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### 21.1.3 Finding out Information about Sparse Matrices

There are a number of functions that allow information concerning
sparse matrices to be obtained. The most basic of these is
*issparse* that identifies whether a particular Octave object is
in fact a sparse matrix.

Another very basic function is *nnz* that returns the number of
non-zero entries there are in a sparse matrix, while the function
*nzmax* returns the amount of storage allocated to the sparse
matrix. Note that Octave tends to crop unused memory at the first
opportunity for sparse objects. There are some cases of user created
sparse objects where the value returned by *nzmax* will not be
the same as *nnz*, but in general they will give the same
result. The function *spstats* returns some basic statistics on
the columns of a sparse matrix including the number of elements, the
mean and the variance of each column.

__Built-in Function:__`scalar`=**nnz***(*`a`)Returns the number of non zero elements in

`a`.**See also:**sparse.

__Built-in Function:__`scalar`=**nzmax***(*`SM`)Return the amount of storage allocated to the sparse matrix

`SM`. Note that Octave tends to crop unused memory at the first opportunity for sparse objects. There are some cases of user created sparse objects where the value returned by*nzmax*will not be the same as*nnz*, but in general they will give the same result.

__Function File:__[`count`,`mean`,`var`] =**spstats***(*`s`)__Function File:__[`count`,`mean`,`var`] =**spstats***(*`s`,`j`)Return the stats for the non-zero elements of the sparse matrix

`s`.`count`is the number of non-zeros in each column,`mean`is the mean of the non-zeros in each column, and`var`is the variance of the non-zeros in each column.Called with two input arguments, if

`s`is the data and`j`is the bin number for the data, compute the stats for each bin. In this case, bins can contain data values of zero, whereas with`spstats (`

the zeros may disappear.`s`)

When solving linear equations involving sparse matrices Octave
determines the means to solve the equation based on the type of the
matrix as discussed in Linear Algebra on Sparse Matrices. Octave probes the
matrix type when the div (/) or ldiv (\) operator is first used with
the matrix and then caches the type. However the *matrix_type*
function can be used to determine the type of the sparse matrix prior
to use of the div or ldiv operators. For example

a = tril (sprandn(1024, 1024, 0.02), -1) ... + speye(1024); matrix_type (a); ans = Lower |

show that Octave correctly determines the matrix type for lower
triangular matrices. *matrix_type* can also be used to force
the type of a matrix to be a particular type. For example

a = matrix_type (tril (sprandn (1024, ... 1024, 0.02), -1) + speye(1024), 'Lower'); |

This allows the cost of determining the matrix type to be avoided. However, incorrectly defining the matrix type will result in incorrect results from solutions of linear equations, and so it is entirely the responsibility of the user to correctly identify the matrix type

There are several graphical means of finding out information about
sparse matrices. The first is the *spy* command, which displays
the structure of the non-zero elements of the
matrix. See fig:spmatrix, for an example of the use of
*spy*. More advanced graphical information can be obtained with the
*treeplot*, *etreeplot* and *gplot* commands.

**Figure 21.1: Structure of simple sparse matrix.
**

One use of sparse matrices is in graph theory, where the
interconnections between nodes are represented as an adjacency
matrix. That is, if the i-th node in a graph is connected to the j-th
node. Then the ij-th node (and in the case of undirected graphs the
ji-th node) of the sparse adjacency matrix is non-zero. If each node
is then associated with a set of coordinates, then the *gplot*
command can be used to graphically display the interconnections
between nodes.

As a trivial example of the use of *gplot*, consider the example

A = sparse([2,6,1,3,2,4,3,5,4,6,1,5], [1,1,2,2,3,3,4,4,5,5,6,6],1,6,6); xy = [0,4,8,6,4,2;5,0,5,7,5,7]'; gplot(A,xy) |

which creates an adjacency matrix `A`

where node 1 is connected
to nodes 2 and 6, node 2 with nodes 1 and 3, etc. The coordinates of
the nodes are given in the n-by-2 matrix `xy`

.
See fig:gplot.

**Figure 21.2: Simple use of the gplot command.
**

The dependencies between the nodes of a Cholesky factorization can be
calculated in linear time without explicitly needing to calculate the
Cholesky factorization by the `etree`

command. This command
returns the elimination tree of the matrix and can be displayed
graphically by the command `treeplot(etree(A))`

if `A`

is
symmetric or `treeplot(etree(A+A'))`

otherwise.

__Function File:__**spy***(*`x`)__Function File:__**spy***(…,*`markersize`)__Function File:__**spy***(…,*`line_spec`)Plot the sparsity pattern of the sparse matrix

`x`. If the argument`markersize`is given as an scalar value, it is used to determine the point size in the plot. If the string`line_spec`is given it is passed to`plot`

and determines the appearance of the plot.**See also:**plot.

__Loadable Function:__`p`=**etree***(*`s`)__Loadable Function:__`p`=**etree***(*`s`,`typ`)__Loadable Function:__[`p`,`q`] =**etree***(*`s`,`typ`)Returns the elimination tree for the matrix

`s`. By default`s`is assumed to be symmetric and the symmetric elimination tree is returned. The argument`typ`controls whether a symmetric or column elimination tree is returned. Valid values of`typ`are 'sym' or 'col', for symmetric or column elimination tree respectivelyCalled with a second argument,

*etree*also returns the postorder permutations on the tree.

__Function File:__**etreeplot***(*`tree`)__Function File:__**etreeplot***(*`tree`,`node_style`,`edge_style`)Plot the elimination tree of the matrix

`s`or

if`s`+`s`'`s`in non-symmetric. The optional parameters`line_style`and`edge_style`define the output style.

__Function File:__**gplot***(*`a`,`xy`)__Function File:__**gplot***(*`a`,`xy`,`line_style`)__Function File:__[`x`,`y`] =**gplot***(*`a`,`xy`)Plot a graph defined by

`A`and`xy`in the graph theory sense.`A`is the adjacency matrix of the array to be plotted and`xy`is an`n`-by-2 matrix containing the coordinates of the nodes of the graph.The optional parameter

`line_style`defines the output style for the plot. Called with no output arguments the graph is plotted directly. Otherwise, return the coordinates of the plot in`x`and`y`.

__Function File:__**treeplot***(*`tree`)__Function File:__**treeplot***(*`tree`,`line_style`,`edge_style`)Produces a graph of tree or forest. The first argument is vector of predecessors, optional parameters

`line_style`and`edge_style`define the output style. The complexity of the algorithm is O(n) in terms of is time and memory requirements.

__Function File:__**treelayout***(*`Tree`)__Function File:__**treelayout***(*`Tree`,`permutation`)treelayout lays out a tree or a forest. The first argument

`Tree`is a vector of predecessors, optional parameter`permutation`is an optional postorder permutation. The complexity of the algorithm is O(n) in terms of time and memory requirements.

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