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<p><a id="X856F202D823280F8" name="X856F202D823280F8"></a></p>
<div class="ChapSects"><a href="chap10_mj.html#X856F202D823280F8">10 <span class="Heading"> Sparse Chain complexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7CFDEEC07F15CF82">10.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X81D5E16D81934320">10.1-1 SparseMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7C85DF92798C625A">10.1-2 TransposeOfSparseMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X8011873278AB2827">10.1-3 ReverseSparseMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X8551FB1D81A85362">10.1-4 SparseRowMult</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X824CD2E6862435EB">10.1-5 SparseRowInterchange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X84E8FA687CABA3CD">10.1-6 SparseRowAdd</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7BE58D2983505606">10.1-7 SparseSemiEchelon</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7A7B47D380A14F28">10.1-8 RankMatDestructive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7B21AE7987D4FB31">10.1-9 RankMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7F10BD65823B1632">10.1-10 SparseChainComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X8156FB007C49C020">10.1-11 SparseChainComplexOfRegularCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7BA17DDC81AA855D">10.1-12 SparseBoundaryMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7C3327917BE532FD">10.1-13 Bettinumbers</a></span>
</div></div>
</div>

<h3>10 <span class="Heading"> Sparse Chain complexes</span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>10.1 <span class="Heading">  </span></h4>

<p><a id="X81D5E16D81934320" name="X81D5E16D81934320"></a></p>

<h5>10.1-1 SparseMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseMat</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a matrix <span class="SimpleMath">\(A\)</span> and returns the matrix in sparse format.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap10.html">1</a></span> </p>

<p><a id="X7C85DF92798C625A" name="X7C85DF92798C625A"></a></p>

<h5>10.1-2 TransposeOfSparseMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransposeOfSparseMat</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a sparse matrix <span class="SimpleMath">\(A\)</span> and returns its transpose sparse format.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X8011873278AB2827" name="X8011873278AB2827"></a></p>

<h5>10.1-3 ReverseSparseMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReverseSparseMat</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a sparse matrix <span class="SimpleMath">\(A\)</span> and modifies it by reversing the order of the columns. This function modifies <span class="SimpleMath">\(A\)</span> and returns no value.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X8551FB1D81A85362" name="X8551FB1D81A85362"></a></p>

<h5>10.1-4 SparseRowMult</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseRowMult</code>( <var class="Arg">A</var>, <var class="Arg">i</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Multiplies the i-th row of a sparse matrix <span class="SimpleMath">\(A\)</span> by <span class="SimpleMath">\(k\)</span>. The sparse matrix <span class="SimpleMath">\(A\)</span> is modified but nothing is returned.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X824CD2E6862435EB" name="X824CD2E6862435EB"></a></p>

<h5>10.1-5 SparseRowInterchange</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseRowInterchange</code>( <var class="Arg">A</var>, <var class="Arg">i</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Interchanges the i-th and j-th rows of a sparse matrix <span class="SimpleMath">\(A\)</span> by <span class="SimpleMath">\(k\)</span>. The sparse matrix <span class="SimpleMath">\(A\)</span> is modified but nothing is returned.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X84E8FA687CABA3CD" name="X84E8FA687CABA3CD"></a></p>

<h5>10.1-6 SparseRowAdd</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseRowAdd</code>( <var class="Arg">A</var>, <var class="Arg">i</var>, <var class="Arg">j</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Adds <span class="SimpleMath">\(k\)</span> times the j-th row to the i-th row of a sparse matrix <span class="SimpleMath">\(A\)</span>. The sparse matrix <span class="SimpleMath">\(A\)</span> is modified but nothing is returned.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7BE58D2983505606" name="X7BE58D2983505606"></a></p>

<h5>10.1-7 SparseSemiEchelon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseSemiEchelon</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Converts a sparse matrix <span class="SimpleMath">\(A\)</span> to semi-echelon form (which means echelon form up to a permutation of rows). The sparse matrix <span class="SimpleMath">\(A\)</span> is modified but nothing is returned.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7A7B47D380A14F28" name="X7A7B47D380A14F28"></a></p>

<h5>10.1-8 RankMatDestructive</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankMatDestructive</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the rank of a sparse matrix <span class="SimpleMath">\(A\)</span>. The sparse matrix <span class="SimpleMath">\(A\)</span> is modified during the calculation.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7B21AE7987D4FB31" name="X7B21AE7987D4FB31"></a></p>

<h5>10.1-9 RankMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RankMat</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the rank of a sparse matrix <span class="SimpleMath">\(A\)</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7F10BD65823B1632" name="X7F10BD65823B1632"></a></p>

<h5>10.1-10 SparseChainComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseChainComplex</code>( <var class="Arg">Y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex <span class="SimpleMath">\(Y\)</span> and returns a sparse chain complex which is chain homotopy equivalent to the cellular chain complex of <span class="SimpleMath">\(Y\)</span>. The function uses discrete vector fields to calculate a smallish chain complex.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap10.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">2</a></span> </p>

<p><a id="X8156FB007C49C020" name="X8156FB007C49C020"></a></p>

<h5>10.1-11 SparseChainComplexOfRegularCWComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseChainComplexOfRegularCWComplex</code>( <var class="Arg">Y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex <span class="SimpleMath">\(Y\)</span> and returns its cellular chain complex as a sparse chain complex. The function SparseChainComplex(Y) will usually return a smaller chain complex.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7BA17DDC81AA855D" name="X7BA17DDC81AA855D"></a></p>

<h5>10.1-12 SparseBoundaryMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SparseBoundaryMatrix</code>( <var class="Arg">C</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a sparse chain complex <span class="SimpleMath">\(C\)</span> and integer <span class="SimpleMath">\(n\)</span>. Returns the n-th boundary matrix of the chain complex in sparse format.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7C3327917BE532FD" name="X7C3327917BE532FD"></a></p>

<h5>10.1-13 Bettinumbers</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bettinumbers</code>( <var class="Arg">C</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a sparse chain complex <span class="SimpleMath">\(C\)</span> and integer <span class="SimpleMath">\(n\)</span>. Returns the n-th Netti number of the chain complex.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTDA.html">2</a></span> </p>


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