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docs/main/ChangeLog.html

@@ -62,6 +62,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>

docs/main/_sources/background.rst.txt

@@ -243,4 +243,28 @@ We define the Matsubara (or imaginary) frequency points as :math:`i \nu_n = (2 n
 i/\beta` for bosonic Green's functions. In ``cppdlr``, Matsubara frequency
 points are represented by specifying the integer ``n``, the inverse temperature
 :math:`\beta`, and whether the point is a fermionic or bosonic Matsubara
-frequency using the ``statistic_t`` specifier.
+frequency using the ``statistic_t`` specifier.
+
+Symmetrized DLR grids
+---------------------
+
+By default, the DLR frequencies :math:`\omega_l` are not chosen to be symmetrized
+about :math:`\omega = 0`, nor are the DLR imaginary time nodes :math:`\tau_k`
+chosen to be symmetrized about :math:`\tau = \beta/2` or the imaginary frequency
+nodes about :math:`i \nu_n = 0`. Indeed, this would represent an additional
+constraint in the pivoted Gram-Schmidt procedure used to select the points, and
+is not necessary in most applications. However, in some cases, it might be
+desirable to have symmetric frequencies and grids, and ``cppdlr`` provides this
+functionality via symmetrization flags. Please see the section :ref:`List of
+other cppdlr capabilities<listofothercapabilities>` on the :ref:`examples
+page<examples>` for a list of `cppdlr` tests which showcase this functionality.
+
+We make a small note about symmetrization for bosonic Green's functions. In this
+case, we always select the DLR frequency :math:`\omega = 0`, the DLR imaginary
+time node :math:`\tau = \beta/2`, and the DLR imaginary frequency node 
+:math:`i \nu_n = 0`. The reason is as follows. A symmetric DLR imaginary frequency grid
+containing the point :math:`i \nu_n = 0` must have an odd number of points.
+Since it is undesirable to disallow this point, and to make sure all grids have
+the same number :math:`r` of points, we force :math:`r` to be odd. This requires
+including in each grid, by hand, a single point of high symmetry, and then allowing the
+pivoted Gram-Schmidt procedure to select the rest of the points symmetrically.

docs/main/_sources/examples.rst.txt

@@ -195,6 +195,15 @@ list several such use cases below.
 - Obtain a DLR expansion by interpolation on the DLR Matsubara frequency nodes:
   see the tests ``imfreq_ops.interp_scalar`` and ``imfreq_ops.interp_matrix`` in
   the file ``test/imfreq_ops.cpp``.
+- Obtain symmetrized DLR grids, and obtain a DLR expansion by interpolation on
+  these grids: see the tests ``imtime_ops.interp_matrix_sym_fer`` and 
+  ``imtime_ops.interp_matrix_sym_bos`` in the file ``test/imtime_ops.cpp`` for
+  fermionic and bosonic Green's functions, respectively, on a symmetric
+  imaginary time grid. See the tests ``imfreq_ops.interp_matrix_sym_fer`` and
+   ``imfreq_ops.interp_matrix_sym_bos`` in the file ``test/imfreq_ops.cpp`` for
+   fermionic and bosonic Green's functions, respectively, on a symmetric
+   Matsubara frequency grid. All of these tests show how to obtain a symmetric
+   set of DLR frequencies.
 - Given a fixed self-energy, solve the Dyson equation in imaginary time to
   obtain the Green's function: see the tests ``dyson_it.dyson_vs_ed_real``,
   ``dyson_it.dyson_vs_ed_cmplx``, and ``dyson_it.dyson_bethe`` in the file

docs/main/background.html

@@ -66,6 +66,7 @@
 <li class="toctree-l2"><a class="reference internal" href="#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>
@@ -294,6 +295,28 @@ <h2>Matsubara frequency point format<a class="headerlink" href="#matsubara-frequ
 <span class="math notranslate nohighlight">\(\beta\)</span>, and whether the point is a fermionic or bosonic Matsubara
 frequency using the <code class="docutils literal notranslate"><span class="pre">statistic_t</span></code> specifier.</p>
 </div>
+<div class="section" id="symmetrized-dlr-grids">
+<h2>Symmetrized DLR grids<a class="headerlink" href="#symmetrized-dlr-grids" title="Permalink to this heading"></a></h2>
+<p>By default, the DLR frequencies <span class="math notranslate nohighlight">\(\omega_l\)</span> are not chosen to be symmetrized
+about <span class="math notranslate nohighlight">\(\omega = 0\)</span>, nor are the DLR imaginary time nodes <span class="math notranslate nohighlight">\(\tau_k\)</span>
+chosen to be symmetrized about <span class="math notranslate nohighlight">\(\tau = \beta/2\)</span> or the imaginary frequency
+nodes about <span class="math notranslate nohighlight">\(i \nu_n = 0\)</span>. Indeed, this would represent an additional
+constraint in the pivoted Gram-Schmidt procedure used to select the points, and
+is not necessary in most applications. However, in some cases, it might be
+desirable to have symmetric frequencies and grids, and <code class="docutils literal notranslate"><span class="pre">cppdlr</span></code> provides this
+functionality via symmetrization flags. Please see the section <a class="reference internal" href="examples.html#listofothercapabilities"><span class="std std-ref">List of
+other cppdlr capabilities</span></a> on the <a class="reference internal" href="examples.html#examples"><span class="std std-ref">examples
+page</span></a> for a list of <cite>cppdlr</cite> tests which showcase this functionality.</p>
+<p>We make a small note about symmetrization for bosonic Green’s functions. In this
+case, we always select the DLR frequency <span class="math notranslate nohighlight">\(\omega = 0\)</span>, the DLR imaginary
+time node <span class="math notranslate nohighlight">\(\tau = \beta/2\)</span>, and the DLR imaginary frequency node
+<span class="math notranslate nohighlight">\(i \nu_n = 0\)</span>. The reason is as follows. A symmetric DLR imaginary frequency grid
+containing the point <span class="math notranslate nohighlight">\(i \nu_n = 0\)</span> must have an odd number of points.
+Since it is undesirable to disallow this point, and to make sure all grids have
+the same number <span class="math notranslate nohighlight">\(r\)</span> of points, we force <span class="math notranslate nohighlight">\(r\)</span> to be odd. This requires
+including in each grid, by hand, a single point of high symmetry, and then allowing the
+pivoted Gram-Schmidt procedure to select the rest of the points symmetrically.</p>
+</div>
 </div>
 
 

docs/main/documentation.html

@@ -64,6 +64,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>

docs/main/examples.html

@@ -66,6 +66,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1 current"><a class="current reference internal" href="#">Examples</a><ul>
@@ -233,26 +234,44 @@ <h2>Example: form a DLR expansion via interpolation, and evaluate it in imaginar
 <p>For <code class="docutils literal notranslate"><span class="pre">cppdlr</span></code> use cases which are not covered by examples in the <code class="docutils literal notranslate"><span class="pre">examples</span></code> directory,
 relevant unit tests in the <code class="docutils literal notranslate"><span class="pre">test</span></code> directory can serve as useful examples. We
 list several such use cases below.</p>
-<ul class="simple">
-<li>Obtain a DLR expansion by fitting to data in imaginary time: see the tests <code class="docutils literal notranslate"><span class="pre">imtime_ops.fit_scalar</span></code>,
-<code class="docutils literal notranslate"><span class="pre">imtime_ops.fit_matrix</span></code>, and <code class="docutils literal notranslate"><span class="pre">imtime_ops.fit_matrix_cmplx</span></code> in the file <code class="docutils literal notranslate"><span class="pre">test/imtime_ops.cpp</span></code>.</li>
-<li>Compute the convolution of two DLR expansions: see the tests
+<ul>
+<li><p class="first">Obtain a DLR expansion by fitting to data in imaginary time: see the tests <code class="docutils literal notranslate"><span class="pre">imtime_ops.fit_scalar</span></code>,
+<code class="docutils literal notranslate"><span class="pre">imtime_ops.fit_matrix</span></code>, and <code class="docutils literal notranslate"><span class="pre">imtime_ops.fit_matrix_cmplx</span></code> in the file <code class="docutils literal notranslate"><span class="pre">test/imtime_ops.cpp</span></code>.</p>
+</li>
+<li><p class="first">Compute the convolution of two DLR expansions: see the tests
 <code class="docutils literal notranslate"><span class="pre">imtime_ops.convolve_scalar_real</span></code>, <code class="docutils literal notranslate"><span class="pre">imtime_ops.convolve_scalar_cmplx</span></code>,
 <code class="docutils literal notranslate"><span class="pre">imtime_ops.convolve_matrix_real</span></code>, and <code class="docutils literal notranslate"><span class="pre">imtime_ops.convolve_matrix_cmplx</span></code>
-in the file <code class="docutils literal notranslate"><span class="pre">test/imtime_ops.cpp</span></code>.</li>
-<li>Perform a “reflection” operation <span class="math notranslate nohighlight">\(G(\tau) \mapsto G(\beta-\tau)\)</span> on a
+in the file <code class="docutils literal notranslate"><span class="pre">test/imtime_ops.cpp</span></code>.</p>
+</li>
+<li><p class="first">Perform a “reflection” operation <span class="math notranslate nohighlight">\(G(\tau) \mapsto G(\beta-\tau)\)</span> on a
 Green’s function: see the test <code class="docutils literal notranslate"><span class="pre">imtime_ops.refl_matrix</span></code> in the file
-<code class="docutils literal notranslate"><span class="pre">test/imtime_ops.cpp</span></code>.</li>
-<li>Obtain a DLR expansion by interpolation on the DLR Matsubara frequency nodes:
+<code class="docutils literal notranslate"><span class="pre">test/imtime_ops.cpp</span></code>.</p>
+</li>
+<li><p class="first">Obtain a DLR expansion by interpolation on the DLR Matsubara frequency nodes:
 see the tests <code class="docutils literal notranslate"><span class="pre">imfreq_ops.interp_scalar</span></code> and <code class="docutils literal notranslate"><span class="pre">imfreq_ops.interp_matrix</span></code> in
-the file <code class="docutils literal notranslate"><span class="pre">test/imfreq_ops.cpp</span></code>.</li>
-<li>Given a fixed self-energy, solve the Dyson equation in imaginary time to
+the file <code class="docutils literal notranslate"><span class="pre">test/imfreq_ops.cpp</span></code>.</p>
+</li>
+<li><p class="first">Obtain symmetrized DLR grids, and obtain a DLR expansion by interpolation on
+these grids: see the tests <code class="docutils literal notranslate"><span class="pre">imtime_ops.interp_matrix_sym_fer</span></code> and
+<code class="docutils literal notranslate"><span class="pre">imtime_ops.interp_matrix_sym_bos</span></code> in the file <code class="docutils literal notranslate"><span class="pre">test/imtime_ops.cpp</span></code> for
+fermionic and bosonic Green’s functions, respectively, on a symmetric
+imaginary time grid. See the tests <code class="docutils literal notranslate"><span class="pre">imfreq_ops.interp_matrix_sym_fer</span></code> and</p>
+<blockquote>
+<div><p><code class="docutils literal notranslate"><span class="pre">imfreq_ops.interp_matrix_sym_bos</span></code> in the file <code class="docutils literal notranslate"><span class="pre">test/imfreq_ops.cpp</span></code> for
+fermionic and bosonic Green’s functions, respectively, on a symmetric
+Matsubara frequency grid. All of these tests show how to obtain a symmetric
+set of DLR frequencies.</p>
+</div></blockquote>
+</li>
+<li><p class="first">Given a fixed self-energy, solve the Dyson equation in imaginary time to
 obtain the Green’s function: see the tests <code class="docutils literal notranslate"><span class="pre">dyson_it.dyson_vs_ed_real</span></code>,
 <code class="docutils literal notranslate"><span class="pre">dyson_it.dyson_vs_ed_cmplx</span></code>, and <code class="docutils literal notranslate"><span class="pre">dyson_it.dyson_bethe</span></code> in the file
-<code class="docutils literal notranslate"><span class="pre">test/dyson_it.cpp</span></code>.</li>
-<li>Solve the Dyson equation self-consistently in imaginary time, given an
+<code class="docutils literal notranslate"><span class="pre">test/dyson_it.cpp</span></code>.</p>
+</li>
+<li><p class="first">Solve the Dyson equation self-consistently in imaginary time, given an
 expression for the self-energy in terms of the Green’s function: see the test
-<code class="docutils literal notranslate"><span class="pre">dyson_it.dyson_bethe_fpi</span></code> in the file <code class="docutils literal notranslate"><span class="pre">test/dyson_it.cpp</span></code>.</li>
+<code class="docutils literal notranslate"><span class="pre">dyson_it.dyson_bethe_fpi</span></code> in the file <code class="docutils literal notranslate"><span class="pre">test/dyson_it.cpp</span></code>.</p>
+</li>
 </ul>
 </div>
 </div>

docs/main/genindex.html

@@ -62,6 +62,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>

docs/main/index.html

@@ -63,6 +63,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>
@@ -166,6 +167,7 @@ <h2>Related libraries<a class="headerlink" href="#related-libraries" title="Perm
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>

docs/main/install.html

@@ -64,6 +64,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>

docs/main/issues.html

@@ -63,6 +63,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>

docs/main/search.html

@@ -65,6 +65,7 @@
 <li class="toctree-l2"><a class="reference internal" href="background.html#operations-in-the-dlr-basis">Operations in the DLR basis</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#imaginary-time-point-format">Imaginary time point format</a></li>
 <li class="toctree-l2"><a class="reference internal" href="background.html#matsubara-frequency-point-format">Matsubara frequency point format</a></li>
+<li class="toctree-l2"><a class="reference internal" href="background.html#symmetrized-dlr-grids">Symmetrized DLR grids</a></li>
 </ul>
 </li>
 <li class="toctree-l1"><a class="reference internal" href="examples.html">Examples</a><ul>