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<li class="toctree-l2"><a class="reference internal" href="../manual/packages.html#publishing-your-package">Publishing Your Package</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../manual/packages.html#fixing-package-requirements">Fixing Package Requirements</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/packages.html#man-package-requirements">Requirements Specification</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../manual/performance-tips.html">Performance Tips</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#avoid-global-variables">Avoid global variables</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#measure-performance-with-time-and-pay-attention-to-memory-allocation">Measure performance with <tt class="docutils literal"><span class="pre">@time</span></tt> and pay attention to memory allocation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#tools">Tools</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#avoid-containers-with-abstract-type-parameters">Avoid containers with abstract type parameters</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#break-functions-into-multiple-definitions">Break functions into multiple definitions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#write-type-stable-functions">Write “type-stable” functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#avoid-changing-the-type-of-a-variable">Avoid changing the type of a variable</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#separate-kernel-functions">Separate kernel functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#access-arrays-in-memory-order-along-columns">Access arrays in memory order, along columns</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#avoid-string-interpolation-for-i-o">Avoid string interpolation for I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#fix-deprecation-warnings">Fix deprecation warnings</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#tweaks">Tweaks</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manual/performance-tips.html#performance-annotations">Performance Annotations</a></li>
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<div class="section" id="sparse-matrices">
<span id="stdlib-sparse"></span><h1>Sparse Matrices<a class="headerlink" href="#sparse-matrices" title="Permalink to this headline">¶</a></h1>
<p>Sparse matrices support much of the same set of operations as dense matrices. The following functions are specific to sparse matrices.</p>
<dl class="function">
<dt id="Base.sparse">
<tt class="descname">sparse</tt><big>(</big><em>I</em>, <em>J</em>, <em>V</em><span class="optional">[</span>, <em>m</em>, <em>n</em>, <em>combine</em><span class="optional">]</span><big>)</big><a class="headerlink" href="#Base.sparse" title="Permalink to this definition">¶</a></dt>
<dd><p>Create a sparse matrix <tt class="docutils literal"><span class="pre">S</span></tt> of dimensions <tt class="docutils literal"><span class="pre">m</span> <span class="pre">x</span> <span class="pre">n</span></tt> such that <tt class="docutils literal"><span class="pre">S[I[k],</span> <span class="pre">J[k]]</span> <span class="pre">=</span> <span class="pre">V[k]</span></tt>. The <tt class="docutils literal"><span class="pre">combine</span></tt> function is used to combine duplicates. If <tt class="docutils literal"><span class="pre">m</span></tt> and <tt class="docutils literal"><span class="pre">n</span></tt> are not specified, they are set to <tt class="docutils literal"><span class="pre">max(I)</span></tt> and <tt class="docutils literal"><span class="pre">max(J)</span></tt> respectively. If the <tt class="docutils literal"><span class="pre">combine</span></tt> function is not supplied, duplicates are added by default.</p>
</dd></dl>
<dl class="function">
<dt id="Base.sparsevec">
<tt class="descname">sparsevec</tt><big>(</big><em>I</em>, <em>V</em><span class="optional">[</span>, <em>m</em>, <em>combine</em><span class="optional">]</span><big>)</big><a class="headerlink" href="#Base.sparsevec" title="Permalink to this definition">¶</a></dt>
<dd><p>Create a sparse matrix <tt class="docutils literal"><span class="pre">S</span></tt> of size <tt class="docutils literal"><span class="pre">m</span> <span class="pre">x</span> <span class="pre">1</span></tt> such that <tt class="docutils literal"><span class="pre">S[I[k]]</span> <span class="pre">=</span> <span class="pre">V[k]</span></tt>. Duplicates are combined using the <tt class="docutils literal"><span class="pre">combine</span></tt> function, which defaults to <tt class="docutils literal"><span class="pre">+</span></tt> if it is not provided. In julia, sparse vectors are really just sparse matrices with one column. Given Julia’s Compressed Sparse Columns (CSC) storage format, a sparse column matrix with one column is sparse, whereas a sparse row matrix with one row ends up being dense.</p>
</dd></dl>
<dl class="function">
<dt>
<tt class="descname">sparsevec</tt><big>(</big><em>D::Dict</em><span class="optional">[</span>, <em>m</em><span class="optional">]</span><big>)</big></dt>
<dd><p>Create a sparse matrix of size <tt class="docutils literal"><span class="pre">m</span> <span class="pre">x</span> <span class="pre">1</span></tt> where the row values are keys from the dictionary, and the nonzero values are the values from the dictionary.</p>
</dd></dl>
<dl class="function">
<dt id="Base.issparse">
<tt class="descname">issparse</tt><big>(</big><em>S</em><big>)</big><a class="headerlink" href="#Base.issparse" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns <tt class="docutils literal"><span class="pre">true</span></tt> if <tt class="docutils literal"><span class="pre">S</span></tt> is sparse, and <tt class="docutils literal"><span class="pre">false</span></tt> otherwise.</p>
</dd></dl>
<dl class="function">
<dt>
<tt class="descname">sparse</tt><big>(</big><em>A</em><big>)</big></dt>
<dd><p>Convert a dense matrix <tt class="docutils literal"><span class="pre">A</span></tt> into a sparse matrix.</p>
</dd></dl>
<dl class="function">
<dt>
<tt class="descname">sparsevec</tt><big>(</big><em>A</em><big>)</big></dt>
<dd><p>Convert a dense vector <tt class="docutils literal"><span class="pre">A</span></tt> into a sparse matrix of size <tt class="docutils literal"><span class="pre">m</span> <span class="pre">x</span> <span class="pre">1</span></tt>. In julia, sparse vectors are really just sparse matrices with one column.</p>
</dd></dl>
<dl class="function">
<dt id="Base.full">
<tt class="descname">full</tt><big>(</big><em>S</em><big>)</big><a class="headerlink" href="#Base.full" title="Permalink to this definition">¶</a></dt>
<dd><p>Convert a sparse matrix <tt class="docutils literal"><span class="pre">S</span></tt> into a dense matrix.</p>
</dd></dl>
<dl class="function">
<dt id="Base.nnz">
<tt class="descname">nnz</tt><big>(</big><em>A</em><big>)</big><a class="headerlink" href="#Base.nnz" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the number of stored (filled) elements in a sparse matrix.</p>
</dd></dl>
<dl class="function">
<dt id="Base.spzeros">
<tt class="descname">spzeros</tt><big>(</big><em>m</em>, <em>n</em><big>)</big><a class="headerlink" href="#Base.spzeros" title="Permalink to this definition">¶</a></dt>
<dd><p>Create an empty sparse matrix of size <tt class="docutils literal"><span class="pre">m</span> <span class="pre">x</span> <span class="pre">n</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="Base.spones">
<tt class="descname">spones</tt><big>(</big><em>S</em><big>)</big><a class="headerlink" href="#Base.spones" title="Permalink to this definition">¶</a></dt>
<dd><p>Create a sparse matrix with the same structure as that of <tt class="docutils literal"><span class="pre">S</span></tt>, but with every nonzero element having the value <tt class="docutils literal"><span class="pre">1.0</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="Base.speye">
<tt class="descname">speye</tt><big>(</big><em>type</em>, <em>m</em><span class="optional">[</span>, <em>n</em><span class="optional">]</span><big>)</big><a class="headerlink" href="#Base.speye" title="Permalink to this definition">¶</a></dt>
<dd><p>Create a sparse identity matrix of specified type of size <tt class="docutils literal"><span class="pre">m</span> <span class="pre">x</span> <span class="pre">m</span></tt>. In case <tt class="docutils literal"><span class="pre">n</span></tt> is supplied, create a sparse identity matrix of size <tt class="docutils literal"><span class="pre">m</span> <span class="pre">x</span> <span class="pre">n</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="Base.spdiagm">
<tt class="descname">spdiagm</tt><big>(</big><em>B</em>, <em>d</em><span class="optional">[</span>, <em>m</em>, <em>n</em><span class="optional">]</span><big>)</big><a class="headerlink" href="#Base.spdiagm" title="Permalink to this definition">¶</a></dt>
<dd><p>Construct a sparse diagonal matrix. <tt class="docutils literal"><span class="pre">B</span></tt> is a tuple of vectors containing the diagonals and <tt class="docutils literal"><span class="pre">d</span></tt> is a tuple containing the positions of the diagonals. In the case the input contains only one diagonaly, <tt class="docutils literal"><span class="pre">B</span></tt> can be a vector (instead of a tuple) and <tt class="docutils literal"><span class="pre">d</span></tt> can be the diagonal position (instead of a tuple), defaulting to 0 (diagonal). Optionally, <tt class="docutils literal"><span class="pre">m</span></tt> and <tt class="docutils literal"><span class="pre">n</span></tt> specify the size of the resulting sparse matrix.</p>
</dd></dl>
<dl class="function">
<dt id="Base.sprand">
<tt class="descname">sprand</tt><big>(</big><em>m</em>, <em>n</em>, <em>p</em><span class="optional">[</span>, <em>rng</em><span class="optional">]</span><big>)</big><a class="headerlink" href="#Base.sprand" title="Permalink to this definition">¶</a></dt>
<dd><p>Create a random <tt class="docutils literal"><span class="pre">m</span></tt> by <tt class="docutils literal"><span class="pre">n</span></tt> sparse matrix, in which the probability of any element being nonzero is independently given by <tt class="docutils literal"><span class="pre">p</span></tt> (and hence the mean density of nonzeros is also exactly <tt class="docutils literal"><span class="pre">p</span></tt>). Nonzero values are sampled from the distribution specified by <tt class="docutils literal"><span class="pre">rng</span></tt>. The uniform distribution is used in case <tt class="docutils literal"><span class="pre">rng</span></tt> is not specified.</p>
</dd></dl>
<dl class="function">
<dt id="Base.sprandn">
<tt class="descname">sprandn</tt><big>(</big><em>m</em>, <em>n</em>, <em>p</em><big>)</big><a class="headerlink" href="#Base.sprandn" title="Permalink to this definition">¶</a></dt>
<dd><p>Create a random <tt class="docutils literal"><span class="pre">m</span></tt> by <tt class="docutils literal"><span class="pre">n</span></tt> sparse matrix with the specified (independent) probability <tt class="docutils literal"><span class="pre">p</span></tt> of any entry being nonzero, where nonzero values are sampled from the normal distribution.</p>
</dd></dl>
<dl class="function">
<dt id="Base.sprandbool">
<tt class="descname">sprandbool</tt><big>(</big><em>m</em>, <em>n</em>, <em>p</em><big>)</big><a class="headerlink" href="#Base.sprandbool" title="Permalink to this definition">¶</a></dt>
<dd><p>Create a random <tt class="docutils literal"><span class="pre">m</span></tt> by <tt class="docutils literal"><span class="pre">n</span></tt> sparse boolean matrix with the specified (independent) probability <tt class="docutils literal"><span class="pre">p</span></tt> of any entry being <tt class="docutils literal"><span class="pre">true</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="Base.etree">
<tt class="descname">etree</tt><big>(</big><em>A</em><span class="optional">[</span>, <em>post</em><span class="optional">]</span><big>)</big><a class="headerlink" href="#Base.etree" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the elimination tree of a symmetric sparse matrix <tt class="docutils literal"><span class="pre">A</span></tt> from <tt class="docutils literal"><span class="pre">triu(A)</span></tt> and, optionally, its post-ordering permutation.</p>
</dd></dl>
<dl class="function">
<dt id="Base.symperm">
<tt class="descname">symperm</tt><big>(</big><em>A</em>, <em>p</em><big>)</big><a class="headerlink" href="#Base.symperm" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the symmetric permutation of A, which is <tt class="docutils literal"><span class="pre">A[p,p]</span></tt>. A should be symmetric and sparse, where only the upper triangular part of the matrix is stored. This algorithm ignores the lower triangular part of the matrix. Only the upper triangular part of the result is returned as well.</p>
</dd></dl>
<dl class="function">
<dt id="Base.nonzeros">
<tt class="descname">nonzeros</tt><big>(</big><em>A</em><big>)</big><a class="headerlink" href="#Base.nonzeros" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a vector of the structural nonzero values in sparse matrix <tt class="docutils literal"><span class="pre">A</span></tt>. This includes zeros that are explicitly stored in the sparse matrix. The returned vector points directly to the internal nonzero storage of <tt class="docutils literal"><span class="pre">A</span></tt>, and any modifications to the returned vector will mutate <tt class="docutils literal"><span class="pre">A</span></tt> as well.</p>
</dd></dl>
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