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  <div class="section" id="introduction">
<h1>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">ΒΆ</a></h1>
<p>When working towards a solution of a linear system <span class="math">\(Ax=b\)</span>, Krylov methods
do not need to know anything structural about the matrix <span class="math">\(A\)</span>; all they
require is the ability to form matrix-vector products <span class="math">\(v \mapsto Av\)</span> and,
possibly, products with the transpose <span class="math">\(u \mapsto A^T u\)</span>. In essence, we
do not even need the <em>operator</em> <span class="math">\(A\)</span> to be represented by a matrix at all;
we simply consider it as a linear function.</p>
<p>In PyKrylov, such linear functions can be conveniently packaged as
<tt class="docutils literal"><span class="pre">LinearOperator</span></tt> objects. If <tt class="docutils literal"><span class="pre">A</span></tt> is an instance of <tt class="docutils literal"><span class="pre">LinearOperator</span></tt> and
represents the &#8220;matrix&#8221; <span class="math">\(A\)</span> above, we may computes matrix-vector products
by simply writing <tt class="docutils literal"><span class="pre">A*v</span></tt>, where <tt class="docutils literal"><span class="pre">v</span></tt> is a Numpy array of appropriate size.</p>
<p>Similarly, if a Krylov method requires access to the transpose operator
<span class="math">\(A^T\)</span>, it is conveniently available as <tt class="docutils literal"><span class="pre">A.T</span></tt> and products may be
computed using, e.g., <tt class="docutils literal"><span class="pre">A.T</span> <span class="pre">*</span> <span class="pre">u</span></tt>. If <tt class="docutils literal"><span class="pre">A</span></tt> represents a symmetric operator
<span class="math">\(A = A^T\)</span>, then <tt class="docutils literal"><span class="pre">A.T</span></tt> is simply a reference to <tt class="docutils literal"><span class="pre">A</span></tt> itself.</p>
<p>More generally, since <span class="math">\((A^T)^T = A\)</span>, the Python statement <tt class="docutils literal"><span class="pre">A.T.T</span> <span class="pre">is</span> <span class="pre">A</span></tt>
always evaluates to <tt class="docutils literal"><span class="pre">True</span></tt>, which means that they are the <em>same</em> object.</p>
<p>In the next two sections, we describe generic linear operators and linear
operators constructed by blocks.</p>
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<li class="toctree-l1 current"><a class="current reference internal" href="">Introduction</a></li>
<li class="toctree-l1"><a class="reference internal" href="linop.html">Linear Operators</a></li>
<li class="toctree-l1"><a class="reference internal" href="blkop.html">Block Linear Operators</a></li>
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