From 3f0510629e422e979b57d3f93791937912a4183a Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Tue, 14 Jun 2022 09:26:07 +0200
Subject: [PATCH] Update to v1.5.

  The update to version 1.5 is rather substantial, and introduces some minor
  backward-incompatibilities:
    * The header "#!symbols" has been replaced by "#!virtual_fields"
    * Multiplying polynomials using the '*' symbol is no longer supported (or,
      rather, the symbolic capabilities of meankondo were enhanced, and the
      syntax has been changed).
    * 'meantools exp' has been removed (its functionality is now handled by
      other means)
    * 'meantoolds derive' has been replaced by 'meantools differentiate'

  * The symbolic capabilities were enhanced: polynomials can now be
    multiplied, added, exponentiated, and their logarithms can be taken
    directly in the configuration file.

  * The flow equation can now be processed after being computed using the
    various "#!postprocess_*" entries.

  * Deprecated kondo_preprocess.

  * Compute the mean using an LU decomposition if possible.

  * More detailed checks for syntax errors in configuration file.

  * Check that different '#!group' entries are indeed uncorrelated.

  * New flags in meankondo: '-p' and '-A'.

  * New tool: meantools expand.

  * Improve conversion to LaTeX using meantools-convert

  * Assign terms randomly to different threads.

  * Created vim files to implement syntax highlighting for configuration
    files.

  * Multiple bug fixes
---
 Changelog                                   |  39 ++
 Makefile                                    |   8 +-
 doc/meankondo-doc.html                      |  64 +-
 editors/vim/ftdetect/meankondo.vim          |   7 +
 editors/vim/ftplugin/meankondo.vim          |   8 +
 editors/vim/syntax/meankondo.vim            | 111 +++
 man/kondo_preprocess.1                      |  35 +-
 man/meankondo.1                             | 185 +++--
 man/meantools-convert.1                     |  14 +-
 man/meantools.1                             | 107 +--
 man/numkondo.1                              |  14 +-
 scripts/config_functions.sh                 |   2 +-
 scripts/meantools-convert                   |  16 +-
 src/array.c                                 |  69 +-
 src/array.h                                 |  14 +-
 src/cli_parser.c                            |  22 +-
 src/cli_parser.h                            |   4 +-
 src/coefficient.c                           | 245 ++++++-
 src/coefficient.h                           |  12 +-
 src/definitions.cpp                         |  10 +-
 src/determinant.c                           |  93 +++
 src/determinant.h                           |  17 +
 src/expansions.c                            |  28 -
 src/expansions.h                            |  34 -
 src/fields.c                                | 273 ++++++--
 src/fields.h                                |  58 +-
 src/flow.c                                  |  73 +-
 src/flow.h                                  |   4 +-
 src/flow_mpfr.c                             |  63 +-
 src/flow_mpfr.h                             |   4 +-
 src/grouped_polynomial.c                    | 138 +++-
 src/grouped_polynomial.h                    |  10 +-
 src/idtable.c                               |   2 +-
 src/idtable.h                               |   2 +-
 src/istring.c                               |   2 +-
 src/istring.h                               |   2 +-
 src/kondo.c                                 |  69 +-
 src/kondo.h                                 |  14 +-
 src/kondo_preprocess.c                      |   7 +-
 src/mean.c                                  | 141 +++-
 src/mean.h                                  |  20 +-
 src/meankondo.c                             | 213 ++++--
 src/meantools.c                             |  29 +-
 src/meantools_deriv.c                       |  14 +-
 src/meantools_deriv.h                       |   4 +-
 src/meantools_eval.c                        |   6 +-
 src/meantools_eval.h                        |   2 +-
 src/meantools_exp.c                         | 130 ----
 src/meantools_expand.c                      | 166 +++++
 src/{meantools_exp.h => meantools_expand.h} |  14 +-
 src/number.c                                | 223 ++++--
 src/number.h                                |  15 +-
 src/numkondo.c                              |  27 +-
 src/parse_file.c                            | 717 +++++++++++++++++---
 src/parse_file.h                            |  31 +-
 src/polynomial.c                            |  77 ++-
 src/polynomial.h                            |  12 +-
 src/rational.h                              |   2 +-
 src/rational_float.c                        |   2 +-
 src/rational_float.h                        |   2 +-
 src/rational_int.c                          |  22 +-
 src/rational_int.h                          |   2 +-
 src/rcc.c                                   |   9 +-
 src/rcc.h                                   |   3 +-
 src/rcc_mpfr.c                              |   9 +-
 src/rcc_mpfr.h                              |   3 +-
 src/symbolic_parser.c                       | 330 +++++++++
 src/symbolic_parser.h                       |  42 ++
 src/tools.c                                 |  12 +-
 src/tools.h                                 |   5 +-
 src/tree.c                                  | 117 ++++
 src/tree.h                                  |  48 ++
 src/types.h                                 |  34 +-
 73 files changed, 3510 insertions(+), 852 deletions(-)
 create mode 100644 editors/vim/ftdetect/meankondo.vim
 create mode 100644 editors/vim/ftplugin/meankondo.vim
 create mode 100644 editors/vim/syntax/meankondo.vim
 create mode 100644 src/determinant.c
 create mode 100644 src/determinant.h
 delete mode 100644 src/expansions.c
 delete mode 100644 src/expansions.h
 delete mode 100644 src/meantools_exp.c
 create mode 100644 src/meantools_expand.c
 rename src/{meantools_exp.h => meantools_expand.h} (67%)
 create mode 100644 src/symbolic_parser.c
 create mode 100644 src/symbolic_parser.h
 create mode 100644 src/tree.c
 create mode 100644 src/tree.h

diff --git a/Changelog b/Changelog
index 2fed1bf..854072a 100644
--- a/Changelog
+++ b/Changelog
@@ -1,3 +1,42 @@
+1.5:
+  The update to version 1.5 is rather substantial, and introduces some minor
+  backward-incompatibilities:
+    * The header "#!symbols" has been replaced by "#!virtual_fields"
+    * Multiplying polynomials using the '*' symbol is no longer supported (or,
+      rather, the symbolic capabilities of meankondo were enhanced, and the
+      syntax has been changed).
+    * 'meantools exp' has been removed (its functionality is now handled by
+      other means)
+    * 'meantoolds derive' has been replaced by 'meantools differentiate'
+
+  * The symbolic capabilities were enhanced: polynomials can now be
+    multiplied, added, exponentiated, and their logarithms can be taken
+    directly in the configuration file.
+
+  * The flow equation can now be processed after being computed using the
+    various "#!postprocess_*" entries.
+
+  * Deprecated kondo_preprocess.
+
+  * Compute the mean using an LU decomposition if possible.
+
+  * More detailed checks for syntax errors in configuration file.
+
+  * Check that different '#!group' entries are indeed uncorrelated.
+
+  * New flags in meankondo: '-p' and '-A'.
+
+  * New tool: meantools expand.
+
+  * Improve conversion to LaTeX using meantools-convert
+
+  * Assign terms randomly to different threads.
+
+  * Created vim files to implement syntax highlighting for configuration
+    files.
+
+  * Multiple bug fixes
+
 1.4:
   * Support MPFR floats in numkondo.
 
diff --git a/Makefile b/Makefile
index d0cd377..5b8c998 100644
--- a/Makefile
+++ b/Makefile
@@ -1,4 +1,4 @@
-## Copyright 2015 Ian Jauslin
+## Copyright 2015-2022 Ian Jauslin
 ## 
 ## Licensed under the Apache License, Version 2.0 (the "License");
 ## you may not use this file except in compliance with the License.
@@ -62,10 +62,10 @@ SRCDIR=./src
 OBJDIR=./objs
 
 # objects
-LIBKONDO_OBJS = $(addprefix $(OBJDIR)/,array.o cli_parser.o coefficient.o fields.o grouped_polynomial.o idtable.o istring.o number.o parse_file.o polynomial.o rational_float.o rational_int.o rcc.o rcc_mpfr.o tools.o)
-MEANKONDO_OBJS = $(addprefix $(OBJDIR)/,meankondo.o mean.o)
+LIBKONDO_OBJS = $(addprefix $(OBJDIR)/,array.o cli_parser.o coefficient.o fields.o grouped_polynomial.o idtable.o istring.o number.o parse_file.o polynomial.o rational_float.o rational_int.o rcc.o rcc_mpfr.o symbolic_parser.o tree.o tools.o)
+MEANKONDO_OBJS = $(addprefix $(OBJDIR)/,meankondo.o determinant.o mean.o)
 NUMKONDO_OBJS = $(addprefix $(OBJDIR)/,numkondo.o flow.o flow_mpfr.o)
-MEANTOOLS_OBJS = $(addprefix $(OBJDIR)/,meantools.o meantools_exp.o meantools_deriv.o meantools_eval.o)
+MEANTOOLS_OBJS = $(addprefix $(OBJDIR)/,meantools.o meantools_deriv.o meantools_eval.o meantools_expand.o)
 KONDO_PP_OBJS = $(addprefix $(OBJDIR)/,kondo_preprocess.o kondo.o)
 
 
diff --git a/doc/meankondo-doc.html b/doc/meankondo-doc.html
index 7074b56..75c9583 100644
--- a/doc/meankondo-doc.html
+++ b/doc/meankondo-doc.html
@@ -1,7 +1,7 @@
 <html>
   <head>
-    <script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> </script>
-    <!--<script type="text/javascript" src="/usr/share/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> </script>-->
+    <!--<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> </script>-->
+    <script type="text/javascript" src="/usr/share/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> </script>
 
     <style>
       body {
@@ -69,10 +69,10 @@
   </head>
 
   <body>
-    <h1 style="margin-bottom:50pt;">meankondo <span style="margin-left:10pt;font-size:18pt">v1.4</span></h1>
+    <h1 style="margin-bottom:50pt;">meankondo <span style="margin-left:10pt;font-size:18pt">v1.5</span></h1>
 
     <p>
-      This is the official documentation for <b>meankondo</b>, version 1.4. The aim of this document is not to give a technical description of how to use the various programs bundled with <b>meankondo</b>, nor is it to explain where hierarchical models come from and what their meaning is, but rather a conceptual overview of how <b>meankondo</b> approaches the computation of flow equations, and how its programs can be made to interact with one another to compute various quantities. For a more technical description, see the man pages included with the <b>meankondo</b> source code. For a more theoretical discussion of Fermionic hierarchical models, see <a href="http://ian.jauslin.org/publications/15bgj">[G.Benfatto, G.Gallavotti, I.Jauslin, 2015]</a>.
+      This is the official documentation for <b>meankondo</b>, version 1.5. The aim of this document is not to give a technical description of how to use the various programs bundled with <b>meankondo</b>, nor is it to explain where hierarchical models come from and what their meaning is, but rather a conceptual overview of how <b>meankondo</b> approaches the computation of flow equations, and how its programs can be made to interact with one another to compute various quantities. For a more technical description, see the man pages included with the <b>meankondo</b> source code. For a more theoretical discussion of Fermionic hierarchical models, see <a href="http://ian.jauslin.org/publications/15bgj">[G.Benfatto, G.Gallavotti, I.Jauslin, 2015]</a>.
     </p>
 
     <h2 style="margin-top:50pt;">Table of contents</h2>
@@ -92,9 +92,9 @@
 	  </ul>
 	<li class="toc_sec"><a href="#operations">Operations on flow equations</a></li>
 	  <ul>
+	    <li class="toc_sub"><a href="#processing">Pre- and post-processing</a></li>
 	    <li class="toc_sub"><a href="#numerical_evaluation">Numerical Evaluation</a></li>
-	    <li class="toc_sub"><a href="#exponentiation">Exponentiation</a></li>
-	    <li class="toc_sub"><a href="#derivation">Derivation</a></li>
+	    <li class="toc_sub"><a href="#differentiation">Differentiation</a></li>
 	  </ul>
 	<li class="toc_sec"><a href="#exactness">Comments on the exactness of the computation</a></li>
 	<li class="toc_sec"><a href="#authors">Authors</a></li>
@@ -107,13 +107,9 @@
       <ul>
 	<li><b>meankondo</b>: computes the flow equation.</li>
 	<li><b>numkondo</b>: iterate the flow equation numerically.</li>
-	<li><b>meantools</b>: tools to exponentiate, derive and evaluate a flow equation.</li>
+	<li><b>meantools</b>: tools to take products, sums, exponentials or logairhtms, differentiate or evaluate a flow equation.</li>
 	<li><b>meantools-convert</b>: python script to convert a flow equation to C, javascript or LaTeX code.</li>
       </ul>
-      as well as <i>pre-processors</i>, whose purpose is to help with writing configuration files for specific models:
-      <ul>
-	<li><b>kondo_preprocess</b>: hierarchical Kondo model.</li>
-      </ul>
       In addition, <b>meankondo</b> includes a library, <b>libkondo</b>, which can either be compiled as a <i>shared</i> or a <i>static</i> object, and contains the various structures and functions <b>meankondo</b> is built with.
     </p>
 
@@ -127,13 +123,13 @@
     </p>
     
     <p>
-      Given a configuration file 'config', the flow equation can be computed by
+      Given a configuration file 'config.mk', the flow equation can be computed by
       <code class="codeblock">
-	meankondo config
+	meankondo config.mk
       </code>
       and it can be iterated for, say, 100 steps starting from \(\ell_0^{[m]}=-0.01\) using
       <code class="codeblock">
-	meankondo -C config | numkondo -N 100 -I "0:-0.01"
+	meankondo -C config.mk | numkondo -N 100 -I "0:-0.01"
       </code>
     </p>
 
@@ -150,7 +146,7 @@
 	<li><b>external</b>: which are organized in pairs, and are denoted by \((\Psi_i^+,\Psi_i^-)\) for \(i\in\{1,\cdots,E\}\).
 	<li><b>super-external</b>: which denoted by \(H_i\) for \(i\in\{1,\cdots,X\}\) (the only difference with external fields is that super-external fields are not in pairs, which is a seemingly innocuous difference; but super-external fields are meant to be used for different purposes as external fields (see <a href="#flow_equation_definition">Definition</a> below)).
       </ul>
-      The fields are used as a basis for a complex algebra, so that we can take products and linear combinations of fields (in other words, the concept of <i>polynomials over the fields</i> is well defined). Some of the fields (<i>Fermions</i>) anti-commute with each other (two fields \(a\) and \(b\) are said to anti-commute if \(ab\equiv-ba\)), and the rest (<i>Bosons</i>) commute. Which fields are Fermions and which are Bosons is specified in the <code>#!fields</code> entry in the configuration file. <b>(Warning: As of version 1.4, all internal fields must be Fermions.)</b>
+      The fields are used as a basis for a complex algebra, so that we can take products and linear combinations of fields (in other words, the concept of <i>polynomials over the fields</i> is well defined). Some of the fields (<i>Fermions</i>) anti-commute with each other (two fields \(a\) and \(b\) are said to anti-commute if \(ab\equiv-ba\)), and the rest (<i>Bosons</i>) commute. Which fields are Fermions and which are Bosons is specified in the <code>#!fields</code> entry in the configuration file. <b>(Warning: As of version 1.5, all internal fields must be Fermions.)</b>
     </p>
     <p>
       In the configuration file of the <b>meankondo</b> program, the fields are specified in the <code>#!fields</code> entry.
@@ -174,6 +170,16 @@
       In the configuration file of the <b>meankondo</b> program, the propagator is specified in the <code>#!propagator</code> entry. Note that <b>meankondo</b> recognizes numeric propagators as well as symbolic ones.
     </p>
 
+    <p>
+      It is convenient to re-expres the Wick rule in determinant form: if \(M\) is the \(n\times n\) matrix whose entries are \(M_{a,b}=\langle\psi_{i_a}^+\psi_{j_b}^-\rangle\), then
+      $$
+      \langle\psi_{i_1}^+\psi_{j_1}^-\cdots\psi_{i_n}^+\psi_{j_n}^-\rangle=
+      \det(M).
+      $$
+      <b>meankondo</b> implements an algorithm, based on an <i>LU decomposition</i>, to compute \(\det(M)\) in \(O(n^3)\) operations. However, when performing the LU decomposition, elements of \(M\) are divided, and since polynomial divisions are not supported in <b>meankondo</b>, the LU decomposition will only be performed if every entry of the propagator is numeric. If the propagator has symbolic entries, then <b>meankondo</b> computes the means summing over permutations, which requires \(O(n!)\) operations but does not require divisions.
+    </p>
+
+
     <h2 class="section" id="flow_equation">Flow equation</h2>
     <p>
       In this section, we discuss what flow equations are, and how <b>meankondo</b> computes them.
@@ -247,7 +253,16 @@
 
     <h2 class="section" id="operations">Operations on flow equations</h2>
     <p>
-      In this section we describe the various operations on flow equations that the tools bundled with <b>meankondo</b> support.
+      In this section we describe the various operations on flow equations that <b>meankondo</b> and the tools bundled with it support.
+    </p>
+
+    <h3 class="subsection" id="processing">Pre- and post-processing</h3>
+    <p>
+      <b>meankondo</b> can perform operations on the effective potential before and after applying the renormalization group transformation. This is useful, for instance, if the effective potential is expressed as an expontential: \(exp(W)\), in which case the input polynomial can be exponentiated before the computation, and the logarithm taken after the computation. To do this, <b>meankondo</b> implements some basic symbolic processing. For the syntax of the symbolic processing, see the <b>man</b> pages bundled with <b>meankondo</b>. The pre-processing is done in the <code>#!input_polynomial</code> configuration entry, and the post-processing can be done in the <code>#!postprocess_operation</code>, <code>#!postprocess_flow_equation</code> or <code>#!numerical_postprocess_operation</code> entries. In addition, <b>meantoolds expand</b> can be used to compute sums, products, exponentials and logarithms of effective potentials.
+    </p>
+
+    <p>
+      There are subtle differences between using <code>#!postprocess_operation</code>, <code>#!postprocess_flow_equation</code> and <code>#!numerical_postprocess_operation</code>. With <code>#!postprocess_operation</code>, the post-processing operation is done immediately after having computed the average. With <code>#!postprocess_flow_equation</code> and <code>#!numerical_postprocess_operation</code>, the avergae is first turned into a flow equation, and then the post-processing is applied to each equation. The main difference is in the handling of the constant term of the polynomials, see the <b>man</b> for details. The <code>numerical</code> entry is to be used for the numerical evaluation of the flow only, using <b>numkondo</b>.
     </p>
 
     <h3 class="subsection" id="numerical_evaluation">Numerical evaluation</h3>
@@ -259,27 +274,18 @@
       Numerical evaluation is handled in a straightforward manner, but for the following consideration. As was mentioned in <a href="#flow_equation_computation">Computation</a>, \(\mathcal R\) is a polynomial in \((\underline\ell,C^{-1}(\underline\ell))\), and when evaluating \(\mathcal R(\underline\ell)\), <b>meankondo</b> first evaluates \(C\) and the computes \(\ell'_n(\underline\ell)\).
     </p>
 
-    <h3 class="subsection" id="exponentiation">Exponentiation</h3>
+    <h3 class="subsection" id="differentiation">Differentiation</h3>
     <p>
-      Oftentimes the renormalization group flow is expressed in terms of an exponential of an effective potential \(\exp(W)\), in which case the exponential must be computed before it can be processed by <b>meankondo</b>:
-      $$
-      \exp(W)=1+V.
-      $$
-      This is handled by <b>meantools exp</b>, which computes the running coupling constants appearing in \(V\) in terms of those in \(W\).
-    </p>
-
-    <h3 class="subsection" id="derivation">Derivation</h3>
-    <p>
-      This feature was introduced to compute the susceptibility in the hierarchical Kondo model. In that case, some of the running coupling constants depend on the field, \(h\), and the susceptibility is expressed as a derivative of \(C(\underline\ell(h))\) with respect to \(h\). To that end, we wrote <b>meantools derive</b> to compute the derivatives of a flow equation with respect to an external variable.
+      This feature was introduced to compute the susceptibility in the hierarchical Kondo model. In that case, some of the running coupling constants depend on the field, \(h\), and the susceptibility is expressed as a derivative of \(C(\underline\ell(h))\) with respect to \(h\). To that end, we wrote <b>meantools differentiate</b> to compute the derivatives of a flow equation with respect to an external variable.
     </p>
 
     <p>
-      The input of <b>meantools derive</b> consists in a flow equation and a collection of variables \(X\subset\{1,\cdots,p\}\). Each running coupling constant \(\ell_i\) for \(i\in X\) is assumed to depend on an external parameter, \(h\). The flow equation is then derived with respect to \(h\): for every \(n\in\{1,\cdots,p\}\), the derivative of \(\ell_n'(\underline\ell)\) with respect to \(h\) in terms of \(\partial_h\ell_i\) for \(i\in X\) is computed. It is then appended to the input flow equation.
+      The input of <b>meantools differentiate</b> consists in a flow equation and a collection of variables \(X\subset\{1,\cdots,p\}\). Each running coupling constant \(\ell_i\) for \(i\in X\) is assumed to depend on an external parameter, \(h\). The flow equation is then differentiated with respect to \(h\): for every \(n\in\{1,\cdots,p\}\), the derivative of \(\ell_n'(\underline\ell)\) with respect to \(h\) in terms of \(\partial_h\ell_i\) for \(i\in X\) is computed. It is then appended to the input flow equation.
     </p>
 
     <h2 class="section" id="exactness">Comments on the exactness of the computation</h2>
     <p>
-      The computation of the flow equation, as well as its exponentiation and derivation, are <i>exact</i> in the sense that they only involve operations on integers and are not subject to truncations. The coefficients appearing in the flow equation are therefore <i>exact</i>. This statement has one major caveat: integer operations are only correct as long as the integers involved are not too large. The precise meaning of "not too large" is system dependent. In the source code, integers relating to flow equation coefficients are declared with the <code>long int</code> type, which, at least using the C library <b>meankondo</b> was tested with (that is <code>glibc 2.21</code>), means integers are encoded on 64 bits on 64-bit systems and 32 bits on 32-bit systems. All operations are therefore exact as long as all integers are in \([-2^{31},2^{31}-1]\) on 64-bit systems and \([-2^{15},2^{15}-1]\) on 32-bit systems.
+      The computation of the flow equation, as well as all the operations done on it, are <i>exact</i> in the sense that they only involve operations on integers and are not subject to truncations. The coefficients appearing in the flow equation are therefore <i>exact</i>. This statement has one major caveat: integer operations are only correct as long as the integers involved are not too large. The precise meaning of "not too large" is system dependent. In the source code, integers relating to flow equation coefficients are declared with the <code>long int</code> type, which, at least using the C library <b>meankondo</b> was tested with (that is <code>glibc 2.21</code>), means integers are encoded on 64 bits on 64-bit systems and 32 bits on 32-bit systems. All operations are therefore exact as long as all integers are in \([-2^{31},2^{31}-1]\) on 64-bit systems and \([-2^{15},2^{15}-1]\) on 32-bit systems.
     </p>
 
     <!--<p>
diff --git a/editors/vim/ftdetect/meankondo.vim b/editors/vim/ftdetect/meankondo.vim
new file mode 100644
index 0000000..3e14be8
--- /dev/null
+++ b/editors/vim/ftdetect/meankondo.vim
@@ -0,0 +1,7 @@
+" Vim ftdetect file
+" Language:     meankondo
+" Maintainer:   Ian Jauslin <ian.jauslin@roma1.infn.it>
+" Last Change:  2015-12-09
+" URL:          http://ian.jauslin.org/software/meankondo
+"
+au BufRead,BufNewFile *.mk set filetype=meankondo
diff --git a/editors/vim/ftplugin/meankondo.vim b/editors/vim/ftplugin/meankondo.vim
new file mode 100644
index 0000000..e4ac2bf
--- /dev/null
+++ b/editors/vim/ftplugin/meankondo.vim
@@ -0,0 +1,8 @@
+" Vim ftplugin file
+" Language:     meankondo
+" Maintainer:   Ian Jauslin <ian.jauslin@roma1.infn.it>
+" Last Change:  2015-12-09
+" URL:          http://ian.jauslin.org/software/meankondo
+"
+
+set mps+=<:>
diff --git a/editors/vim/syntax/meankondo.vim b/editors/vim/syntax/meankondo.vim
new file mode 100644
index 0000000..bf94c0d
--- /dev/null
+++ b/editors/vim/syntax/meankondo.vim
@@ -0,0 +1,111 @@
+" Vim syntax file
+" Language:     meankondo
+" Maintainer:   Ian Jauslin <ian.jauslin@roma1.infn.it>
+" Last Change:  2022-06-06
+" URL:          http://ian.jauslin.org/software/meankondo
+"
+
+if exists("b:current_syntax")
+  finish
+endif
+
+" Comments
+syn match Comment '#.*'
+
+" Regions
+syn region FieldsRegion        matchgroup=configHeader start='#!fields'                 end='&' transparent contains=Comment,Separator,FieldHeader,FieldInField,Equal
+syn region VariablesRegion    matchgroup=configHeader start='#!preprocessor_variables' end='&' transparent contains=@Globals,VariableHeader
+syn region IdTableRegion      matchgroup=configHeader start='#!id_table'               end='&' transparent contains=@Globals,RegionHeader
+syn region SymbolsRegion      matchgroup=configHeader start='#!symbols'                end='&' transparent contains=@Globals,VariableHeader
+syn region PolynomialRegion   matchgroup=configHeader start='#!input_polynomial'       end='&' transparent
+syn region PolynomialRegion   matchgroup=configHeader start='#!postprocess_operation'       end='&' transparent
+syn region PolynomialRegion   matchgroup=configHeader start='#!postprocess_flow_equation'       end='&' transparent
+syn region PolynomialRegion   matchgroup=configHeader start='#!numerical_postprocess_operation'       end='&' transparent
+syn region PropagatorRegion   matchgroup=configHeader start='#!propagator'             end='&' transparent contains=Comment,Number,NumberBase,Separator,FieldInProp,Equal,NumberInProp
+syn region IdentitiesRegion   matchgroup=configHeader start='#!identities'             end='&' transparent
+syn region GroupsRegion       matchgroup=configHeader start='#!groups'                 end='&' transparent contains=Comment,Separator,FieldHeader,FieldInField,Equal,GroupParenthesis
+syn region LabelsRegion       matchgroup=configHeader start='#!labels'                 end='&' transparent contains=String,RegionHeader,Separator,Comment,Equal
+syn region FlowEquationRegion matchgroup=configHeader start='#!flow_equation'          end='&' transparent
+syn region InitCdRegion       matchgroup=configHeader start='#!initial_condition'      end='&' transparent contains=Comment,Separator,RegionHeader,Equal,NumberInInit,InitHeader
+syn region DerivVarsRegion    matchgroup=configHeader start='#!variables'              end='&' transparent  contains=Comment,Separator,FieldHeader,FieldInField,Equal,VarAll
+
+" Generic syntax
+syn match Field '\[f[-0-9]\+\]'
+syn match Factor '\[l[-0-9]\+\]'
+syn match Factor '\[d*% *[-0-9]\+\]' contains=Deriv
+syn match ConstantF '\[d*C *[0-9]\+\]' contains=Deriv
+syn match ConstantD '\[/C *[0-9]\+ *\^ *[0-9]\+\]'
+syn match Deriv 'd' contained
+syn match Variables '\$[^>]*'
+syn match Number '<[-/0-9]*>'
+syn match Number '([-/0-9]*)'
+syn match NumberBase 's{[-/0-9]*}'
+syn match NumberBase '(s{[-/0-9]*})'
+syn match Operator '[*+]'
+syn match Operator '%exp'
+syn match Operator '%log_1'
+syn match Separator ','
+syn match Equal '[:=]'
+
+syn cluster Globals contains=Comment,Field,Factor,Constant,Variables,Number,NumberBase,Operator,Separator,Equal
+
+" Fields
+syn keyword FieldHeader i contained
+syn keyword FieldHeader x contained
+syn keyword FieldHeader h contained
+syn keyword FieldHeader f contained
+syn keyword FieldHeader a contained
+syn match FieldInField '[-0-9]\+' contained
+
+" Variables
+syn match VariableHeader '^[^=,]*='he=e-1 contained contains=Equal
+
+" Groups
+syn match GroupParenthesis '[()]' contained
+
+" Propagator
+syn match FieldInProp '[0-9]\+ *[;:]'he=e-1 contained contains=Equal,NumberInProp
+syn match NumberInProp ': *[-/()0-9s{}]*'hs=s+1 contained
+
+" IdTable
+syn match RegionHeader '[0-9]\+:'he=e-1 contained contains=Equal
+
+" Labels
+syn match String '"[^"]*"'hs=s+1,he=e-1 contained
+
+" Initial condition
+syn match InitHeader '[0-9]\+:'he=e-1 contained contains=Equal,NumberInInit
+syn match NumberInInit ': *[-.e0-9]*'hs=s+1 contained
+
+" Variables
+syn keyword VarAll all contained
+
+
+" Colors
+hi Comment         ctermfg=cyan
+hi configHeader    ctermfg=red
+hi Field           ctermfg=Lightblue
+hi Factor          ctermfg=Lightblue
+hi ConstantF       ctermfg=Lightblue
+hi ConstantD       ctermfg=Lightgreen
+hi Deriv           ctermfg=red
+hi Header          ctermfg=Lightgreen
+hi Variables       ctermfg=yellow
+hi Number          ctermfg=green
+hi NumberBase      ctermfg=green
+hi Separator       ctermfg=red
+hi Equal           ctermfg=Lightred
+hi Operator        ctermfg=magenta
+hi String          ctermfg=magenta
+hi NumberInInit    ctermfg=magenta
+
+hi def link FieldInField     Field
+hi def link VariableHeader   Variables
+hi def link FieldInProp      Field
+hi def link FieldHeader      Header
+hi def link NumberInProp     Number
+hi def link SeparatorInProp  Separator
+hi def link RegionHeader     Header
+hi def link InitHeader       Header
+hi def link GroupParenthesis Separator
+hi def link VarAll           Field
diff --git a/man/kondo_preprocess.1 b/man/kondo_preprocess.1
index 1e22cae..414a44a 100644
--- a/man/kondo_preprocess.1
+++ b/man/kondo_preprocess.1
@@ -1,5 +1,5 @@
-.Dd $Mdocdate: September 22 2015 $
-.Dt kondo_preprocess 1.4
+.Dd $Mdocdate: February 3 2016 $
+.Dt kondo_preprocess 1.5
 .Os
 .Sh NAME
 .Nm kondo_preprocess
@@ -13,11 +13,21 @@ for the Kondo model
 .Pp
 .Nm
 .Fl v
+.Sh DEPRECATION NOTICE
+The use of
+.Nm
+is deprecated as of
+.Sy meankondo
+v1.5.
+.Pp
+Similar functionality can be obtained rather easily using the enhanced symbolic capabilities introduced in v1.5, which are more flexible than
+.Nm .
+.Pp
 .Sh DESCRIPTION
 .Nm
 generates a configuration file to be read by
 .Sy meankondo
-for the Kondo model. It generates the '#!fields', '#!symbols', '#!identities', '#!groups', '#!propagator', '#!input_polynomial' and '#!id_table' entries from special '#!propagator', '#!input_polynomial' and '#!id_table' entries, which are much more synthetic than those needed for the Kondo model.
+for the Kondo model. It generates the '#!fields', '#!virtual_fields', '#!identities', '#!groups', '#!propagator', '#!input_polynomial' and '#!id_table' entries from special '#!propagator', '#!input_polynomial' and '#!id_table' entries, which are much more synthetic than those needed for the Kondo model.
 .Pp
 The quantities in the configuration file are expressed in terms of the observables A and B, which we do not define here, as well as the magnetic field h.
 .Pp
@@ -32,14 +42,7 @@ is part of a set of tools to compute and manipulate Fermionic hierarchical flows
 : numerical evaluation of flow equations.
 .It
 .Sy meantools, meantools-convert
-: perform various operations on flow equations (derivation, exponentiation, evaluation and conversion to other formats).
-.El
-.Pp
-as well as the following pre-processors, which generate configuration files for their associated model:
-.Bl -bullet
-.It
-.Sy kondo_proprocess
-:  Kondo model
+: perform various operations on flow equations (differentiation, products, sums, exponentials and logarithms of flow equations, evaluation and conversion to other formats).
 .El
 .Pp
 .Sh COMMAND-LINE ARGUMENTS
@@ -73,13 +76,13 @@ up to the following differences.
 .It
 The fields can be specified as scalar products of A's and B's. For each n in {1,...,dimension},
 .Nm
-defines An and Bn, as well as symbols for scalar products of the form
+defines An and Bn, as well as virtual fields for scalar products of the form
 .D1 [f An.An]
 .D1 [f Bn.Bn]
 .D1 [f An.Bn]
 .D1 [f An.h]
 .D1 [f Bn.h]
-In addition, a vector product symbol is defined for (AnxBn).h :
+In addition, a vector product virtual field is defined for (AnxBn).h :
 .D1 [f AnxBn.h]
 .Pp
 .It
@@ -101,7 +104,7 @@ Scalar products of A's and B's may also be specified using the '<#.#>' syntax:
 .D1 <An.h>
 .D1 <Bn.h>
 .Pp
-The difference between '[f #.#]' and '<#.#>' is that the former corresponds to a '#!symbols' entry whereas the latter is replaced by its corresponding polynomial when
+The difference between '[f #.#]' and '<#.#>' is that the former corresponds to a '#!virtual_fields' entry whereas the latter is replaced by its corresponding polynomial when
 .Nm
 reads it (see
 .Sx meankondo Ns (1)).
@@ -114,7 +117,7 @@ A vector 't=(t1,t2,t3)' of Pauli matrices (satisfying the Pauli commutation rela
 .D1 <a.t>
 .D1 <b.t>
 .Pp
-Note that the '<#,#>' must be used since these scalar products do not commute whereas '#!symbols' entries must commute (see
+Note that the '<#,#>' must be used since these scalar products do not commute whereas '#!virtual_fiields' entries must commute (see
 .Sx meankondo Ns (1)).
 .Pp
 .It
@@ -161,7 +164,7 @@ Example:
 .D1 A1;A2: 1 , A2;A1: -1 , B1;B2: s{-1} , B2;B1: (-1)s{-1}
 .Pp
 .It Sy extra entries
-If there is a '#!symbols' or an '#!identities' entry in the configuration file, then they are appended to the end of those entries in the new configuration file.
+If there is a '#!virtual_fields' or an '#!identities' entry in the configuration file, then they are appended to the end of those entries in the new configuration file.
 .Pp
 Any other entry is appended to the new configuration file. This can be useful to pipe the output to tools other than
 .Sy meankondo
diff --git a/man/meankondo.1 b/man/meankondo.1
index 599be08..086bc52 100644
--- a/man/meankondo.1
+++ b/man/meankondo.1
@@ -1,5 +1,5 @@
-.Dd $Mdocdate: September 22 2015 $
-.Dt meankondo 1.4
+.Dd $Mdocdate: June 6 2022 $
+.Dt meankondo 1.5
 .Os
 .Sh NAME
 .Nm meankondo
@@ -8,6 +8,8 @@
 .Nm
 .Op Fl t Ar threads
 .Op Fl C
+.Op Fl p
+.Op Fl A
 .Op Ar config_file
 .Pp
 .Nm
@@ -29,14 +31,7 @@ is part of a set of tools to compute and manipulate Fermionic hierarchical flows
 : numerical evaluation of flow equations.
 .It
 .Sy meantools, meantools-convert
-: perform various operations on flow equations (derivation, exponentiation, evaluation and conversion to other formats).
-.El
-.Pp
-as well as the following pre-processors, which generate configuration files for their associated model:
-.Bl -bullet
-.It
-.Sy kondo_proprocess
-:  Kondo model
+: perform various operations on flow equations (differentiation, products, sums, exponentials and logarithms of flow equations, evaluation and conversion to other formats).
 .El
 .Pp
 .Sh COMMAND-LINE ARGUMENTS
@@ -47,6 +42,10 @@ The number of threads to use for the computation.
 Format the output so it can be piped to
 .Sy numkondo ,
 that is, instead of printing the flow equation, print a full configuration file containing the flow equation as well as all the other entries of the configuration file that do not pertain to the computation of the flow equation.
+.It Fl p
+Print the progress of the computation.
+.It Fl A
+Compute the average of the effective potential, but do not write the result as a flow equation.
 .It Fl v
 Print version information and exit.
 .El
@@ -70,13 +69,15 @@ A list of the fields of the model.
 The fields entry contains 5 lines which start with 'i:', 'x:', 'h:', 'f:' and 'a:'. Each of these is followed by a ',' separated list of field indices, which are positive integers.
 .Bl -bullet
 .It
-The indices following 'i' correspond to internal fields, which are integrated out using the Wick rule and the propagator provided in the '#!propagator' entry. Each internal field is associated a conjugate field, whose index is the opposite of the field's index (e.g. 'i:101' defines a field whose index is -101)
+The indices following 'i' correspond to internal fields, which are integrated out using the Wick rule and the propagator provided in the '#!propagator' entry. Each internal field is associated a conjugate field, whose index is the opposite of the field's index (e.g. 'i:101' defines two fields whose indices are 101 and -101).
 .It
-The indices following 'x' correspond to external fields that are associated conjugate field (e.g. 'x:100' defines a field whose index is -100). External indices may not appear as internal indices.
+The indices following 'x' correspond to external fields that are associated conjugate field (e.g. 'x:100' defines two fields whose indices are 100 and -100). External indices may not appear as internal indices.
 .It
 The indices following 'h' correspond to external fields that are not associated a conjugate field. External indices may not appear as internal indices.
 .It
-The 'f' line specifies which of the internal and external indices are Fermions, i.e. which fields anti-commute. The fields appearing in the 'f' line should also either appear in the 'i' or 'x' line. WARNING: for the moment, only cases in which all of the internal fields are Fermions are supported.
+The 'f' line specifies which of the internal and external indices are Fermions, i.e. which fields anti-commute. The fields appearing in the 'f' line should also either appear in the 'i' or 'x' line.
+.Pp
+WARNING: only cases in which all of the internal fields are Fermions are supported.
 .It
 The 'a' line specifies a list of external fields listed in the 'h' entry that do not commute with each other. Specifying fields in this entry will prevent
 .Nm
@@ -96,14 +97,43 @@ The propagator of the model.
 .Pp
 The propagator entry is a ',' separated list whose elements are of the form
 .D1 index1;index2: polynomial
-where index1 and index2 are internal indices, and polynomial is a polynomial (see the POLYNOMIALS section below for information on how to format polynomials). The polynomial must not depend on the internal fields. Note that a number is a special type of polynomial, so propagators with numerical entries are handled by
+where 'index1' and 'index2' are internal indices, and 'polynomial' is a polynomial (see the POLYNOMIALS section below for information on how to format polynomials). The polynomial must not depend on the internal fields. Note that a number is a special type of polynomial, so propagators with numerical entries are handled by
 .Nm
 just as easily as propagators with symbolic entries. Such an entry means that
 .D1 <psi_{index1}^-psi_{index2}^+> = polynomial.
 .Pp
+Note that if the entries of the propagator are numbers instead of polynomials, then the Wick rule is implemented using a determinant instead of a sum over permutations, which is a lot faster for large monomials. Since the efficient computation of determinants requires divisions, polynomial entries in the propagator make the computation awkward, and
+.Nm
+falls back to implementing the Wick rule as a sum over permutations.
+.Pp
 Example:
 .D1 101;102: 1 , 102;101: -1 , 201;202: s{-1} + (-1)[l10] , 202;201: (-1)s{-1} + [l10]
 .Pp
+.It Sy #!input_polynomial
+The polynomial whose mean we wish to compute in order to calculate the flow equation.
+.Pp
+The format of the polynomial is that specified in the POLYNOMIALS section.
+.Pp
+.It Sy #!preprocessor_variables
+In order to simplify configuration files, symbolic variables can be defined. When
+.Nm
+reads the configuration file, it replaces every variable with its value (a process which is referred to as "preprocessing").
+.Pp
+The preprocessor_variables entry is a ',' separated list, whose elements are of the form
+.D1 variable_name=value
+where 'variable_name' is a string that is not 'OUT', 'FLOW' or 'RCC', and may not contain any of the following characters: '$', '<', '>', '*', '+', '%'; and 'value' is a polynomial, formatted as described in the POLYNOMIALS section. The variable names 'OUT' 'FLOW' and 'RCC' are reserved and cannot be used. Note that 'value' can contain other preprocessor variables. There is no safeguard against self-referencing definitions that may cause infinite loops.
+.Pp
+A variable can be used throughout the configuration file by using the format '<$variable_name>'. Whenever '<$variable_name>' is encountered, it is replaced by its corresponding value. The order in which the variables are defined is irrelevant, since
+.Nm
+reads all variables definitions before replacing variables in the configuration file.
+.Pp
+Spaces surrounding the variable name are ignored.
+.Pp
+Example:
+.D1 psi1 = [f1]+[f-1],
+.D1 psi2 = [f2]+[f-2],
+.D1 A = <$psi1>*<$psi2>
+.Pp
 .It Sy #!identities
 Identities satisfied by some of the fields (optional entry).
 .Pp
@@ -111,7 +141,9 @@ In some cases, some of the quantities involved in a model will satisfy an identi
 .Pp
 The identities entry is a ',' separated list, whose elements are of the form
 .D1 monomial=polynomial
-where monomial represents the left side of the identity and is a sequence of field indices of the form '[f index1][f index2]...' and polynomial represents the right side of the identity (see the POLYNOMIALS section below for information on how to format polynomials).
+where 'monomial' represents the left side of the identity and is a sequence of field indices of the form '[f index1][f index2]...' and 'polynomial' represents the right side of the identity (see the POLYNOMIALS section below for information on how to format polynomials).
+.Pp
+Identities could be used to reproduce the functionality of preprocessor variables, though it is less convenient (see COMMENT ON PREPROCESSOR VARIABLES, IDENTITIES AND VIRTUAL FIELDS).
 .Pp
 Example:
 .D1 [f301][f301]=(1)+(-1)[f302][f302]+(-1)[f303][f303],
@@ -121,37 +153,30 @@ Example:
 .Pp
 This entry is optional.
 .Pp
-.It Sy #!symbols
-Symbolic variables used as shortcuts for more complicated expressions (optional entry).
-.Pp
-In order to simplify long expressions, symbolic variables can be defined in this entry. Each variable is assigned an index, which is a positive integer that must be different from any of the internal and external indices defined in the '#!fields' entry.
-.Pp
-Seemingly similar functionality can be achieved using an '#!identity' entry (see above), though symbols are handled differently from identities. Indeed, while identities are simplified out of the polynomials as soon as they occur, symbols are only resolved when
+.It Sy #!virtual_fields
+Virtual fields are used to keep the memory footprint of
 .Nm
-computes the mean of the input polynomial. Using symbols can thereby be a lot faster than using identities. However, as is mentioned below, symbols must commute with each other and all other fields, whereas identities can be made to be fermionic or non-commuting.
+small, even when the input polynomial contains many terms.
 .Pp
-The symbols entry is a ',' separated list, whose elements are of the form
+Every term of the input polynomial is parsed and stored when the configuration file is read, which may require large amounts of memory. In order to avoid this, some terms of the input polynomial can be grouped together and stored as a single virtual field, which only occupies as much memory as a single field.
+.Pp
+A virtual field is identified by an index, which must be different from those of the fields defined in the '#!fields' entry.
+.Pp
+The virtual_fields entry is a ',' separated list, whose elements are of the form
 .D1 index= polynomial
-where index is the index of the variable and polynomial is the expression it stands for (see the POLYNOMIALS section below for information on how to format polynomials). Note that polynomial can contain other symbolic variables. There is no safeguard against self-referencing definitions that may cause infinite loops.
+where 'index' is the index of the virtual_field and 'polynomial' is the expression it stands for (see the POLYNOMIALS section below for information on how to format polynomials). Note that 'polynomial' can contain other virtual fields. There is no safeguard against self-referencing definitions that may cause infinite loops.
 .Pp
-WARNING: Symbols are assumed to commute with each other and all other Fermions. They should therefore not represent quantities that do not commute (e.g. odd monomials of fermions or non-commuting objects specified in the 'a:' entry in the '#!fields' entry).
+WARNING: Virtual fields are assumed to commute with each other and all other Fermions. They should therefore not represent quantities that do not commute (e.g. odd monomials of Fermions or non-commuting objects specified in the 'a:' entry in the '#!fields' entry).
+.Pp
+Virtual fields could be used to reproduce the functionality of commuting preprocessor variables (see COMMENT ON PREPROCESSOR VARIABLES, IDENTITIES AND VIRTUAL FIELDS).
 .Pp
 Example:
 .D1 1001= (-1)[f-100][f100] + (-1)[f-101][f101] , 2001=[f-100][f100] + [f-201][f201]
 .Pp
 This entry is optional.
 .Pp
-.It Sy #!input_polynomial
-The polynomial whose mean we wish to compute in order to calculate the flow equation.
-.Pp
-The format of the polynomial is that specified in the POLYNOMIALS section. In addition, the polynomial can be specified as the product of other polynomials:
-.D1 polynomial1 * polynomial2 * ...
-Note that there are no parentheses, and therefore, products cannot be nested, nor can a product of polynomials be summed with another polynomial.
-.Pp
-Example:
-.D1 (1) + (1/2)[l1][f1001] * (1) + (1/2)[l2][f2001]
 .It Sy #!id_table
-The idtable used to identify the running coupling constants.
+The idtable is used to identify the running coupling constants.
 .Pp
 Once the mean of the input polynomial has been computed, we are left with a polynomial of the external fields and the running coupling constants that were in the input polynomial. In order to compute a flow equation from this average,
 .Nm
@@ -159,7 +184,7 @@ uses an idtable to identify which of the monomials of the average contribute to
 .Pp
 The id_table entry is a ',' separated list, whose elements are of the form
 .D1 rcc: polynomial
-where rcc is the index of the corresponding running coupling constant, which is a non-negative integer, and polynomial is the polynomial to which rcc refers to (which is a polynomial of the external fields).
+where 'rcc' is the index of the corresponding running coupling constant, which is a non-negative integer, and 'polynomial' is the polynomial to which 'rcc' refers to (which is a polynomial of the external fields).
 .Pp
 Example:
 .D1 1:(-1)[f-100][f100] , 2:[f-200][f200]
@@ -172,10 +197,6 @@ computes the mean of a monomial containing elements of different groups, it fact
 .Nm
 does not repeatedly try to pair independent fields.
 .Pp
-WARNING:
-.Nm
-assumes that the symbols and fields in each group are independent but does not check that they are. If symbols or fields that are not independent are put in different groups, or if some are in a group while others are not in any group, then the resulting flow equation may be wrong.
-.Pp
 The groups entry is a list of collections of fields or symbols of the following form
 .D1 (index1,index2,...)
 .Pp
@@ -183,15 +204,80 @@ Example:
 .D1 (1001,1002) (2001,2002)
 .Pp
 This entry is optional.
+.It Sy #!postprocess_operation
+An operation that is done after having computed the mean of the input polynomial. (optional entry)
+.Pp
+The format of this entry is a polynomial, as specified in the POLYNOMIALS section.
+.Pp
+When this entry is present in the configuration, 
+.Nm
+creates a preprocessor variable named 'OUT', which contains the mean of the input polynomial, and can be used in the postprocessing.
+.Pp
+Example:
+To multiply the mean of the input polynomial by 8:
+.D1 <<8>*<$OUT>>
+.Pp
+This entry is optional.
+.It Sy #!postprocess_flow_equation
+This entry is similar to 'posprocess_operation', except that the operation is performed on the flow equation, that is, it is performed after having grouped the polynomial. (optional entry)
+.Pp
+The main difference with carrying out the operation in this way is that the constant term gets handles differently. Whereas 'postprocess_operation' gets applied to the entire polynomial, 'postprocess_flow_equation' is applied to each running coupling constant in the flow equation except the constant term. This is quite useful when the operation is not a polynomial function, such as log_1 or exp.
+.Pp
+The format of this entry is a polynomial, as specified in the POLYNOMIALS section.
+.Pp
+When this entry is present in the configuration, 
+.Nm
+creates a preprocessor variable named 'FLOW', which contains the polynomial obtained by adding each term in the id_table, and can be used in the postprocessing.
+.Pp
+Example:
+To take the logarithm of the flow equation:
+.D1 <%log_1<$FLOW>>
+.Pp
+This entry is optional.
+.It Sy #!numerical_postprocess_operation
+An operation that is done at each step of an eventual numerical computation done with numkondo. (optional entry)
+.Pp
+This is similar in spirit to the 'postprocess_flow_equation' entry, except that the postprocessing is entirely numerical (no symbolic operations are performed).
+.Pp
+The format of this entry is a polynomial, as specified in the POLYNOMIALS section.
+.Pp
+When this entry is present in the configuration, 
+.Nm
+creates a preprocessor variable named 'RCC', which contains the polynomial obtained by adding each term in the id_table, and can be used in the postprocessing.
+.Pp
+When this entry is present in the configuration along with the -C option,
+.Nm
+will add a 'preprocessor_operation' entry in the configuration file to be piped to numkondo.
+.Pp
+Example:
+To take the logarithm of the polynomial:
+.D1 <%log_1<$RCC>>
+.Pp
+This entry is optional.
+.Pp
 .El
 .Pp
+.Sh COMMENT ON PREPROCESSOR VARIABLES, IDENTITIES AND VIRTUAL FIELDS
+On the surface, preprocessor variables, identities and virtual fields can be used to perform similar tasks, but
+.Nm
+handles them in very different ways.
+.Pp
+A preprocessor variable could be replaced by an identity by introducing an extra field corresponding to the variable, and using an identity to make
+.Nm
+replace the extra field by its definition. Using a preprocessor variable will, however, be more convenient since no extra field needs to be introduced.
+.Pp
+Virtual fields could also play the role of preprocessor variables, in that they can be used to simplify the configuration file. However, virtual fields are not replaced by their corresponding expression until their average is computed, and, in the various manipulations required to carry out the computations, the monomials containing virtual fields will be manipulated and virtual fields commuted with other fields. As a consequence, virtual fields must commute with all other fields, which severely limits their potential role as a preprocessor variable.
+.Pp
+In short, preprocessor variables are meant to be used to simplify the configuration file, identities, to implement identities between fields, and virtual fields to reduce the memory footprint of
+.Nm .
+.Pp
 .Sh NUMBERS
 .Nm
 can parse rational numbers and linear combinations of square roots of integers (positive or negative (which is how complex numbers are implemented)) with rational coefficients (i.e. elements of the field extension of Q generated by sqrt(Z)).
 .Pp
 A number is a '+' separated list whose elements are of the form
 .D1 (a/b)s{r}
-where a and r are integers and b is a positive integer. s{r} stands for sqrt(r).
+where 'a' and 'r' are integers and 'b' is a positive integer. 's{r}' stands for 'sqrt(r)'.
 .Pp
 If a=b, then the number may be written as 's{r}'. If b=1, then it can be '(a)s{r}'. If r=1, then it can be 'a/b'. If b=r=1, then it can be 'a'.
 .Pp
@@ -199,7 +285,10 @@ Example:
 .D1 (1/2)s{2} + (-1)s{-1} + 3/2
 .Pp
 .Sh POLYNOMIALS
-Polynomials are '+' separated lists of monomials. Each monomial is a sequence of numbers, rccs and fields.
+.Nm
+implements some elementary symbolic algebra in order to parse polynomials.
+.Pp
+The format of polynomials is defined recursively. If the polynomial contains no '<', then it is said to be 'terminal'. Terminal polynomials are '+' separated lists of monomials. Each monomial is a sequence of numbers, rccs and fields.
 .Bl -bullet
 .It
 Numbers are enclosed between '(' and ')'. If there are several numbers in a monomial, then they are multiplied.
@@ -211,8 +300,17 @@ Fields are non-vanishing indices enclosed between '[f' and ']'. Fields must eith
 .Pp
 If the numerical factor of a monomial is 1, then it can be dropped. However, even if the numerical factor is a single integer, its '(' and ')' delimiters cannot be omitted.
 .Pp
+If the polynomial is not terminal, then it is of the form
+.D1 <polynomial>operation<polynomial>
+where 'operation' is either '+' or '*', or
+.D1 <%func<polynomial>>
+where 'func' is 'exp' or 'log_1'. '<%exp<P>>' returns the exponential of P, whereas '<%log_1<P>>' returns log(1+P).
+.Pp
+.Nm
+parses polynomials by recursing through this structure, adding and multiplying terminal polynomials when '+' and '*' operations are encountered, and taking exponentials and logarithms when '%exp' or '%log_1' are encountered.
+.Pp
 Example:
-.D1 (1) + ((3/2)s{2} + (-1)s{-1} + 3)[l1][l2][f100][f1001][f101] + [l1][f101] + (3)[l2]
+.D1 <(1)+((3/2)s{2}+(-1)s{-1}+3)[l1][l2][f100][f1001][f101]>*<%exp<[l1][f101]+(3)[l2]>>
 .Pp
 .Sh OUTPUT
 .Nm
@@ -232,5 +330,4 @@ returns 0 on success and -1 on error.
 .Sx numkondo Ns (1) ,
 .Sx meantools Ns (1) ,
 .Sx meantools-convert Ns (1) ,
-.Sx kondo_preprocess Ns (1)
 .Pp
diff --git a/man/meantools-convert.1 b/man/meantools-convert.1
index 8393b28..bb585a8 100644
--- a/man/meantools-convert.1
+++ b/man/meantools-convert.1
@@ -1,5 +1,5 @@
-.Dd $Mdocdate: September 22 2015 $
-.Dt meantools-convert 1.4
+.Dd $Mdocdate: June 6 2022 $
+.Dt meantools-convert 1.5
 .Os
 .Sh NAME
 .Nm meantools-convert
@@ -43,14 +43,7 @@ is part of a set of tools to compute and manipulate Fermionic hierarchical flows
 : numerical evaluation of flow equations.
 .It
 .Sy meantools, meantools-convert
-: perform various operations on flow equations (derivation, exponentiation, evaluation and conversion to other formats).
-.El
-.Pp
-as well as the following pre-processors, which generate configuration files for their associated model:
-.Bl -bullet
-.It
-.Sy kondo_proprocess
-:  Kondo model
+: perform various operations on flow equations (differentiation, products, sums, exponentials and logarithms of flow equations, evaluation and conversion to other formats).
 .El
 .Pp
 .Sh COMMAND-LINE ARGUMENTS
@@ -133,5 +126,4 @@ returns 0 on success and -1 on error.
 .Sx meankondo Ns (1)
 .Sx numkondo Ns (1) ,
 .Sx meantools Ns (1) ,
-.Sx kondo_preprocess Ns (1)
 .Pp
diff --git a/man/meantools.1 b/man/meantools.1
index c4c73c6..8feb0c4 100644
--- a/man/meantools.1
+++ b/man/meantools.1
@@ -1,16 +1,12 @@
-.Dd $Mdocdate: September 22 2015 $
-.Dt meantools 1.4
+.Dd $Mdocdate: June 6 2022 $
+.Dt meantools 1.5
 .Os
 .Sh NAME
 .Nm meantools
 .Nd A tool to manipulate flow equations
 .Sh SYNOPSIS
 .Nm
-.Sy exp
-.Op Ar config_file
-.Pp
-.Nm
-.Sy derive
+.Sy differentiate
 .Op Fl d Ar nderivs
 .Op Fl V Ar variables
 .Op Fl C
@@ -23,11 +19,16 @@
 .Op Fl E Ar max_exponent
 .Op Ar config_file
 .Pp
+.Nm
+.Sy expand
+.Op Fl N Ar namespace
+.OpAr config_file
+.Pp
 .Sh DESCRIPTION
 .Nm
 performs various operations on flow equations generated by
 .Sy meankondo.
-Namely, it can exponentiate, derive and evaluate flow equations.
+Namely, it can differentiate and evaluate flow equations, as well as perform elementary operations on polynomials.
 .Pp
 .Nm
 is part of a set of tools to compute and manipulate Fermionic hierarchical flows:
@@ -40,45 +41,11 @@ is part of a set of tools to compute and manipulate Fermionic hierarchical flows
 : numerical evaluation of flow equations.
 .It
 .Sy meantools, meantools-convert
-: perform various operations on flow equations (derivation, exponentiation, evaluation and conversion to other formats).
+: perform various operations on flow equations (differentiation, products, sums, exponentials and logarithms of flow equations, evaluation and conversion to other formats).
 .El
 .Pp
-as well as the following pre-processors, which generate configuration files for their associated model:
-.Bl -bullet
-.It
-.Sy kondo_proprocess
-:  Kondo model
-.El
-.Pp
-.Sh EXP
-When run with the 'exp' command,
-.Nm
-computes the exponential of a flow equation. All the required parameters are set in the configuration file, which it either reads from the file provided on the command line, or from stdin.
-.Pp
-The syntax for the configuration file is the same as for
-.Sx meankondo Ns (1) ,
-and will not be belaboured here. The supported entries are
-.Bl -tag -width Ds
-.It Sy #!input_polynomial
-The polynomial whose exponential is to be computed.
-.Pp
-.It Sy #!fields
-The fields appearing in the polynomial
-.Pp
-.It Sy #!symbols
-Symbolic variables (optional entry).
-.Pp
-.It Sy #!identities
-identities between fields (optional entry).
-.Pp
-.It Sy #!id_table
-The idtable used to compute a flow equation from the polynomial.
-.El
-.Pp
-The resulting flow equation is written to stdout.
-.Pp
-.Sh DERIVE
-When run with the 'derive' command,
+.Sh DIFFERENTIATE
+When run with the 'differentiate' command,
 .Nm
 computes derivatives of a flow equation provided in the configuration file, which can either be passed as a command-line argument or through stdin.
 .Pp
@@ -86,7 +53,7 @@ The derivatives are derivatives with respect to an extra virtual parameter, whic
 .Pp
 When multiple derivatives are taken, the flow equation becomes a flow equation for the rccs, their derivatives, second derivatives, and so forth...
 .Pp
-This operation can be useful, for instance, to compute moments in an interacting system, in which the generating functional can be expressed as an effective potential depending on a parameter with respect to which the result of the integration should be derived. The 'derive' command writes the flow equation for the derived rccs, from which the quantities of interest can be computed.
+This operation can be useful, for instance, to compute moments in an interacting system, in which the generating functional can be expressed as an effective potential depending on a parameter with respect to which the result of the integration should be differentiated. The 'differentiate' command writes the flow equation for the differentiated rccs, from which the quantities of interest can be computed.
 .Pp
 .Sy Command-line arguments:
 .Bl -tag -width Ds
@@ -97,7 +64,7 @@ The variables that depend on the extra virtual parameter (defaults to all) (WARN
 .Nm
 would interpret the argument as being a flag, for example, write '-V "0,-1"' instead of '-V "-1,0"').
 .Pp
-Can either be a ',' separated list if indices or 'all' to derive with respect to all available variables.
+Can either be a ',' separated list if indices or 'all' to differentiate with respect to all available variables.
 .It Fl C
 Format the output so it can be piped to
 .Sy numkondo ,
@@ -106,10 +73,10 @@ that is, instead of printing the flow equation, print a full configuration file
 .Pp
 .Sy Configuration file:
 .Pp
-The configuration file contains the flow equation to derive, and optionally a list of variables (similar to the '-V' flag). The following entries are supported:
+The configuration file contains the flow equation to differentiate, and optionally a list of variables (similar to the '-V' flag). The following entries are supported:
 .Bl -tag -width Ds
 .It Sy #!flow_equation
-The flow equation to derive.
+The flow equation to differentiate.
 .Pp
 The syntax is identical to that in
 .Sx numkondo Ns (1) .
@@ -169,9 +136,49 @@ If the '-R' flag is provided on the command-line, this entry is ignored.
 .Pp
 The result of the evaluation is written to stdout, and is formatted is such a way that it can be used as an initial condition for
 .Pp
+.Sh EXPAND
+When run with the 'expand' command,
+.Nm
+expands the preprocessor variables in the input polynomial, provided in the configuration file, which can either be passed as a command-line argument or through stdin, and prints the result.
+.Pp
+.Sy Command-line arguments:
+.Bl -tag -width Ds
+.It Fl N Ar namespace
+If the configuration file is to be used to perform other operations, it may be convenient to specify the input polynomial for the 'expand' command alongside another '#!input_polynomial' entry, used for some other computation. This is made possible by namespaces.
+.Pp
+If a namespace is provided to
+.Nm
+on the command line, then it will search for the entries in the configuration file in the form
+.D1 #!namespace:header
+and default to #!header if no such header is present.
+.Pp
+In this way, the configuration file can, for instance, contain a '#!namspace:input_polynomial' entry for this computation, as well as a '#!input_polynomial' entry, to be used for some other purpose, all the while using the same '#!fields', '#!preprocessor_variables', '#!virtual_fields' and '#!identities' entries.
+.El
+.Pp
+.Sy Configuration file:
+.Pp
+The supported entries are
+.Bl -tag -width Ds
+.It Sy #!input_polynomial
+The polynomial whose exponential is to be computed.
+.Pp
+.It Sy #!fields
+The fields appearing in the polynomial.
+.Pp
+.It Sy #!preprocessor_variables
+Preprocessor variables (optional entry).
+.Pp
+.It Sy #!virtual_fields
+Virtual fields (optional entry).
+.Pp
+.It Sy #!identities
+identities between fields (optional entry).
+.El
+.Pp
+The result is written to stdout.
+.Pp
 .Sh SEE ALSO
 .Sx meankondo Ns (1) ,
 .Sx numkondo Ns (1) ,
 .Sx meantools-convert Ns (1) ,
-.Sx kondo_preprocess Ns (1)
 .Pp
diff --git a/man/numkondo.1 b/man/numkondo.1
index 3af84df..58d834f 100644
--- a/man/numkondo.1
+++ b/man/numkondo.1
@@ -1,5 +1,5 @@
-.Dd $Mdocdate: September 22 2015 $
-.Dt numkondo 1.4
+.Dd $Mdocdate: June 6 2022 $
+.Dt numkondo 1.5
 .Os
 .Sh NAME
 .Nm numkondo
@@ -29,14 +29,7 @@ is part of a set of tools to compute and manipulate Fermionic hierarchical flows
 : numerical evaluation of flow equations.
 .It
 .Sy meantools, meantools-convert
-: perform various operations on flow equations (derivation, exponentiation, evaluation and conversion to other formats).
-.El
-.Pp
-as well as the following pre-processors, which generate configuration files for their associated model:
-.Bl -bullet
-.It
-.Sy kondo_proprocess
-:  Kondo model
+: perform various operations on flow equations (differentiation, products, sums, exponentials and logarithms of flow equations, evaluation and conversion to other formats).
 .El
 .Pp
 .Sh COMMAND-LINE ARGUMENTS
@@ -167,5 +160,4 @@ returns 0 on success and -1 on error.
 .Sx meankondo Ns (1) ,
 .Sx meantools Ns (1) ,
 .Sx meantools-convert Ns (1) ,
-.Sx kondo_preprocess Ns (1)
 .Pp
diff --git a/scripts/config_functions.sh b/scripts/config_functions.sh
index 593ca94..1fffbce 100644
--- a/scripts/config_functions.sh
+++ b/scripts/config_functions.sh
@@ -1,4 +1,4 @@
-## Copyright 2015 Ian Jauslin
+## Copyright 2015-2022 Ian Jauslin
 ## 
 ## Licensed under the Apache License, Version 2.0 (the "License");
 ## you may not use this file except in compliance with the License.
diff --git a/scripts/meantools-convert b/scripts/meantools-convert
index ef87227..3931418 100755
--- a/scripts/meantools-convert
+++ b/scripts/meantools-convert
@@ -1,6 +1,6 @@
 #!/usr/bin/env python
 
-## Copyright 2015 Ian Jauslin
+## Copyright 2015-2022 Ian Jauslin
 ## 
 ## Licensed under the Apache License, Version 2.0 (the "License");
 ## you may not use this file except in compliance with the License.
@@ -76,8 +76,8 @@ def latex_engine(argv,text):
 
     i=1
     # defaults
-    lsym="\\ell"
-    Lsym="\\ell'"
+    lsym=r'\\ell'
+    Lsym=r'\\ell'
     Csym="C"
     oneline=1
     while (i<argc):
@@ -171,6 +171,10 @@ def convert_latex(text, lsym, Lsym, Csym, oneline):
     # remove extra space
     text=re.sub(r' {2,}',' ',text)
 
+    # remove unnecessary 1's
+    text=re.sub(r'\(-1\)\[',r'-[',text)
+    text=re.sub(r'\(1\)\[',r'[',text)
+
     # replace left hand side variables
     text=re.sub(r'\[(d*)C *([0-9]*)\] =',r'@\1'+Csym+r'_{\2} =', text)
     text=re.sub(r'\[(d*)% *([0-9]*)\] =',r'@\1'+Lsym+r'_{\2} =', text)
@@ -213,7 +217,11 @@ def convert_latex(text, lsym, Lsym, Csym, oneline):
     # fractions
     text=re.sub(r'\((-?)([0-9]*)/([0-9]*)\)',r'\1\\frac{\2}{\3}',text)
     # numbers
-    text=re.sub(r'\(([-0-9]*)\)',r'\1',text)
+    #text=re.sub(r'\(([-0-9]*)\)',r'\1',text)
+
+    # fix signs
+    text=re.sub(r'\+-',r'-',text)
+
 
 
     return(text)
diff --git a/src/array.c b/src/array.c
index 11d14f7..7fb6bd8 100644
--- a/src/array.c
+++ b/src/array.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -72,6 +72,14 @@ int int_array_append(int val, Int_Array* output){
   return(0);
 }
 
+// add a value only if it is not already present
+int int_array_append_unique(int val, Int_Array* output){
+  if(int_array_find(val,*output)<0){
+    int_array_append(val,output);
+  }
+  return(0);
+}
+
 // concatenate
 int int_array_concat(Int_Array input, Int_Array* output){
   int i;
@@ -87,6 +95,15 @@ int int_array_concat(Int_Array input, Int_Array* output){
   return(0);
 }
 
+// concat but only add values that are not already present in the array
+int int_array_concat_unique(Int_Array input, Int_Array* output){
+  int i;
+  for(i=0;i<input.length;i++){
+    int_array_append_unique(input.values[i],output);
+  }
+  return(0);
+}
+
 // find (does not assume the array is sorted)
 int int_array_find(int val, Int_Array array){
   int i;
@@ -199,7 +216,12 @@ int int_array_print(Int_Array array){
   for(i=0;i<array.length-1;i++){
     printf("%d,",array.values[i]);
   }
-  printf("%d)",array.values[array.length-1]);
+  if(array.length>0){
+    printf("%d)",array.values[array.length-1]);
+  }
+  else{
+    printf(")");
+  }
   return(0);
 }
 
@@ -334,6 +356,19 @@ int char_array_concat(Char_Array input, Char_Array* output){
   return(0);
 }
 
+// substring
+int char_array_substring(Char_Array str, int begin, int end, Char_Array* substr){
+  int i;
+  if(begin>end || begin<0 || end>=str.length){
+    fprintf(stderr,"error: cannot extract a substring [%d,%d] from a string of length %d\n", begin, end, str.length);
+    exit(-1);
+  }
+  init_Char_Array(substr,end-begin);
+  for(i=begin;i<=end;i++){
+    char_array_append(str.str[i],substr);
+  }
+  return(0);
+}
 
 
 // convert to char*
@@ -343,7 +378,7 @@ int char_array_to_str(Char_Array input, char** output){
   for(i=0;i<input.length;i++){
     (*output)[i]=input.str[i];
   }
-  if((*output)[input.length-1]!='\0'){
+  if(input.length==0 ||  (*output)[input.length-1]!='\0'){
     (*output)[input.length]='\0';
   }
   return(0);
@@ -371,6 +406,34 @@ int str_to_char_array(char* str, Char_Array* output){
   return(0);
 }
 
+// compare char_array's
+int char_array_cmp(Char_Array char_array1, Char_Array char_array2){
+  int j;
+  if(char_array1.length!=char_array2.length){
+    return(0);
+  }
+  for(j=0;j<char_array1.length && j<char_array2.length;j++){
+    if(char_array1.str[j]!=char_array2.str[j]){
+      return(0);
+    }
+  }
+  return(1);
+}
+
+// compare a char_array and a char*
+int char_array_cmp_str(Char_Array char_array, char* str){
+  int j;
+  for(j=0;j<char_array.length && str[j]!='\0';j++){
+    if(char_array.str[j]!=str[j]){
+      return(0);
+    }
+  }
+  if(j==char_array.length && str[j]=='\0'){
+    return(1);
+  }
+  return(0);
+}
+
 
 // format strings
 int char_array_snprintf(Char_Array* output, char* fmt, ...){
diff --git a/src/array.h b/src/array.h
index fb74e67..480b589 100644
--- a/src/array.h
+++ b/src/array.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -34,8 +34,12 @@ int int_array_resize(Int_Array* array, int newsize);
 
 // add a value
 int int_array_append(int val, Int_Array* output);
+// add a value only if it is not already present
+int int_array_append_unique(int val, Int_Array* output);
 // concatenate
 int int_array_concat(Int_Array input, Int_Array* output);
+// concat but only add values that are not already present in the array
+int int_array_concat_unique(Int_Array input, Int_Array* output);
 
 // find (does not assume the array is sorted)
 int int_array_find(int val, Int_Array array);
@@ -75,6 +79,9 @@ int char_array_append_str(char* str, Char_Array* output);
 // concatenate
 int char_array_concat(Char_Array input, Char_Array* output);
 
+// substring
+int char_array_substring(Char_Array str, int begin, int end, Char_Array* substr);
+
 // convert to char*
 int char_array_to_str(Char_Array input, char** output);
 // noinit (changes the size of input if needed)
@@ -82,6 +89,11 @@ char* char_array_to_str_noinit(Char_Array* input);
 // convert from char*
 int str_to_char_array(char* str, Char_Array* output);
 
+// compare char_array's
+int char_array_cmp(Char_Array char_array1, Char_Array char_array2);
+// compare a char_array and a char*
+int char_array_cmp_str(Char_Array char_array, char* str);
+
 // format strings
 int char_array_snprintf(Char_Array* output, char* fmt, ...);
 
diff --git a/src/cli_parser.c b/src/cli_parser.c
index bdca700..6012e8a 100644
--- a/src/cli_parser.c
+++ b/src/cli_parser.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -121,3 +121,23 @@ int find_str_arg(char* title, Str_Array str_args){
   }
 }
 
+// find a string argument with the specified title
+// namespace support
+int find_str_arg_ns(char* title, Char_Array namespace, Str_Array str_args){
+  Char_Array buffer;
+  int ret;
+
+  // append namespace to title
+  char_array_cpy(namespace,&buffer);
+  char_array_append(':',&buffer);
+  char_array_append_str(title,&buffer);
+
+  // check whether the namespace entry exists
+  ret=find_str_arg(char_array_to_str_noinit(&buffer), str_args);
+  free_Char_Array(buffer);
+  // if not, use global entry
+  if(ret==-1){
+    return(find_str_arg(title, str_args));
+  }
+  return(ret);
+}
diff --git a/src/cli_parser.h b/src/cli_parser.h
index b050e37..b66608c 100644
--- a/src/cli_parser.h
+++ b/src/cli_parser.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -30,6 +30,8 @@ int read_config_file(Str_Array* str_args, const char* file, int read_from_stdin)
 int get_str_arg_title(Char_Array str_arg, Char_Array* out);
 // find a string argument with the specified title
 int find_str_arg(char* title, Str_Array str_args);
+// namespace support
+int find_str_arg_ns(char* title, Char_Array namespace, Str_Array str_args);
 
 
 #endif
diff --git a/src/coefficient.c b/src/coefficient.c
index d4cb6eb..1efe8fd 100644
--- a/src/coefficient.c
+++ b/src/coefficient.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -202,6 +202,235 @@ int coefficient_simplify(Coefficient* coefficient){
   return(0);
 }
 
+// put all terms under a common denominator and simplify the resulting fraction
+int coefficient_simplify_rational(Coefficient constant, Coefficient* coefficient){
+  int ret;
+  Coefficient remainder;
+  Coefficient quotient;
+  Coefficient quotient_prev;
+  Coefficient out;
+  int power;
+  int max_power;
+
+  // common denominator
+  coefficient_common_denominator(constant, coefficient);
+
+  // init
+  init_Coefficient(&out, COEF_SIZE);
+
+  // simplify, one power at a time
+  // largest power (larger powers are at the end)
+  max_power=(*coefficient).denoms[(*coefficient).length-1].power;
+
+  quotient_prev=*coefficient;
+  for(power=max_power;power>=1;power--){
+    ret=coefficient_simplify_fraction(constant, quotient_prev, &remainder, &quotient);
+
+    // if fail to simplify, stop
+    if(ret<0){
+      if(power<max_power){
+	coefficient_concat_noinit(quotient_prev, &out);
+      }
+      else{
+	coefficient_concat(quotient_prev, &out);
+      }
+      break;
+    }
+
+    // add to output
+    coefficient_concat_noinit(remainder, &out);
+  }
+  // if the factorization always succeeded
+  if(max_power>=1 && power==0){
+    coefficient_concat_noinit(quotient, &out);
+  }
+
+  coefficient_simplify(&out);
+
+  // set coefficient to out
+  free_Coefficient(*coefficient);
+  *coefficient=out;
+
+  return 0;
+}
+
+// put all terms under a common denominator
+// only supports coefficients with only one constant
+int coefficient_common_denominator(Coefficient constant, Coefficient* coefficient){
+  int max_power;
+  int i,j;
+  Coefficient tmp;
+  Coefficient out;
+  Coefficient* C_n;
+
+  init_Coefficient(&out, COEF_SIZE);
+
+  // largest power (larger powers are at the end)
+  max_power=(*coefficient).denoms[(*coefficient).length-1].power;
+
+  // store powers of the constant
+  C_n=calloc(sizeof(Coefficient), max_power-1);
+  for(i=0;i<max_power-1;i++){
+    // start from previous product
+    if(i==0){
+      coefficient_cpy(constant, C_n+i);
+    }
+    else{
+      coefficient_cpy(C_n[i-1], C_n+i);
+    }
+    // multiply by constant
+    coefficient_prod_chain(constant, C_n+i);
+  }
+
+  // multiply each term
+  for (i=0;i<(*coefficient).length;i++){
+    init_Coefficient(&tmp, COEF_SIZE);
+    // start with numerator
+    coefficient_append_noinit((*coefficient).factors[i], (*coefficient).nums[i], (*coefficient).denoms[i], &tmp);
+    // multiply
+    if((*coefficient).denoms[i].power<max_power){
+      if((*coefficient).denoms[i].power==max_power-1){
+	coefficient_prod_chain(constant, &tmp);
+      }
+      else{
+	coefficient_prod_chain(C_n[max_power-(*coefficient).denoms[i].power-2], &tmp);
+      }
+    }
+
+    // set denom
+    for(j=0;j<tmp.length;j++){
+      tmp.denoms[j].power=max_power;
+    }
+
+    // add to out
+    coefficient_concat_noinit(tmp, &out);
+  }
+
+  // free C_n
+  for(i=0;i<max_power-1;i++){
+    free_Coefficient(C_n[i]);
+  }
+  free(C_n);
+
+  // free coefficient vectors
+  free((*coefficient).factors);
+  free((*coefficient).nums);
+  free((*coefficient).denoms);
+
+  // set output
+  *coefficient=out;
+
+  coefficient_simplify(coefficient);
+
+  return(0);
+}
+
+// simplify coefficient / constant
+// returns both the remainder and the quotient
+// assumes both coefficient and constant are ordered with the highest order terms last
+int coefficient_simplify_fraction(Coefficient constant, Coefficient coefficient, Coefficient* remainder, Coefficient* out){
+  Coefficient tmp;
+  int step_counter=0;
+  int max_order;
+  int i,j,k;
+  Int_Array rfactors;
+
+  if(constant.length==0){
+    // nothing to do
+    return 0;
+  }
+
+  coefficient_cpy(coefficient, remainder);
+  init_Coefficient(out, COEF_SIZE);
+
+  // continue until (*remainder) is of lower order than constant
+  while((*remainder).length>0 && (*remainder).factors[(*remainder).length-1].length>=constant.factors[constant.length-1].length){
+    step_counter++;
+
+    // interrupt if too long
+    if(step_counter>=coefficient.length*100){
+      free_Coefficient(*remainder);
+      free_Coefficient(*out);
+      return -1;
+    }
+
+    // try to find a term in the constant that divides the last term of the (*remainder)
+    rfactors=(*remainder).factors[(*remainder).length-1];
+
+    // highest order in constant
+    max_order=constant.factors[constant.length-1].length;
+
+    // start from one of the highest order term and look for a common factor
+    for(i=constant.length-1; i>=0; i--){
+      // fail: no highest order terms have been matched
+      if(constant.factors[i].length<max_order){
+	free_Coefficient(*remainder);
+	free_Coefficient(*out);
+	return -2;
+      }
+
+      // check whether the term can be a factor of the last term of the (*remainder)
+      if(int_array_is_subarray_ordered(constant.factors[i], rfactors)==1){
+	// extract the factors that are not in constant
+	init_Coefficient(&tmp, constant.length);
+
+	// init with one term
+	tmp.length=1;
+	init_Int_Array(tmp.factors,MONOMIAL_SIZE);
+
+	for(j=0,k=0;j<rfactors.length;j++){
+	  // check that index is not in constant
+	  if(k<constant.factors[i].length){
+	    if(rfactors.values[j]!=constant.factors[i].values[k]){
+	      int_array_append(rfactors.values[j],tmp.factors);
+	    }
+	    else{
+	      // move to next term in constant
+	      k++;
+	    }
+	  }
+	}
+
+	// numerical prefactor: term in the (*remainder) / term in the constant
+	number_quot((*remainder).nums[(*remainder).length-1], constant.nums[i], tmp.nums);
+	// denominator (dummy)
+	tmp.denoms[0]=(*remainder).denoms[(*remainder).length-1];
+
+	// add to out
+	coefficient_concat(tmp, out);
+
+	// multiply by -1
+	Q minus_1;
+	minus_1.numerator=-1;
+	minus_1.denominator=1;
+	number_Qprod_chain(minus_1, tmp.nums);
+
+	// multiply by constant
+	coefficient_prod_chain(constant, &tmp);
+
+	// add to remainder
+	coefficient_concat(tmp, remainder);
+
+	// free memory
+	free_Coefficient(tmp);
+
+	// simplify
+	coefficient_simplify(remainder);
+
+	break;
+      }
+    }
+  }
+
+  // success!
+  // decrease power of constant
+  for(i=0;i<(*out).length;i++){
+    (*out).denoms[i].power=(*out).denoms[i].power-1;
+  }
+
+  return(0);
+}
+
 // sort the terms in an equation (quicksort algorithm)
 int sort_coefficient(Coefficient* coefficient, int begin, int end){
   int i;
@@ -251,7 +480,7 @@ int exchange_coefficient_terms(int i, int j, Coefficient* coefficient){
   return(0);
 }
 
-// derive a coefficient with respect to an index
+// differentiate a coefficient with respect to an index
 int coefficient_deriv_noinit(Coefficient input, int index, Coefficient* output){
   int i,j;
   // temp list of indices
@@ -325,7 +554,7 @@ int coefficient_deriv(Coefficient input, int index, Coefficient* output){
 }
 
 /*
-// derive a coefficient with respect to an index (as a polynomial) (does not derive the 1/(1+C)^p )
+// differentiate a coefficient with respect to an index (as a polynomial) (does not differentiate the 1/(1+C)^p )
 int coefficient_deriv_noinit(Coefficient input, int index, Coefficient* output){
   int i;
   // temp list of indices
@@ -365,7 +594,7 @@ int coefficient_deriv_noinit(Coefficient input, int index, Coefficient* output){
   return(0);
 }
 
-// derive a monomial with respect to an index
+// differentiate a monomial with respect to an index
 int monomial_deriv(Int_Array factor, int index, Int_Array* out_factor, int* match_count){
   int j;
 
@@ -414,14 +643,14 @@ int coefficient_prod(Coefficient coef1, Coefficient coef2, Coefficient* output){
       int_array_concat(coef2.factors[j],&factor);
       
       // don't throw an error if the power is 0
-      if(coef2.denoms[i].power==0){
-	coef2.denoms[i].index=coef1.denoms[i].index;
+      if(coef2.denoms[j].power==0){
+	coef2.denoms[j].index=coef1.denoms[i].index;
       }
       else if(coef1.denoms[i].power==0){
-	coef1.denoms[i].index=coef2.denoms[i].index;
+	coef1.denoms[i].index=coef2.denoms[j].index;
       }
       if(coef1.denoms[i].index!=coef2.denoms[j].index){
-	fprintf(stderr,"error: cannot multiply flow equations with different constants\n");
+	fprintf(stderr,"error: cannot multiply flow equations with different constants: got %d and %d\n", coef1.denoms[i].index, coef2.denoms[j].index);
 	exit(-1);
       }
       denom=coef1.denoms[i];
diff --git a/src/coefficient.h b/src/coefficient.h
index 43dd313..f8a813a 100644
--- a/src/coefficient.h
+++ b/src/coefficient.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -48,12 +48,20 @@ int coefficient_concat_noinit(Coefficient input, Coefficient* output);
 
 // simplify a Coefficient
 int coefficient_simplify(Coefficient* coefficient);
+
+// put all terms under a common denominator and simplify the resulting fraction
+int coefficient_simplify_rational(Coefficient constant, Coefficient* coefficient);
+// put all terms under a common denominator
+int coefficient_common_denominator(Coefficient constant, Coefficient* coefficient);
+// simplify coefficient / constant
+int coefficient_simplify_fraction(Coefficient constant, Coefficient coefficient, Coefficient* remainder, Coefficient* out);
+
 // sort the terms in an equation (quicksort algorithm)
 int sort_coefficient(Coefficient* coefficient, int begin, int end);
 // exchange two terms (for the sorting algorithm)
 int exchange_coefficient_terms(int i, int j, Coefficient* coefficient);
 
-// derive a coefficient with respect to an index (as a polynomial) (does not derive the 1/(1+C)^p )
+// differentiate a coefficient with respect to an index (as a polynomial) (does not differentiate the 1/(1+C)^p )
 int coefficient_deriv_noinit(Coefficient input, int index, Coefficient* output);
 int coefficient_deriv(Coefficient input, int index, Coefficient* output);
 
diff --git a/src/definitions.cpp b/src/definitions.cpp
index 30f021f..db09e57 100644
--- a/src/definitions.cpp
+++ b/src/definitions.cpp
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -33,10 +33,16 @@ limitations under the License.
 #define EQUATION_SIZE 20
 // number of fields
 #define FIELDS_SIZE 50
+// number of variables
+#define VARIABLES_SIZE 10
 // number of elements in numbers
 #define NUMBER_SIZE 5
 // number of elements in a group
 #define GROUP_SIZE 5
+// number of children per node in a symbol_tree
+#define SYMBOL_TREE_SIZE 2
+// size of character string in symbol tree
+#define SYMBOL_TREE_LABEL_SIZE 10
 
 
 // display options
@@ -55,6 +61,6 @@ limitations under the License.
 #define FIELD_PARAMETER 1
 #define FIELD_EXTERNAL 2
 #define FIELD_INTERNAL 3
-#define FIELD_SYMBOL 4
+#define FIELD_VIRTUAL 4
 
 #endif
diff --git a/src/determinant.c b/src/determinant.c
new file mode 100644
index 0000000..906c75f
--- /dev/null
+++ b/src/determinant.c
@@ -0,0 +1,93 @@
+#include "determinant.h"
+
+#include "number.h"
+#include "rational.h"
+#include "definitions.cpp"
+
+// determinant of a matrix
+// replaces the matrix by its LU decomposition
+int determinant_inplace(Number_Matrix M, Number* out){
+  int i;
+  int sign_correction;
+
+  LU_dcmp_inplace(M, &sign_correction);
+
+  if(sign_correction==0){
+    *out=number_zero();
+    return(0);
+  }
+
+  *out=number_one();
+  if(sign_correction==-1){
+    number_Qprod_chain(quot(-1,1), out);
+  }
+
+  for(i=0;i<M.length;i++){
+    number_prod_chain(M.matrix[i][i], out);
+  }
+
+  return(0);
+}
+
+// LU decomposition
+// uses pivoting to avoid dividing by 0
+// the sign_correction should be multiplied to the determinant to obtain the right value
+// if dividing by 0 is unavoidable, then the determinant is 0, and sign_correction is set to 0
+int LU_dcmp_inplace(Number_Matrix M, int* sign_correction){
+  int i,j,k,pivot;
+  Number tmp;
+
+  *sign_correction=1;
+
+  for(j=0;j<M.length;j++){
+    for(i=0;i<=j;i++){
+      for(k=0;k<i;k++){
+	// -M[i][k]*M[k][j]
+	number_prod(M.matrix[i][k], M.matrix[k][j], &tmp);
+	number_Qprod_chain(quot(-1,1), &tmp);
+	number_add_chain(tmp, M.matrix[i]+j);
+	free_Number(tmp);
+      }
+    }
+    for(i=j+1;i<M.length;i++){
+      for(k=0;k<j;k++){
+	// -M[i][k]*M[k][j]
+	number_prod(M.matrix[i][k], M.matrix[k][j], &tmp);
+	number_Qprod_chain(quot(-1,1), &tmp);
+	number_add_chain(tmp, M.matrix[i]+j);
+	free_Number(tmp);
+      }
+    }
+
+    // pivot if M[j][j]==0
+    // find first M[j][j] that is not 0
+    for(pivot=j;pivot<M.length && number_is_zero(M.matrix[pivot][j])==1;pivot++){}
+
+    // no non-zero M[j][j] left: return
+    if(pivot>=M.length){
+      *sign_correction=0;
+      return(0);
+    }
+    // pivot if needed
+    if(pivot!=j){
+      for(k=0;k<M.length;k++){
+	tmp=M.matrix[j][k];
+	M.matrix[j][k]=M.matrix[pivot][k];
+	M.matrix[pivot][k]=tmp;
+      }
+      *sign_correction*=-1;
+
+    }
+
+    for(i=j+1;i<M.length;i++){
+      // do not use the inplace algorithm if M[j][j] has more than one terms, since it would be modified by the inplace function
+      if(M.matrix[j][j].length<=1){
+	number_quot_inplace(M.matrix[i]+j, M.matrix[j]+j);
+      }
+      else{
+	number_quot_chain(M.matrix[i]+j, M.matrix[j][j]);
+      }
+    }
+  }
+  return(0);
+}
diff --git a/src/determinant.h b/src/determinant.h
new file mode 100644
index 0000000..613be87
--- /dev/null
+++ b/src/determinant.h
@@ -0,0 +1,17 @@
+/*
+  Compute the determinant of a number matrix
+*/
+
+#ifndef DETERMINANT_H
+#define DETERMINANT_H
+
+#include "types.h"
+
+// determinant of a matrix
+int determinant_inplace(Number_Matrix M, Number* out);
+
+// LU decomposition
+int LU_dcmp_inplace(Number_Matrix M, int* sign_correction);
+
+
+#endif
diff --git a/src/expansions.c b/src/expansions.c
deleted file mode 100644
index c889829..0000000
--- a/src/expansions.c
+++ /dev/null
@@ -1,28 +0,0 @@
-/*
-Copyright 2015 Ian Jauslin
-
-Licensed under the Apache License, Version 2.0 (the "License");
-you may not use this file except in compliance with the License.
-You may obtain a copy of the License at
-
-    http://www.apache.org/licenses/LICENSE-2.0
-
-Unless required by applicable law or agreed to in writing, software
-distributed under the License is distributed on an "AS IS" BASIS,
-WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
-See the License for the specific language governing permissions and
-limitations under the License.
-*/
-
-#include "expansions.h"
-
-#include <stdio.h>
-#include <stdlib.h>
-#include "definitions.cpp"
-#include "tools.h"
-#include "array.h"
-#include "polynomial.h"
-#include "number.h"
-#include "rational.h"
-
-
diff --git a/src/expansions.h b/src/expansions.h
deleted file mode 100644
index 85d6c2b..0000000
--- a/src/expansions.h
+++ /dev/null
@@ -1,34 +0,0 @@
-/*
-Copyright 2015 Ian Jauslin
-
-Licensed under the Apache License, Version 2.0 (the "License");
-you may not use this file except in compliance with the License.
-You may obtain a copy of the License at
-
-    http://www.apache.org/licenses/LICENSE-2.0
-
-Unless required by applicable law or agreed to in writing, software
-distributed under the License is distributed on an "AS IS" BASIS,
-WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
-See the License for the specific language governing permissions and
-limitations under the License.
-*/
-
-/*
-Compute exp(V) and log(1+W)
-*/
-
-#ifndef EXPANSIONS_H
-#define EXPANSIONS_H
-
-#include "polynomial.h"
-#include "fields.h"
-
-// exp(V)
-int expand_exponential(Polynomial input_polynomial,Polynomial* output, Fields_Table fields);
-
-// log(1+W)
-int expand_logarithm(Polynomial input_polynomial, Polynomial* output, Fields_Table fields);
-
-
-#endif
diff --git a/src/fields.c b/src/fields.c
index bfd1f39..7be84df 100644
--- a/src/fields.c
+++ b/src/fields.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -23,6 +23,13 @@ limitations under the License.
 #include "polynomial.h"
 #include "array.h"
 #include "rational.h"
+#include "tree.h"
+
+//---------------------------------------------------------------------
+//
+//                              Fields_Table
+//
+//---------------------------------------------------------------------
 
 // init and free for Fields_Table
 int init_Fields_Table(Fields_Table* fields){
@@ -30,7 +37,7 @@ int init_Fields_Table(Fields_Table* fields){
   init_Int_Array(&((*fields).external),FIELDS_SIZE);
   init_Int_Array(&((*fields).internal),FIELDS_SIZE);
   init_Identities(&((*fields).ids), FIELDS_SIZE);
-  init_Symbols(&((*fields).symbols), FIELDS_SIZE);
+  init_Virtual_fields(&((*fields).virtual_fields), FIELDS_SIZE);
   init_Int_Array(&((*fields).fermions),FIELDS_SIZE);
   init_Int_Array(&((*fields).noncommuting),FIELDS_SIZE);
   return(0);
@@ -40,7 +47,7 @@ int free_Fields_Table(Fields_Table fields){
   free_Int_Array(fields.external);
   free_Int_Array(fields.internal);
   free_Identities(fields.ids);
-  free_Symbols(fields.symbols);
+  free_Virtual_fields(fields.virtual_fields);
   free_Int_Array(fields.fermions);
   free_Int_Array(fields.noncommuting);
   return(0);
@@ -57,11 +64,11 @@ int field_type(int index, Fields_Table fields){
   else if(int_array_find(abs(index), fields.internal)>=0){
     return(FIELD_INTERNAL);
   }
-  else if(intlist_find(fields.symbols.indices, fields.symbols.length, index)>=0){
-    return(FIELD_SYMBOL);
+  else if(intlist_find(fields.virtual_fields.indices, fields.virtual_fields.length, index)>=0){
+    return(FIELD_VIRTUAL);
   }
 
-  fprintf(stderr,"error: index %d is neither a parameter nor an external or an internal field, nor a symbol\n",index);
+  fprintf(stderr,"error: index %d is neither a parameter nor an external or an internal field, nor a virtual field\n",index);
   exit(-1);
 }
 
@@ -86,7 +93,11 @@ int is_noncommuting(int index, Fields_Table fields){
 }
 
 
-// ------------------ Identities --------------------
+//---------------------------------------------------------------------
+//
+//                               Identities
+//
+//---------------------------------------------------------------------
 
 // allocate memory
 int init_Identities(Identities* identities,int size){
@@ -314,6 +325,11 @@ int int_array_is_subarray_noncommuting(Int_Array input, Int_Array test_array, Fi
   int match_nc;
   int first=-1;
 
+  // cannot fit
+  if(test_array.length<input.length){
+    return(-1);
+  }
+
   // bound noncommuting elements
   while(is_noncommuting(input.values[post_nc], fields)==1){
     post_nc++;
@@ -324,7 +340,7 @@ int int_array_is_subarray_noncommuting(Int_Array input, Int_Array test_array, Fi
       match_nc=1;
     }
     for(j=1;j<post_nc;j++){
-      if(test_array.values[i+j]!=input.values[j]){
+      if(i+j>=test_array.length || test_array.values[i+j]!=input.values[j]){
 	match_nc=0;
       }
     }
@@ -359,57 +375,61 @@ int int_array_is_subarray_noncommuting(Int_Array input, Int_Array test_array, Fi
 }
 
 
-// ------------------ Symbols --------------------
+//---------------------------------------------------------------------
+//
+//                                Virtual_fields
+//
+//---------------------------------------------------------------------
 
 // allocate memory
-int init_Symbols(Symbols* symbols,int size){
-  (*symbols).indices=calloc(size,sizeof(int));
-  (*symbols).expr=calloc(size,sizeof(Polynomial));
-  (*symbols).length=0;
-  (*symbols).memory=size;
+int init_Virtual_fields(Virtual_fields* virtual_fields,int size){
+  (*virtual_fields).indices=calloc(size,sizeof(int));
+  (*virtual_fields).expr=calloc(size,sizeof(Polynomial));
+  (*virtual_fields).length=0;
+  (*virtual_fields).memory=size;
   return(0);
 }
 
 // free memory
-int free_Symbols(Symbols symbols){
+int free_Virtual_fields(Virtual_fields virtual_fields){
   int i;
-  for(i=0;i<symbols.length;i++){
-    free_Polynomial(symbols.expr[i]);
+  for(i=0;i<virtual_fields.length;i++){
+    free_Polynomial(virtual_fields.expr[i]);
   }
-  free(symbols.indices);
-  free(symbols.expr);
+  free(virtual_fields.indices);
+  free(virtual_fields.expr);
   return(0);
 }
 
 // resize
-int resize_symbols(Symbols* symbols,int new_size){
-  Symbols new_symbols;
+int resize_virtual_fields(Virtual_fields* virtual_fields,int new_size){
+  Virtual_fields new_virtual_fields;
   int i;
 
-  init_Symbols(&new_symbols,new_size);
-  for(i=0;i<(*symbols).length;i++){
-    new_symbols.indices[i]=(*symbols).indices[i];
-    new_symbols.expr[i]=(*symbols).expr[i];
+  init_Virtual_fields(&new_virtual_fields,new_size);
+  for(i=0;i<(*virtual_fields).length;i++){
+    new_virtual_fields.indices[i]=(*virtual_fields).indices[i];
+    new_virtual_fields.expr[i]=(*virtual_fields).expr[i];
   }
-  new_symbols.length=(*symbols).length;
+  new_virtual_fields.length=(*virtual_fields).length;
 
-  free((*symbols).indices);
-  free((*symbols).expr);
+  free((*virtual_fields).indices);
+  free((*virtual_fields).expr);
 
-  *symbols=new_symbols;
+  *virtual_fields=new_virtual_fields;
   return(0);
 }
 
 // copy
-int symbols_cpy(Symbols input, Symbols* output){
-  init_Symbols(output,input.length);
-  symbols_cpy_noinit(input,output);
+int virtual_fields_cpy(Virtual_fields input, Virtual_fields* output){
+  init_Virtual_fields(output,input.length);
+  virtual_fields_cpy_noinit(input,output);
   return(0);
 }
-int symbols_cpy_noinit(Symbols input, Symbols* output){
+int virtual_fields_cpy_noinit(Virtual_fields input, Virtual_fields* output){
   int i;
   if((*output).memory<input.length){
-    fprintf(stderr,"error: trying to copy a symbols collection of length %d to another with memory %d\n",input.length,(*output).memory);
+    fprintf(stderr,"error: trying to copy a virtual fields collection of length %d to another with memory %d\n",input.length,(*output).memory);
     exit(-1);
   }
   for(i=0;i<input.length;i++){
@@ -421,12 +441,12 @@ int symbols_cpy_noinit(Symbols input, Symbols* output){
   return(0);
 }
 
-// append an element to a symbols
-int symbols_append(int index, Polynomial expr, Symbols* output){
+// append an element to a virtual_fields
+int virtual_fields_append(int index, Polynomial expr, Virtual_fields* output){
   int offset=(*output).length;
 
   if((*output).length>=(*output).memory){
-    resize_symbols(output,2*(*output).memory+1);
+    resize_virtual_fields(output,2*(*output).memory+1);
   }
 
   // copy and allocate
@@ -436,12 +456,12 @@ int symbols_append(int index, Polynomial expr, Symbols* output){
   (*output).length++;
   return(0);
 }
-// append an element to a symbols without allocating memory
-int symbols_append_noinit(int index, Polynomial expr, Symbols* output){
+// append an element to a virtual_fields without allocating memory
+int virtual_fields_append_noinit(int index, Polynomial expr, Virtual_fields* output){
   int offset=(*output).length;
 
   if((*output).length>=(*output).memory){
-    resize_symbols(output,2*(*output).memory+1);
+    resize_virtual_fields(output,2*(*output).memory+1);
   }
 
   // copy without allocating
@@ -452,18 +472,22 @@ int symbols_append_noinit(int index, Polynomial expr, Symbols* output){
   return(0);
 }
 
-// concatenate two symbolss
-int symbols_concat(Symbols input, Symbols* output){
+// concatenate two virtual_fields
+int virtual_fields_concat(Virtual_fields input, Virtual_fields* output){
   int i;
   for(i=0;i<input.length;i++){
-    symbols_append(input.indices[i],input.expr[i],output);
+    virtual_fields_append(input.indices[i],input.expr[i],output);
   }
   return(0);
 }
 
 
 
-// ------------------ Groups --------------------
+//---------------------------------------------------------------------
+//
+//                                 Groups
+//
+//---------------------------------------------------------------------
 
 // allocate memory
 int init_Groups(Groups* groups,int size){
@@ -570,3 +594,164 @@ int find_group(int index, Groups groups){
   }
   return(-1);
 }
+
+
+//---------------------------------------------------------------------
+//
+//                               Variables
+//
+//---------------------------------------------------------------------
+
+// allocate memory
+int init_Variables(Variables* variables,int size){
+  (*variables).var_names=calloc(size,sizeof(Char_Array));
+  (*variables).symbol_trees=calloc(size,sizeof(Tree));
+  (*variables).length=0;
+  (*variables).memory=size;
+  return(0);
+}
+
+// free memory
+int free_Variables(Variables variables){
+  int i;
+  for(i=0;i<variables.length;i++){
+    free_Char_Array(variables.var_names[i]);
+    free_Tree(variables.symbol_trees[i]);
+  }
+  free(variables.var_names);
+  free(variables.symbol_trees);
+  return(0);
+}
+
+// resize
+int resize_variables(Variables* variables,int new_size){
+  Variables new_variables;
+  int i;
+
+  init_Variables(&new_variables,new_size);
+  for(i=0;i<(*variables).length;i++){
+    new_variables.var_names[i]=(*variables).var_names[i];
+    new_variables.symbol_trees[i]=(*variables).symbol_trees[i];
+  }
+  new_variables.length=(*variables).length;
+
+  free((*variables).var_names);
+  free((*variables).symbol_trees);
+
+  *variables=new_variables;
+  return(0);
+}
+
+// copy
+int variables_cpy(Variables input, Variables* output){
+  init_Variables(output,input.length);
+  variables_cpy_noinit(input,output);
+  return(0);
+}
+int variables_cpy_noinit(Variables input, Variables* output){
+  int i;
+  if((*output).memory<input.length){
+    fprintf(stderr,"error: trying to copy a variables collection of length %d to another with memory %d\n",input.length,(*output).memory);
+    exit(-1);
+  }
+  for(i=0;i<input.length;i++){
+    char_array_cpy(input.var_names[i], (*output).var_names+i);
+    tree_cpy(input.symbol_trees[i],(*output).symbol_trees+i);
+  }
+  (*output).length=input.length;
+  
+  return(0);
+}
+
+// append an element to a variables collection
+int variables_append(Char_Array var_name, Tree symbol_tree, Variables* output){
+  int offset=(*output).length;
+
+  if((*output).length>=(*output).memory){
+    resize_variables(output,2*(*output).memory+1);
+  }
+
+  // copy and allocate
+  char_array_cpy(var_name,(*output).var_names+offset);
+  tree_cpy(symbol_tree,(*output).symbol_trees+offset);
+  // increment length
+  (*output).length++;
+  return(0);
+}
+// append an element to a variables collection without allocating memory
+int variables_append_noinit(Char_Array var_name, Tree symbol_tree, Variables* output){
+  int offset=(*output).length;
+
+  if((*output).length>=(*output).memory){
+    resize_variables(output,2*(*output).memory+1);
+  }
+
+  // copy without allocating
+  (*output).var_names[offset]=var_name;
+  (*output).symbol_trees[offset]=symbol_tree;
+  // increment length
+  (*output).length++;
+  return(0);
+}
+
+// concatenate two variables collections
+int variables_concat(Variables input, Variables* output){
+  int i;
+  for(i=0;i<input.length;i++){
+    variables_append(input.var_names[i], input.symbol_trees[i], output);
+  }
+  return(0);
+}
+
+
+// find a variable matching a var_name
+int variables_find_var(Char_Array name, Variables variables, Tree* output){
+  Char_Array varname;
+  int i;
+
+  // drop inital '$'
+  char_array_substring(name, 1, name.length-1, &varname);
+  for(i=0;i<variables.length;i++){
+    if(char_array_cmp(varname, variables.var_names[i])==1){
+      tree_cpy(variables.symbol_trees[i], output);
+      break;
+    }
+  }
+
+  // error if no variable was found
+  if(i==variables.length){
+    fprintf(stderr, "error: variable '$%s' not found\n",char_array_to_str_noinit(&varname));
+    exit(-1);
+  }
+  free_Char_Array(varname);
+
+  return(0);
+}
+
+
+// add a polynomials as a new named variable
+int add_polynomial_to_variables(char* name, Polynomial polynomial, Variables* variables){
+  // save polynomial to string (to convert it to a variable, it must first be a string)
+  Char_Array poly_str;
+  Char_Array out_name;
+  Tree out_tree;
+
+  init_Char_Array(&poly_str, STR_SIZE);
+  polynomial_sprint(polynomial, &poly_str);
+
+  // convert name to Char_Array
+  init_Char_Array(&out_name,STR_SIZE);
+  char_array_append_str(name, &out_name);
+
+  // trivial tree containing the polynomial
+  init_Tree(&out_tree,0,poly_str.length);
+  tree_set_label(poly_str, &out_tree);
+  free_Char_Array(poly_str);
+
+  // add variable
+  variables_append(out_name, out_tree, variables);
+  free_Tree(out_tree);
+  free_Char_Array(out_name);
+
+  return 0;
+}
diff --git a/src/fields.h b/src/fields.h
index b6f05c2..7b7a07a 100644
--- a/src/fields.h
+++ b/src/fields.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -21,6 +21,9 @@ limitations under the License.
 
 #include "types.h"
 
+
+// Fields_Table
+
 // init
 int init_Fields_Table(Fields_Table* fields);
 int free_Fields_Table(Fields_Table fields);
@@ -33,6 +36,8 @@ int is_fermion(int index, Fields_Table fields);
 int is_noncommuting(int index, Fields_Table fields);
 
 
+// Identities
+
 // init
 int init_Identities(Identities* identities,int size);
 int free_Identities(Identities identities);
@@ -57,25 +62,29 @@ int resolve_ids(Polynomial* polynomial, Fields_Table fields);
 int int_array_is_subarray_noncommuting(Int_Array input, Int_Array test_array, Fields_Table fields);
 
 
+// Virtual_fields
+
 // init
-int init_Symbols(Symbols* symbols,int size);
-int free_Symbols(Symbols symbols);
+int init_Virtual_fields(Virtual_fields* virtual_fields,int size);
+int free_Virtual_fields(Virtual_fields virtual_fields);
 
 // resize
-int resize_symbols(Symbols* symbols,int new_size);
+int resize_virtual_fields(Virtual_fields* virtual_fields,int new_size);
 
 // copy
-int symbols_cpy(Symbols input, Symbols* output);
-int symbols_cpy_noinit(Symbols input, Symbols* output);
+int virtual_fields_cpy(Virtual_fields input, Virtual_fields* output);
+int virtual_fields_cpy_noinit(Virtual_fields input, Virtual_fields* output);
 
-// append an element to a symbols
-int symbols_append(int index, Polynomial expr, Symbols* output);
-int symbols_append_noinit(int index, Polynomial expr, Symbols* output);
+// append an element to a virtual_fields
+int virtual_fields_append(int index, Polynomial expr, Virtual_fields* output);
+int virtual_fields_append_noinit(int index, Polynomial expr, Virtual_fields* output);
 
-// concatenate two symbolss
-int symbols_concat(Symbols input, Symbols* output);
+// concatenate two virtual_fields
+int virtual_fields_concat(Virtual_fields input, Virtual_fields* output);
 
 
+// Groups
+
 // init
 int init_Groups(Groups* groups,int size);
 int free_Groups(Groups groups);
@@ -98,5 +107,32 @@ int groups_concat(Groups input, Groups* output);
 int find_group(int index, Groups groups);
 
 
+// Variables
+
+// allocate memory
+int init_Variables(Variables* variables,int size);
+// free memory
+int free_Variables(Variables variables);
+
+// resize
+int resize_variables(Variables* variables,int new_size);
+
+// copy
+int variables_cpy(Variables input, Variables* output);
+int variables_cpy_noinit(Variables input, Variables* output);
+
+// append an element to a variables collection
+int variables_append(Char_Array var_name, Tree symbol_tree, Variables* output);
+int variables_append_noinit(Char_Array var_name, Tree symbol_tree, Variables* output);
+
+// concatenate two variables collections
+int variables_concat(Variables input, Variables* output);
+
+// find a variable matching a var_name
+int variables_find_var(Char_Array name, Variables variables, Tree* output);
+
+// add a polynomials as a new named variable
+int add_polynomial_to_variables(char* name, Polynomial polynomial, Variables* variables);
+
 #define FIELDS_H_DONE
 #endif
diff --git a/src/flow.c b/src/flow.c
index b294bb0..10c5ae0 100644
--- a/src/flow.c
+++ b/src/flow.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -25,14 +25,20 @@ limitations under the License.
 #include "array.h"
 #include "coefficient.h"
 #include "rcc.h"
+#include "grouped_polynomial.h"
 
 
 // compute flow numerically, no exponentials
-int numerical_flow(Grouped_Polynomial flow_equation, RCC init, Labels labels, int niter, int display_mode){
+int numerical_flow(Grouped_Polynomial flow_equation, RCC init, Grouped_Polynomial postprocess_flow_equation, Labels labels, int niter, int display_mode){
   // running coupling contants
   RCC rccs=init;
+  // for printing
+  RCC rcc_print;
   int i,j;
 
+  // init printing rcc
+  init_RCC(&rcc_print, rccs.length);
+
   if(display_mode==DISPLAY_NUMERICAL){
     // print labels
     printf("%5s  ","n");
@@ -42,9 +48,25 @@ int numerical_flow(Grouped_Polynomial flow_equation, RCC init, Labels labels, in
     printf("\n\n");
 
     // print initial values
+    RCC_cpy_noinit(rccs,&rcc_print);
+    if(postprocess_flow_equation.length>0){
+      // ignore constants
+      for(j=0;j<rcc_print.length;j++){
+	if(rcc_print.indices[j]<0){
+	  rcc_print.values[j]=1.;
+	}
+      }
+      evaleq(rcc_print, rcc_print, postprocess_flow_equation);
+    }
     printf("%5d  ",0);
-    for(j=0;j<rccs.length;j++){
-      printf("% 14.7Le  ",rccs.values[j]);
+    for(j=0;j<rcc_print.length;j++){
+      // use constants from rcc
+      if(rcc_print.indices[j]<0){
+	printf("% 14.7Le  ",rccs.values[j]);
+      }
+      else{
+	printf("% 14.7Le  ",rcc_print.values[j]);
+      }
     }
     printf("\n");
   }
@@ -52,12 +74,32 @@ int numerical_flow(Grouped_Polynomial flow_equation, RCC init, Labels labels, in
   for(i=0;i<niter;i++){
     // compute a single step
     step_flow(&rccs, flow_equation);
-    // convert ls to alphas
+
+    // print
+    if(postprocess_flow_equation.length>0){
+      RCC_cpy_noinit(rccs,&rcc_print);
+      // ignore constants
+      for(j=0;j<rcc_print.length;j++){
+	if(rcc_print.indices[j]<0){
+	  rcc_print.values[j]=1.;
+	}
+      }
+      evaleq(rcc_print, rcc_print, postprocess_flow_equation);
+    }
+    else{
+      RCC_cpy_noinit(rccs,&rcc_print);
+    }
     if(display_mode==DISPLAY_NUMERICAL){
       // print the result
       printf("%5d  ",i+1);
-      for(j=0;j<rccs.length;j++){
-	printf("% 14.7Le  ",rccs.values[j]);
+      for(j=0;j<rcc_print.length;j++){
+	// use constants from rcc
+	if(rcc_print.indices[j]<0){
+	  printf("% 14.7Le  ",rccs.values[j]);
+	}
+	else{
+	  printf("% 14.7Le  ",rcc_print.values[j]);
+	}
       }
       printf("\n");
     }
@@ -74,9 +116,24 @@ int numerical_flow(Grouped_Polynomial flow_equation, RCC init, Labels labels, in
   }
 
   if(display_mode==DISPLAY_FINAL){
-    RCC_print(rccs);
+    if(postprocess_flow_equation.length>0){
+      RCC_cpy_noinit(rccs,&rcc_print);
+      // ignore constants
+      for(j=0;j<rcc_print.length;j++){
+	if(rcc_print.indices[j]<0){
+	  rcc_print.values[j]=1.;
+	}
+      }
+      evaleq(rcc_print, rcc_print, postprocess_flow_equation);
+    }
+    else{
+      RCC_cpy_noinit(rccs,&rcc_print);
+    }
+    RCC_print(rcc_print);
   }
 
+  free_RCC(rcc_print);
+
   return(0);
 }
 
diff --git a/src/flow.h b/src/flow.h
index 9491323..ad43b72 100644
--- a/src/flow.h
+++ b/src/flow.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -24,7 +24,7 @@ Compute flow numerically
 #include "types.h"
 
 // compute flow
-int numerical_flow(Grouped_Polynomial flow_equation, RCC init, Labels labels, int niter, int display_mode);
+int numerical_flow(Grouped_Polynomial flow_equation, RCC init, Grouped_Polynomial postprocess_flow_equation, Labels labels, int niter, int display_mode);
 // single step
 int step_flow(RCC* rccs, Grouped_Polynomial flow_equation);
 
diff --git a/src/flow_mpfr.c b/src/flow_mpfr.c
index e03c130..1701b15 100644
--- a/src/flow_mpfr.c
+++ b/src/flow_mpfr.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -32,14 +32,21 @@ limitations under the License.
 #include "coefficient.h"
 #include "flow.h"
 #include "rcc_mpfr.h"
+#include "grouped_polynomial.h"
 
 
 
 // compute flow numerically
-int numerical_flow_mpfr(Grouped_Polynomial flow_equation, RCC_mpfr init, Labels labels, int niter, int display_mode){
+int numerical_flow_mpfr(Grouped_Polynomial flow_equation, RCC_mpfr init, Grouped_Polynomial postprocess_flow_equation, Labels labels, int niter, int display_mode){
   // running coupling contants
   RCC_mpfr rccs=init;
   int i,j;
+  // for printing
+  RCC_mpfr rcc_print;
+
+
+  // init printing rcc
+  init_RCC_mpfr(&rcc_print, rccs.length);
 
   if(display_mode==DISPLAY_NUMERICAL){
     // print labels
@@ -50,9 +57,25 @@ int numerical_flow_mpfr(Grouped_Polynomial flow_equation, RCC_mpfr init, Labels
     printf("\n\n");
 
     // print initial values
+    RCC_mpfr_cpy_noinit(rccs,&rcc_print);
+    if(postprocess_flow_equation.length>0){
+      // ignore constants
+      for(j=0;j<rcc_print.length;j++){
+	if(rcc_print.indices[j]<0){
+	  mpfr_set_ui(rcc_print.values[j], 1, MPFR_RNDN);
+	}
+      }
+      evaleq_mpfr(rcc_print, rccs, postprocess_flow_equation);
+    }
     printf("%5d  ",0);
-    for(j=0;j<rccs.length;j++){
-      mpfr_printf("% 14.7Re  ",rccs.values[j]);
+    for(j=0;j<rcc_print.length;j++){
+      // use constants from rcc
+      if(rcc_print.indices[j]<0){
+	mpfr_printf("% 14.7Re  ",rccs.values[j]);
+      }
+      else{
+	mpfr_printf("% 14.7Re  ",rcc_print.values[j]);
+      }
     }
     printf("\n");
   }
@@ -60,12 +83,29 @@ int numerical_flow_mpfr(Grouped_Polynomial flow_equation, RCC_mpfr init, Labels
   for(i=0;i<niter;i++){
     // compute a single step
     step_flow_mpfr(&rccs, flow_equation);
-    // convert ls to alphas
+
+    // print
+    RCC_mpfr_cpy_noinit(rccs,&rcc_print);
+    if(postprocess_flow_equation.length>0){
+      // ignore constants
+      for(j=0;j<rcc_print.length;j++){
+	if(rcc_print.indices[j]<0){
+	  mpfr_set_ui(rcc_print.values[j], 1, MPFR_RNDN);
+	}
+      }
+      evaleq_mpfr(rcc_print, rccs, postprocess_flow_equation);
+    }
     if(display_mode==DISPLAY_NUMERICAL){
       // print the result
       printf("%5d  ",i+1);
-      for(j=0;j<rccs.length;j++){
-	mpfr_printf("% 14.7Re  ",rccs.values[j]);
+      for(j=0;j<rcc_print.length;j++){
+	// use constants from rcc
+	if(rcc_print.indices[j]<0){
+	  mpfr_printf("% 14.7Re  ",rccs.values[j]);
+	}
+	else{
+	  mpfr_printf("% 14.7Re  ",rcc_print.values[j]);
+	}
       }
       printf("\n");
     }
@@ -82,9 +122,16 @@ int numerical_flow_mpfr(Grouped_Polynomial flow_equation, RCC_mpfr init, Labels
   }
 
   if(display_mode==DISPLAY_FINAL){
-    RCC_mpfr_print(rccs);
+    if(postprocess_flow_equation.length>0){
+      evaleq_mpfr(rcc_print, rccs, postprocess_flow_equation);
+    }
+    else{
+      rcc_print=rccs;
+    }
+    RCC_mpfr_print(rcc_print);
   }
 
+  free_RCC_mpfr(rcc_print);
   return(0);
 }
 
diff --git a/src/flow_mpfr.h b/src/flow_mpfr.h
index e06cbd8..203770d 100644
--- a/src/flow_mpfr.h
+++ b/src/flow_mpfr.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -25,7 +25,7 @@ Compute flow numerically
 #include "types.h"
 
 // compute flow
-int numerical_flow_mpfr(Grouped_Polynomial flow_equation, RCC_mpfr init, Labels labels, int niter, int display_mode);
+int numerical_flow_mpfr(Grouped_Polynomial flow_equation, RCC_mpfr init, Grouped_Polynomial postprocess_flow_equation, Labels labels, int niter, int display_mode);
 // single step
 int step_flow_mpfr(RCC_mpfr* rccs, Grouped_Polynomial flow_equation);
 
diff --git a/src/grouped_polynomial.c b/src/grouped_polynomial.c
index 0d4d75d..9e245a8 100644
--- a/src/grouped_polynomial.c
+++ b/src/grouped_polynomial.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -205,9 +205,9 @@ int group_polynomial(Polynomial polynomial, Grouped_Polynomial* grouped_polynomi
 
     if(index==-2){
       fprintf(stderr,"error: monomial (");
-      for(j=0;j<polynomial.monomials[i].length;j++){
-	fprintf(stderr,"%d", polynomial.monomials[i].values[j]);
-	if(j<polynomial.monomials[i].length-1){
+      for(j=0;j<remainder.monomials[i].length;j++){
+	fprintf(stderr,"%d", remainder.monomials[i].values[j]);
+	if(j<remainder.monomials[i].length-1){
 	  fprintf(stderr,",");
 	}
       }
@@ -436,7 +436,7 @@ int simplify_grouped_polynomial(Grouped_Polynomial* polynomial){
 }
 
 
-// derive a flow equation with respect to an unknown variable
+// differentiate a flow equation with respect to an unknown variable
 // equivalent to DB.dl where dl are symbols for the derivatives of the indices in the flow equation with respect to the unknown variable
 // indices specifies the list of indices that depend on the variable
 int flow_equation_derivx(Grouped_Polynomial flow_equation, Int_Array indices, Grouped_Polynomial* dflow){
@@ -487,7 +487,7 @@ int flow_equation_derivx(Grouped_Polynomial flow_equation, Int_Array indices, Gr
 
 
 /*
-// derive a flow equation with respect to an index
+// differentiate a flow equation with respect to an index
 int flow_equation_deriv(Grouped_Polynomial flow_equation, int index, Grouped_Polynomial* output){
   int i,k;
   // temp list of indices
@@ -603,7 +603,7 @@ int grouped_polynomial_print(Grouped_Polynomial grouped_polynomial, char lhs_pre
     init_Char_Array(&buffer, STR_SIZE);
     coefficient_sprint(grouped_polynomial.coefs[i],&buffer,9,rhs_pre);
     if(buffer.length>0){
-      printf("%s",buffer.str);
+      printf("%s",char_array_to_str_noinit(&buffer));
     }
     free_Char_Array(buffer);
 
@@ -732,29 +732,33 @@ int char_array_to_Grouped_Polynomial(Char_Array str, Grouped_Polynomial* output)
 }
 
 
-// eValuate an equation on a vector
-int evaleq(RCC* rccs, Grouped_Polynomial poly){
+// evaluate an equation on a vector
+int evaleq(RCC out, RCC in, Grouped_Polynomial poly){
   int i;
-  long double* res=calloc((*rccs).length,sizeof(long double));
+  long double* res=calloc(out.length,sizeof(long double));
 
-  if((*rccs).length!=poly.length){
-    fprintf(stderr, "error: trying to evaluate an flow equation with %d components on an rcc with %d\n",poly.length,(*rccs).length);
+  if(in.length!=poly.length){
+    fprintf(stderr, "error: trying to evaluate a flow equation with %d components on an rcc with %d\n",poly.length,in.length);
+    exit(-1);
+  }
+  if(out.length!=poly.length){
+    fprintf(stderr, "error: trying to write the output of a flow equation with %d components on an rcc with %d\n",poly.length,out.length);
     exit(-1);
   }
 
-  // initialize vectors to 0
-  for(i=0;i<(*rccs).length;i++){
+  // initialize vectors to 0 in an auxiliary vector (to allow for out=in without interference)
+  for(i=0;i<in.length;i++){
     res[i]=0.;
   }
 
   // for each equation
   for(i=0;i<poly.length;i++){
-    evalcoef(*rccs, poly.coefs[i], res+i);
+    evalcoef(in, poly.coefs[i], res+i);
   }
 
   // copy res to rccs
-  for(i=0;i<(*rccs).length;i++){
-    (*rccs).values[i]=res[i];
+  for(i=0;i<out.length;i++){
+    out.values[i]=res[i];
   }
 
   // free memory
@@ -763,25 +767,29 @@ int evaleq(RCC* rccs, Grouped_Polynomial poly){
 
 }
 // evaluate an equation on a vector (using mpfr floats)
-int evaleq_mpfr(RCC_mpfr* rccs, Grouped_Polynomial poly){
+int evaleq_mpfr(RCC_mpfr out, RCC_mpfr in, Grouped_Polynomial poly){
   int i;
   mpfr_t* res;
 
-  if((*rccs).length!=poly.length){
-    fprintf(stderr, "error: trying to evaluate an flow equation with %d components on an rcc with %d\n",poly.length,(*rccs).length);
+  if(in.length!=poly.length){
+    fprintf(stderr, "error: trying to evaluate a flow equation with %d components on an rcc with %d\n",poly.length,in.length);
+    exit(-1);
+  }
+  if(out.length!=poly.length){
+    fprintf(stderr, "error: trying to write the output of a flow equation with %d components on an rcc with %d\n",poly.length,out.length);
     exit(-1);
   }
 
-  res=calloc((*rccs).length,sizeof(mpfr_t));
+  res=calloc(out.length,sizeof(mpfr_t));
 
   // for each equation
   for(i=0;i<poly.length;i++){
-    evalcoef_mpfr(*rccs, poly.coefs[i], res[i]);
+    evalcoef_mpfr(in, poly.coefs[i], res[i]);
   }
 
   // copy res to rccs
-  for(i=0;i<(*rccs).length;i++){
-    mpfr_set((*rccs).values[i], res[i], MPFR_RNDN);
+  for(i=0;i<out.length;i++){
+    mpfr_set(out.values[i], res[i], MPFR_RNDN);
     mpfr_clear(res[i]);
   }
 
@@ -791,3 +799,85 @@ int evaleq_mpfr(RCC_mpfr* rccs, Grouped_Polynomial poly){
 }
 
 
+// compose two flow equations (replace the rcc's of flow1 by the right hand side of flow2)
+int compose_flow_equations(Grouped_Polynomial flow1, Grouped_Polynomial flow2, Grouped_Polynomial* out){
+  if(flow1.length!=flow2.length){
+    fprintf(stderr, "error: trying to compose two flow equations of different size\n");
+    exit(-1);
+  }
+  int i,j,k;
+  Coefficient constant;
+
+  // init
+  init_Grouped_Polynomial(out, flow1.length);
+  (*out).length=flow1.length;
+
+  // init constant (so we can tell when the constant was not found)
+  constant.length=0;
+
+  // loop over rcc's
+  for(i=0;i<flow1.length;i++){
+    // set indices
+    (*out).indices[i]=flow1.indices[i];
+
+    // passthrough constant terms
+    if((*out).indices[i]<0){
+      int index=intlist_find_err(flow2.indices,flow2.length,(*out).indices[i]);
+      coefficient_cpy(flow2.coefs[index], (*out).coefs+i);
+      constant=flow2.coefs[index];
+      continue;
+    }
+
+    // init
+    init_Coefficient((*out).coefs+i, COEF_SIZE);
+
+    // loop over terms
+    for(j=0;j<flow1.coefs[i].length;j++){
+      Coefficient tmp_coef;
+
+      // init
+      init_Coefficient(&tmp_coef, COEF_SIZE);
+      // init factor
+      Int_Array tmp_factor;
+      init_Int_Array(&tmp_factor, MONOMIAL_SIZE);
+      // init denom
+      coef_denom denom;
+      // index should be that appearing in flow2
+      if(flow2.coefs[i].length<1){
+	fprintf(stderr,"error: composing two flow equations: the %d-th term in the flow equation is empty\n",flow1.indices[i]);
+	exit(-1);
+      }
+      denom.index=flow2.coefs[i].denoms[0].index;
+      denom.power=0;
+      // init num
+      Number tmp_num;
+      number_cpy(flow1.coefs[i].nums[j], &tmp_num);
+
+      // init coefficient with numerical prefactor
+      coefficient_append_noinit(tmp_factor, tmp_num, denom, &tmp_coef);
+
+      // loop over factors
+      for(k=0;k<flow1.coefs[i].factors[j].length;k++){
+	// multiply factors together
+	coefficient_prod_chain(flow2.coefs[intlist_find_err(flow2.indices,flow2.length,flow1.coefs[i].factors[j].values[k])], &tmp_coef);
+      }
+
+      // add to out
+      coefficient_concat_noinit(tmp_coef, (*out).coefs+i);
+    }
+
+  }
+
+  // simplify fractions
+  if(constant.length!=0){
+    for(i=0;i<(*out).length;i++){
+      if((*out).indices[i]>=0){
+	// reduce them to a common denominator (not much is gained from trying to simplify them)
+	coefficient_common_denominator(constant, (*out).coefs+i);
+	//coefficient_simplify_rational(constant, (*out).coefs+i);
+      }
+    }
+  }
+
+  return(0);
+}
diff --git a/src/grouped_polynomial.h b/src/grouped_polynomial.h
index e369721..7d95a2e 100644
--- a/src/grouped_polynomial.h
+++ b/src/grouped_polynomial.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -59,7 +59,7 @@ int find_id(Int_Array monomial, Id_Table idtable, int start);
 // simplify grouped polynomial
 int simplify_grouped_polynomial(Grouped_Polynomial* polynomial);
 
-// derive a flow equation with respect to an unknown variable
+// differentiate a flow equation with respect to an unknown variable
 int flow_equation_derivx(Grouped_Polynomial flow_equation, Int_Array indices, Grouped_Polynomial* dflow);
 
 // print a grouped polynomial
@@ -69,8 +69,10 @@ int grouped_polynomial_print(Grouped_Polynomial grouped_polynomial, char lhs_pre
 int char_array_to_Grouped_Polynomial(Char_Array str, Grouped_Polynomial* output);
 
 // evaluate an equation on an RCC
-int evaleq(RCC* rccs, Grouped_Polynomial poly);
+int evaleq(RCC out, RCC in, Grouped_Polynomial poly);
 // evaluate an equation on a vector (using mpfr floats)
-int evaleq_mpfr(RCC_mpfr* rccs, Grouped_Polynomial poly);
+int evaleq_mpfr(RCC_mpfr out, RCC_mpfr in, Grouped_Polynomial poly);
 
+// compose two flow equations (replace the rcc's of flow1 by the right hand side of flow2)
+int compose_flow_equations(Grouped_Polynomial flow1, Grouped_Polynomial flow2, Grouped_Polynomial* out);
 #endif
diff --git a/src/idtable.c b/src/idtable.c
index fe409c2..fc72f56 100644
--- a/src/idtable.c
+++ b/src/idtable.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/idtable.h b/src/idtable.h
index 38ab249..aa99dca 100644
--- a/src/idtable.h
+++ b/src/idtable.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/istring.c b/src/istring.c
index 663c06a..d78f67b 100644
--- a/src/istring.c
+++ b/src/istring.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/istring.h b/src/istring.h
index 5b47b9b..5027981 100644
--- a/src/istring.h
+++ b/src/istring.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/kondo.c b/src/kondo.c
index 79b7c56..93318d5 100644
--- a/src/kondo.c
+++ b/src/kondo.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -35,7 +35,7 @@ limitations under the License.
 #define KONDO_SPIN 2
 
 // offsets for indices of A, B, h and t
-// order matters for symbols table
+// order matters for virtual_fields table
 #define KONDO_A_OFFSET 1
 #define KONDO_B_OFFSET 2
 #define KONDO_H_OFFSET 3
@@ -72,6 +72,7 @@ limitations under the License.
 int kondo_generate_conf(Str_Array* str_args, int box_count){
   Str_Array new_args;
   Fields_Table fields;
+  Variables variables;
   Char_Array tmp_str;
   int arg_index;
   int i;
@@ -83,16 +84,19 @@ int kondo_generate_conf(Str_Array* str_args, int box_count){
   kondo_fields_table(box_count, &tmp_str, &fields);
   str_array_append_noinit(tmp_str, &new_args);
 
-  // symbols
-  kondo_symbols(&tmp_str, box_count, &fields);
-  arg_index=find_str_arg("symbols", *str_args);
+  // dummy variables
+  init_Variables(&variables,1);
+
+  // virtual fields
+  kondo_virtual_fields(&tmp_str, box_count, &fields);
+  arg_index=find_str_arg("virtual_fields", *str_args);
   if(arg_index>=0){
     if(tmp_str.length>0){
       char_array_snprintf(&tmp_str,",\n");
     }
     char_array_concat((*str_args).strs[arg_index], &tmp_str);
   }
-  parse_input_symbols(tmp_str, &fields);
+  parse_input_virtual_fields(tmp_str, &fields, variables);
   str_array_append_noinit(tmp_str, &new_args);
 
   // identities
@@ -104,7 +108,7 @@ int kondo_generate_conf(Str_Array* str_args, int box_count){
     }
     char_array_concat((*str_args).strs[arg_index], &tmp_str);
   }
-  parse_input_identities(tmp_str, &fields);
+  parse_input_identities(tmp_str, &fields, variables);
   str_array_append_noinit(tmp_str, &new_args);
 
   // groups
@@ -136,7 +140,7 @@ int kondo_generate_conf(Str_Array* str_args, int box_count){
   // copy remaining entries
   for(i=0;i<(*str_args).length;i++){
     get_str_arg_title((*str_args).strs[i], &title);
-    if(str_cmp(title.str, "symbols")==0 &&\
+    if(str_cmp(title.str, "virtual_fields")==0 &&\
        str_cmp(title.str, "identities")==0 &&\
        str_cmp(title.str, "propagator")==0 &&\
        str_cmp(title.str, "input_polynomial")==0 &&\
@@ -149,6 +153,7 @@ int kondo_generate_conf(Str_Array* str_args, int box_count){
   }
 
   free_Fields_Table(fields);
+  free_Variables(variables);
   free_Str_Array(*str_args);
   *str_args=new_args;
 
@@ -250,15 +255,15 @@ int kondo_fields_table(int box_count, Char_Array* str_fields, Fields_Table* fiel
 }
 
 
-// generate Kondo symbols
-int kondo_symbols(Char_Array* str_symbols, int box_count, Fields_Table* fields){
+// generate Kondo virtual_fields
+int kondo_virtual_fields(Char_Array* str_virtual_fields, int box_count, Fields_Table* fields){
   int i,j,k,l;
   Char_Array tmp_str;
   Polynomial poly;
   char letters[3]={'A','B','h'};
 
-  init_Char_Array(str_symbols, STR_SIZE);
-  char_array_snprintf(str_symbols, "#!symbols\n");
+  init_Char_Array(str_virtual_fields, STR_SIZE);
+  char_array_snprintf(str_virtual_fields, "#!virtual_fields\n");
 
   // loop over box index
   for(i=1;i<=box_count;i++){
@@ -267,7 +272,7 @@ int kondo_symbols(Char_Array* str_symbols, int box_count, Fields_Table* fields){
       // loop over space dimension
       for(k=0;k<KONDO_DIM;k++){
 	// write index
-	char_array_snprintf(str_symbols, "%d=", 100*(10*(KONDO_A_OFFSET+j)+k)+i);
+	char_array_snprintf(str_virtual_fields, "%d=", 100*(10*(KONDO_A_OFFSET+j)+k)+i);
 	// write the name of the scalar product
 	init_Char_Array(&tmp_str, 6);
 	char_array_snprintf(&tmp_str, "%c%d%d", letters[j], k, i);
@@ -275,10 +280,10 @@ int kondo_symbols(Char_Array* str_symbols, int box_count, Fields_Table* fields){
 	kondo_resolve_ABht(tmp_str.str, &poly, *fields);
 	free_Char_Array(tmp_str);
 	// write to output
-	polynomial_sprint(poly, str_symbols);
+	polynomial_sprint(poly, str_virtual_fields);
 	free_Polynomial(poly);
 	// add ,
-	char_array_snprintf(str_symbols,",\n");
+	char_array_snprintf(str_virtual_fields,",\n");
       }
     }
   }
@@ -290,46 +295,46 @@ int kondo_symbols(Char_Array* str_symbols, int box_count, Fields_Table* fields){
     for(j=0;j<3;j++){
       for(k=0;k<3;k++){
 	// write index
-	char_array_snprintf(str_symbols, "%d=", 1000*(10*(KONDO_A_OFFSET+j)+KONDO_A_OFFSET+k)+i);
+	char_array_snprintf(str_virtual_fields, "%d=", 1000*(10*(KONDO_A_OFFSET+j)+KONDO_A_OFFSET+k)+i);
 	for(l=0;l<KONDO_DIM;l++){
-	  char_array_snprintf(str_symbols, "(1)");
+	  char_array_snprintf(str_virtual_fields, "(1)");
 	  if(j<2){
-	    char_array_snprintf(str_symbols,"[f%d]", 100*(10*(KONDO_A_OFFSET+j)+l)+i);
+	    char_array_snprintf(str_virtual_fields,"[f%d]", 100*(10*(KONDO_A_OFFSET+j)+l)+i);
 	  }
 	  else{
-	    char_array_snprintf(str_symbols,"[f%d]", 10*(10*(KONDO_A_OFFSET+j)+l));
+	    char_array_snprintf(str_virtual_fields,"[f%d]", 10*(10*(KONDO_A_OFFSET+j)+l));
 	  }
 	  if(k<2){
-	    char_array_snprintf(str_symbols,"[f%d]", 100*(10*(KONDO_A_OFFSET+k)+l)+i);
+	    char_array_snprintf(str_virtual_fields,"[f%d]", 100*(10*(KONDO_A_OFFSET+k)+l)+i);
 	  }
 	  else{
-	    char_array_snprintf(str_symbols,"[f%d]", 10*(10*(KONDO_A_OFFSET+k)+l));
+	    char_array_snprintf(str_virtual_fields,"[f%d]", 10*(10*(KONDO_A_OFFSET+k)+l));
 	  }
 	    
 	  if(l<KONDO_DIM-1){
-	    char_array_append('+',str_symbols);
+	    char_array_append('+',str_virtual_fields);
 	  }
 	}
 	// add ,
-	char_array_snprintf(str_symbols,",\n");
+	char_array_snprintf(str_virtual_fields,",\n");
       }
     }
   }
 
   // vector products
   for(i=1;i<=box_count;i++){
-    char_array_snprintf(str_symbols, "%d=", 100*(100*(KONDO_A_OFFSET)+10*KONDO_B_OFFSET+KONDO_H_OFFSET)+i);
+    char_array_snprintf(str_virtual_fields, "%d=", 100*(100*(KONDO_A_OFFSET)+10*KONDO_B_OFFSET+KONDO_H_OFFSET)+i);
     for(l=0;l<KONDO_DIM;l++){
       // remember (-1 %3 = -1)
-      char_array_snprintf(str_symbols, "(1)[f%d][f%d][f%d]+(-1)[f%d][f%d][f%d]", 100*(10*KONDO_A_OFFSET+((l+1)%KONDO_DIM))+i, 100*(10*KONDO_B_OFFSET+((l+2)%KONDO_DIM))+i, 10*(10*KONDO_H_OFFSET+l), 100*(10*KONDO_A_OFFSET+((l+2)%KONDO_DIM))+i, 100*(10*KONDO_B_OFFSET+((l+1)%KONDO_DIM))+i, 10*(10*KONDO_H_OFFSET+l));
+      char_array_snprintf(str_virtual_fields, "(1)[f%d][f%d][f%d]+(-1)[f%d][f%d][f%d]", 100*(10*KONDO_A_OFFSET+((l+1)%KONDO_DIM))+i, 100*(10*KONDO_B_OFFSET+((l+2)%KONDO_DIM))+i, 10*(10*KONDO_H_OFFSET+l), 100*(10*KONDO_A_OFFSET+((l+2)%KONDO_DIM))+i, 100*(10*KONDO_B_OFFSET+((l+1)%KONDO_DIM))+i, 10*(10*KONDO_H_OFFSET+l));
       if(l<KONDO_DIM-1){
-	char_array_append('+',str_symbols);
+	char_array_append('+',str_virtual_fields);
       }
     }
 
     // add ,
     if(i<box_count){
-      char_array_snprintf(str_symbols,",\n");
+      char_array_snprintf(str_virtual_fields,",\n");
     }
   }
 
@@ -778,13 +783,13 @@ int parse_kondo_polynomial_str(char* str_polynomial, Polynomial* output, Fields_
 	  // if polynomial exists, add to each monomial
 	  if(tmp_poly.length>0){
 	    for(i=0;i<tmp_poly.length;i++){
-	      int_array_append(get_symbol_index(buffer), tmp_poly.monomials+i);
+	      int_array_append(get_virtual_field_index(buffer), tmp_poly.monomials+i);
 	    }
 	  }
 	  // if not, create a new term in the polynomial
 	  else{
 	    init_Int_Array(&tmp_monomial, MONOMIAL_SIZE);
-	    int_array_append(get_symbol_index(buffer), &tmp_monomial);
+	    int_array_append(get_virtual_field_index(buffer), &tmp_monomial);
 	    init_Int_Array(&dummy_factor, 1);
 	    polynomial_append_noinit(tmp_monomial, dummy_factor, number_one(), &tmp_poly);
 	  }
@@ -1387,8 +1392,8 @@ int get_offsets_index(char* str, int* offset1, int* offset2, int* index){
   return(0);
 }
 
-// get the index of the symbol corresponding to a given string
-int get_symbol_index(char* str){
+// get the index of the virtual_field corresponding to a given string
+int get_virtual_field_index(char* str){
   char* ptr;
   int offset=-1;
   int index=0;
@@ -1439,7 +1444,7 @@ int get_symbol_index(char* str){
   if(offset==-1){
     return(-1);
   }
-  // no symbol for h or t
+  // no virtual field for h or t
   if(offset==KONDO_H_OFFSET || offset==KONDO_T_OFFSET){
     return(10*(10*offset+dim));
   }
diff --git a/src/kondo.h b/src/kondo.h
index c534145..18488ae 100644
--- a/src/kondo.h
+++ b/src/kondo.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -27,10 +27,10 @@ int kondo_generate_conf(Str_Array* str_args, int box_count);
 // generate the Kondo fields table
 int kondo_fields_table(int box_count, Char_Array* str_fields, Fields_Table* fields);
 
-// generate Kondo symbols
-int kondo_symbols(Char_Array* str_symbols, int box_count, Fields_Table* fields);
-// generate Kondo symbols (older method: one symbol for each scalar product)
-int kondo_symbols_scalarprod(Char_Array* str_symbols, int box_count, Fields_Table* fields);
+// generate Kondo virtual_fields
+int kondo_virtual_fields(Char_Array* str_virtual_fields, int box_count, Fields_Table* fields);
+// generate Kondo virtual_fields (older method: one virtual_field for each scalar product)
+int kondo_virtual_fields_scalarprod(Char_Array* str_virtual_fields, int box_count, Fields_Table* fields);
 
 // generate Kondo groups (groups of independent variables)
 int kondo_groups(Char_Array* str_groups, int box_count);
@@ -71,6 +71,6 @@ int kondo_resolve_scalar_prod_symbols(char* str, Polynomial* output);
 int get_offset_index(char* str, int* offset, int* index);
 // get the offsets and index of a scalar product
 int get_offsets_index(char* str, int* offset1, int* offset2, int* index);
-// get the index of the symbol corresponding to a given string
-int get_symbol_index(char* str);
+// get the index of the virtual_field corresponding to a given string
+int get_virtual_field_index(char* str);
 #endif
diff --git a/src/kondo_preprocess.c b/src/kondo_preprocess.c
index 3371014..c472e9d 100644
--- a/src/kondo_preprocess.c
+++ b/src/kondo_preprocess.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -104,6 +104,11 @@ for(i=1;i<argc;i++){
     // number of dimensions
     else if (flag==CP_FLAG_DIMENSION){
       sscanf(argv[i],"%d",&((*opts).dimension));
+      // check value of the dimension
+      if((*opts).dimension<=0 || (*opts).dimension>=4){
+	fprintf(stderr,"error: kondo_preprocess only supports dimensions 1, 2 and 3 (got %d)\n",(*opts).dimension);
+	exit(-1);
+      }
       flag=0;
     }
     // read file name from command-line
diff --git a/src/mean.c b/src/mean.c
index 39ece56..0c5707e 100644
--- a/src/mean.c
+++ b/src/mean.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -30,6 +30,7 @@ As of version 1.0, the mean of a monomial is computed directly
 #include "array.h"
 #include "fields.h"
 #include "number.h"
+#include "determinant.h"
 
 // mean of a monomial
 int mean(Int_Array monomial, Polynomial* out, Fields_Table fields, Polynomial_Matrix propagator){
@@ -61,10 +62,79 @@ int mean(Int_Array monomial, Polynomial* out, Fields_Table fields, Polynomial_Ma
 
 // compute the mean of a monomial of internal fields (with split + and -)
 int mean_internal(Int_Array internal_plus, Int_Array internal_minus, Polynomial* out, Polynomial_Matrix propagator, Fields_Table fields){
+  int ret;
+  Number num;
+
   if(internal_plus.length!=internal_minus.length){
     fprintf(stderr,"error: monomial contains unmatched +/- fields\n");
     exit(-1);
   }
+
+  ret=mean_determinant(internal_plus, internal_minus, &num, propagator, fields);
+  // cannot compute the mean as a determinant, use permutations
+  // can be because some fields are not Fermions
+  // can be because the propagator has non-numeric values (inverting polynomials is not implemented, and would be required for the computation of the determinant)
+  if(ret==-1){
+    mean_permutations(internal_plus, internal_minus, out, propagator, fields);
+  }
+  else{
+    polynomial_multiply_scalar(*out, num);
+    free_Number(num);
+  }
+
+  return(0);
+}
+
+// compute the mean of a monomial by computing a determinant
+// can only be used if all of the propagators are numbers
+int mean_determinant(Int_Array internal_plus, Int_Array internal_minus, Number* out, Polynomial_Matrix propagator, Fields_Table fields){
+  Number_Matrix M;
+  int n=internal_minus.length;
+  int i,j;
+  int a,b;
+  int sign;
+
+  init_Number_Matrix(&M,n);
+  
+  // extra sign: the monomial is sorted in such a way that minus fields are on the left of plus fields, but the determinant formula requires the fields to be alternated +-
+  if((n+1)/2%2==1){
+    sign=-1;
+  }
+  else{
+    sign=1;
+  }
+
+  // construct matrix
+  for(i=0;i<n;i++){
+    a=intlist_find_err(propagator.indices, propagator.length, internal_plus.values[i]);
+    for(j=0;j<n;j++){
+      b=intlist_find_err(propagator.indices, propagator.length, -internal_minus.values[j]);
+      // ignore 0
+      if(propagator.matrix[a][b].length!=0){
+	// check whether the fields are Fermions, and whether the entry is a number
+	if(is_fermion(internal_plus.values[i], fields)==0 || is_fermion(internal_minus.values[j], fields)==0 || polynomial_is_number(propagator.matrix[a][b])==0){
+	  free_Number_Matrix(M);
+	  return(-1);
+	}
+
+	number_add_chain(propagator.matrix[a][b].nums[0], M.matrix[i]+j);
+      }
+    }
+  }
+
+  // compute determinant
+  determinant_inplace(M, out);
+
+  number_Qprod_chain(quot(sign,1), out);
+
+  free_Number_Matrix(M);
+
+  return(0);
+}
+
+
+// compute the mean of a monomial by summing over permutations
+int mean_permutations(Int_Array internal_plus, Int_Array internal_minus, Polynomial* out, Polynomial_Matrix propagator, Fields_Table fields){
   int n=internal_minus.length;
   // pairing as an array of positions
   int* pairing=calloc(n,sizeof(int));
@@ -118,7 +188,7 @@ int mean_internal(Int_Array internal_plus, Int_Array internal_minus, Polynomial*
     pairing_sign=permutation_signature(pairing,n);
   }
 
-  // only simplify in mean_symbols
+  // only simplify in mean_virtual_fields
   polynomial_prod_chain_nosimplify(num_summed,out,fields);
   free_Polynomial(num_summed);
   free(pairing);
@@ -346,10 +416,10 @@ int get_internals(Int_Array monomial, Int_Array* internal_plus, Int_Array* inter
 }
 
 
-// compute the mean of a monomial containing symbolic expressions
+// compute the mean of a monomial containing virtual fields
 // keep track of which means were already computed
-int mean_symbols(Int_Array monomial, Polynomial* output, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, Identities* computed){
-  Int_Array symbol_list;
+int mean_virtual_fields(Int_Array monomial, Polynomial* output, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, Identities* computed){
+  Int_Array virtual_field_list;
   int i;
   int power;
   int* current_term;
@@ -375,43 +445,43 @@ int mean_symbols(Int_Array monomial, Polynomial* output, Fields_Table fields, Po
     }
   }
 
-  init_Int_Array(&symbol_list, monomial.length);
+  init_Int_Array(&virtual_field_list, monomial.length);
   init_Int_Array(&base_monomial, monomial.length);
 
-  // generate symbols list
+  // generate virtual_fields list
   for(i=0;i<monomial.length;i++){
-    if(field_type(monomial.values[i], fields)==FIELD_SYMBOL){
-      int_array_append(intlist_find_err(fields.symbols.indices, fields.symbols.length, monomial.values[i]), &symbol_list);
+    if(field_type(monomial.values[i], fields)==FIELD_VIRTUAL){
+      int_array_append(intlist_find_err(fields.virtual_fields.indices, fields.virtual_fields.length, monomial.values[i]), &virtual_field_list);
     }
     else{
       int_array_append(monomial.values[i], &base_monomial);
     }
   }
-  power=symbol_list.length;
+  power=virtual_field_list.length;
 
   // trivial case
   if(power==0){
     mean(monomial, &mean_num, fields, propagator);
     polynomial_concat_noinit(mean_num, output);
     
-    free_Int_Array(symbol_list);
+    free_Int_Array(virtual_field_list);
     free_Int_Array(base_monomial);
     return(0);
   }
   else{
     // initialize current term to a position that has no repetitions
     current_term=calloc(power,sizeof(int));
-    exists_next=init_prod(current_term, symbol_list, fields, power, base_monomial)+1;
+    exists_next=init_prod(current_term, virtual_field_list, fields, power, base_monomial)+1;
   }
 
-  // loop over terms; the loop stops when all the pointers are at the end of the first symbol
+  // loop over terms; the loop stops when all the pointers are at the end of the first virtual field
   while(exists_next==1){
     // construct monomial
     int_array_cpy(base_monomial, &tmp_monomial);
     tmp_num=number_one();
     for(i=0;i<power;i++){
-      int_array_concat(fields.symbols.expr[symbol_list.values[i]].monomials[current_term[i]], &tmp_monomial);
-      number_prod_chain(fields.symbols.expr[symbol_list.values[i]].nums[current_term[i]], &tmp_num);
+      int_array_concat(fields.virtual_fields.expr[virtual_field_list.values[i]].monomials[current_term[i]], &tmp_monomial);
+      number_prod_chain(fields.virtual_fields.expr[virtual_field_list.values[i]].nums[current_term[i]], &tmp_num);
     }
     // check whether the monomial vanishes
     if(check_monomial_match(tmp_monomial, fields)==1){
@@ -432,7 +502,7 @@ int mean_symbols(Int_Array monomial, Polynomial* output, Fields_Table fields, Po
     free_Int_Array(tmp_monomial);
 
     // next term
-    exists_next=next_prod(current_term, symbol_list, fields, power, base_monomial)+1;
+    exists_next=next_prod(current_term, virtual_field_list, fields, power, base_monomial)+1;
 
     
     // simplfiy every 25 steps (improves both memory usage and performance)
@@ -451,13 +521,13 @@ int mean_symbols(Int_Array monomial, Polynomial* output, Fields_Table fields, Po
 
   // free memory
   free(current_term);
-  free_Int_Array(symbol_list);
+  free_Int_Array(virtual_field_list);
   free_Int_Array(base_monomial);
   return(0);
 }
 
 // first term in product with no repetitions
-int init_prod(int* current_term, Int_Array symbol_list, Fields_Table fields, int power, Int_Array base_monomial){
+int init_prod(int* current_term, Int_Array virtual_field_list, Fields_Table fields, int power, Int_Array base_monomial){
   // index we want to increment
   int move=0;
   // tmp monomial
@@ -474,7 +544,7 @@ int init_prod(int* current_term, Int_Array symbol_list, Fields_Table fields, int
   // loop until move is out of range
   while(move>=0 && move<power){
     // move
-    current_term[move]=next_term_norepeat(current_term[move], fields.symbols.expr[symbol_list.values[move]], &monomial, fields);
+    current_term[move]=next_term_norepeat(current_term[move], fields.virtual_fields.expr[virtual_field_list.values[move]], &monomial, fields);
     // if the next term does not exist, then move previous index
     if(current_term[move]==-1){
       move--;
@@ -495,7 +565,7 @@ int init_prod(int* current_term, Int_Array symbol_list, Fields_Table fields, int
 }
 
 // next term in product with no repetitions
-int next_prod(int* current_term, Int_Array symbol_list, Fields_Table fields, int power, Int_Array base_monomial){
+int next_prod(int* current_term, Int_Array virtual_field_list, Fields_Table fields, int power, Int_Array base_monomial){
   // index we want to increment
   int move=power-1;
   // tmp monomial
@@ -506,13 +576,13 @@ int next_prod(int* current_term, Int_Array symbol_list, Fields_Table fields, int
   init_Int_Array(&monomial, base_monomial.length+5*power);
   int_array_cpy_noinit(base_monomial, &monomial);
   for(i=0;i<=move;i++){
-    int_array_concat(fields.symbols.expr[symbol_list.values[i]].monomials[current_term[i]],&monomial);
+    int_array_concat(fields.virtual_fields.expr[virtual_field_list.values[i]].monomials[current_term[i]],&monomial);
   }
 
   // loop until move is out of range
   while(move>=0 && move<power){
     // move
-    current_term[move]=next_term_norepeat(current_term[move], fields.symbols.expr[symbol_list.values[move]], &monomial, fields);
+    current_term[move]=next_term_norepeat(current_term[move], fields.virtual_fields.expr[virtual_field_list.values[move]], &monomial, fields);
     // if the next term does not exist, then move previous index
     if(current_term[move]==-1){
       move--;
@@ -623,13 +693,13 @@ int mean_groups(Int_Array monomial, Polynomial* output, Fields_Table fields, Pol
   int group=-2;
   int next_group=-2;
   Polynomial tmp_poly;
-  int sign;
+  int sign=1;
 
   init_Polynomial(output, MONOMIAL_SIZE);
 
-  // check whether there are symbols
-  // IMPORTANT: the symbols must be at the end of the monomial
-  if(monomial.length==0 || field_type(monomial.values[monomial.length-1], fields)!=FIELD_SYMBOL){
+  // check whether there are virtual fields
+  // IMPORTANT: the virtual fields must be at the end of the monomial
+  if(monomial.length==0 || field_type(monomial.values[monomial.length-1], fields)!=FIELD_VIRTUAL){
     // mean
     mean(monomial, &num_mean, fields, propagator);
     // add to output
@@ -650,7 +720,7 @@ int mean_groups(Int_Array monomial, Polynomial* output, Fields_Table fields, Pol
       }
       // if group changes, take mean
       if((i>0 && next_group!=group) || i==monomial.length){
-	mean_symbols(tmp_monomial, &tmp_poly, fields, propagator, groups, computed);
+	mean_virtual_fields(tmp_monomial, &tmp_poly, fields, propagator, groups, computed);
 	// if zero
 	if(polynomial_is_zero(tmp_poly)==1){
 	  // set output to 0
@@ -702,10 +772,11 @@ struct mean_args{
   Fields_Table fields;
   Polynomial_Matrix propagator;
   Groups groups;
+  int print_progress;
 };
 
 // multithreaded
-int polynomial_mean_multithread(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads){
+int polynomial_mean_multithread(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads, int print_progress){
   int i;
   Polynomial thread_polys[threads];
   pthread_t thread_ids[threads];
@@ -720,11 +791,15 @@ int polynomial_mean_multithread(Polynomial* polynomial, Fields_Table fields, Pol
     args[i].fields=fields;
     args[i].propagator=propagator;
     args[i].groups=groups;
+    args[i].print_progress=print_progress;
   }
 
   // split polynomial
+  // randomly choose the thread
+  // see random number generator
+  srand(time(NULL));
   for(i=0;i<len;i++){
-    polynomial_append((*polynomial).monomials[i], (*polynomial).factors[i], (*polynomial).nums[i], thread_polys+(i % threads));
+    polynomial_append((*polynomial).monomials[i], (*polynomial).factors[i], (*polynomial).nums[i], thread_polys+(rand() % threads));
   }
 
   // start threads
@@ -750,12 +825,12 @@ int polynomial_mean_multithread(Polynomial* polynomial, Fields_Table fields, Pol
 // mean for one of the threads
 void* polynomial_mean_thread(void* mean_args){
   struct mean_args *args=mean_args;
-  polynomial_mean((*args).polynomial,(*args).fields,(*args).propagator,(*args).groups);
+  polynomial_mean((*args).polynomial,(*args).fields,(*args).propagator,(*args).groups, (*args).print_progress);
   return(NULL);
 }
 
 // single threaded version
-int polynomial_mean(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups){
+int polynomial_mean(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int print_progress){
   int i,j;
   Polynomial output;
   Polynomial tmp_poly;
@@ -769,7 +844,9 @@ int polynomial_mean(Polynomial* polynomial, Fields_Table fields, Polynomial_Matr
 
   // mean of each monomial
   for(i=0;i<(*polynomial).length;i++){
-    fprintf(stderr,"computing %d of %d means\n",i,(*polynomial).length-1);
+    if(print_progress==1){
+      fprintf(stderr,"computing %d of %d means\n",i,(*polynomial).length-1);
+    }
     mean_groups((*polynomial).monomials[i], &tmp_poly, fields, propagator, groups, &computed);
 
     // write factors
diff --git a/src/mean.h b/src/mean.h
index 34ec1d3..2726992 100644
--- a/src/mean.h
+++ b/src/mean.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -28,6 +28,12 @@ int mean(Int_Array monomial, Polynomial* out, Fields_Table fields, Polynomial_Ma
 
 // compute the mean of a monomial of internal fields (with split + and -)
 int mean_internal(Int_Array internal_plus, Int_Array internal_minus, Polynomial* out, Polynomial_Matrix propagator, Fields_Table fields);
+
+// compute the mean of a monomial by computing a determinant
+int mean_determinant(Int_Array internal_plus, Int_Array internal_minus, Number* out, Polynomial_Matrix propagator,Fields_Table fields);
+
+// compute the mean of a monomial by summing over permutations
+int mean_permutations(Int_Array internal_plus, Int_Array internal_minus, Polynomial* out, Polynomial_Matrix propagator, Fields_Table fields);
 // first pairing with a non-vanishing propagator
 int init_pairing(int* pairing, int* mask, int n, Polynomial_Matrix propagator, int* indices_plus, int* indices_minus);
 // next pairing with a non-vanishing propagator
@@ -43,12 +49,12 @@ int mean_internal_slow(Int_Array internal_plus, Int_Array internal_minus, Number
 // requires the monomial to be sorted (for the sign to be correct)
 int get_internals(Int_Array monomial, Int_Array* internal_plus, Int_Array* internal_minus, Int_Array* others, Fields_Table fields);
 
-// compute the mean of a monomial containing symbolic expressions
-int mean_symbols(Int_Array monomial, Polynomial* output, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, Identities* computed);
+// compute the mean of a monomial containing virtual fields
+int mean_virtual_fields(Int_Array monomial, Polynomial* output, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, Identities* computed);
 // first term in product with no repetitions
-int init_prod(int* current_term, Int_Array symbol_list, Fields_Table fields, int power, Int_Array base_monomial);
+int init_prod(int* current_term, Int_Array virtual_field_list, Fields_Table fields, int power, Int_Array base_monomial);
 // next term in product with no repetitions
-int next_prod(int* current_term, Int_Array symbol_list, Fields_Table fields, int power, Int_Array base_monomial);
+int next_prod(int* current_term, Int_Array virtual_field_list, Fields_Table fields, int power, Int_Array base_monomial);
 // find the next term in a polynomial that can be multiplied to a given monomial and add it to the monomial
 int next_term_norepeat(int start, Polynomial polynomial, Int_Array* monomial, Fields_Table fields);
 
@@ -62,9 +68,9 @@ int sort_fermions(int* array, int begin, int end, int* sign);
 int mean_groups(Int_Array monomial, Polynomial* output, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, Identities* computed);
 
 // compute the mean of a polynomial
-int polynomial_mean(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups);
+int polynomial_mean(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int print_progress);
 // multithreaded
-int polynomial_mean_multithread(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads);
+int polynomial_mean_multithread(Polynomial* polynomial, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads, int print_progress);
 // single thread mean
 void* polynomial_mean_thread(void* mean_args);
 #endif
diff --git a/src/meankondo.c b/src/meankondo.c
index 1c976d4..8b5c197 100644
--- a/src/meankondo.c
+++ b/src/meankondo.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -17,7 +17,7 @@ limitations under the License.
 /*
 meankondo
 
-A simple tool to compute the renormalization group flow for Fermionic hierarchical models
+A tool to compute the renormalization group flow for Fermionic hierarchical models
 
 */
 
@@ -49,15 +49,19 @@ A simple tool to compute the renormalization group flow for Fermionic hierarchic
 #include "mean.h"
 // various string operations
 #include "istring.h"
+// symbolic trees
+# include "tree.h"
 
 // read cli arguments
 int read_args_meankondo(int argc,const char* argv[], Str_Array* str_args, Meankondo_Options* opts);
 // print usage message
 int print_usage_meankondo();
+// check consistency of options
+int check_meankondo_opts(Meankondo_Options opts);
 // compute flow
 int compute_flow(Str_Array str_args, Meankondo_Options opts);
-// compute the flow equation
-int compute_flow_equation(Polynomial init_poly, Id_Table idtable, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads, Grouped_Polynomial* flow_equation);
+// compute average
+int compute_average(Polynomial init_poly, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads, int print_progress, Polynomial* exp_poly);
 
 
 int main (int argc, const char* argv[]){
@@ -69,6 +73,9 @@ int main (int argc, const char* argv[]){
   // read command-line arguments
   read_args_meankondo(argc,argv,&str_args,&opts);
 
+  // check command-line arguments
+  check_meankondo_opts(opts);
+
   // warning message if representing rational numbers as floats
 #ifdef RATIONAL_AS_FLOAT
   fprintf(stderr,"info: representing rational numbers using floats\n");
@@ -101,6 +108,10 @@ int read_args_meankondo(int argc,const char* argv[], Str_Array* str_args, Meanko
   (*opts).threads=1;
   // do not chain
   (*opts).chain=0;
+  // do not print progress
+  (*opts).print_progress=0;
+  // print the flow equation
+  (*opts).group_poly=1;
 
   // loop over arguments
   for(i=1;i<argc;i++){
@@ -116,6 +127,13 @@ int read_args_meankondo(int argc,const char* argv[], Str_Array* str_args, Meanko
 	case 'C':
 	  (*opts).chain=1;
 	  break;
+	// print progress
+	case 'p':
+	  (*opts).print_progress=1;
+	  break;
+	case 'A':
+	  (*opts).group_poly=0;
+	  break;
 	// print version
 	case 'v':
 	  printf("meankondo " VERSION "\n");
@@ -147,7 +165,16 @@ int read_args_meankondo(int argc,const char* argv[], Str_Array* str_args, Meanko
 
 // print usage message
 int print_usage_meankondo(){
-  printf("\nusage:\n   meankondo [-t threads] [-C] <filename>\n\n");
+  printf("\nusage:\n   meankondo [-t threads] [-C] [-p] [-A] <filename>\n\n");
+  return(0);
+}
+
+// check consistency of options
+int check_meankondo_opts(Meankondo_Options opts){
+  if(opts.chain==1 && opts.group_poly==0){
+    fprintf(stderr,"aborting: the '-C' and '-A' options are incompatible\n");
+    exit(-1);
+  }
   return(0);
 }
 
@@ -163,6 +190,8 @@ int compute_flow(Str_Array str_args, Meankondo_Options opts){
   Fields_Table fields;
   // their propagator
   Polynomial_Matrix propagator;
+  // preprocessor variables
+  Variables variables;
   // initial polynomial
   Polynomial init_poly;
   // list of rccs
@@ -171,6 +200,8 @@ int compute_flow(Str_Array str_args, Meankondo_Options opts){
   Groups groups;
   // flow equation
   Grouped_Polynomial flow_equation;
+  // polynomial produced by the averaging operation
+  Polynomial exp_poly;
 
 
   // parse fields
@@ -183,29 +214,38 @@ int compute_flow(Str_Array str_args, Meankondo_Options opts){
     parse_input_fields(str_args.strs[arg_index],&fields);
   }
 
-  // parse id table
-  arg_index=find_str_arg("id_table", str_args);
-  if(arg_index<0){
-    fprintf(stderr,"error: no id table entry in the configuration file\n");
-    exit(-1);
+  // parse variables
+  // must precede id_table, virtual_fields, identities and input_polynomial
+  arg_index=find_str_arg("preprocessor_variables", str_args);
+  if(arg_index>=0){
+    parse_input_variables(str_args.strs[arg_index],&variables);
   }
   else{
-    parse_input_id_table(str_args.strs[arg_index],&idtable, fields);
+    init_Variables(&variables,1);
   }
 
-  // parse symbols
-  arg_index=find_str_arg("symbols", str_args);
-  if(arg_index>=0){
-    parse_input_symbols(str_args.strs[arg_index],&fields);
+  // parse id table
+  if(opts.group_poly==1){
+    arg_index=find_str_arg("id_table", str_args);
+    if(arg_index<0){
+      fprintf(stderr,"error: no id table entry in the configuration file\n");
+      exit(-1);
+    }
+    else{
+      parse_input_id_table(str_args.strs[arg_index],&idtable, fields, variables);
+    }
   }
-  else{
-    init_Symbols(&(fields.symbols),1);
+
+  // parse virtual_fields
+  arg_index=find_str_arg("virtual_fields", str_args);
+  if(arg_index>=0){
+    parse_input_virtual_fields(str_args.strs[arg_index], &fields, variables);
   }
 
   // parse input polynomial
   arg_index=find_str_arg("input_polynomial", str_args);
   if(arg_index>=0){
-    parse_input_polynomial(str_args.strs[arg_index],&init_poly, fields);
+    parse_input_polynomial(str_args.strs[arg_index],&init_poly, fields, variables);
   }
   else{
     fprintf(stderr,"error: no input polynomial entry in the configuration file\n");
@@ -225,41 +265,90 @@ int compute_flow(Str_Array str_args, Meankondo_Options opts){
   // parse identities
   arg_index=find_str_arg("identities", str_args);
   if(arg_index>=0){
-    parse_input_identities(str_args.strs[arg_index],&fields);
-  }
-  else{
-    init_Identities(&(fields.ids),1);
+    parse_input_identities(str_args.strs[arg_index],&fields, variables);
   }
 
-  // parse groups
+  // parse groups (must come after virtual_fields and propagator)
   arg_index=find_str_arg("groups", str_args);
   if(arg_index>=0){
-    parse_input_groups(str_args.strs[arg_index],&groups);
+    parse_input_groups(str_args.strs[arg_index],&groups, propagator, fields);
   }
   else{
     init_Groups(&groups, 1);
   }
 
-  // flow equation
-  compute_flow_equation(init_poly, idtable, fields, propagator, groups, opts.threads, &flow_equation);
+  // compute the average
+  compute_average(init_poly, fields, propagator, groups, opts.threads, opts.print_progress, &exp_poly);
+
   free_Polynomial(init_poly);
   free_Polynomial_Matrix(propagator);
-  free_Fields_Table(fields);
   free_Groups(groups);
 
+  // parse postprocessing entry
+  arg_index=find_str_arg("postprocess_operation", str_args);
+  if(arg_index>=0){
+    add_polynomial_to_variables("OUT", exp_poly, &variables);
+    // parse postprocess entry
+    Polynomial postprocess_operation;
+    parse_input_polynomial(str_args.strs[arg_index], &postprocess_operation, fields, variables);
+    // replace exp_poly
+    free_Polynomial(exp_poly);
+    exp_poly=postprocess_operation;
+  }
+
+  if(opts.group_poly==1){
+    // flow equation
+    group_polynomial(exp_poly, &flow_equation, idtable, fields);
+  }
+
+  // postprocess flow equation
+  arg_index=find_str_arg("postprocess_flow_equation", str_args);
+  if(arg_index>=0){
+    Polynomial flow_polynomial;
+    // polynomial made of the rcc's multiplied by the corresponding fields (parsed from idtable)
+    idtable_to_polynomial(idtable, &flow_polynomial);
+
+    // add to variables
+    add_polynomial_to_variables("FLOW", flow_polynomial, &variables);
+    free_Polynomial(flow_polynomial);
+
+    // parse postprocess entry
+    Polynomial postprocess_polynomial;
+    parse_input_polynomial(str_args.strs[arg_index], &postprocess_polynomial, fields, variables);
+
+    // convert to flow equation
+    Grouped_Polynomial postprocess_flow_equation;
+    group_polynomial(postprocess_polynomial, &postprocess_flow_equation, idtable, fields);
+    free_Polynomial(postprocess_polynomial);
+
+    // apply postprocessing to flow equation
+    Grouped_Polynomial new_flow;
+    compose_flow_equations(postprocess_flow_equation, flow_equation, &new_flow);
+    free_Grouped_Polynomial(postprocess_flow_equation);
+
+    // replace flow_equation
+    free_Grouped_Polynomial(flow_equation);
+    flow_equation=new_flow;
+  }
+
+
   // if chain then print config file
   if(opts.chain==1){
     for(i=0;i<str_args.length;i++){
       // check whether to print the str_arg
       get_str_arg_title(str_args.strs[i], &arg_header);
       if (\
-	str_cmp(arg_header.str, "symbols")==0 &&\
+	str_cmp(arg_header.str, "variables")==0 &&\
+	str_cmp(arg_header.str, "virtual_fields")==0 &&\
 	str_cmp(arg_header.str, "groups")==0 &&\
 	str_cmp(arg_header.str, "fields")==0 &&\
 	str_cmp(arg_header.str, "identities")==0 &&\
 	str_cmp(arg_header.str, "propagator")==0 &&\
 	str_cmp(arg_header.str, "input_polynomial")==0 &&\
-	str_cmp(arg_header.str, "id_table")==0 ){
+	str_cmp(arg_header.str, "id_table")==0 &&\
+	str_cmp(arg_header.str, "postprocess_operation")==0 &&\
+	str_cmp(arg_header.str, "numerical_postprocess_operation")==0
+	){
 	  printf("%s\n&\n",str_args.strs[i].str);
       }
       free_Char_Array(arg_header);
@@ -268,34 +357,66 @@ int compute_flow(Str_Array str_args, Meankondo_Options opts){
     printf("#!flow_equation\n");
   }
 
-  // print flow equation
-  grouped_polynomial_print(flow_equation,'%','%');
+  // print result
+  if(opts.group_poly==1){
+    grouped_polynomial_print(flow_equation,'%','%');
+
+    // free memory
+    free_Grouped_Polynomial(flow_equation);
+  }
+  else{
+    polynomial_print(exp_poly);
+  }
 
   // free memory
-  free_Id_Table(idtable);
-  free_Grouped_Polynomial(flow_equation);
+  free_Polynomial(exp_poly);
+
+  // parse numerical_postprocessing entry
+  arg_index=find_str_arg("numerical_postprocess_operation", str_args);
+  if(arg_index>=0){
+    Polynomial rcc_polynomial;
+    // polynomial made of the rcc's multiplied by the corresponding fields (parsed from idtable)
+    idtable_to_polynomial(idtable, &rcc_polynomial);
+
+    // add to variables
+    add_polynomial_to_variables("RCC", rcc_polynomial, &variables);
+    free_Polynomial(rcc_polynomial);
+
+    // parse postprocess entry
+    Polynomial numerical_postprocess_operation;
+    parse_input_polynomial(str_args.strs[arg_index], &numerical_postprocess_operation, fields, variables);
+
+    // convert to flow equation
+    Grouped_Polynomial numerical_postprocess_flow_equation;
+    group_polynomial(numerical_postprocess_operation, &numerical_postprocess_flow_equation, idtable, fields);
+    free_Polynomial(numerical_postprocess_operation);
+
+    // print postprocessing flow equation
+    printf("\n&\n#!postprocess_operation\n");
+    grouped_polynomial_print(numerical_postprocess_flow_equation,'%','%');
+    free_Grouped_Polynomial(numerical_postprocess_flow_equation);
+  }
+
+  if(opts.group_poly==1){
+    free_Id_Table(idtable);
+  }
+  free_Fields_Table(fields);
+  free_Variables(variables);
+
   return(0);
 }
 
 
-// compute the flow equation
-int compute_flow_equation(Polynomial init_poly, Id_Table idtable, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads, Grouped_Polynomial* flow_equation){
-  // expectation
-  Polynomial exp_poly;
-
-  polynomial_cpy(init_poly,&exp_poly);
+// compute average
+int compute_average(Polynomial init_poly, Fields_Table fields, Polynomial_Matrix propagator, Groups groups, int threads, int print_progress, Polynomial* exp_poly){
+  polynomial_cpy(init_poly,exp_poly);
 
   // average
   if(threads>1){
-    polynomial_mean_multithread(&exp_poly, fields, propagator, groups, threads);
+    polynomial_mean_multithread(exp_poly, fields, propagator, groups, threads, print_progress);
   }
   else{
-    polynomial_mean(&exp_poly, fields, propagator, groups);
+    polynomial_mean(exp_poly, fields, propagator, groups, print_progress);
   }
-    
-  // grouped representation of expanded_poly
-  group_polynomial(exp_poly,flow_equation,idtable, fields);
-  free_Polynomial(exp_poly);
-
   return(0);
 }
diff --git a/src/meantools.c b/src/meantools.c
index e3e7247..64cf3f1 100644
--- a/src/meantools.c
+++ b/src/meantools.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -39,13 +39,13 @@ Utility to perform various operations on flow equations
 // string functions
 #include "istring.h"
 // tools
-#include "meantools_exp.h"
 #include "meantools_deriv.h"
 #include "meantools_eval.h"
+#include "meantools_expand.h"
 
-#define EXP_COMMAND 1
-#define DERIV_COMMAND 2
-#define EVAL_COMMAND 3
+#define DERIV_COMMAND 1
+#define EVAL_COMMAND 2
+#define EXPAND_COMMAND 3
 
 // read cli arguments
 int read_args_meantools(int argc,const char* argv[], Str_Array* str_args, Meantools_Options* opts);
@@ -63,12 +63,12 @@ int main (int argc, const char* argv[]){
   read_args_meantools(argc,argv,&str_args, &opts);
 
   switch(opts.command){
-  case EXP_COMMAND: tool_exp(str_args);
-    break;
   case DERIV_COMMAND: tool_deriv(str_args,opts);
     break;
   case EVAL_COMMAND: tool_eval(str_args,opts);
     break;
+  case EXPAND_COMMAND: tool_expand(str_args,opts);
+    break;
   }
 
   //free memory
@@ -86,11 +86,7 @@ int read_args_meantools(int argc,const char* argv[], Str_Array* str_args, Meanto
     exit(-1);
   }
 
-  if(str_cmp((char*)argv[1],"exp")==1){
-    (*opts).command=EXP_COMMAND;
-    tool_exp_read_args(argc, argv, str_args);
-  }
-  else if(str_cmp((char*)argv[1],"derive")==1){
+  if(str_cmp((char*)argv[1],"differentiate")==1){
     (*opts).command=DERIV_COMMAND;
     tool_deriv_read_args(argc, argv, str_args, opts);
   }
@@ -98,6 +94,10 @@ int read_args_meantools(int argc,const char* argv[], Str_Array* str_args, Meanto
     (*opts).command=EVAL_COMMAND;
     tool_eval_read_args(argc, argv, str_args, opts);
   }
+  else if(str_cmp((char*)argv[1],"expand")==1){
+    (*opts).command=EXPAND_COMMAND;
+    tool_expand_read_args(argc, argv, str_args, opts);
+  }
   else{
     print_usage_meantools();
     exit(-1);
@@ -108,9 +108,6 @@ int read_args_meantools(int argc,const char* argv[], Str_Array* str_args, Meanto
 
 // print usage message
 int print_usage_meantools(){
-  printf("\nusage:\n   meantools exp [config_file]\n   meantools derive [-d derivatives] [-V variables] [-C] [config_file]\n   meantools eval [-R values] [-P precision] [-E max_exponent] [config_file]\n\n");
+  printf("\nusage:\n   meantools differentiate [-d derivatives] [-V variables] [-C] [config_file]\n   meantools eval [-R values] [-P precision] [-E max_exponent] [config_file]\n   meantools expand [-N namespace] [config_file]\n\n");
   return(0);
 }
-
-
-
diff --git a/src/meantools_deriv.c b/src/meantools_deriv.c
index b096380..3530808 100644
--- a/src/meantools_deriv.c
+++ b/src/meantools_deriv.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -42,9 +42,9 @@ int tool_deriv_read_args(int argc, const char* argv[], Str_Array* str_args, Mean
   char* ptr;
 
   // defaults
-  // derive once
+  // differentiate once
   (*opts).deriv_derivs=1;
-  // derive with respect to all variables
+  // differentiate with respect to all variables
   (*opts).deriv_vars.length=-1;
   // do not chain
   (*opts).chain=0;
@@ -77,7 +77,7 @@ int tool_deriv_read_args(int argc, const char* argv[], Str_Array* str_args, Mean
     }
     // variables
     else if(flag==CP_FLAG_VARS){
-      // if the argument is "all" then derive wrt all variables
+      // if the argument is "all" then differentiate wrt all variables
       if(str_cmp((char*)argv[i],"all")){
 	(*opts).deriv_vars.length=-2;
       }
@@ -101,7 +101,7 @@ int tool_deriv_read_args(int argc, const char* argv[], Str_Array* str_args, Mean
 }
 
 
-// derive a flow equation
+// differentiate a flow equation
 int tool_deriv(Str_Array str_args, Meantools_Options opts){
   // index of the entry in the input file
   int arg_index;
@@ -158,7 +158,7 @@ int tool_deriv(Str_Array str_args, Meantools_Options opts){
       get_str_arg_title(str_args.strs[i], &arg_header);
       if (\
 	str_cmp(arg_header.str, "flow_equation")==0 &&\
-	str_cmp(arg_header.str, "symbols")==0 &&\
+	str_cmp(arg_header.str, "virtual_fields")==0 &&\
 	str_cmp(arg_header.str, "groups")==0 &&\
 	str_cmp(arg_header.str, "fields")==0 &&\
 	str_cmp(arg_header.str, "identities")==0 &&\
@@ -208,7 +208,7 @@ int flow_equation_derivative(int n, Int_Array variables, Grouped_Polynomial flow
     // free tmpflow
     free_Grouped_Polynomial(tmpflow);
 
-    // add the derived indices as variables for the next derivative
+    // add the differentiated indices as variables for the next derivative
     for(i=0;i<variables.length;i++){
       if(variables.values[i]>=0){
 	int_array_append((j+1)*DOFFSET+variables.values[i], &indices);
diff --git a/src/meantools_deriv.h b/src/meantools_deriv.h
index 4ea36a9..7425fd1 100644
--- a/src/meantools_deriv.h
+++ b/src/meantools_deriv.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -21,7 +21,7 @@ limitations under the License.
 
 // read arguments
 int tool_deriv_read_args(int argc, const char* argv[], Str_Array* str_args, Meantools_Options* opts);
-// derive a flow equation
+// differentiate a flow equation
 int tool_deriv(Str_Array str_args, Meantools_Options opts);
 // n first derivatives of a flow equation wrt to variables
 int flow_equation_derivative(int n, Int_Array variables, Grouped_Polynomial flow_equation, Grouped_Polynomial* flow_equation_derivs);
diff --git a/src/meantools_eval.c b/src/meantools_eval.c
index 4f166cd..027d18d 100644
--- a/src/meantools_eval.c
+++ b/src/meantools_eval.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -163,12 +163,12 @@ int tool_eval(Str_Array str_args, Meantools_Options opts){
 
   // evaluate
   if(mpfr_flag==0){
-    evaleq(&rccs, flow_equation);
+    evaleq(rccs, rccs, flow_equation);
     RCC_print(rccs);
     free_RCC(rccs);
   }
   else{
-    evaleq_mpfr(&rccs_mpfr, flow_equation);
+    evaleq_mpfr(rccs_mpfr, rccs_mpfr, flow_equation);
     RCC_mpfr_print(rccs_mpfr);
     free_RCC_mpfr(rccs_mpfr);
   }
diff --git a/src/meantools_eval.h b/src/meantools_eval.h
index 518049d..54f6e51 100644
--- a/src/meantools_eval.h
+++ b/src/meantools_eval.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/meantools_exp.c b/src/meantools_exp.c
deleted file mode 100644
index 1aca928..0000000
--- a/src/meantools_exp.c
+++ /dev/null
@@ -1,130 +0,0 @@
-/*
-Copyright 2015 Ian Jauslin
-
-Licensed under the Apache License, Version 2.0 (the "License");
-you may not use this file except in compliance with the License.
-You may obtain a copy of the License at
-
-    http://www.apache.org/licenses/LICENSE-2.0
-
-Unless required by applicable law or agreed to in writing, software
-distributed under the License is distributed on an "AS IS" BASIS,
-WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
-See the License for the specific language governing permissions and
-limitations under the License.
-*/
-
-#include "meantools_exp.h"
-
-#include <stdio.h>
-#include <stdlib.h>
-#include "parse_file.h"
-#include "cli_parser.h"
-#include "polynomial.h"
-#include "fields.h"
-#include "grouped_polynomial.h"
-#include "idtable.h"
-
-// read command line arguments
-int tool_exp_read_args(int argc, const char* argv[], Str_Array* str_args){
-  // file to read the polynomial from in flow mode
-  const char* file="";
-  // whether a file was specified on the command-line
-  int exists_file=0;
-
-  if(argc>=3){
-    file=argv[2];
-    exists_file=1;
-  }
-  read_config_file(str_args, file, 1-exists_file);
-
-  return(0);
-}
-
-
-// compute the exponential of the input polynomial
-int tool_exp(Str_Array str_args){
-  // index of the entry in the input file
-  int arg_index;
-  // list of fields
-  Fields_Table fields;
-  // input polynomial
-  Polynomial poly;
-  // exp as a polynomial
-  Polynomial exp_poly;
-  // list of rccs
-  Id_Table idtable;
-  // exp
-  Grouped_Polynomial exp;
-  int i,j;
-
-  // parse fields
-  arg_index=find_str_arg("fields", str_args);
-  if(arg_index<0){
-    fprintf(stderr,"error: no fields entry in the configuration file\n");
-    exit(-1);
-  }
-  else{
-    parse_input_fields(str_args.strs[arg_index],&fields);
-  }
-
-  // parse id table
-  arg_index=find_str_arg("id_table", str_args);
-  if(arg_index<0){
-    fprintf(stderr,"error: no id table entry in the configuration file\n");
-    exit(-1);
-  }
-  else{
-    parse_input_id_table(str_args.strs[arg_index],&idtable, fields);
-  }
-
-  // parse input polynomial
-  arg_index=find_str_arg("input_polynomial", str_args);
-  if(arg_index>=0){
-    parse_input_polynomial(str_args.strs[arg_index],&poly, fields);
-  }
-  else{
-    fprintf(stderr,"error: no input polynomial entry in the configuration file\n");
-    exit(-1);
-  }
-
-  // parse symbols
-  arg_index=find_str_arg("symbols", str_args);
-  if(arg_index>=0){
-    parse_input_symbols(str_args.strs[arg_index],&fields);
-  }
-  else{
-    init_Symbols(&(fields.symbols),1);
-  }
-
-  // parse identities
-  arg_index=find_str_arg("identities", str_args);
-  if(arg_index>=0){
-    parse_input_identities(str_args.strs[arg_index],&fields);
-  }
-  else{
-    init_Identities(&(fields.ids),1);
-  }
-
-  // exp(V)
-  polynomial_exponential(poly,&exp_poly, fields);
-  // grouped representation
-  group_polynomial(exp_poly, &exp, idtable, fields);
-  free_Polynomial(exp_poly);
-  free_Polynomial(poly);
-
-  // no denominators
-  for(i=0;i<exp.length;i++){
-    for(j=0;j<exp.coefs[i].length;j++){
-      exp.coefs[i].denoms[j].power=0;
-    }
-  }
-
-  grouped_polynomial_print(exp,'%','%');
-
-  // free memory
-  free_Fields_Table(fields);
-  free_Id_Table(idtable);
-  free_Grouped_Polynomial(exp);
-  return(0);
-}
diff --git a/src/meantools_expand.c b/src/meantools_expand.c
new file mode 100644
index 0000000..2586ff5
--- /dev/null
+++ b/src/meantools_expand.c
@@ -0,0 +1,166 @@
+/*
+Copyright 2015-2022 Ian Jauslin
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+#include "meantools_expand.h"
+
+#include <stdio.h>
+#include <stdlib.h>
+#include "cli_parser.h"
+#include "parse_file.h"
+#include "polynomial.h"
+#include "array.h"
+#include "fields.h"
+
+
+#define CP_FLAG_NAMESPACE 1
+// read command line arguments
+int tool_expand_read_args(int argc, const char* argv[], Str_Array* str_args, Meantools_Options* opts){
+  // file to read the polynomial from in flow mode
+  const char* file="";
+  // whether a file was specified on the command-line
+  int exists_file=0;
+  // flag
+  int flag=0;
+  int i;
+  char* ptr;
+
+  // defaults
+  // no namespace
+  (*opts).namespace.length=-1;
+
+
+  // loop over arguments
+  for(i=2;i<argc;i++){
+    // flag
+    if(argv[i][0]=='-'){
+      for(ptr=((char*)argv[i])+1;*ptr!='\0';ptr++){
+	switch(*ptr){
+	// number of derivatives
+	case 'N':
+	  flag=CP_FLAG_NAMESPACE;
+	  break;
+	}
+      }
+    }
+    // namespace
+    else if(flag==CP_FLAG_NAMESPACE){
+      str_to_char_array((char*)argv[i], &(opts->namespace));
+      flag=0;
+    }
+    // read file name from command-line
+    else{
+      file=argv[i];
+      exists_file=1;
+    }
+  }
+
+  read_config_file(str_args, file, 1-exists_file);
+
+  return(0);
+}
+
+
+// expand a polynomial
+int tool_expand(Str_Array str_args, Meantools_Options opts){
+  // index of the entry in the input file
+  int arg_index;
+  // input polynomial
+  Polynomial polynomial;
+  // fields table
+  Fields_Table fields;
+  // preprocessor variables
+  Variables variables;
+
+  // parse fields
+  if(opts.namespace.length>=0){
+    arg_index=find_str_arg_ns("fields", opts.namespace, str_args);
+  }
+  else{
+    arg_index=find_str_arg("fields", str_args);
+  }
+  if(arg_index<0){
+    fprintf(stderr,"error: no fields entry in the configuration file\n");
+    exit(-1);
+  }
+  else{
+    parse_input_fields(str_args.strs[arg_index],&fields);
+  }
+
+  // parse variables
+  // must precede id_table, virtual_fields, identities and input_polynomial
+  if(opts.namespace.length>=0){
+    arg_index=find_str_arg_ns("preprocessor_variables", opts.namespace, str_args);
+  }
+  else{
+    arg_index=find_str_arg("preprocessor_variables", str_args);
+  }
+  if(arg_index>=0){
+    parse_input_variables(str_args.strs[arg_index],&variables);
+  }
+  else{
+    init_Variables(&variables,1);
+  }
+
+  // parse virtual_fields
+  if(opts.namespace.length>=0){
+    arg_index=find_str_arg_ns("virtual_fields", opts.namespace, str_args);
+  }
+  else{
+    arg_index=find_str_arg("virtual_fields", str_args);
+  }
+  if(arg_index>=0){
+    parse_input_virtual_fields(str_args.strs[arg_index], &fields, variables);
+  }
+
+  // parse identities
+  if(opts.namespace.length>=0){
+    arg_index=find_str_arg_ns("identities", opts.namespace, str_args);
+  }
+  else{
+    arg_index=find_str_arg("identities", str_args);
+  }
+  if(arg_index>=0){
+    parse_input_identities(str_args.strs[arg_index],&fields, variables);
+  }
+
+  // parse input polynomial
+  if(opts.namespace.length>=0){
+    arg_index=find_str_arg_ns("input_polynomial", opts.namespace, str_args);
+  }
+  else{
+    arg_index=find_str_arg("input_polynomial", str_args);
+  }
+  if(arg_index>=0){
+    parse_input_polynomial(str_args.strs[arg_index], &polynomial, fields, variables);
+  }
+  else{
+    fprintf(stderr,"error: no input polynomial entry in the configuration file\n");
+    exit(-1);
+  }
+
+  // print polynomial
+  polynomial_print(polynomial);
+
+  // free memory
+  free_Variables(variables);
+  free_Fields_Table(fields);
+  free_Polynomial(polynomial);
+
+  if(opts.namespace.length>=0){
+    free_Char_Array(opts.namespace);
+  }
+  return(0);
+}
diff --git a/src/meantools_exp.h b/src/meantools_expand.h
similarity index 67%
rename from src/meantools_exp.h
rename to src/meantools_expand.h
index f3f6666..9d988b1 100644
--- a/src/meantools_exp.h
+++ b/src/meantools_expand.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -14,14 +14,16 @@ See the License for the specific language governing permissions and
 limitations under the License.
 */
 
-#ifndef MEANTOOLS_EXP_H
-#define MEANTOOLS_EXP_H
+#ifndef MEANTOOLS_EXPAND_H
+#define MEANTOOLS_EXPAND_H
 
 #include "types.h"
 
 // read arguments
-int tool_exp_read_args(int argc, const char* argv[], Str_Array* str_args);
-// compute the exponential of the input polynomial
-int tool_exp(Str_Array str_args);
+int tool_expand_read_args(int argc, const char* argv[], Str_Array* str_args, Meantools_Options* opts);
+// expand a flow equation
+int tool_expand(Str_Array str_args, Meantools_Options opts);
 
 #endif
+
+
diff --git a/src/number.c b/src/number.c
index e63427f..5bc87de 100644
--- a/src/number.c
+++ b/src/number.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -27,6 +27,7 @@ limitations under the License.
 #include "tools.h"
 #include "rational.h"
 #include "array.h"
+#include "parse_file.h"
 
 // init
 int init_Number(Number* number, int memory){
@@ -271,50 +272,107 @@ Number number_Qprod_ret(Q q, Number x){
   return(ret);
 }
 
-// inverse
-int number_inverse_inplace(Number* inout){
-  int i;
-  for(i=0;i<(*inout).length;i++){
-    if((*inout).base[i]>0){
-      (*inout).scalars[i]=Q_inverse((*inout).scalars[i]);
-      (*inout).scalars[i].denominator*=(*inout).base[i];
+// quotient of two numbers
+// recursively simplify numerator/denominator until denominator only has one term
+// the output is set to 'numerator'
+// both numerator and denominator may be changed by this function
+// this algorithm is not optimal in cases where denominator has several terms, in particular if their bases are large
+// it is optimal if denominator only has one term
+int number_quot_inplace(Number* numerator, Number* denominator){
+  Number tmp;
+  int i,factor;
+
+  switch((*denominator).length){
+  // error
+  case 0:
+    fprintf(stderr,"error: attempting to invert 0\n");
+    exit(-1);
+    break;
+
+  // trivial case
+  case 1:
+    // trivial base
+    if((*denominator).base[0]==1){
+      number_Qprod_chain(Q_inverse((*denominator).scalars[0]), numerator);
     }
-    else if((*inout).base[i]<0){
-      (*inout).scalars[i]=Q_inverse((*inout).scalars[i]);
-      (*inout).scalars[i].denominator*=-(*inout).base[i];
-      (*inout).scalars[i].numerator*=-1;
+    // non-trivial base
+    else{
+      // set tmp=1/denominator
+      tmp=number_one();
+      tmp.base[0]=(*denominator).base[0];
+      tmp.scalars[0].denominator=(*denominator).scalars[0].numerator*abs((*denominator).base[0]);
+      if((*denominator).base[0]>0){
+	tmp.scalars[0].numerator=(*denominator).scalars[0].denominator;
+      }
+      else if((*denominator).base[0]<0){
+	tmp.scalars[0].numerator=-(*denominator).scalars[0].denominator;
+      }
+      else{
+	fprintf(stderr,"error: attempting to invert 0\n");
+	exit(-1);
+      }
+      number_prod_chain(tmp, numerator);
+    }
+    break;
+
+  default:
+    // find non-trivial basis
+    for(i=0;(*denominator).base[i]==1;i++){}
+
+    // smallest prime factor of the base
+    if((*denominator).base[i]<0){
+      factor=-1;
+    }
+    else if((*denominator).base[i]==2){
+      factor=2;
     }
     else{
-      fprintf(stderr,"error: attempting to invert 0\n");
-      exit(-1);
+      // COMMENT: this is not optimal, but, provided the basis is not too large, this should not be a problem
+      for(factor=3;is_factor(factor, (*denominator).base[i])==0;factor=factor+2){}
     }
+
+    // tmp is set to ((terms that k does not divide)-(terms that k divides))
+    // tmp will be multiplied to numerator and denominator
+    init_Number(&tmp,(*denominator).length);
+
+    // find all terms whose base is a multiple of k
+    for(i=0;i<(*denominator).length;i++){
+      if(is_factor(factor, (*denominator).base[i])){
+	// add to tmp with a - sign
+	number_add_elem(quot(-(*denominator).scalars[i].numerator,(*denominator).scalars[i].denominator), (*denominator).base[i], &tmp);
+      }
+      else{
+	// add to tmp
+	number_add_elem((*denominator).scalars[i], (*denominator).base[i], &tmp);
+      }
+    }
+
+    number_prod_chain(tmp, numerator);
+    number_prod_chain(tmp, denominator);
+    free_Number(tmp);
+
+    // recurse
+    number_quot_inplace(numerator, denominator);
   }
+
   return(0);
 }
-// write to output
-int number_inverse(Number input, Number* output){
-  number_cpy(input,output);
-  number_inverse_inplace(output);
-  return(0);
-}
-// return result
-Number number_inverse_ret(Number x){
-  Number ret;
-  number_inverse(x,&ret);
-  return(ret);
-}
 
-// quotient
+// not inplace
 int number_quot(Number x1, Number x2, Number* output){
-  Number inv;
-  number_inverse(x2, &inv);
-  number_prod(x1, inv, output);
-  free_Number(inv);
+  Number numerator, denominator;
+  number_cpy(x1, &numerator);
+  number_cpy(x2, &denominator);
+  number_quot_inplace(&numerator, &denominator);
+  *output=numerator;
+  free_Number(denominator);
   return(0);
 }
-int number_quot_chain(Number x1, Number* inout){
-  number_inverse_inplace(inout);
-  number_prod_chain(x1, inout);
+int number_quot_chain(Number* inout, Number x2){
+  Number tmp;
+  number_quot(*inout,x2,&tmp);
+  free_Number(*inout);
+  *inout=tmp;
   return(0);
 }
 Number number_quot_ret(Number x1, Number x2){
@@ -419,7 +477,7 @@ int number_print(Number number){
   Char_Array buffer;
   init_Char_Array(&buffer,5*number.length);
   number_sprint(number, &buffer);
-  printf("%s",buffer.str);
+  printf("%s",char_array_to_str_noinit(&buffer));
   return(0);
 }
 
@@ -434,18 +492,54 @@ int str_to_Number(char* str, Number* number){
   char* buffer_ptr=buffer;
   Q num;
   int base;
-  // whether there are parentheses in the string
-  int exist_parenthesis=0;
+  int ret;
+  int j;
+  char* aux_str;
+  int aux_free=0;
 
   init_Number(number, NUMBER_SIZE);
 
+  // check whether the string is blank (return 0 in that case)
+  for(ptr=str;*ptr!='\0';ptr++){
+    if(*ptr!=' ' && *ptr!='\n'){
+      break;
+    }
+  }
+  // blank string
+  if(*ptr=='\0'){
+    number_add_elem(quot(0,1), 1, number);
+    free(buffer);
+    return(0);
+  }
+
   // init num and base
-  // init to 0 so that if str is empty, then the number is set to 0
-  num=quot(0,1);
+  num=quot(1,1);
   base=1;
 
+  // check whether the str only contains a rational number, and add parentheses
+  // keep rtack of the length of str
+  for(j=0,ptr=str;*ptr!='\0';j++,ptr++){
+    if((*ptr<'0' || *ptr>'9') && *ptr!='-' && *ptr!='/'){
+      break;
+    }
+  }
+  // only rational
+  if(*ptr=='\0'){
+    aux_str=calloc(j+3,sizeof(char));
+    aux_str[0]='(';
+    for(j=0,ptr=str;*ptr!='\0';ptr++,j++){
+      aux_str[j+1]=*ptr;
+    }
+    aux_str[j+1]=')';
+    aux_str[j+2]='\0';
+    aux_free=1;
+  }
+  else{
+    aux_str=str;
+  }
+
   mode=PP_NULL_MODE;
-  for(ptr=str;*ptr!='\0';ptr++){
+  for(ptr=aux_str;*ptr!='\0';ptr++){
     switch(*ptr){
     // read number
     case '(':
@@ -453,7 +547,10 @@ int str_to_Number(char* str, Number* number){
 	// init base
 	base=1;
 	mode=PP_NUM_MODE;
-	exist_parenthesis=1;
+      }
+      else{
+	fprintf(stderr,"syntax error: misplaced '(' in number '%s'\n",str);
+	exit(-1);
       }
       break;
     case ')':
@@ -463,6 +560,10 @@ int str_to_Number(char* str, Number* number){
 	*buffer_ptr='\0';
 	mode=PP_NULL_MODE;
       }
+      else{
+	fprintf(stderr,"syntax error: mismatched ')' in number '%s'\n",str);
+	exit(-1);
+      }
       break;
 
     // read sqrt
@@ -474,16 +575,26 @@ int str_to_Number(char* str, Number* number){
       if(mode==PP_NULL_MODE){
 	mode=PP_SQRT_MODE;
       }
-      // if there is a square root, then do not read a fraction (see end of loop)
-      exist_parenthesis=1;
+      else{
+	fprintf(stderr,"syntax error: misplaced '{' in number '%s'\n",str);
+	exit(-1);
+      }
       break;
     case '}':
       if(mode==PP_SQRT_MODE){
-	sscanf(buffer,"%d",&base);
+	ret=read_int(buffer,&base);
+	if(ret<0){
+	  fprintf(stderr,"syntax error: number base should be an integer, got '%s' in '%s'",buffer,str);
+	  exit(-1);
+	}
 	buffer_ptr=buffer;
 	*buffer_ptr='\0';
 	mode=PP_NULL_MODE;
       }
+      else{
+	fprintf(stderr,"syntax error: mismatched '}' in number '%s'\n",str);
+	exit(-1);
+      }
       break;
 
     // write num
@@ -494,24 +605,40 @@ int str_to_Number(char* str, Number* number){
 	num=quot(0,1);
 	base=1;
 	}
+      else{
+	fprintf(stderr,"syntax error: misplaced '+' in number '%s'\n",str);
+	exit(-1);
+      }
       break;
 
+    // ignore 's', ' ' and '\n'
+    case 's':break;
+    case ' ':break;
+    case '\n':break;
     default:
       if(mode!=PP_NULL_MODE){
 	buffer_ptr=str_addchar(buffer_ptr,*ptr);
       }
+      else{
+	fprintf(stderr,"syntax error: unrecognized character '%c' in number '%s'\n",*ptr,str);
+	exit(-1);
+      }
       break;
     }
   }
 
-  // last step
   if(mode==PP_NULL_MODE){
-    if(exist_parenthesis==0){
-      str_to_Q(str, &num);
-    }
     number_add_elem(num, base, number);
   }
+  else{
+    fprintf(stderr,"syntax error: mismatched '(' in number '%s'\n",str);
+    exit(-1);
+  }
 
+
+  if(aux_free==1){
+    free(aux_str);
+  }
   free(buffer);
   return(0);
 }
diff --git a/src/number.h b/src/number.h
index a33d425..5eacdaa 100644
--- a/src/number.h
+++ b/src/number.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -78,16 +78,11 @@ int number_Qprod(Q q, Number x, Number* inout);
 // return result
 Number number_Qprod_ret(Q q, Number x);
 
-// inverse
-int number_inverse_inplace(Number* inout);
-// write to output
-int number_inverse(Number input, Number* output);
-// return result
-Number number_inverse_ret(Number x);
-
-// quotient
+// quotient of two numbers
+int number_quot_inplace(Number* numerator, Number* denominator);
+// not inplace
 int number_quot(Number x1, Number x2, Number* output);
-int number_quot_chain(Number x1, Number* inout);
+int number_quot_chain(Number* inout, Number x2);
 Number number_quot_ret(Number x1, Number x2);
 
 // remove 0's
diff --git a/src/numkondo.c b/src/numkondo.c
index 0568861..91e6be2 100644
--- a/src/numkondo.c
+++ b/src/numkondo.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -86,12 +86,6 @@ int read_args_numkondo(int argc,const char* argv[], Str_Array* str_args, Numkond
   // whether a file was specified on the command-line
   int exists_file=0;
 
-  // if there are no arguments
-  if(argc==1){
-    print_usage_numkondo();
-    exit(-1);
-  }
-
   // defaults
   // display entire flow
   (*opts).display_mode=DISPLAY_NUMERICAL;
@@ -193,6 +187,8 @@ int numflow(Str_Array str_args, Numkondo_Options opts){
   Grouped_Polynomial flow_equation;
   // whether or not to use mpfr floats
   int mpfr_flag=0;
+  // postprocess flow equation
+  Grouped_Polynomial postprocess_flow_equation;
 
   // set mpfr defaults
   if(opts.mpfr_prec!=0){
@@ -205,7 +201,7 @@ int numflow(Str_Array str_args, Numkondo_Options opts){
   }
 
 
-  // parse id table
+  // parse labels
   arg_index=find_str_arg("labels", str_args);
   if(arg_index<0){
     fprintf(stderr,"error: no labels entry in the configuration file\n");
@@ -225,6 +221,16 @@ int numflow(Str_Array str_args, Numkondo_Options opts){
     char_array_to_Grouped_Polynomial(str_args.strs[arg_index], &flow_equation);
   }
 
+  // parse postprocess operation
+  arg_index=find_str_arg("postprocess_operation", str_args);
+  if(arg_index>=0){
+    char_array_to_Grouped_Polynomial(str_args.strs[arg_index], &postprocess_flow_equation);
+  }
+  else{
+    init_Grouped_Polynomial(&postprocess_flow_equation,1);
+  }
+
+
   // initial conditions
   // check they were not specified on the command line
   if(opts.eval_rccstring.length==-1){
@@ -252,17 +258,18 @@ int numflow(Str_Array str_args, Numkondo_Options opts){
   }
 
   if(mpfr_flag==0){
-    numerical_flow(flow_equation, init_cd, labels, opts.niter, opts.display_mode);
+    numerical_flow(flow_equation, init_cd, postprocess_flow_equation, labels, opts.niter, opts.display_mode);
     free_RCC(init_cd);
   }
   else{
-    numerical_flow_mpfr(flow_equation, init_cd_mpfr, labels, opts.niter, opts.display_mode);
+    numerical_flow_mpfr(flow_equation, init_cd_mpfr, postprocess_flow_equation, labels, opts.niter, opts.display_mode);
     free_RCC_mpfr(init_cd_mpfr);
   }
 
 
   // free memory
   free_Labels(labels);
+  free_Grouped_Polynomial(postprocess_flow_equation);
   free_Grouped_Polynomial(flow_equation);
   return(0);
 }
diff --git a/src/parse_file.c b/src/parse_file.c
index ae3a9af..e998803 100644
--- a/src/parse_file.c
+++ b/src/parse_file.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -30,6 +30,8 @@ limitations under the License.
 #include "istring.h"
 #include "tools.h"
 #include "idtable.h"
+#include "tree.h"
+#include "symbolic_parser.h"
 
 
 // parsing modes
@@ -59,6 +61,70 @@ limitations under the License.
 #define PP_GROUP_MODE 16
 
 
+// read a positive integer from a string
+int read_positive_int(char* str, int* out){
+  char* ptr;
+  int res;
+  for(ptr=str;*ptr!='\0';ptr++){
+    if(*ptr-'0'>=10 || *ptr-'0'<0){
+      return(-1);
+    }
+  }
+  res=sscanf(str,"%d",out);
+  if(res!=1){
+    return(-2);
+  }
+  return(0);
+}
+// read an integer from a string
+int read_int(char* str, int* out){
+  char* ptr;
+  int res;
+  for(ptr=str;*ptr!='\0';ptr++){
+    if((*ptr-'0'>=10 || *ptr-'0'<0) && (*ptr!='-')){
+      return(-1);
+    }
+  }
+  res=sscanf(str,"%d",out);
+  if(res!=1){
+    return(-2);
+  }
+  return(0);
+}
+// read an long int from a string
+int read_long_int(char* str, long int* out){
+  char* ptr;
+  int res;
+  for(ptr=str;*ptr!='\0';ptr++){
+    if((*ptr-'0'>=10 || *ptr-'0'<0) && (*ptr!='-')){
+      return(-1);
+    }
+  }
+  res=sscanf(str,"%ld",out);
+  if(res!=1){
+    return(-2);
+  }
+  return(0);
+}
+
+// read a long double
+int read_long_double(char* str, long double* out){
+  char* ptr;
+  int res;
+  for(ptr=str;*ptr!='\0';ptr++){
+    if((*ptr-'0'>=10 || *ptr-'0'<0) && (*ptr!='-') && (*ptr!='e') && (*ptr!='E') && (*ptr!='.')){
+      return(-1);
+    }
+  }
+  res=sscanf(str,"%Lf",out);
+  if(res!=1){
+    return(-2);
+  }
+  return(0);
+}
+
+
+
 // parse fields list
 int parse_input_fields(Char_Array str_fields, Fields_Table* fields){
   // buffer
@@ -67,6 +133,8 @@ int parse_input_fields(Char_Array str_fields, Fields_Table* fields){
   int i,j;
   int mode;
   int comment=0;
+  int res;
+  int expect_colon=0;
 
   // allocate memory
   init_Fields_Table(fields);
@@ -86,39 +154,100 @@ int parse_input_fields(Char_Array str_fields, Fields_Table* fields){
 	if(mode==PP_NULL_MODE){
 	  mode=PP_PARAMETER_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: fields entry (%d): 'h' should be at the beginning of the line\n",j);
+	  exit(-1);
+	}
+	// flag: colon is expected immediately after letter
+	expect_colon=1;
 	break;
       // external fields
       case 'x':
 	if(mode==PP_NULL_MODE){
 	  mode=PP_EXTERNAL_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: fields entry (%d): 'x' should be at the beginning of the line\n",j);
+	  exit(-1);
+	}
+	// flag: colon is expected immediately after letter
+	expect_colon=1;
 	break;
       // internal fields
       case 'i':
 	if(mode==PP_NULL_MODE){
 	  mode=PP_INTERNAL_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: fields entry (%d): 'i' should be at the beginning of the line\n",j);
+	  exit(-1);
+	}
+	// flag: colon is expected immediately after letter
+	expect_colon=1;
 	break;
       case 'f':
 	if(mode==PP_NULL_MODE){
 	  mode=PP_FERMIONS_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: fields entry (%d): 'f' should be at the beginning of the line\n",j);
+	  exit(-1);
+	}
+	// flag: colon is expected immediately after letter
+	expect_colon=1;
 	break;
       case 'a':
 	if(mode==PP_NULL_MODE){
 	  mode=PP_NONCOMMUTING_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: fields entry (%d): 'a' should be at the beginning of the line\n",j);
+	  exit(-1);
+	}
+	// flag: colon is expected immediately after letter
+	expect_colon=1;
 	break;
 
       // reset buffer
       case ':':
+	// check mode
+	if(mode==PP_NULL_MODE || expect_colon==0){
+	  fprintf(stderr,"syntax error: fields entry (%d): ':' must be preceded by 'h', 'x', 'i', 'f', or 'a' on the same line\n",j);
+	  exit(-1);
+	}
+	expect_colon=0;
 	buffer_ptr=buffer;
 	*buffer_ptr='\0';
 	break;
 
       // write to fields
       case ',':
-	sscanf(buffer,"%d",&i);
+	res=read_positive_int(buffer,&i);
+
+	// check i
+	if(res<0){
+	  fprintf(stderr,"syntax error: fields entry (%d): fields must be non-negative integers, got '%s'\n",j,buffer);
+	  exit(-1);
+	}
+	if(mode==PP_PARAMETER_MODE || mode==PP_EXTERNAL_MODE || mode==PP_INTERNAL_MODE){
+	  if(int_array_find(i,fields->parameter)>=0 || int_array_find(i,fields->external)>=0 || int_array_find(i,fields->internal)>=0){
+	    fprintf(stderr,"error: fields entry (%d): field %d is already present in the fields table\n",j,i);
+	    exit(-1);
+	  }
+	}
+	else if(mode==PP_FERMIONS_MODE){
+	  if(int_array_find(i,fields->fermions)>=0){
+	    fprintf(stderr,"error: fields entry (%d): field %d is already present in the Fermions table\n",j,i);
+	    exit(-1);
+	  }
+	}    
+	else if(mode==PP_NONCOMMUTING_MODE){
+	  if(int_array_find(i,fields->noncommuting)>=0){
+	    fprintf(stderr,"error: fields entry (%d): field %d is already present in the noncommuting table\n",j,i);
+	    exit(-1);
+	  }
+	}    
+
 	if(mode==PP_PARAMETER_MODE){
 	  int_array_append(i,&((*fields).parameter));
 	}
@@ -134,31 +263,65 @@ int parse_input_fields(Char_Array str_fields, Fields_Table* fields){
 	else if(mode==PP_NONCOMMUTING_MODE){
 	  int_array_append(i,&((*fields).noncommuting));
 	}
+	else{
+	  fprintf(stderr,"syntax error: fields entry (%d): misplaced ','\n",j);
+	  exit(-1);
+	}
 	buffer_ptr=buffer;
 	*buffer_ptr='\0';
 	break;
 
       // back to null mode
       case '\n':
-	sscanf(buffer,"%d",&i);
-	if(mode==PP_PARAMETER_MODE){
-	  int_array_append(i,&((*fields).parameter));
+	// ignore if in NULL_MODE
+	if(mode!=PP_NULL_MODE){
+	  res=read_positive_int(buffer,&i);
+	  // check i
+	  if(res<0){
+	    fprintf(stderr,"syntax error: fields entry (%d): fields must be non-negative integers, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
+	  if(mode==PP_PARAMETER_MODE || mode==PP_EXTERNAL_MODE || mode==PP_INTERNAL_MODE){
+	    if(int_array_find(i,fields->parameter)>=0 || int_array_find(i,fields->external)>=0 || int_array_find(i,fields->internal)>=0){
+	      fprintf(stderr,"error: fields entry (%d): field %d is already present in the fields table\n",j,i);
+	      exit(-1);
+	    }
+	  }
+	  else if(mode==PP_FERMIONS_MODE){
+	    if(int_array_find(i,fields->fermions)>=0){
+	      fprintf(stderr,"error: fields entry (%d): field %d is already present in the Fermions table\n",j,i);
+	      exit(-1);
+	    }
+	  }    
+	  else if(mode==PP_NONCOMMUTING_MODE){
+	    if(int_array_find(i,fields->noncommuting)>=0){
+	      fprintf(stderr,"error: fields entry (%d): field %d is already present in the noncommuting table\n",j,i);
+	      exit(-1);
+	    }
+	  }    
+
+	  if(mode==PP_PARAMETER_MODE){
+	    int_array_append(i,&((*fields).parameter));
+	  }
+	  else if(mode==PP_EXTERNAL_MODE){
+	    int_array_append(i,&((*fields).external));
+	  }
+	  else if(mode==PP_INTERNAL_MODE){
+	    int_array_append(i,&((*fields).internal));
+	  }
+	  else if(mode==PP_FERMIONS_MODE){
+	    int_array_append(i,&((*fields).fermions));
+	  }
+	  else if(mode==PP_NONCOMMUTING_MODE){
+	    int_array_append(i,&((*fields).noncommuting));
+	  }
+	  mode=PP_NULL_MODE;
 	}
-	else if(mode==PP_EXTERNAL_MODE){
-	  int_array_append(i,&((*fields).external));
-	}
-	else if(mode==PP_INTERNAL_MODE){
-	  int_array_append(i,&((*fields).internal));
-	}
-	else if(mode==PP_FERMIONS_MODE){
-	  int_array_append(i,&((*fields).fermions));
-	}
-	else if(mode==PP_NONCOMMUTING_MODE){
-	  int_array_append(i,&((*fields).noncommuting));
-	}
-	mode=PP_NULL_MODE;
 	break;
 
+  
+      // ignore spaces
+      case ' ':break;
       // comment
       case '#':
 	comment=1;
@@ -168,57 +331,85 @@ int parse_input_fields(Char_Array str_fields, Fields_Table* fields){
 	if(mode!=PP_NULL_MODE){
 	  buffer_ptr=str_addchar(buffer_ptr,str_fields.str[j]);
 	}
+	else{
+	  fprintf(stderr,"syntax error: fields entry (%d): unrecognized character '%c'\n",j,str_fields.str[j]);
+	  exit(-1);
+	}
+	break;
+
 	break;
       }
     }
   }
   free(buffer);
+
+  // check whether there are non-Fermionic internal fields
+  for(i=0;i<(*fields).internal.length;i++){
+    if(is_fermion((*fields).internal.values[i], *fields)==0){
+      fprintf(stderr,"warning: some internal fields are not Fermions: means may be computed using sums over permutations rather than the more efficient LU decomposition\n");
+      break;
+    }
+  }
+
   return(0);
 }
 
 
-// parse symbols list
+// parse virtual_fields list
 // write result to fields
-int parse_input_symbols(Char_Array str_symbols, Fields_Table* fields){
+int parse_input_virtual_fields(Char_Array str_virtual_fields, Fields_Table* fields, Variables variables){
   // buffer
-  char* buffer=calloc(str_symbols.length+1,sizeof(char));
+  char* buffer=calloc(str_virtual_fields.length+1,sizeof(char));
   char* buffer_ptr=buffer;
   Polynomial polynomial;
   int index;
   int i,j;
   int mode;
   int comment=0;
+  int res;
 
   // loop over input
   mode=PP_INDEX_MODE;
-  for(j=0;j<str_symbols.length;j++){
+  for(j=0;j<str_virtual_fields.length;j++){
     if(comment==1){
-      if(str_symbols.str[j]=='\n'){
+      if(str_virtual_fields.str[j]=='\n'){
 	comment=0;
       }
     }
     // stay in polynomial mode until ','
     else if(mode==PP_POLYNOMIAL_MODE){
-      if(str_symbols.str[j]==','){
+      if(str_virtual_fields.str[j]==','){
 	// parse polynomial
-	str_to_Polynomial(buffer, &polynomial);
+	parse_symbolic_expression_str(buffer, *fields, variables, &polynomial);
 	// write index and polynomial
-	symbols_append_noinit(index, polynomial, &((*fields).symbols));
+	virtual_fields_append_noinit(index, polynomial, &((*fields).virtual_fields));
 	mode=PP_INDEX_MODE;
 	buffer_ptr=buffer;
 	*buffer_ptr='\0';
       }
+      // check there are no '=' signs in polynomial (forgotten ',')
+      else if(str_virtual_fields.str[j]=='='){
+	fprintf(stderr,"syntax error: virtual fields entry (%d): '=' in polynomial '%s'\n",j, buffer);
+	exit(-1);
+      }
+      else if(str_virtual_fields.str[j]=='#'){
+	comment=1;
+      }
       else{
-	buffer_ptr=str_addchar(buffer_ptr,str_symbols.str[j]);
+	buffer_ptr=str_addchar(buffer_ptr,str_virtual_fields.str[j]);
       }
     }
     else{
-      switch(str_symbols.str[j]){
+      switch(str_virtual_fields.str[j]){
       // polynomial mode
       case '=':
 	if(mode==PP_INDEX_MODE){
 	  // read index
-	  sscanf(buffer,"%d",&index);
+	  res=read_positive_int(buffer, &index);
+	  if(res<0){
+	    fprintf(stderr,"syntax error: virtual fields entry (%d): virtual field index must be a non-negative integer, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
 	  // reset buffer
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
@@ -226,43 +417,53 @@ int parse_input_symbols(Char_Array str_symbols, Fields_Table* fields){
 	}
 	break;
 
+      // misplaced ','
+      case ',':
+	fprintf(stderr,"syntax error: virtual fields entry (%d): misplaced ',' or forgotten virtual field index\n",j);
+	exit(-1);
+	break;
+
+      // ignore spaces and newlines
+      case ' ':break;
+      case '\n':break;
+
       // comment
       case '#':
 	comment=1;
 	break;
 
       default:
-	buffer_ptr=str_addchar(buffer_ptr,str_symbols.str[j]);
+	buffer_ptr=str_addchar(buffer_ptr,str_virtual_fields.str[j]);
 	break;
       }
     }
   }
 
   // last step
-  if(polynomial.length>0){
-    str_to_Polynomial(buffer, &polynomial);
-    symbols_append_noinit(index, polynomial, &((*fields).symbols));
-  }
+  parse_symbolic_expression_str(buffer, *fields, variables, &polynomial);
+  virtual_fields_append_noinit(index, polynomial, &((*fields).virtual_fields));
 
   // simplify
-  for(i=0;i<(*fields).symbols.length;i++){
-    polynomial_simplify((*fields).symbols.expr+i, *fields);
+  for(i=0;i<(*fields).virtual_fields.length;i++){
+    polynomial_simplify((*fields).virtual_fields.expr+i, *fields);
   }
 
   free(buffer);
   return(0);
 }
 
+
 // parse groups of independent fields
-int parse_input_groups(Char_Array str_groups, Groups* groups){
+int parse_input_groups(Char_Array str_groups, Groups* groups, Polynomial_Matrix propagator, Fields_Table fields){
   // buffer
   char* buffer=calloc(str_groups.length+1,sizeof(char));
   char* buffer_ptr=buffer;
   int index;
-  int j;
+  int i,j;
   Int_Array group;
   int mode;
   int comment=0;
+  int res;
 
   // alloc
   init_Groups(groups, GROUP_SIZE);
@@ -287,26 +488,60 @@ int parse_input_groups(Char_Array str_groups, Groups* groups){
 	  init_Int_Array(&group, GROUP_SIZE);
 	  mode=PP_GROUP_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: groups entry (%d): nested parentheses are not allowed in this entry\n",j);
+	  exit(-1);
+	}
 	break;
       case')':
 	if(mode==PP_GROUP_MODE){
-	  sscanf(buffer,"%d",&index);
+	  res=read_positive_int(buffer,&index);
+	  if(res<0){
+	    fprintf(stderr,"syntax error: groups entry (%d): index must be a non-negative integer, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
+	  for(i=0;i<groups->length;i++){
+	    if(int_array_find(index,groups->indices[i])>=0){
+	      fprintf(stderr,"syntax error: groups entry (%d): index %d is already present in group %d\n",j,index,i);
+	      exit(-1);
+	    }
+	  }
+
 	  int_array_append(index, &group);
 	  groups_append_noinit(group, groups);
 	  mode=PP_NULL_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: groups entry (%d): extra ')'\n",j);
+	  exit(-1);
+	}
 	break;
 
       // read index
       case',':
 	if(mode==PP_GROUP_MODE){
-	  sscanf(buffer,"%d",&index);
+	  res=read_positive_int(buffer,&index);
+	  if(res<0){
+	    fprintf(stderr,"syntax errorL groups entry (%d): index must be a non-negative integer, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
+	  for(i=0;i<groups->length;i++){
+	    if(int_array_find(index,groups->indices[i])>=0){
+	      fprintf(stderr,"syntax error: groups entry (%d): index %d is already present in group %d\n",j,index,i);
+	      exit(-1);
+	    }
+	  }
+
 	  int_array_append(index, &group);
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	}
 	break;
 
+      // skip spaces and newlines
+      case ' ':break;
+      case '\n':break;
+
       // comment
       case '#':
 	comment=1;
@@ -316,11 +551,204 @@ int parse_input_groups(Char_Array str_groups, Groups* groups){
 	if(mode!=PP_NULL_MODE){
 	  buffer_ptr=str_addchar(buffer_ptr,str_groups.str[j]);
 	}
+	else{
+	  fprintf(stderr,"syntax error: groups entry (%d): unrecognized character '%c'\n",j,str_groups.str[j]);
+	  exit(-1);
+	}
 	break;
       }
     }
   }
 
+  // check fields in groups
+  check_groups(*groups, propagator, fields);
+
+  free(buffer);
+  return(0);
+}
+
+// check that the members of groups are independent (assuming the virtual fields and propagator were already parsed)
+int check_groups(Groups groups, Polynomial_Matrix propagator, Fields_Table fields){
+  Int_Array* group_fields=calloc(groups.length,sizeof(Int_Array));
+  int i,j,k,l,a,b;
+
+  // get the lists of fields involved in each group
+  for(i=0;i<groups.length;i++){
+    // expand virtual_fields
+    fields_in_virtual_field_list(groups.indices[i], fields, group_fields+i);
+  }
+
+  // check whether pairs of fields are present in the propagator
+  for(i=0;i<groups.length;i++){
+    for(j=0;j<group_fields[i].length;j++){
+      if(field_type(group_fields[i].values[j], fields)==FIELD_INTERNAL){
+	for(k=0;k<groups.length;k++){
+	  if(k!=i){
+	    for(l=0;l<group_fields[k].length;l++){
+	      if(field_type(group_fields[k].values[l], fields)==FIELD_INTERNAL){
+		if(group_fields[i].values[j]==group_fields[k].values[l]){
+		  fprintf(stderr,"error: groups %d and %d both contain internal field %d\n",i,k,group_fields[i].values[j]);
+		  exit(-1);
+		}
+		a=intlist_find(propagator.indices, propagator.length, group_fields[i].values[j]);
+		b=intlist_find(propagator.indices, propagator.length, group_fields[k].values[l]);
+		if(a>=0 && b>=0 && polynomial_is_zero(propagator.matrix[a][b])==0){
+		  fprintf(stderr,"error: groups %d and %d are not independent: they contain %d and %d which have a non-vanishing propagator\n",i,k,group_fields[i].values[j], group_fields[k].values[l]);
+		  exit(-1);
+		}
+	      }
+	    }
+	  }
+	}
+      }
+    }
+  }
+
+  for(i=0;i<groups.length;i++){
+    free_Int_Array(group_fields[i]);
+  }
+  free(group_fields);
+
+  return(0);
+}
+
+// list of fields involved in a list of virtual_fields
+int fields_in_virtual_field_list(Int_Array indices, Fields_Table fields, Int_Array* output){
+  int i,j,k;
+  Int_Array tmp_array;
+
+  init_Int_Array(output,FIELDS_SIZE);
+  for(k=0;k<indices.length;k++){
+    if(field_type(indices.values[k],fields)!=FIELD_VIRTUAL){
+      int_array_append(abs(indices.values[k]), output);
+    }
+    else{
+      i=intlist_find_err(fields.virtual_fields.indices, fields.virtual_fields.length, indices.values[k]);
+      for(j=0;j<fields.virtual_fields.expr[i].length;j++){
+	fields_in_virtual_field_list(fields.virtual_fields.expr[i].monomials[j], fields, &tmp_array);
+	int_array_concat_unique(tmp_array, output);
+	free_Int_Array(tmp_array);
+      }
+    }
+  }
+
+  return(0);
+}
+
+
+
+// parse variables list
+int parse_input_variables(Char_Array str_variables, Variables* variables){
+  // buffer
+  char* buffer=calloc(str_variables.length+1,sizeof(char));
+  char* buffer_ptr=buffer;
+  Tree symbol_tree;
+  Char_Array varname;
+  int j;
+  int mode;
+  int comment=0;
+  char* ptr;
+
+  init_Variables(variables,VARIABLES_SIZE);
+
+  // loop over input
+  mode=PP_INDEX_MODE;
+  for(j=0;j<str_variables.length;j++){
+    if(comment==1){
+      if(str_variables.str[j]=='\n'){
+	comment=0;
+      }
+    }
+    // stay in polynomial mode until ','
+    else if(mode==PP_POLYNOMIAL_MODE){
+      if(str_variables.str[j]==','){
+	// parse symbol tree
+	str_to_symbol_tree(buffer, &symbol_tree);
+	variables_append_noinit(varname , symbol_tree, variables);
+	mode=PP_INDEX_MODE;
+	buffer_ptr=buffer;
+	*buffer_ptr='\0';
+      }
+      // check there are no '=' signs in polynomial (forgotten ',')
+      else if(str_variables.str[j]=='='){
+	fprintf(stderr,"syntax error: preprocessor_variables entry (%d): '=' in polynomial '%s'\n",j, buffer);
+	exit(-1);
+      }
+      else if(str_variables.str[j]=='#'){
+	comment=1;
+      }
+      else{
+	buffer_ptr=str_addchar(buffer_ptr,str_variables.str[j]);
+      }
+    }
+    else{
+      switch(str_variables.str[j]){
+      // polynomial mode
+      case '=':
+	if(mode==PP_INDEX_MODE){
+	  if(*buffer=='\0'){
+	    fprintf(stderr,"syntax error: preprocessor_variables entry (%d): empty variable name\n",j);
+	    exit(-1);
+	  }
+	  for(ptr=buffer;*ptr!='\0';ptr++){
+	    if(*ptr=='<' || *ptr=='>'){
+	      fprintf(stderr,"syntax error: preprocessor_variables entry (%d): forbidden '%c' character in variable name '%s'\n",j,*ptr,buffer);
+	      exit(-1);
+	    }
+	  }
+	  str_to_char_array(buffer,&varname);
+	  // check that the variable name is not 'OUT' or 'FLOW' or 'RCC'
+	  if(char_array_cmp_str(varname, "OUT")==1){
+	    fprintf(stderr,"error: preprocessor_variables entry (%d): variable name cannot be 'OUT' (this name is reserved)\n",j);
+	    exit(-1);
+	  }
+	  if(char_array_cmp_str(varname, "FLOW")==1){
+	    fprintf(stderr,"error: preprocessor_variables entry (%d): variable name cannot be 'FLOW' (this name is reserved)\n",j);
+	    exit(-1);
+	  }
+	  if(char_array_cmp_str(varname, "RCC")==1){
+	    fprintf(stderr,"error: preprocessor_variables entry (%d): variable name cannot be 'RCC' (this name is reserved)\n",j);
+	    exit(-1);
+	  }
+	  // reset buffer
+	  buffer_ptr=buffer;
+	  *buffer_ptr='\0';
+	  mode=PP_POLYNOMIAL_MODE;
+	}
+	break;
+
+      // misplaced ','
+      case ',':
+	fprintf(stderr,"syntax error: preprocessor_variables entry (%d): misplaced ',' or forgotten variable name\n",j);
+	exit(-1);
+	break;
+
+      // ignore characters
+      case ' ':break;
+      case '\n': break;
+
+      // comment
+      case '#':
+	comment=1;
+	break;
+
+      default:
+	buffer_ptr=str_addchar(buffer_ptr,str_variables.str[j]);
+	break;
+      }
+    }
+  }
+
+  // last step
+  if(mode==PP_POLYNOMIAL_MODE){
+    str_to_symbol_tree(buffer, &symbol_tree);
+    variables_append_noinit(varname , symbol_tree, variables);
+  }
+  else if(*buffer!='\0'){
+    fprintf(stderr,"syntax error: preprocessor_variables entry (%d): mismatched variable/polynomial pair, got '%s'\n",j,buffer);
+    exit(-1);
+  }
+
   free(buffer);
   return(0);
 }
@@ -328,7 +756,7 @@ int parse_input_groups(Char_Array str_groups, Groups* groups){
 
 // parse identities between fields
 // write result to fields
-int parse_input_identities(Char_Array str_identities, Fields_Table* fields){
+int parse_input_identities(Char_Array str_identities, Fields_Table* fields, Variables variables){
   // buffer
   char* buffer=calloc(str_identities.length+1,sizeof(char));
   char* buffer_ptr=buffer;
@@ -339,6 +767,7 @@ int parse_input_identities(Char_Array str_identities, Fields_Table* fields){
   int tmp;
   int mode;
   int comment=0;
+  int ret;
 
   init_Int_Array(&monomial, MONOMIAL_SIZE);
 
@@ -354,13 +783,21 @@ int parse_input_identities(Char_Array str_identities, Fields_Table* fields){
     else if(mode==PP_POLYNOMIAL_MODE){
       if(str_identities.str[j]==','){
 	// parse polynomial
-	str_to_Polynomial(buffer, &polynomial);
+	parse_symbolic_expression_str(buffer, *fields, variables, &polynomial);
 	// write monomial and polynomial
 	identities_append_noinit(monomial, polynomial, &((*fields).ids));
 	// realloc
 	init_Int_Array(&monomial, MONOMIAL_SIZE);
 	mode=PP_NULL_MODE;
       }
+      // check there are no '=' signs in polynomial (forgotten ',')
+      else if(str_identities.str[j]=='='){
+	fprintf(stderr,"syntax error: identities entry (%d): '=' in polynomial '%s'\n",j, buffer);
+	exit(-1);
+      }
+      else if(str_identities.str[j]=='#'){
+	comment=1;
+      }
       else{
 	buffer_ptr=str_addchar(buffer_ptr,str_identities.str[j]);
       }
@@ -377,20 +814,36 @@ int parse_input_identities(Char_Array str_identities, Fields_Table* fields){
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	}
+	else{
+	  fprintf(stderr,"syntax error: identities entry (%d): misplaced '['\n",j);
+	  exit(-1);
+	}
 	break;
       // for notational homogeneity
       case 'f':
 	if(mode==PP_BRACKET_MODE){
 	  mode=PP_FIELD_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: identities entry (%d): misplaced 'f'\n",j);
+	  exit(-1);
+	}
 	break;
       // write monomial term
       case ']':
 	if(mode==PP_FIELD_MODE){
-	  sscanf(buffer,"%d",&i);
+	  ret=read_int(buffer,&i);
+	  if(ret<0){
+	    fprintf(stderr,"syntax error: identities entry (%d): field indices must be integers, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
 	  int_array_append(i,&monomial);
 	  mode=PP_INDEX_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: identities entry (%d): mismatched ']'\n",j);
+	  exit(-1);
+	}
 	break;
 
       // polynomial mode
@@ -400,8 +853,22 @@ int parse_input_identities(Char_Array str_identities, Fields_Table* fields){
 	  *buffer_ptr='\0';
 	  mode=PP_POLYNOMIAL_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: identities entry (%d): misplaced '='\n",j);
+	  exit(-1);
+	}
 	break;
 
+      // misplaced ','
+      case ',':
+	fprintf(stderr,"syntax error: identities entry (%d): misplaced ',' or forgotten left hand side\n",j);
+	exit(-1);
+	break;
+
+      // ignore spaces and newlines
+      case ' ':break;
+      case '\n':break;
+
       // comment
       case '#':
 	comment=1;
@@ -411,6 +878,10 @@ int parse_input_identities(Char_Array str_identities, Fields_Table* fields){
 	if(mode!=PP_NULL_MODE){
 	  buffer_ptr=str_addchar(buffer_ptr,str_identities.str[j]);
 	}
+	else{
+	  fprintf(stderr,"syntax error: identities entry (%d): unrecognized character '%c'\n",j,str_identities.str[j]);
+	  exit(-1);
+	}
 	break;
       }
     }
@@ -418,9 +889,13 @@ int parse_input_identities(Char_Array str_identities, Fields_Table* fields){
 
   // last step
   if(mode==PP_POLYNOMIAL_MODE){
-    str_to_Polynomial(buffer, &polynomial);
+    parse_symbolic_expression_str(buffer, *fields, variables, &polynomial);
     identities_append_noinit(monomial, polynomial, &((*fields).ids));
   }
+  else if(*buffer!='\0'){
+    fprintf(stderr,"syntax error: identities entry (%d): mismatched left/right hand side, got '%s'\n",j,buffer);
+    exit(-1);
+  }
   else{
     free_Int_Array(monomial);
   }
@@ -451,6 +926,7 @@ int parse_input_propagator(Char_Array str_propagator, Polynomial_Matrix* propaga
   int index2=-1;
   int mode;
   int comment=0;
+  int ret;
 
   // allocate memory
   init_Polynomial_Matrix(propagator, fields.internal.length);
@@ -473,20 +949,36 @@ int parse_input_propagator(Char_Array str_propagator, Polynomial_Matrix* propaga
       // indices
       case ';':
 	if(mode==PP_INDEX_MODE){
-	  sscanf(buffer,"%d",&i);
+	  ret=read_positive_int(buffer,&i);
+	  if(ret<0){
+	    fprintf(stderr,"syntax error: propagator entry (%d): indices must be non-negative integers, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
 	  index1=intlist_find_err((*propagator).indices, (*propagator).length, i);
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	}
+	else{
+	  fprintf(stderr,"syntax error: propagator entry (%d): misplaced ';'\n",j);
+	  exit(-1);
+	}
 	break;
       case ':':
 	if(mode==PP_INDEX_MODE){
-	  sscanf(buffer,"%d",&i);
+	  ret=read_positive_int(buffer,&i);
+	  if(ret<0){
+	    fprintf(stderr,"syntax error: propagator entry (%d): indices must be non-negative integers, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
 	  index2=intlist_find_err((*propagator).indices, (*propagator).length, i);
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	  mode=PP_POLYNOMIAL_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: propagator entry (%d): misplaced ':'\n",j);
+	  exit(-1);
+	}
 	break;
 
       // num
@@ -494,12 +986,22 @@ int parse_input_propagator(Char_Array str_propagator, Polynomial_Matrix* propaga
 	if(mode==PP_POLYNOMIAL_MODE && index1>=0 && index2>=0){
 	  free_Polynomial((*propagator).matrix[index1][index2]);
 	  str_to_Polynomial(buffer,(*propagator).matrix[index1]+index2);
+	  index1=-1;
+	  index2=-1;
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	  mode=PP_INDEX_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: propagator entry (%d): mismatched index/polynomial pair\n",j);
+	  exit(-1);
+	}
 	break;
 
+      // ignore ' 'and newline
+      case ' ':break;
+      case '\n':break;
+
       // comment
       case '#':
 	comment=1;
@@ -517,6 +1019,15 @@ int parse_input_propagator(Char_Array str_propagator, Polynomial_Matrix* propaga
     free_Polynomial((*propagator).matrix[index1][index2]);
     str_to_Polynomial(buffer,(*propagator).matrix[index1]+index2);
   }
+  else if (*buffer!='\0'){
+    fprintf(stderr,"syntax error: propagator entry (%d): mismatched index/polynomial pair, got '%s'\n",j,buffer);
+    exit(-1);
+  }
+
+  // check whether the polynomial is numeric
+  if(polynomial_matrix_is_numeric(*propagator)==0){
+    fprintf(stderr,"warning: the propagator contains polynomial entries: means may be computed using sums over permutations rather than the more efficient LU decomposition\n");
+  }
 
   free(buffer);
   return(0);
@@ -524,49 +1035,9 @@ int parse_input_propagator(Char_Array str_propagator, Polynomial_Matrix* propaga
 
 
 // parse input polynomial
-int parse_input_polynomial(Char_Array str_polynomial, Polynomial* output, Fields_Table fields){
-  int j;
-  // buffer
-  char* buffer=calloc(str_polynomial.length+1,sizeof(char));
-  char* buffer_ptr=buffer;
-  Polynomial tmp_poly;
-
-  // allocate memory
-  init_Polynomial(output,POLY_SIZE);
-
-  for(j=0;j<str_polynomial.length;j++){
-    switch(str_polynomial.str[j]){
-    case '*':
-      str_to_Polynomial(buffer, &tmp_poly);
-      if((*output).length==0){
-	polynomial_concat(tmp_poly, output);
-      }
-      else{
-	polynomial_prod_chain(tmp_poly, output, fields);
-      }
-      free_Polynomial(tmp_poly);
-
-      buffer_ptr=buffer;
-      *buffer_ptr='\0';
-      break;
-
-    default:
-      buffer_ptr=str_addchar(buffer_ptr,str_polynomial.str[j]);
-      break;
-    }
-  }
-
-  //last step
-  str_to_Polynomial(buffer, &tmp_poly);
-  if((*output).length==0){
-    polynomial_concat(tmp_poly, output);
-  }
-  else{
-    polynomial_prod_chain(tmp_poly, output, fields);
-  }
-  free_Polynomial(tmp_poly);
-
-  free(buffer);
+int parse_input_polynomial(Char_Array str_polynomial, Polynomial* output, Fields_Table fields, Variables variables){
+  parse_symbolic_expression(str_polynomial, fields, variables, output);
+  polynomial_simplify(output, fields);
   return(0);
 }
 
@@ -574,7 +1045,7 @@ int parse_input_polynomial(Char_Array str_polynomial, Polynomial* output, Fields
 
 // parse id table
 // fields argument for sorting
-int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table fields){
+int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table fields, Variables variables){
   // buffer
   char* buffer=calloc(str_idtable.length+1,sizeof(char));
   char* buffer_ptr=buffer;
@@ -583,6 +1054,7 @@ int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table
   int j;
   int mode;
   int comment=0;
+  int ret;
 
   // allocate memory
   init_Id_Table(idtable,EQUATION_SIZE);
@@ -601,24 +1073,40 @@ int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table
       case ',':
 	// write polynomial
 	if(mode==PP_POLYNOMIAL_MODE){
-	  str_to_Polynomial(buffer,&polynomial);
+	  parse_symbolic_expression_str(buffer, fields, variables, &polynomial);
 	  // add to idtable
 	  idtable_append_noinit(index,polynomial,idtable);
 	  mode=PP_INDEX_MODE;
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	  }
+	else{
+	  fprintf(stderr,"syntax error: id_table entry (%d): misplaced ','\n",j);
+	  exit(-1);
+	}
 	break;
 
       case ':':
 	if(mode==PP_INDEX_MODE){
-	  sscanf(buffer,"%d",&index);
+	  ret=read_positive_int(buffer,&index);
+	  if(ret<0){
+	    fprintf(stderr,"syntax error: id_table entry (%d): index must be a non-negative integer, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
 	  mode=PP_POLYNOMIAL_MODE;
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	}
+	else{
+	  fprintf(stderr,"syntax error: id_table entry (%d): misplaced ':'\n",j);
+	  exit(-1);
+	}
 	break;
 
+      // ignore ' ' and '\n'
+      case ' ':break;
+      case '\n':break;
+
       // comment
       case '#':
 	comment=1;
@@ -628,6 +1116,10 @@ int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table
 	if(mode!=PP_NULL_MODE){
 	  buffer_ptr=str_addchar(buffer_ptr,str_idtable.str[j]);
 	}
+	else{
+	  fprintf(stderr,"syntax error: id_table entry (%d): unrecognized character '%c'\n",j, str_idtable.str[j]);
+	  exit(-1);
+	}
 	break;
       }
     }
@@ -635,9 +1127,13 @@ int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table
 
   //last step
   if(mode==PP_POLYNOMIAL_MODE){
-    str_to_Polynomial(buffer,&polynomial);
+    parse_symbolic_expression_str(buffer, fields, variables, &polynomial);
     idtable_append_noinit(index,polynomial,idtable);
   }
+  else if(*buffer!='\0'){
+    fprintf(stderr,"syntax error: id_table entry (%d): mismatched index/polynomial pair, got '%s'\n",j,buffer);
+    exit(-1);
+  }
 
   // sort
   for(j=0;j<(*idtable).length;j++){
@@ -658,6 +1154,7 @@ int parse_labels(Char_Array str_labels, Labels* labels){
   int j;
   int mode;
   int comment=0;
+  int ret;
 
   // allocate memory
   init_Labels(labels,EQUATION_SIZE);
@@ -686,18 +1183,30 @@ int parse_labels(Char_Array str_labels, Labels* labels){
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	}
+	else{
+	  fprintf(stderr,"syntax error: labels entry (%d): mismatched '\"'\n",j);
+	  exit(-1);
+	}
 	break;
 
       case ':':
 	// write
 	if(mode==PP_INDEX_MODE){
 	  sscanf(buffer,"%d",&index);
+	  ret=read_positive_int(buffer,&index);
+	  if(ret<0){
+	    fprintf(stderr,"syntax error: labels entry (%d): index must be a non-negative integer, got '%s'\n",j,buffer);
+	    exit(-1);
+	  }
+	}
+	else{
+	  fprintf(stderr,"syntax error: labels entry (%d): misplaced ':'\n",j);
+	  exit(-1);
 	}
 	break;
 
       // characters to ignore
       case ' ':break;
-      case '&':break;
       case '\n':break;
       case ',':break;
 
@@ -726,6 +1235,7 @@ int parse_init_cd(Char_Array init_cd, RCC* init, RCC_mpfr* init_mpfr, int mpfr_f
   int i,j;
   int comment_mode=0;
   int dcount=0;
+  int ret;
 
   *buffer_ptr='\0';
   // loop over the input
@@ -741,7 +1251,11 @@ int parse_init_cd(Char_Array init_cd, RCC* init, RCC_mpfr* init_mpfr, int mpfr_f
       case ',':
 	// write init
 	if(mpfr_flag==0){
-	  sscanf(buffer,"%Lf",(*init).values+intlist_find_err((*init).indices,(*init).length,index));
+	  ret=read_long_double(buffer,(*init).values+intlist_find_err((*init).indices,(*init).length,index));
+	  if(ret<0){
+	    fprintf(stderr,"syntax error: initial_condition entry (%d): failed to extract long double from '%s'\n",j,buffer);
+	    exit(-1);
+	  }
 	}
 	else{
 	  mpfr_strtofr((*init_mpfr).values[intlist_find_err((*init_mpfr).indices,(*init_mpfr).length,index)], buffer, &buffer_ptr, 10, MPFR_RNDN);
@@ -755,7 +1269,11 @@ int parse_init_cd(Char_Array init_cd, RCC* init, RCC_mpfr* init_mpfr, int mpfr_f
       // separator
       case ':':
 	// write index
-	sscanf(buffer,"%d",&i);
+	ret=read_int(buffer,&i);
+	if(ret<0){
+	  fprintf(stderr,"syntax error: initial_condition entry (%d): index must be an integer, got '%s'\n",j,buffer);
+	  exit(-1);
+	}
 	if(i<0){
 	  index=i-dcount*DOFFSET;
 	}
@@ -772,7 +1290,6 @@ int parse_init_cd(Char_Array init_cd, RCC* init, RCC_mpfr* init_mpfr, int mpfr_f
 
       // characters to ignore
       case ' ':break;
-      case '&':break;
       case '\n':break;
 
       // comments
@@ -790,7 +1307,11 @@ int parse_init_cd(Char_Array init_cd, RCC* init, RCC_mpfr* init_mpfr, int mpfr_f
 
   // write init
   if(mpfr_flag==0){
-    sscanf(buffer,"%Lf",(*init).values+intlist_find_err((*init).indices,(*init).length,index));
+    ret=read_long_double(buffer,(*init).values+intlist_find_err((*init).indices,(*init).length,index));
+    if(ret<0){
+      fprintf(stderr,"syntax error: initial_condition entry (%d): failed to extract long double from '%s'\n",j,buffer);
+      exit(-1);
+    }
   }
   else{
     mpfr_strtofr((*init_mpfr).values[intlist_find_err((*init_mpfr).indices,(*init_mpfr).length,index)], buffer, &buffer_ptr, 10, MPFR_RNDN);
diff --git a/src/parse_file.h b/src/parse_file.h
index 0430763..e22393e 100644
--- a/src/parse_file.h
+++ b/src/parse_file.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -23,26 +23,43 @@ Parse the input file
 
 #include "types.h"
 
+// read a positive integer from a string
+int read_positive_int(char* str, int* out);
+// read an integer from a string
+int read_int(char* str, int* out);
+// read an long int from a string
+int read_long_int(char* str, long int* out);
+// read a long double
+int read_long_double(char* str, long double* out);
+
+
 // parse fields list
 int parse_input_fields(Char_Array str_fields, Fields_Table* fields);
 
-// parse symbols list
-int parse_input_symbols(Char_Array str_symbols, Fields_Table* fields);
+// parse virtual_fields list
+int parse_input_virtual_fields(Char_Array str_virtual_fields, Fields_Table* fields, Variables variables);
 
 // parse groups of independent fields
-int parse_input_groups(Char_Array str_groups, Groups* groups);
+int parse_input_groups(Char_Array str_groups, Groups* groups, Polynomial_Matrix propagator, Fields_Table fields);
+// check that the members of groups are independent (assuming the virtual_fields and propagator were already parsed)
+int check_groups(Groups groups, Polynomial_Matrix propagator, Fields_Table fields);
+// list of fields involved in a list of virtual_fields
+int fields_in_virtual_field_list(Int_Array indices, Fields_Table fields, Int_Array* output);
+
+// parse variables list
+int parse_input_variables(Char_Array str_variables, Variables* variables);
 
 // parse identities between fields
-int parse_input_identities(Char_Array str_identities, Fields_Table* fields);
+int parse_input_identities(Char_Array str_identities, Fields_Table* fields, Variables variables);
 
 // parse propagator
 int parse_input_propagator(Char_Array str_propagator, Polynomial_Matrix* propagator, Fields_Table fields);
 
 // parse input polynomial
-int parse_input_polynomial(Char_Array str_polynomial, Polynomial* output, Fields_Table fields);
+int parse_input_polynomial(Char_Array str_polynomial, Polynomial* output, Fields_Table fields, Variables variables);
 
 // parse id table
-int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table fields);
+int parse_input_id_table(Char_Array str_idtable, Id_Table* idtable, Fields_Table fields, Variables variables);
 
 // parse a list of labels
 int parse_labels(Char_Array str_labels, Labels* labels);
diff --git a/src/polynomial.c b/src/polynomial.c
index d082b60..fae0c08 100644
--- a/src/polynomial.c
+++ b/src/polynomial.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -27,6 +27,7 @@ limitations under the License.
 #include "array.h"
 #include "number.h"
 #include "fields.h"
+#include "parse_file.h"
 
 
 // allocate memory
@@ -368,7 +369,7 @@ int polynomial_prod_chain_nosimplify(Polynomial input, Polynomial* inout, Fields
   Int_Array out_monomial;
   Int_Array out_factor;
   Number out_num;
-  // save length of inout (which changes during the loop
+  // save length of inout (which changes during the loop)
   int inout_length=(*inout).length;
   // first position in input which can multiply a term of inout without vanishing
   int firstpos;
@@ -630,8 +631,8 @@ int remove_unmatched_plusminus(Polynomial* polynomial, Fields_Table fields){
 	  match_internals--;
 	}
       }
-      // don't remove a term containing symbols
-      else if(type==FIELD_SYMBOL){
+      // don't remove a term containing virtual_field
+      else if(type==FIELD_VIRTUAL){
 	match_internals=0;
 	break;
       }
@@ -1093,7 +1094,7 @@ int polynomial_print(Polynomial polynomial){
   Char_Array buffer;
   init_Char_Array(&buffer, STR_SIZE);
   polynomial_sprint(polynomial, &buffer);
-  printf("%s",buffer.str);
+  printf("%s",char_array_to_str_noinit(&buffer));
   free_Char_Array(buffer);
   return(0);
 }
@@ -1104,7 +1105,7 @@ int polynomial_print(Polynomial polynomial){
 #define PP_MONOMIAL_MODE 2
 #define PP_FACTOR_MODE 3
 #define PP_NUMBER_MODE 4
-int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
+int Char_Array_to_Polynomial(Char_Array str_polynomial, Polynomial* output){
   // buffer
   char* buffer=calloc(str_polynomial.length+1,sizeof(char));
   char* buffer_ptr=buffer;
@@ -1115,6 +1116,7 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
   int comment=0;
   int i,j;
   int parenthesis_count=0;
+  int ret;
 
   // allocate memory
   init_Polynomial(output,POLY_SIZE);
@@ -1174,6 +1176,10 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 	  init_Int_Array(&factor, MONOMIAL_SIZE);
 	  num=number_one();
 	}
+	else{
+	  fprintf(stderr,"syntax error: misplaced '+' in polynomial\n");
+	  exit(-1);
+	}
 	break;
 
       // enter monomial or factor mode
@@ -1181,6 +1187,10 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 	if(mode==PP_NULL_MODE){
 	  mode=PP_BRACKET_MODE;
 	}
+	else{
+	  fprintf(stderr,"syntax error: misplaced '[' in polynomial\n");
+	  exit(-1);
+	}
 	break;
       // factor mode
       case 'l':
@@ -1189,6 +1199,10 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	}
+	else{
+	  fprintf(stderr,"syntax error: misplaced 'l' in polynomial\n");
+	  exit(-1);
+	}
 	break;
       // monomial mode
       case 'f':
@@ -1197,16 +1211,30 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 	  buffer_ptr=buffer;
 	  *buffer_ptr='\0';
 	}
+	else{
+	  fprintf(stderr,"syntax error: misplaced 'j' in polynomial\n");
+	  exit(-1);
+	}
 	break;
       // read monomial or factor
       case ']':
 	sscanf(buffer,"%d",&i);
+	ret=read_int(buffer,&i);
+	if(ret<0){
+	  fprintf(stderr,"syntax error: in polynomial, expected integer field or factor index, got '%s'\n",buffer);
+	  exit(-1);
+	}
+	  
 	if(mode==PP_FACTOR_MODE){
 	  int_array_append(i,&factor);
 	}
 	else if(mode==PP_MONOMIAL_MODE){
 	  int_array_append(i,&monomial);
 	}
+	else{
+	  fprintf(stderr,"syntax error: mismatched ']' in polynomial\n");
+	  exit(-1);
+	}
 	// switch back to null mode
 	mode=PP_NULL_MODE;
 	break;
@@ -1224,6 +1252,10 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 	  parenthesis_count++;
 	  buffer_ptr=str_addchar(buffer_ptr,str_polynomial.str[j]);
 	}
+	else{
+	  fprintf(stderr,"syntax error: misplaced '(' in polynomial\n");
+	  exit(-1);
+	}
 	break;
       case ')':
 	if(mode==PP_NUMBER_MODE){
@@ -1240,11 +1272,14 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 	    buffer_ptr=str_addchar(buffer_ptr,str_polynomial.str[j]);
 	  }
 	}
+	else{
+	  fprintf(stderr,"syntax error: mismatched ')' in polynomial\n");
+	  exit(-1);
+	}
 	break;
 
       // characters to ignore
       case ' ':break;
-      case '&':break;
       case '\n':break;
       
       // comments
@@ -1257,6 +1292,10 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 	  // write to buffer
 	  buffer_ptr=str_addchar(buffer_ptr,str_polynomial.str[j]);
 	}
+	else{
+	  fprintf(stderr,"syntax error: in polynomial, unrecognized character '%c'\n",str_polynomial.str[j]);
+	  exit(-1);
+	}
 	break;
       }
     }
@@ -1270,7 +1309,7 @@ int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output){
 }
 
 // with str input
-int str_to_Polynomial(char* str_polynomial,Polynomial* output){
+int str_to_Polynomial(char* str_polynomial, Polynomial* output){
   Char_Array buffer;
   str_to_char_array(str_polynomial, &buffer);
   Char_Array_to_Polynomial(buffer, output);
@@ -1278,6 +1317,15 @@ int str_to_Polynomial(char* str_polynomial,Polynomial* output){
   return(0);
 }
 
+// check whether the polynomial is a constant
+int polynomial_is_number(Polynomial poly){
+  if(poly.length==0 || (poly.length==1 && poly.monomials[0].length==0 && poly.factors[0].length==0)){
+    return(1);
+  }
+  else{
+    return(0);
+  }
+}
 
 // -------------------- Polynomial_Matrix ---------------------
 
@@ -1307,3 +1355,16 @@ int free_Polynomial_Matrix(Polynomial_Matrix matrix){
   free(matrix.indices);
   return(0);
 }
+
+// check whether the entries are numbers
+int polynomial_matrix_is_numeric(Polynomial_Matrix matrix){
+  int i,j;
+  for(i=0;i<matrix.length;i++){
+    for(j=0;j<matrix.length;j++){
+      if(polynomial_is_number(matrix.matrix[i][j])==0){
+	return(0);
+      }
+    }
+  }
+  return(1);
+}
diff --git a/src/polynomial.h b/src/polynomial.h
index 6aff76d..f3488f5 100644
--- a/src/polynomial.h
+++ b/src/polynomial.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -124,12 +124,18 @@ int replace_factors(Grouped_Polynomial equations, Polynomial* polynomial);
 int polynomial_sprint(Polynomial polynomial, Char_Array* output);
 int polynomial_print(Polynomial polynomial);
 // read a polynomial
-int Char_Array_to_Polynomial(Char_Array str_polynomial,Polynomial* output);
-int str_to_Polynomial(char* str_polynomial,Polynomial* output);
+int Char_Array_to_Polynomial(Char_Array str_polynomial, Polynomial* output);
+int str_to_Polynomial(char* str_polynomial, Polynomial* output);
+
+// check whether the polynomial is a constant
+int polynomial_is_number(Polynomial poly);
 
 //------------------------ Polynomial_Matrix --------------------------
 // init
 int init_Polynomial_Matrix(Polynomial_Matrix* matrix, int length);
 int free_Polynomial_Matrix(Polynomial_Matrix matrix);
 
+// check whether the entries are numbers
+int polynomial_matrix_is_numeric(Polynomial_Matrix matrix);
+
 #endif
diff --git a/src/rational.h b/src/rational.h
index 5beab8d..aea0625 100644
--- a/src/rational.h
+++ b/src/rational.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/rational_float.c b/src/rational_float.c
index d6a6a4c..2538d41 100644
--- a/src/rational_float.c
+++ b/src/rational_float.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/rational_float.h b/src/rational_float.h
index 02c4a22..ef5541b 100644
--- a/src/rational_float.h
+++ b/src/rational_float.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/rational_int.c b/src/rational_int.c
index 7aab561..f3e5475 100644
--- a/src/rational_int.c
+++ b/src/rational_int.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -25,6 +25,7 @@ limitations under the License.
 #include <mpfr.h>
 #include "istring.h"
 #include "array.h"
+#include "parse_file.h"
 
 Q quot(long int p, long int q){
   Q ret;
@@ -171,13 +172,18 @@ int str_to_Q(char* str, Q* num){
   int mode;
   char* buffer=calloc(str_len(str)+1,sizeof(char));
   char* buffer_ptr=buffer;
+  int ret;
 
   *num=quot(0,1);
 
   mode=PP_NUMERATOR_MODE;
   for(ptr=str;*ptr!='\0';ptr++){
     if(*ptr=='/'){
-      sscanf(buffer,"%ld",&((*num).numerator));
+      ret=read_long_int(buffer,&((*num).numerator));
+      if(ret<0){
+	fprintf(stderr,"syntax error: numerator should be an integer, got '%s' in '%s'\n",buffer,str);
+	exit(-1);
+      }
       buffer_ptr=buffer;
       *buffer_ptr='\0';
       mode=PP_DENOMINATOR_MODE;
@@ -189,10 +195,18 @@ int str_to_Q(char* str, Q* num){
 
   // last step
   if(mode==PP_NUMERATOR_MODE){
-    sscanf(buffer,"%ld",&((*num).numerator));
+    ret=read_long_int(buffer,&((*num).numerator));
+    if(ret<0){
+      fprintf(stderr,"syntax error: numerator should be an integer, got '%s' in '%s'\n",buffer,str);
+      exit(-1);
+    }
   }
   else if(mode==PP_DENOMINATOR_MODE){
-    sscanf(buffer,"%ld",&((*num).denominator));
+    ret=read_long_int(buffer,&((*num).denominator));
+    if(ret<0){
+      fprintf(stderr,"syntax error: numerator should be an integer, got '%s' in '%s'\n",buffer,str);
+      exit(-1);
+    }
   }
 
   free(buffer);
diff --git a/src/rational_int.h b/src/rational_int.h
index a4dd000..1aba15d 100644
--- a/src/rational_int.h
+++ b/src/rational_int.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
diff --git a/src/rcc.c b/src/rcc.c
index 484e20a..89d2c96 100644
--- a/src/rcc.c
+++ b/src/rcc.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -49,6 +49,13 @@ int RCC_cpy(RCC input,RCC* output){
   }
   return(0);
 }
+int RCC_cpy_noinit(RCC input,RCC* output){
+  int i;
+  for(i=0;i<input.length;i++){
+    RCC_set_elem(input.values[i], input.indices[i], output, i);
+  }
+  return(0);
+}
 
 // concatenate rccs
 int RCC_concat(RCC rccs1, RCC rccs2, RCC* output){
diff --git a/src/rcc.h b/src/rcc.h
index 770309c..6beeee2 100644
--- a/src/rcc.h
+++ b/src/rcc.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -28,6 +28,7 @@ int free_RCC(RCC rccs);
 int RCC_set_elem(long double value, int index, RCC* rcc, int pos);
 // copy
 int RCC_cpy(RCC input,RCC* output);
+int RCC_cpy_noinit(RCC input,RCC* output);
 // concatenate 2 rccs
 int RCC_concat(RCC rccs1, RCC rccs2, RCC* output);
 // append an rcc to another
diff --git a/src/rcc_mpfr.c b/src/rcc_mpfr.c
index 8a362e3..7703b1e 100644
--- a/src/rcc_mpfr.c
+++ b/src/rcc_mpfr.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -65,6 +65,13 @@ int RCC_mpfr_cpy(RCC_mpfr input,RCC_mpfr* output){
   }
   return(0);
 }
+int RCC_mpfr_cpy_noinit(RCC_mpfr input,RCC_mpfr* output){
+  int i;
+  for(i=0;i<input.length;i++){
+    RCC_mpfr_set_elem(input.values[i], input.indices[i], output, i);
+  }
+  return(0);
+}
 
 // concatenate rcc_mpfr
 int RCC_mpfr_concat(RCC_mpfr rcc_mpfr1, RCC_mpfr rcc_mpfr2, RCC_mpfr* output){
diff --git a/src/rcc_mpfr.h b/src/rcc_mpfr.h
index 18ebed3..940fbb1 100644
--- a/src/rcc_mpfr.h
+++ b/src/rcc_mpfr.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -32,6 +32,7 @@ int free_RCC_mpfr(RCC_mpfr rcc_mpfr);
 int RCC_mpfr_set_elem(mpfr_t value, int index, RCC_mpfr* rcc_mpfr, int pos);
 // copy
 int RCC_mpfr_cpy(RCC_mpfr input,RCC_mpfr* output);
+int RCC_mpfr_cpy_noinit(RCC_mpfr input,RCC_mpfr* output);
 // concatenate 2 rcc_mpfr_mpfr
 int RCC_mpfr_concat(RCC_mpfr rcc_mpfr_mpfr1, RCC_mpfr rcc_mpfr_mpfr2, RCC_mpfr* output);
 // append an rcc_mpfr to another
diff --git a/src/symbolic_parser.c b/src/symbolic_parser.c
new file mode 100644
index 0000000..08cae88
--- /dev/null
+++ b/src/symbolic_parser.c
@@ -0,0 +1,330 @@
+/*
+Copyright 2015-2022 Ian Jauslin
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+#include "symbolic_parser.h"
+#include <stdlib.h>
+#include <stdio.h>
+#include "tree.h"
+#include "definitions.cpp"
+#include "array.h"
+#include "istring.h"
+#include "fields.h"
+#include "polynomial.h"
+
+#define SP_NULL_MODE 0
+#define SP_FUNCTION_MODE 1
+
+// parse a symbolic expression from a char_array
+int parse_symbolic_expression(Char_Array str, Fields_Table fields, Variables variables, Polynomial* polynomial){
+  Tree symbol_tree;
+  char_array_to_symbol_tree(str, &symbol_tree);
+  resolve_symbol_tree(symbol_tree, fields, variables, polynomial);
+  free_Tree(symbol_tree);
+  return(0);
+}
+// from char*
+int parse_symbolic_expression_str(char* str, Fields_Table fields, Variables variables, Polynomial* polynomial){
+  Char_Array char_array;
+  str_to_char_array(str,&char_array);
+  parse_symbolic_expression(char_array, fields, variables, polynomial);
+  free_Char_Array(char_array);
+  return(0);
+}
+
+// compute the symbol tree from a string
+int char_array_to_symbol_tree(Char_Array str, Tree* symbol_tree){
+  // buffer
+  char* buffer=calloc(str.length+1,sizeof(char));
+  char* buffer_ptr=buffer;
+  Tree child;
+  int match;
+  Char_Array nodestr;
+  Char_Array label;
+  Char_Array str_clean;
+  int mode;
+  int comment=0;
+  int j;
+  int gotanode=0;
+
+  // allocate memory
+  init_Tree(symbol_tree,SYMBOL_TREE_SIZE, SYMBOL_TREE_LABEL_SIZE);
+
+  // remove comments, ' ' and '\n'
+  init_Char_Array(&str_clean,str.length);
+  for(j=0;j<str.length;j++){
+    if(comment==1){
+      if(str.str[j]=='\n'){
+	comment=0;
+      }
+    }
+    else{
+      switch(str.str[j]){
+      case ' ':break;
+      case '\n':break;
+      // comments
+      case '#':
+	comment=1;
+	break;
+      default:
+	char_array_append(str.str[j],&str_clean);
+	break;
+      }
+    }
+  }
+
+  // if the string contains no '<', then trivial tree
+  for(j=0;j<str_clean.length;j++){
+    if(str_clean.str[j]=='<'){
+      break;
+    }
+  }
+  // no '<': trivial tree
+  if(j==str_clean.length){
+    tree_set_label(str_clean, symbol_tree);
+    free(buffer);
+    free_Char_Array(str_clean);
+    return(0);
+  }
+
+  *buffer_ptr='\0';
+  // loop over the input string
+  // start in null mode
+  mode=SP_NULL_MODE;
+  for(j=0;j<str_clean.length;j++){
+    switch(str_clean.str[j]){
+    // new node
+    case '<':
+      // find matching bracket
+      match=matching_bracket(str_clean,j);
+      // check whether it exists
+      if(match<0){
+	fprintf(stderr,"syntax error: unmatched brackets in %s\n",char_array_to_str_noinit(&str));
+	exit(-1);
+      }
+      // extract substring until bracket
+      char_array_substring(str_clean,j+1,match-1,&nodestr);
+
+      // check whether node is trivial
+      if(j==0 && match==str_clean.length-1){
+	free_Tree(*symbol_tree);
+	char_array_to_symbol_tree(nodestr, symbol_tree);
+	free_Char_Array(nodestr);
+	j=match;
+	break;
+      }
+
+      // parse subexpression
+      char_array_to_symbol_tree(nodestr, &child);
+      free_Char_Array(nodestr);
+      // add child to tree
+      tree_append_child_noinit(child, symbol_tree);
+      // boolean indicating a node has been found
+      gotanode=1;
+      // set next position after the node
+      j=match;
+
+      // if function mode, then check that the match is at the end of the node
+      if(mode==SP_FUNCTION_MODE){
+	if(match<str_clean.length-1){
+	  fprintf(stderr,"syntax error: functions must occupy an entire node (e.g. <%%exp<...>>), got %s\n",char_array_to_str_noinit(&str));
+	  exit(-1);
+	}
+	else{
+	  // set label
+	  str_to_char_array(buffer,&label);
+	  tree_set_label(label,symbol_tree);
+	  free_Char_Array(label);
+	}
+      }
+      break;
+
+    // function
+    case '%':
+      if(j>0){
+	fprintf(stderr,"syntax error: functions must occupy an entire node (e.g. <%%exp<...>>), got %s\n",char_array_to_str_noinit(&str));
+	exit(-1);
+      }
+      mode=SP_FUNCTION_MODE;
+      break;
+
+    // product
+    case '*':
+      if(gotanode==0){
+	fprintf(stderr,"syntax error: '*' is not preceded by a node in %s\n",char_array_to_str_noinit(&str));
+	exit(-1);
+      }
+      if(j>=str_clean.length-1){
+	fprintf(stderr,"syntax error: '*' cannot be at the end of an expression, got %s\n",char_array_to_str_noinit(&str));
+	exit(-1);
+      }
+      // set label
+      init_Char_Array(&label,1);
+      char_array_append('*',&label);
+      tree_set_label(label,symbol_tree);
+      free_Char_Array(label);
+      // next child
+      char_array_substring(str_clean,j+1,str_clean.length-1,&nodestr);
+      // parse subexpression
+      char_array_to_symbol_tree(nodestr, &child);
+      free_Char_Array(nodestr);
+      // append next child
+      tree_append_child_noinit(child, symbol_tree);
+
+      // make it stop
+      j=str_clean.length-1;
+      break;
+
+    // sum
+    case '+':
+      if(gotanode==0){
+	fprintf(stderr,"syntax error: '+' is not preceded by a node in %s\n",char_array_to_str_noinit(&str));
+	exit(-1);
+      }
+      if(j>=str_clean.length-1){
+	fprintf(stderr,"syntax error: '+' cannot be at the end of an expression, got %s\n",char_array_to_str_noinit(&str));
+	exit(-1);
+      }
+      // set label
+      init_Char_Array(&label,1);
+      char_array_append('+',&label);
+      tree_set_label(label,symbol_tree);
+      free_Char_Array(label);
+      // next child
+      char_array_substring(str_clean,j+1,str_clean.length-1,&nodestr);
+      // parse subexpression
+      char_array_to_symbol_tree(nodestr, &child);
+      free_Char_Array(nodestr);
+      // append next child
+      tree_append_child_noinit(child, symbol_tree);
+
+      // make it stop
+      j=str_clean.length-1;
+      break;
+
+    default:
+      if(mode!=SP_NULL_MODE){
+	// write to buffer
+	buffer_ptr=str_addchar(buffer_ptr,str_clean.str[j]);
+      }
+      break;
+    }
+  }
+
+  free_Char_Array(str_clean);
+  free(buffer);
+  return(0);
+}
+// from char*
+int str_to_symbol_tree(char* str, Tree* symbol_tree){
+  Char_Array char_array;
+  str_to_char_array(str,&char_array);
+  char_array_to_symbol_tree(char_array,symbol_tree);
+  free_Char_Array(char_array);
+  return(0);
+}
+
+
+// find matching '<' and '>'
+int matching_bracket(Char_Array str, int start){
+  int bracket_count=0;
+  int i;
+  for(i=start;i<str.length;i++){
+    if(str.str[i]=='<'){
+      bracket_count++;
+    }
+    else if(str.str[i]=='>'){
+      bracket_count--;
+      if(bracket_count==0){
+	return(i);
+      }
+    }
+  }
+  // if the function has not returned, then no matching bracket
+  return(-1);
+}
+
+
+// resolve a symbol tree to its corresponding polynomial
+int resolve_symbol_tree(Tree symbol_tree, Fields_Table fields, Variables variables, Polynomial* output){
+  Polynomial poly;
+  Tree variable_tree;
+
+  // trivial tree
+  if(symbol_tree.length==0){
+    // variable
+    if(symbol_tree.root_label.length>0 && symbol_tree.root_label.str[0]=='$'){
+      variables_find_var(symbol_tree.root_label, variables, &variable_tree);
+      resolve_symbol_tree(variable_tree, fields, variables, output);
+      free_Tree(variable_tree);
+    }
+    //polynomial
+    else{
+      Char_Array_to_Polynomial(symbol_tree.root_label, output);
+    }
+  }
+
+  // exp
+  else if (char_array_cmp_str(symbol_tree.root_label,"exp")==1){
+    if(symbol_tree.length!=1){
+      fprintf(stderr,"syntax error: exp must have 1 argument\n");
+      exit(-1);
+    }
+    resolve_symbol_tree(symbol_tree.children[0], fields, variables, &poly);
+    polynomial_exponential(poly, output, fields);
+    free_Polynomial(poly);
+  }
+
+  // log
+  else if (char_array_cmp_str(symbol_tree.root_label,"log_1")==1){
+    if(symbol_tree.length!=1){
+      fprintf(stderr,"syntax error: log_1 must have 1 argument\n");
+      exit(-1);
+    }
+    resolve_symbol_tree(symbol_tree.children[0], fields, variables, &poly);
+    polynomial_logarithm(poly, output, fields);
+    free_Polynomial(poly);
+  }
+
+  // product
+  else if (char_array_cmp_str(symbol_tree.root_label,"*")==1){
+    if(symbol_tree.length!=2){
+      fprintf(stderr,"syntax error: '*' must have 2 arguments\n");
+      exit(-1);
+    }
+    resolve_symbol_tree(symbol_tree.children[0], fields, variables, output);
+    resolve_symbol_tree(symbol_tree.children[1], fields, variables, &poly);
+    polynomial_prod_chain(poly, output, fields);
+    free_Polynomial(poly);
+  }
+
+  // sum
+  else if (char_array_cmp_str(symbol_tree.root_label,"+")==1){
+    if(symbol_tree.length!=2){
+      fprintf(stderr,"syntax error: '+' must have 2 arguments\n");
+      exit(-1);
+    }
+    resolve_symbol_tree(symbol_tree.children[0], fields, variables, output);
+    resolve_symbol_tree(symbol_tree.children[1], fields, variables, &poly);
+    polynomial_add_chain_noinit(poly, output, fields);
+  }
+
+  else{
+    fprintf(stderr,"syntax error: unrecognized operation '%s'\n",char_array_to_str_noinit(&(symbol_tree.root_label)));
+    exit(-1);
+  }
+
+  return(0);
+}
diff --git a/src/symbolic_parser.h b/src/symbolic_parser.h
new file mode 100644
index 0000000..e91c902
--- /dev/null
+++ b/src/symbolic_parser.h
@@ -0,0 +1,42 @@
+/*
+Copyright 2015-2022 Ian Jauslin
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+/*
+  parse symbolic expressions
+*/
+
+#ifndef SYMBOLIC_PARSER_H
+#define SYMBOLIC_PARSER_H
+
+#include "types.h"
+
+// parse a symbolic expression from a char_array
+int parse_symbolic_expression(Char_Array str, Fields_Table fields, Variables variables, Polynomial* polynomial);
+// from char*
+int parse_symbolic_expression_str(char* str, Fields_Table fields, Variables variables, Polynomial* polynomial);
+
+// compute the symbol tree from a string
+int char_array_to_symbol_tree(Char_Array str, Tree* symbol_tree);
+// from char*
+int str_to_symbol_tree(char* str, Tree* symbol_tree);
+
+// find matching '<' and '>'
+int matching_bracket(Char_Array str, int start);
+
+// resolve a symbol tree to its corresponding polynomial
+int resolve_symbol_tree(Tree symbol_tree, Fields_Table fields, Variables variables, Polynomial* output);
+
+#endif
diff --git a/src/tools.c b/src/tools.c
index 2b7e85d..5912819 100644
--- a/src/tools.c
+++ b/src/tools.c
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -140,3 +140,13 @@ int min(int x1, int x2){
     return(x2);
   }
 }
+
+// check whether a divides b
+int is_factor(int a, int b){
+  if(b-a*(b/a)==0){
+    return(1);
+  }
+  else{
+    return(0);
+  }
+}
diff --git a/src/tools.h b/src/tools.h
index e5f319f..8f29cf8 100644
--- a/src/tools.h
+++ b/src/tools.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -46,4 +46,7 @@ int intlist_find_err(int* list, int size, int x);
 int max(int x1, int x2);
 int min(int x1, int x2);
 
+// check whether a divides b
+int is_factor(int a, int b);
+
 #endif
diff --git a/src/tree.c b/src/tree.c
new file mode 100644
index 0000000..0edf558
--- /dev/null
+++ b/src/tree.c
@@ -0,0 +1,117 @@
+/*
+Copyright 2015-2022 Ian Jauslin
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+#include "tree.h"
+#include <stdlib.h>
+#include <stdio.h>
+#include "array.h"
+
+// init
+int init_Tree(Tree* tree, int memory_children, int memory_label){
+  init_Char_Array(&(tree->root_label),memory_label);
+  (*tree).children=calloc(memory_children,sizeof(Tree));
+  (*tree).memory=memory_children;
+  (*tree).length=0;
+  return(0);
+}
+int free_Tree(Tree tree){
+  int i;
+  free_Char_Array(tree.root_label);
+  for(i=0;i<tree.length;i++){
+    free_Tree(tree.children[i]);
+  }
+  free(tree.children);
+  return(0);
+}
+
+// copy
+int tree_cpy(Tree input, Tree* output){
+  init_Tree(output,input.length, input.root_label.length);
+  tree_cpy_noinit(input,output);
+  return(0);
+}
+int tree_cpy_noinit(Tree input, Tree* output){
+  int i;
+  if((*output).memory<input.length){
+    fprintf(stderr,"error: trying to copy a tree of length %d to another with memory %d\n",input.length, (*output).memory);
+    exit(-1);
+  }
+  char_array_cpy_noinit(input.root_label,&(output->root_label));
+  for(i=0;i<input.length;i++){
+    tree_cpy(input.children[i],(*output).children+i);
+  }
+
+  (*output).length=input.length;
+  return(0);
+}
+
+// resize memory
+int tree_resize(Tree* tree, int newsize){
+  Tree new_tree;
+  init_Tree(&new_tree,newsize,tree->root_label.memory);
+  tree_cpy_noinit(*tree,&new_tree);
+  free_Tree(*tree);
+  *tree=new_tree;
+  return(0);
+}
+
+// set label
+int tree_set_label(Char_Array label, Tree* tree){
+  if(label.length > tree->root_label.memory){
+    char_array_resize(&(tree->root_label),label.length);
+  }
+  char_array_cpy_noinit(label,&(tree->root_label));
+  return(0);
+}
+
+// add a child to a tree
+int tree_append_child(Tree child, Tree* output){
+  if((*output).length>=(*output).memory){
+    tree_resize(output,2*(*output).memory+1);
+  }
+  tree_cpy(child,(*output).children+(*output).length);
+  (*output).length++;
+  return(0);
+}
+// add a child to a tree without allocating memory for the new child
+int tree_append_child_noinit(Tree child, Tree* output){
+  if((*output).length>=(*output).memory){
+    tree_resize(output,2*(*output).memory+1);
+  }
+  (*output).children[(*output).length]=child;
+  (*output).length++;
+  return(0);
+}
+
+// concatenate the children of two trees
+int tree_concat_children(Tree input, Tree* output){
+  int i;
+  for(i=0;i<input.length;i++){
+    tree_append_child(input.children[i],output);
+  }
+  return(0);
+}
+// noinit
+int tree_concat_children_noinit(Tree input, Tree* output){
+  int i;
+  for(i=0;i<input.length;i++){
+    tree_append_child_noinit(input.children[i],output);
+  }
+  // free input array
+  free(input.children);
+  return(0);
+}
+
diff --git a/src/tree.h b/src/tree.h
new file mode 100644
index 0000000..60b222c
--- /dev/null
+++ b/src/tree.h
@@ -0,0 +1,48 @@
+/*
+Copyright 2015-2022 Ian Jauslin
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+/* Trees */
+
+#ifndef TREE_H
+#define TREE_H
+
+#include "types.h"
+
+int init_Tree(Tree* tree, int memory_children, int memory_label);
+int free_Tree(Tree tree);
+
+// copy
+int tree_cpy(Tree input, Tree* output);
+int tree_cpy_noinit(Tree input, Tree* output);
+
+// resize memory
+int tree_resize(Tree* tree, int newsize);
+
+// set label
+int tree_set_label(Char_Array label, Tree* tree);
+
+// add a child to a tree
+int tree_append_child(Tree tree, Tree* output);
+// add a child to a tree without allocating memory for the new child
+int tree_append_child_noinit(Tree tree, Tree* output);
+
+// concatenate the children of two trees
+int tree_concat_children(Tree input, Tree* output);
+// noinit
+int tree_concat_children_noinit(Tree input, Tree* output);
+
+#endif
+
diff --git a/src/types.h b/src/types.h
index d30840e..00a34ac 100644
--- a/src/types.h
+++ b/src/types.h
@@ -1,5 +1,5 @@
 /*
-Copyright 2015 Ian Jauslin
+Copyright 2015-2022 Ian Jauslin
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
@@ -71,6 +71,15 @@ typedef struct Str_Array{
   int memory;
 } Str_Array;
 
+// tree
+typedef struct Tree Tree;
+struct Tree{
+  Char_Array root_label;
+  Tree* children;
+  int length;
+  int memory;
+};
+
 // polynomial
 typedef struct Polynomial{
   Int_Array* monomials;
@@ -133,13 +142,21 @@ typedef struct Identities{
   int memory;
 } Identities;
 
-// symbolic expressions
-typedef struct Symbols{
+// virtual_fields
+typedef struct Virtual_fields{
   int* indices;
   Polynomial* expr;
   int length;
   int memory;
-} Symbols;
+} Virtual_fields;
+
+// variables used in symbolic expressions
+typedef struct Variables{
+  Char_Array* var_names;
+  Tree* symbol_trees;
+  int length;
+  int memory;
+} Variables;
 
 // groups of independent fields
 typedef struct Groups{
@@ -159,10 +176,10 @@ typedef struct Fields_Table{
   // identities between fields
   Identities ids;
   // symbolic expressions (commuting)
-  Symbols symbols;
-  // list of anti-commuting variables (fields or symbols)
+  Virtual_fields virtual_fields;
+  // list of anti-commuting variables
   Int_Array fermions;
-  // list of non-commuting variables (fields or symbols)
+  // list of non-commuting variables
   Int_Array noncommuting;
 } Fields_Table;
 
@@ -187,6 +204,8 @@ typedef struct Id_Table{
 typedef struct Meankondo_Options{
   int threads;
   int chain;
+  int print_progress;
+  int group_poly;
 } Meankondo_Options;
 
 typedef struct Numkondo_Options{
@@ -203,6 +222,7 @@ typedef struct Meantools_Options{
   Int_Array deriv_vars;
   Char_Array eval_rccstring;
   int chain;
+  Char_Array namespace;
   mpfr_prec_t mpfr_prec;
   mpfr_exp_t mpfr_emax;
 } Meantools_Options;