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<h4 class="subsection">22.1.2 Creating Sparse Matrices</h4>
<p>There are several means to create sparse matrix.
<dl>
<dt>Returned from a function<dd>There are many functions that directly return sparse matrices. These include
<dfn>speye</dfn>, <dfn>sprand</dfn>, <dfn>diag</dfn>, etc.
<br><dt>Constructed from matrices or vectors<dd>The function <dfn>sparse</dfn> allows a sparse matrix to be constructed from
three vectors representing the row, column and data. Alternatively, the
function <dfn>spconvert</dfn> uses a three column matrix format to allow easy
importation of data from elsewhere.
<br><dt>Created and then filled<dd>The function <dfn>sparse</dfn> or <dfn>spalloc</dfn> can be used to create an empty
matrix that is then filled by the user
<br><dt>From a user binary program<dd>The user can directly create the sparse matrix within an oct-file.
</dl>
<p>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
<code>speye (</code><var>n</var><code>)</code> or <code>speye (</code><var>r</var><code>, </code><var>c</var><code>)</code>, which
creates an <var>n</var>-by-<var>n</var> or <var>r</var>-by-<var>c</var> sparse identity
matrix.
<p>Another typical sparse matrix that is often needed is a random distribution
of random elements. The functions <dfn>sprand</dfn> and <dfn>sprandn</dfn> perform
this for uniform and normal random distributions of elements. They have exactly
the same calling convention, where <code>sprand (</code><var>r</var><code>, </code><var>c</var><code>, </code><var>d</var><code>)</code>,
creates an <var>r</var>-by-<var>c</var> sparse matrix with a density of filled
elements of <var>d</var>.
<p>Other functions of interest that directly create sparse matrices, are
<dfn>diag</dfn> or its generalization <dfn>spdiags</dfn>, that can take the
definition of the diagonals of the matrix and create the sparse matrix
that corresponds to this. For example,
<pre class="example"> s = diag (sparse(randn(1,n)), -1);
</pre>
<p class="noindent">creates a sparse (<var>n</var>+1)-by-(<var>n</var>+1) sparse matrix with a single
diagonal defined.
<!-- spdiags scripts/sparse/spdiags.m -->
<p><a name="doc_002dspdiags"></a>
<div class="defun">
— Function File: [<var>b</var>, <var>c</var>] = <b>spdiags</b> (<var>A</var>)<var><a name="index-spdiags-2176"></a></var><br>
— Function File: <var>b</var> = <b>spdiags</b> (<var>A, c</var>)<var><a name="index-spdiags-2177"></a></var><br>
— Function File: <var>b</var> = <b>spdiags</b> (<var>v, c, A</var>)<var><a name="index-spdiags-2178"></a></var><br>
— Function File: <var>b</var> = <b>spdiags</b> (<var>v, c, m, n</var>)<var><a name="index-spdiags-2179"></a></var><br>
<blockquote><p>A generalization of the function <code>diag</code>. Called with a single
input argument, the non-zero diagonals <var>c</var> of <var>A</var> are extracted.
With two arguments the diagonals to extract are given by the vector
<var>c</var>.
<p>The other two forms of <code>spdiags</code> modify the input matrix by
replacing the diagonals. They use the columns of <var>v</var> to replace
the columns represented by the vector <var>c</var>. If the sparse matrix
<var>A</var> is defined then the diagonals of this matrix are replaced.
Otherwise a matrix of <var>m</var> by <var>n</var> is created with the
diagonals given by <var>v</var>.
<p>Negative values of <var>c</var> represent diagonals below the main
diagonal, and positive values of <var>c</var> diagonals above the main
diagonal.
<p>For example:
<pre class="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
</pre>
</blockquote></div>
<!-- speye scripts/sparse/speye.m -->
<p><a name="doc_002dspeye"></a>
<div class="defun">
— Function File: <var>y</var> = <b>speye</b> (<var>m</var>)<var><a name="index-speye-2180"></a></var><br>
— Function File: <var>y</var> = <b>speye</b> (<var>m, n</var>)<var><a name="index-speye-2181"></a></var><br>
— Function File: <var>y</var> = <b>speye</b> (<var>sz</var>)<var><a name="index-speye-2182"></a></var><br>
<blockquote><p>Return a sparse identity matrix. This is significantly more
efficient than <code>sparse (eye (</code><var>m</var><code>))</code> as the full matrix
is not constructed.
<p>Called with a single argument a square matrix of size <var>m</var> by
<var>m</var> is created. Otherwise a matrix of <var>m</var> by <var>n</var> is
created. If called with a single vector argument, this argument
is taken to be the size of the matrix to create.
</p></blockquote></div>
<!-- spones scripts/sparse/spones.m -->
<p><a name="doc_002dspones"></a>
<div class="defun">
— Function File: <var>r</var> = <b>spones</b> (<var>S</var>)<var><a name="index-spones-2183"></a></var><br>
<blockquote><p>Replace the non-zero entries of <var>S</var> with ones. This creates a
sparse matrix with the same structure as <var>S</var>.
</p></blockquote></div>
<!-- sprand scripts/sparse/sprand.m -->
<p><a name="doc_002dsprand"></a>
<div class="defun">
— Function File: <b>sprand</b> (<var>m, n, d</var>)<var><a name="index-sprand-2184"></a></var><br>
— Function File: <b>sprand</b> (<var>s</var>)<var><a name="index-sprand-2185"></a></var><br>
<blockquote><p>Generate a random sparse matrix. The size of the matrix will be
<var>m</var> by <var>n</var>, with a density of values given by <var>d</var>.
<var>d</var> should be between 0 and 1. Values will be uniformly
distributed between 0 and 1.
<p>If called with a single matrix argument, a random sparse matrix is
generated wherever the matrix <var>S</var> is non-zero.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dsprandn.html#doc_002dsprandn">sprandn</a>, <a href="doc_002dsprandsym.html#doc_002dsprandsym">sprandsym</a>.
</p></blockquote></div>
<!-- sprandn scripts/sparse/sprandn.m -->
<p><a name="doc_002dsprandn"></a>
<div class="defun">
— Function File: <b>sprandn</b> (<var>m, n, d</var>)<var><a name="index-sprandn-2186"></a></var><br>
— Function File: <b>sprandn</b> (<var>s</var>)<var><a name="index-sprandn-2187"></a></var><br>
<blockquote><p>Generate a random sparse matrix. The size of the matrix will be
<var>m</var> by <var>n</var>, with a density of values given by <var>d</var>.
<var>d</var> should be between 0 and 1. Values will be normally
distributed with mean of zero and variance 1.
<p>If called with a single matrix argument, a random sparse matrix is
generated wherever the matrix <var>S</var> is non-zero.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dsprand.html#doc_002dsprand">sprand</a>, <a href="doc_002dsprandsym.html#doc_002dsprandsym">sprandsym</a>.
</p></blockquote></div>
<!-- sprandsym scripts/sparse/sprandsym.m -->
<p><a name="doc_002dsprandsym"></a>
<div class="defun">
— Function File: <b>sprandsym</b> (<var>n, d</var>)<var><a name="index-sprandsym-2188"></a></var><br>
— Function File: <b>sprandsym</b> (<var>s</var>)<var><a name="index-sprandsym-2189"></a></var><br>
<blockquote><p>Generate a symmetric random sparse matrix. The size of the matrix will be
<var>n</var> by <var>n</var>, with a density of values given by <var>d</var>.
<var>d</var> should be between 0 and 1. Values will be normally
distributed with mean of zero and variance 1.
<p>If called with a single matrix argument, a random sparse matrix is
generated wherever the matrix <var>S</var> is non-zero in its lower
triangular part.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dsprand.html#doc_002dsprand">sprand</a>, <a href="doc_002dsprandn.html#doc_002dsprandn">sprandn</a>.
</p></blockquote></div>
<p>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,
<pre class="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);
</pre>
<p class="noindent">creates an <var>r</var>-by-<var>c</var> sparse matrix with a random distribution
of <var>n</var> (<<var>r</var>) 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.
<p>The function <dfn>spconvert</dfn> 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:
<pre class="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
</pre>
<p>An example of creating and filling a matrix might be
<pre class="example"> 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
</pre>
<p>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 <dfn>spalloc</dfn> function ignores the <var>nz</var> 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.
<!-- full src/data.cc -->
<p><a name="doc_002dfull"></a>
<div class="defun">
— Loadable Function: <var>FM</var> = <b>full</b> (<var>SM</var>)<var><a name="index-full-2190"></a></var><br>
<blockquote><p>Return a full storage matrix from a sparse, diagonal, permutation matrix
or a range.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dsparse.html#doc_002dsparse">sparse</a>.
</p></blockquote></div>
<!-- spalloc src/sparse.cc -->
<p><a name="doc_002dspalloc"></a>
<div class="defun">
— Loadable Function: <var>s</var> = <b>spalloc</b> (<var>m, n, nz</var>)<var><a name="index-spalloc-2191"></a></var><br>
<blockquote><p>Create an <var>m</var>-by-<var>n</var> sparse matrix with pre-allocated space for at
most <var>nz</var> nonzero elements. This is useful for building the matrix
incrementally by a sequence of indexed assignments. Subsequent indexed
assignments will reuse the pre-allocated memory, provided they are of one of
the simple forms
<ul>
<li><var>s</var><code>(I:J) = </code><var>x</var>
<li><var>s</var><code>(:,I:J) = </code><var>x</var>
<li><var>s</var><code>(K:L,I:J) = </code><var>x</var>
</ul>
<p><b>and</b> that the following conditions are met:
<ul>
<li>the assignment does not decrease nnz(<var>S</var>).
<li>after the assignment, nnz(<var>S</var>) does not exceed <var>nz</var>.
<li>no index is out of bounds.
</ul>
<p>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
contiguous block of columns.
<p>The amount of pre-allocated memory for a given matrix may be queried using
the function <code>nzmax</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dnzmax.html#doc_002dnzmax">nzmax</a>, <a href="doc_002dsparse.html#doc_002dsparse">sparse</a>.
</p></blockquote></div>
<!-- sparse src/sparse.cc -->
<p><a name="doc_002dsparse"></a>
<div class="defun">
— Loadable Function: <var>s</var> = <b>sparse</b> (<var>a</var>)<var><a name="index-sparse-2192"></a></var><br>
— Loadable Function: <var>s</var> = <b>sparse</b> (<var>i, j, sv, m, n, nzmax</var>)<var><a name="index-sparse-2193"></a></var><br>
— Loadable Function: <var>s</var> = <b>sparse</b> (<var>i, j, sv</var>)<var><a name="index-sparse-2194"></a></var><br>
— Loadable Function: <var>s</var> = <b>sparse</b> (<var>i, j, s, m, n, "unique"</var>)<var><a name="index-sparse-2195"></a></var><br>
— Loadable Function: <var>s</var> = <b>sparse</b> (<var>m, n</var>)<var><a name="index-sparse-2196"></a></var><br>
<blockquote><p>Create a sparse matrix from the full matrix or row, column, value triplets.
If <var>a</var> is a full matrix, convert it to a sparse matrix representation,
removing all zero values in the process.
<p>Given the integer index vectors <var>i</var> and <var>j</var>, a 1-by-<code>nnz</code> vector
of real of complex values <var>sv</var>, overall dimensions <var>m</var> and <var>n</var>
of the sparse matrix. The argument <code>nzmax</code> is ignored but accepted for
compatibility with <span class="sc">matlab</span>. If <var>m</var> or <var>n</var> are not specified
their values are derived from the maximum index in the vectors <var>i</var> and
<var>j</var> as given by <var>m</var><code> = max (</code><var>i</var><code>)</code>,
<var>n</var><code> = max (</code><var>j</var><code>)</code>.
<p><strong>Note</strong>: if multiple values are specified with the same
<var>i</var>, <var>j</var> indices, the corresponding values in <var>s</var> will
be added.
<p>The following are all equivalent:
<pre class="example"> s = sparse (i, j, s, m, n)
s = sparse (i, j, s, m, n, "summation")
s = sparse (i, j, s, m, n, "sum")
</pre>
<p>Given the option "unique". if more than two values are specified for the
same <var>i</var>, <var>j</var> indices, the last specified value will be used.
<p><code>sparse(</code><var>m</var><code>, </code><var>n</var><code>)</code> is equivalent to
<code>sparse ([], [], [], </code><var>m</var><code>, </code><var>n</var><code>, 0)</code>
<p>If any of <var>sv</var>, <var>i</var> or <var>j</var> are scalars, they are expanded
to have a common size.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dfull.html#doc_002dfull">full</a>.
</p></blockquote></div>
<!-- spconvert scripts/sparse/spconvert.m -->
<p><a name="doc_002dspconvert"></a>
<div class="defun">
— Function File: <var>x</var> = <b>spconvert</b> (<var>m</var>)<var><a name="index-spconvert-2197"></a></var><br>
<blockquote><p>This function converts for a simple sparse matrix format easily
produced by other programs into Octave's internal sparse format. The
input <var>x</var> 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.
</p></blockquote></div>
<p>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 <a href="Dynamically-Linked-Functions.html#Dynamically-Linked-Functions">Dynamically Linked Functions</a>, to have
a full description of the techniques involved.
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