eqsolver.html - Multidimensional Equation Solver
  Copyright (C) 2009  Claire Xenia Wolf <claire@clairexen.net>

  This program is free software; you can redistribute it and/or modify
  it under the terms of the GNU General Public License as published by
  the Free Software Foundation; either version 2 of the License, or
  (at your option) any later version.

  This program is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  GNU General Public License for more details.

  ---------------------------------------------------------------------

  This scripts implements a simple equation solver based on the
  Gauss-Newton algorithm for non-linear least square fitting. The
  algorithm has been altered to improve it for some corner cases
  and add support for weighted equations.

  The Jacobi Matrices are creating using a rather simple numerical
  differentiation approach.

  Normal equations can spezified using the '=' operator. Inequations
  can be written using the '<' and '>' operators.

  Variable aliases, macros or static constants can be declared using
  special ':=' equations, with the macro name on the left side.

  The start value for all variables is '1'. Another start value can
  be defined using special '?=' equations, with the variable name
  on the left and a numerical value on the right side.

  Since this program is based on a least square fitter it can also
  'solve' overdefined quation systems with no real solution. For this
  cases the "Remaining error: X" line has been included in the output.

  It is also possible to use this script for solving underdefined
  quation systems. In this cases any numerically valid solution is
  returned.

  Some special commands can be used to modify the behavior of the
  equation solver:

    $tryhard
        Deactivates the close-to-singularity detection. In some
        scenarios the equation solver returns better results with
        this option set. But it also incrases the solver runtime.

    $scalehack
        Setting this option enables a small hack in the Gauss-Newton
        solver that helps finding solutions in equations that are
        otherwise problematic with Gauss-Newton solvers (such as
	the equation 'pow(x,3)-2*x+2 = 0') and speeds up solving of
        some equation systems.

    $showindep
        Also print the independent variables/expressions.

    $range:X:Min:Max
        This command can be used to specify the valid range for
        fitted variable. That e.g. usefull when using a variable as
        argument sqrt() and thus only positive values are valid. Open
	ranges can be implemented by leaving the Min or Max field empty.

    $weight:X
        This command sets the weight for all (in)equations following
        this statement. The default weight is '1'. A higher weight
        for an (in)equation results in a higher impact on the
        solution.

    $function:avg(a, b)
      return (a+b) / 2;
    $endfunction
        Such a code block can be used to define a javascript function
        which can be used in the equation system.

  Note that all JavaScript Math.* functions can be used in the
  equations, but without the "Math." prefix. I.e. '0.8 = sin(x)' is
  a valid equation.

  Beware of bugs in this program. I haven't tested it much yet..