparse***(*`i`,`j`,`s`,`m`,`n`, "unique")__Loadable Function:__`s`=**sparse***(*`m`,`n`)Create a sparse matrix from the full matrix 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.Given the integer index vectors

`i`and`j`, a 1-by-`nnz`

vector of real of complex values`sv`, overall dimensions`m`and`n`of the sparse matrix. The argument`nzmax`

is ignored but accepted for compatibility with MATLAB. If`m`or`n`are not specified 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 values in`s`will be added.The following are all equivalent:

s = sparse (i, j, s, m, n) s = sparse (i, j, s, m, n, "summation") s = sparse (i, j, s, m, n, "sum")

Given the option "unique". if more than two values are specified for the same

`i`,`j`indices, the last specified value will be used.`sparse(`

is equivalent to`m`,`n`)`sparse ([], [], [],`

`m`,`n`, 0)If any of

`sv`,`i`or`j`are scalars, they are expanded to have a common size.**See also:**full.

__Function File:__`x`=**spconvert***(*`m`)This function converts for a simple sparse matrix format easily produced by other programs into Octave's internal sparse format. The input

`x`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.

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, and you are referred to chapter Dynamically Linked Functions, to have a full description of the techniques involved.

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