Peter J. Gawthrop (peter@gawthrop.net)

This collection of functions for use with GNU Octave is hereby granted to
the public domain.

THIS COLLECTION IS PROVIDED BY THE CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR
IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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The individual functions are copyrighted by their respective authors and
released under different licenses.  See the corresponding source files
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To the best of our knowledge, all files included in this distribution
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                    GNU GENERAL PUBLIC LICENSE
                       Version 2, June 1991

 Copyright (C) 1989, 1991 Free Software Foundation, Inc.
                          675 Mass Ave, Cambridge, MA 02139, USA
 Everyone is permitted to copy and distribute verbatim copies
 of this license document, but changing it is not allowed.

                            Preamble

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=======================================================================
Prerequisites

Octave and ginsh must be installed before installing gino


=======================================================================
Simple system installation

$ tar xzf gino-<ver>.tar.gz
$ cd gino-<ver>
$ make clean
$ make 

As superuser
# make install

## Makefile for gino library

all: g_funs

g_funs:
	./create_gino.sh

clean:
	rm -rf g_funs

install: g_funs
	cp -rf ../gino /usr/share/octave/site/m

Gino is a symbolic toolbox for octave (www.octave.org) using ginsh -
the shell interface to ginac (www.ginac.de).

ginsh.m provides a low-level octave interface to ginsh. All of the
ginsh functions are implemented by octave functions of the form
g_fun(arglist).

All in and out arguments of the toolbox functions are octave strings
containing valid ginsh expressions. Where appropriate, these can be
turned into matlab scalars, matrices and lists using octave's eval
function. Spurious output is avoided by setting
default_eval_print_flag=0.

This provides a clear division between symbolic expressions (always
contained in strings) and octave expressions. It also allows the
powerful octave string machinery to be used to manipulate symbolic
expressions.

gino_examples.m contains some introductory examples.

P.J.Gawthrop. 19th December 2002.

#! /bin/sh
## Creates gino functions by querying ginsh
# Copyright (C) 2002 by Peter J. Gawthrop

prefix="g_"			# Prefix function names by this
subdir=${prefix}funs		# Where the generated functions live
ginsh="/usr/bin/ginsh"		# Location of ginsh

create_fun() {
    funname=${prefix}$1
    filename=${funname}.m

    ## Creates an octave function
    echo Creating ${subdir}/${filename}

    ## Find usage string

    usage=`echo "?$1" | ginsh | head -1`

    ## The octave function
    cat<<EOF > ${subdir}/${filename}
function res = ${funname} (arg1,arg2,arg3)

  ## usage:  res = ${prefix}${usage}

  ## Generated by create_gino.sh on `date`
  ## DO NOT EDIT ME

  if nargin<1
    arg1 = "";
  endif

  if nargin<2
    arg2 = "";
  endif

  if nargin<3
    arg3 = "";
  endif

  res = gino("$1", arg1, arg2, arg3);

endfunction
EOF
}

create_readme() {
    cat > ${subdir}/README <<EOF
These files were generated automatically by create_gino.sh
on `date`.

DO NOT EDIT THEM
EOF
}

## Find available ops
ops=`echo "??" | ginsh | gawk '{if(i++==1) {gsub(",",""); print $0}}'`

## Does ginsh exist?
if [ -x "${ginsh}" ]; then
    echo $ginsh exists - ok
else
    echo $ginsh does not exist, exiting
    exit
fi

## Create the corresponding octave funs.
mkdir -p ${subdir}
create_readme;
for op in ${ops}; do
    create_fun ${op}
done

function o_pol = g2o_pol (g_pol, object)

  ## usage:  o_pol = g2o_pol (g_pol)
  ##
  ## Converts a ginsh polynomial to an octave polynomial
  ## Both o_pol and g_pol are strings
  ## o_pol can be an argument to eval

  if nargin<1
    error("usage:  o_pol = g2o_pol (g_pol)");
  endif
  
  if nargin<2
    object="s";
  endif
  
  g_pol = g_expand(g_pol);
  n = g_degree(g_pol, object);
  n = eval([n,";"]);
  o_pol = "";
  for i=n:-1:0
    pol_i = g_coeff(g_pol,object,int2str(i));
    o_pol = sprintf("%s %s", o_pol, pol_i);
  endfor
  o_pol = sprintf("[%s ];", o_pol);

endfunction

function res = gino (op,arg1,arg2,arg3)

  ## Gino is the octave interface to ginsh - the GiNaC shell interface.
  ## The simplest way to use gino is via the octave functions  go_fun.
  ## where "fun" is a ginsh function - type gino("??") for a list.
  ## usage:  res = gino (op,arg1,arg2,arg3)
  ## 
  ## Please set default_eval_print_flag = 0
  ##
  ## Part of the gino (ginsh-octave) toolbox
  ## Copyright (C) 2002 by Peter J. Gawthrop

  if nargin<1
    error("usage: gino(op,[arg1,arg2,arg3])p");
  elseif (nargin==1)||(length(arg1)==0)
    go_in = sprintf("%s", op);
  elseif (nargin==2)||(length(arg2)==0)
    go_in = sprintf("%s(%s)", op, arg1);
  elseif (nargin==3)||(length(arg3)==0)
    go_in = sprintf("%s(%s,%s)", op, arg1, arg2);
  elseif (nargin==4)
    go_in = sprintf("%s(%s,%s,%s)", op, arg1, arg2, arg3);
  else
    error("usage: gino(op,[arg1,arg2,arg3]");
  endif

  res = ginsh(go_in);

endfunction

## Simple example of creating an octave polynomial symbolically
P = g2o_pol("s^2*(s+a)*(s+b)^3") # Create symbolic polynomial in octave form
a = 1; b = 2;			# Give a and b numerical values
p = eval(P)			# Create numerical polynomial
roots(p)			# Use p in any appropriate octave
				# function

## More complicated example of symbolic linearisation

## ODE dx/dt = f(x,u); y = g(x,u)
f_1 = "x_2 - beta*tanh(u_1)";
f_2 = "-x_1 + x_2^2 + u_1";
g_1   = "x_1";
f = go_list(f_1,f_2)
g = go_list(g_1)

## Linearise via symbolic differentiation
[A,B,C,D] = go_lin(f,g)

## Find the transfer function

I = "[[1,0],[0,1]]";		# Unit matrix
## Create G as a string: C(sI-A)^-1B + D
G_str = sprintf("%s*((s*%s-%s)^(-1))*%s + %s", C,I,A,B,D)

## Find G itself
G = g_evalm(G_str);		

## Substitute steady-state values
G = g_subs(G, "{u_1,x_1,x_2}", "{0,0,0}")

## Extract the (only) matrix element
G = g_op(G,"0")

## Find numerator and denominator
a = g2o_pol(g_denom(G))
b = g2o_pol(g_numer(G))

## Evaluate numerically
beta = 0.5;
b = eval(b)
a = eval(a)

function go_out = ginsh (go_in)

  ## usage:  go_out = ginsh (go_in)
  ##         go_out, go_in strings containing valid ginsh code
  ## Interface to ginsh to provide symbolic manipulation within octave
  ## Part of the gino (ginsh-octave) toolbox

  ## Copyright (C) 2002 by Peter J. Gawthrop

  ##   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.
  
  ##    You should have received a copy of the GNU General Public License
  ##    along with this program; if not, write to the Free Software
  ##    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 

  command = sprintf("echo ""%s;"" | ginsh ", go_in); 
  go_out = system(command);

  ## Strip trailing NL
  go_out = substr(go_out,1,length(go_out)-1);
  

endfunction

function new_list = go_append (list,element)

  ## usage:  new_list = go_append (list,element)
  ##
  ## Appends element to list to give new list
  ## This is missing from ginsh
  ## Copyright (C) 2002 by Peter J. Gawthrop

  new_list = deblank(list);	# Remove trailing blanks
  new_list = new_list(1:length(new_list)-1); # Zap final }
  if (g_nops(list)=="0")
    new_list = sprintf("%s%s}", new_list, element);
  else
    new_list = sprintf("%s,%s}", new_list, element);
  endif
  

endfunction

function [dfdx_l, dfdx_m] = go_diff (f,x)

  ## usage:  [dfdx_l dfdx_m] = go_diff (f,x)
  ##
  ## dfdx_l is a list with ith element df/dx
  ## dfdx_m the corresponing matrix (in ginsh form)
  ## f may itself be a list
  ## The elements of x must be symbols

  N = eval(g_nops(f));
  M = eval(g_nops(x));

  dfdx = "{}";

  ## Find derivatives of f for each x_i and append to list
  for j = 1:M
    dfdx = go_append(dfdx, g_diff(f,g_op(x,int2str(j-1))));
  endfor

  ## Convert to matrix and transpose
  dfdx_m = g_transpose(go_lst_to_matrix(dfdx));

  ## And back to a list
  dfdx_l = go_matrix_to_lst(dfdx_m);

endfunction

function M_m = go_gmatrix2mmatrix (M-g)

  ## usage:  M_m = go_gmatrix2mmatrix (M-g)
  ##
  ## Converts a ginsh format matrix to a matlab format matrix

  M_m = strrep(M_g,"],[","];[");

endfunction

function [A,B,C,D] = go_lin (f,g)

  ## usage:  [A,B,C,D] = go_lin (f,g)
  ##
  ## 

  n_x = eval(g_nops(f));		# Number of states
  n_y = eval(g_nops(g));		# Number of outputs

  n_u = n_y;

  ## Create symbolic state list
  x = "{}";
  for i=1:n_x
    x = go_append(x,sprintf("x_%i", i));
  endfor


  ## Create symbolic input list
  u = "{}";
  for i=1:n_u
    u = go_append(u,sprintf("u_%i", i));
  endfor
		  
  [A_l,A] = go_diff(f,x);
  [B_l,B] = go_diff(f,u);
  [C_l,C] = go_diff(g,x);
  [D_l,D] = go_diff(g,u);

endfunction

function list = go_list (exp0,exp1,exp2,exp3,exp4,exp5,exp6,exp7,exp8,exp9)

  ## usage:  list = go_list
  ## (exp0[,exp1,exp2,exp3,exp4,exp5,exp6,exp7,exp8,exp9])
  ## Creates a list containing the expressions in the argument list
  ## 

  ## Copyright (C) 2002 by Peter J. Gawthrop
  if nargin>10
    error("Only 10  expressions allowed in argument list");
  endif

  N = nargin;

  list=exp0;
  for i=1:N-1
    exp = eval(sprintf("exp%i;",i));
    list=sprintf("%s, %s", list, exp);
  endfor

  list = sprintf("{%s}", list);

endfunction

function M = go_lst_to_matrix (L)

  ## usage:  M = go_lst_to_matrix (L)
  ##
  ## Converts a list to a matrix
  ## Assumes list contains equal-length lists as elements
  ## Not in ginsh
  ## Copyright (C) 2002 by Peter J. Gawthrop

  n = eval(g_nops(L));
  m = eval(g_nops(g_op(L,"0")));

  M = "";
  for i = 1:n
    M = sprintf("%s[",M);
    m_i = g_op(L,int2str(i-1));
    if (eval(g_nops(m_i))!=m)
      error("All list elements must be lists of same length");
    endif
    
    for j = 1:m
      m_ij = g_op(m_i,int2str(j-1));
      if j>1 
	jsep = ", ";
      else
	jsep = "";
      endif
      M = sprintf("%s%s%s", M, jsep, m_ij);
    endfor
    if i<n 
      isep=", ";
    else
      isep="";
    endif
    
    M = sprintf("%s]%s", M, isep);
  endfor

  M = sprintf("[%s]",M);
  
endfunction

function L = go_matrix_to_lst (M)

  ## usage:  L = go_matrix_to_lst (M)
  ##
  ## Converts matrix to list (in ginsh format)

  L = strrep(strrep(M,"[","{"), "]","}");

endfunction