deploy: 6deab0d

Author: lorenzotomada

_sources/pyclassify.rst.txt

@@ -22,7 +22,7 @@ Submodule: utils
 Submodule: helpers_secular
 ================
 
-.. automodule:: pyclassify.helpers_secular
+.. automodule:: pyclassify.zero_finder
    :members:
    :undoc-members:
    :show-inheritance:

genindex.html

@@ -36,25 +36,40 @@
 <h1 id="index">Index</h1>
 
 <div class="genindex-jumpbox">
- <a href="#C"><strong>C</strong></a>
+ <a href="#B"><strong>B</strong></a>
+ | <a href="#C"><strong>C</strong></a>
  | <a href="#E"><strong>E</strong></a>
+ | <a href="#F"><strong>F</strong></a>
  | <a href="#I"><strong>I</strong></a>
  | <a href="#L"><strong>L</strong></a>
  | <a href="#M"><strong>M</strong></a>
  | <a href="#P"><strong>P</strong></a>
  | <a href="#R"><strong>R</strong></a>
+ | <a href="#S"><strong>S</strong></a>
 
 </div>
-<h2 id="C">C</h2>
+<h2 id="B">B</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
-      <li><a href="pyclassify.html#pyclassify.utils.check_square_matrix">check_square_matrix() (in module pyclassify.utils)</a>
+      <li><a href="pyclassify.html#pyclassify.zero_finder.bisection">bisection() (in module pyclassify.zero_finder)</a>
 </li>
   </ul></td>
+</tr></table>
+
+<h2 id="C">C</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="pyclassify.html#pyclassify.utils.check_square_matrix">check_square_matrix() (in module pyclassify.utils)</a>
+</li>
       <li><a href="pyclassify.html#pyclassify.utils.check_symm_square">check_symm_square() (in module pyclassify.utils)</a>
 </li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li><a href="pyclassify.html#pyclassify.EigenSolver.compute_eigenval">compute_eigenval() (pyclassify.EigenSolver method)</a>
+</li>
+      <li><a href="pyclassify.html#pyclassify.zero_finder.compute_outer_zero">compute_outer_zero() (in module pyclassify.zero_finder)</a>
+</li>
+      <li><a href="pyclassify.html#pyclassify.zero_finder.compute_Psi">compute_Psi() (in module pyclassify.zero_finder)</a>
 </li>
   </ul></td>
 </tr></table>
@@ -75,6 +90,14 @@ <h2 id="E">E</h2>
   </ul></td>
 </tr></table>
 
+<h2 id="F">F</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="pyclassify.html#pyclassify.zero_finder.find_root">find_root() (in module pyclassify.zero_finder)</a>
+</li>
+  </ul></td>
+</tr></table>
+
 <h2 id="I">I</h2>
 <table style="width: 100%" class="indextable genindextable"><tr>
   <td style="width: 33%; vertical-align: top;"><ul>
@@ -98,15 +121,15 @@ <h2 id="M">M</h2>
 </li>
       <li><a href="pyclassify.html#pyclassify.utils.max_iteration_warning">max_iteration_warning() (in module pyclassify.utils)</a>
 </li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li>
     module
 
       <ul>
         <li><a href="pyclassify.html#module-pyclassify">pyclassify</a>
 </li>
         <li><a href="pyclassify.html#module-pyclassify.utils">pyclassify.utils</a>
+</li>
+        <li><a href="pyclassify.html#module-pyclassify.zero_finder">pyclassify.zero_finder</a>
 </li>
       </ul></li>
   </ul></td>
@@ -121,20 +144,27 @@ <h2 id="P">P</h2>
 </li>
       <li><a href="pyclassify.html#pyclassify.power_method_numba">power_method_numba() (in module pyclassify)</a>
 </li>
-  </ul></td>
-  <td style="width: 33%; vertical-align: top;"><ul>
       <li>
     pyclassify
 
       <ul>
         <li><a href="pyclassify.html#module-pyclassify">module</a>
 </li>
       </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
       <li>
     pyclassify.utils
 
       <ul>
         <li><a href="pyclassify.html#module-pyclassify.utils">module</a>
+</li>
+      </ul></li>
+      <li>
+    pyclassify.zero_finder
+
+      <ul>
+        <li><a href="pyclassify.html#module-pyclassify.zero_finder">module</a>
 </li>
       </ul></li>
   </ul></td>
@@ -148,6 +178,14 @@ <h2 id="R">R</h2>
   </ul></td>
 </tr></table>
 
+<h2 id="S">S</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="pyclassify.html#pyclassify.zero_finder.secular_solver_python">secular_solver_python() (in module pyclassify.zero_finder)</a>
+</li>
+  </ul></td>
+</tr></table>
+
 
 
           </div>

index.html

@@ -65,7 +65,14 @@ <h1>Table of Contents<a class="headerlink" href="#table-of-contents" title="Link
 <li class="toctree-l2"><a class="reference internal" href="pyclassify.html#pyclassify.utils.read_config"><code class="docutils literal notranslate"><span class="pre">read_config()</span></code></a></li>
 </ul>
 </li>
-<li class="toctree-l1"><a class="reference internal" href="pyclassify.html#submodule-helpers-secular">Submodule: helpers_secular</a></li>
+<li class="toctree-l1"><a class="reference internal" href="pyclassify.html#submodule-helpers-secular">Submodule: helpers_secular</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="pyclassify.html#pyclassify.zero_finder.bisection"><code class="docutils literal notranslate"><span class="pre">bisection()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="pyclassify.html#pyclassify.zero_finder.compute_Psi"><code class="docutils literal notranslate"><span class="pre">compute_Psi()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="pyclassify.html#pyclassify.zero_finder.compute_outer_zero"><code class="docutils literal notranslate"><span class="pre">compute_outer_zero()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="pyclassify.html#pyclassify.zero_finder.find_root"><code class="docutils literal notranslate"><span class="pre">find_root()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="pyclassify.html#pyclassify.zero_finder.secular_solver_python"><code class="docutils literal notranslate"><span class="pre">secular_solver_python()</span></code></a></li>
+</ul>
+</li>
 </ul>
 </div>
 </section>

objects.inv

@@ -57,6 +57,11 @@ <h1>Python Module Index</h1>
        <td>&#160;&#160;&#160;
        <a href="pyclassify.html#module-pyclassify.utils"><code class="xref">pyclassify.utils</code></a></td><td>
        <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="pyclassify.html#module-pyclassify.zero_finder"><code class="xref">pyclassify.zero_finder</code></a></td><td>
+       <em></em></td></tr>
    </table>
 
 

py-modindex.html

@@ -385,6 +385,106 @@ <h1>PyClassify Module Documentation<a class="headerlink" href="#pyclassify-modul
 </section>
 <section id="submodule-helpers-secular">
 <h1>Submodule: helpers_secular<a class="headerlink" href="#submodule-helpers-secular" title="Link to this heading"></a></h1>
+<dl class="py function" id="module-pyclassify.zero_finder">
+<dt class="sig sig-object py" id="pyclassify.zero_finder.bisection">
+<span class="sig-prename descclassname"><span class="pre">pyclassify.zero_finder.</span></span><span class="sig-name descname"><span class="pre">bisection</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">f</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_iter</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#pyclassify.zero_finder.bisection" title="Link to this definition"></a></dt>
+<dd><p>Standard bisection method to find a root of the function f in the interval [a, b].</p>
+<p>This implementation is used in <cite>compute_outer_zero</cite> to locate the outer eigenvalue.</p>
+<p>Parameters:
+f (callable): A continuous function for which f(a) * f(b) &lt; 0.
+a (float): Left endpoint of the interval.
+b (float): Right endpoint of the interval.
+tol (float): Tolerance for convergence. The method stops when the interval is smaller than tol or when f(c) is sufficiently small.
+max_iter (int): Maximum number of iterations before stopping.</p>
+<p>Returns:
+float: Approximation of the root within the specified tolerance.</p>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="pyclassify.zero_finder.compute_Psi">
+<span class="sig-prename descclassname"><span class="pre">pyclassify.zero_finder.</span></span><span class="sig-name descname"><span class="pre">compute_Psi</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">i</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">d</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rho</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#pyclassify.zero_finder.compute_Psi" title="Link to this definition"></a></dt>
+<dd><p>This function computes the functions Psi1, and Psi2 and their derivative, accordingly to the equation described
+by the notes chapter 5-6 contained in the shared folders.
+The equation utilized are the equation 5.27, 5.25 and 5.29.
+d is assumed to be already a vector whose elements are already sorted.
+Inputs:</p>
+<blockquote>
+<div><p>-i: It is the index of the i-th smallest eigenvalue that is being computed
+-v: Rank one correction
+-d: Diagonal element of the Diagonal matrix described in equation 5.7
+-rho: off diagonal element used for the splitting.</p>
+</div></blockquote>
+<dl class="simple">
+<dt>Output:</dt><dd><p>Psi1:  Lamoba function that returns the value of Psi1 at the point x
+Psi2:  Lamoba function that returns the value of Psi2 at the point x
+dPsi1: Lamoba function that returns the value of the derivative dPsi1/dx at the point x
+dPsi2: Lamoba function that returns the value of the derivative dPsi2/dx at the point x</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="pyclassify.zero_finder.compute_outer_zero">
+<span class="sig-prename descclassname"><span class="pre">pyclassify.zero_finder.</span></span><span class="sig-name descname"><span class="pre">compute_outer_zero</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">v</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">d</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rho</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">interval_end</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-12</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_iter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1000</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#pyclassify.zero_finder.compute_outer_zero" title="Link to this definition"></a></dt>
+<dd><p>Computes the outer eigenvalue (lambda[0] if rho &lt; 0, lambda[n-1] if rho &gt; 0) of a rank-one modified diagonal matrix.</p>
+<dl class="simple">
+<dt>The secular function  behaves such that:</dt><dd><ul class="simple">
+<li><p>If rho &gt; 0, the outer eigenvalue lies in (d[n-1], infty), and f is increasing in this interval.</p></li>
+<li><p>If rho &lt; 0, the outer eigenvalue lies in (-infty, d[0]), and f is decreasing in this interval.</p></li>
+</ul>
+</dd>
+</dl>
+<p>This function:
+1. Determines the direction to search based on the sign of rho.
+2. Finds an upper bound (or lower bound for rho &lt; 0) where the secular function changes sign.
+3. Uses the bisection method to find the root in the determined interval.</p>
+<p>Returns:
+float: Approximation of the outer eigenvalue.</p>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="pyclassify.zero_finder.find_root">
+<span class="sig-prename descclassname"><span class="pre">pyclassify.zero_finder.</span></span><span class="sig-name descname"><span class="pre">find_root</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">i</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">left_center</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">d</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rho</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">lam_0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-15</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">maxiter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#pyclassify.zero_finder.find_root" title="Link to this definition"></a></dt>
+<dd><p>Find the roots of the secular equation contained inside the interval min(d)=d[0] and max(d)=d[-1].
+Inputs:</p>
+<blockquote>
+<div><p>-i: It is the index of the i-th smallest eigenvalue that is being computed
+-left_center: boolen variable. True if the left extreme is the center od the new - shifted reference system.
+-v: Rank one correction
+-d: Diagonal element of the Diagonal matrix described in equation 5.7
+-rho: off diagonal element used for the splitting.
+-lam_0: initial guess of the i-th root of the secular equation
+-tol: absolute or relative tollerence. For lamba value below 1e-6, tols refers to the absolute tolerance (avoiding division by zero). For</p>
+<blockquote>
+<div><p>lambda value greater than 1e-6, tol is the absolute tolerance.</p>
+</div></blockquote>
+<p>-maxiter: Maximum number of iteration of the  iterative solver.</p>
+</div></blockquote>
+<dl class="simple">
+<dt>Outputs:</dt><dd><p>-shift + lam_0: i-th smallest eigenvalue
+-lam_0: difference between the i-th eigenvalue and the nearest diagonal element.</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="pyclassify.zero_finder.secular_solver_python">
+<span class="sig-prename descclassname"><span class="pre">pyclassify.zero_finder.</span></span><span class="sig-name descname"><span class="pre">secular_solver_python</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rho</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">d</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#pyclassify.zero_finder.secular_solver_python" title="Link to this definition"></a></dt>
+<dd><p>Computes all the roots of the secular equation.
+Inputs:</p>
+<blockquote>
+<div><p>-v: Rank one correction
+-d: Diagonal element of the Diagonal matrix described in equation 5.7
+-rho: off diagonal element used for the splitting.</p>
+</div></blockquote>
+<dl class="simple">
+<dt>Outputs:</dt><dd><p>-eig_val: vector of the roots of the secular equation
+-index: vector of the index of the center of the shifted frame of reference associated to the i-ith root
+-delta: vector of difference between the i-th eigenvalue and the nearest diagonal element.</p>
+</dd>
+</dl>
+</dd></dl>
+
 </section>
 
 
@@ -434,7 +534,14 @@ <h1 class="logo"><a href="index.html">final_project</a></h1>
 <li class="toctree-l2"><a class="reference internal" href="#pyclassify.utils.read_config"><code class="docutils literal notranslate"><span class="pre">read_config()</span></code></a></li>
 </ul>
 </li>
-<li class="toctree-l1"><a class="reference internal" href="#submodule-helpers-secular">Submodule: helpers_secular</a></li>
+<li class="toctree-l1"><a class="reference internal" href="#submodule-helpers-secular">Submodule: helpers_secular</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="#pyclassify.zero_finder.bisection"><code class="docutils literal notranslate"><span class="pre">bisection()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#pyclassify.zero_finder.compute_Psi"><code class="docutils literal notranslate"><span class="pre">compute_Psi()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#pyclassify.zero_finder.compute_outer_zero"><code class="docutils literal notranslate"><span class="pre">compute_outer_zero()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#pyclassify.zero_finder.find_root"><code class="docutils literal notranslate"><span class="pre">find_root()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#pyclassify.zero_finder.secular_solver_python"><code class="docutils literal notranslate"><span class="pre">secular_solver_python()</span></code></a></li>
+</ul>
+</li>
 </ul>
 
 <div class="relations">