[ginac.git] / ginac / normal.cpp

/** @file normal.cpp
 *
 *  This file implements several functions that work on univariate and
 *  multivariate polynomials and rational functions.
 *  These functions include polynomial quotient and remainder, GCD and LCM
 *  computation, square-free factorization and rational function normalization. */

/*
 *  GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
 *
 *  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
 */

#include <stdexcept>
#include <algorithm>
#include <map>

#include "normal.h"
#include "basic.h"
#include "ex.h"
#include "add.h"
#include "constant.h"
#include "expairseq.h"
#include "fail.h"
#include "indexed.h"
#include "inifcns.h"
#include "lst.h"
#include "mul.h"
#include "ncmul.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
#include "utils.h"

#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
#endif // ndef NO_NAMESPACE_GINAC

// If comparing expressions (ex::compare()) is fast, you can set this to 1.
// Some routines like quo(), rem() and gcd() will then return a quick answer
// when they are called with two identical arguments.
#define FAST_COMPARE 1

// Set this if you want divide_in_z() to use remembering
#define USE_REMEMBER 0

// Set this if you want divide_in_z() to use trial division followed by
// polynomial interpolation (usually slower except for very large problems)
#define USE_TRIAL_DIVISION 0

// Set this to enable some statistical output for the GCD routines
#define STATISTICS 0


#if STATISTICS
// Statistics variables
static int gcd_called = 0;
static int sr_gcd_called = 0;
static int heur_gcd_called = 0;
static int heur_gcd_failed = 0;

// Print statistics at end of program
static struct _stat_print {
        _stat_print() {}
        ~_stat_print() {
                cout << "gcd() called " << gcd_called << " times\n";
                cout << "sr_gcd() called " << sr_gcd_called << " times\n";
                cout << "heur_gcd() called " << heur_gcd_called << " times\n";
                cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
        }
} stat_print;
#endif


/** Return pointer to first symbol found in expression.  Due to GiNaC´s
 *  internal ordering of terms, it may not be obvious which symbol this
 *  function returns for a given expression.
 *
 *  @param e  expression to search
 *  @param x  pointer to first symbol found (returned)
 *  @return "false" if no symbol was found, "true" otherwise */
static bool get_first_symbol(const ex &e, const symbol *&x)
{
    if (is_ex_exactly_of_type(e, symbol)) {
        x = static_cast<symbol *>(e.bp);
        return true;
    } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
        for (unsigned i=0; i<e.nops(); i++)
            if (get_first_symbol(e.op(i), x))
                return true;
    } else if (is_ex_exactly_of_type(e, power)) {
        if (get_first_symbol(e.op(0), x))
            return true;
    }
    return false;
}


/*
 *  Statistical information about symbols in polynomials
 */

/** This structure holds information about the highest and lowest degrees
 *  in which a symbol appears in two multivariate polynomials "a" and "b".
 *  A vector of these structures with information about all symbols in
 *  two polynomials can be created with the function get_symbol_stats().
 *
 *  @see get_symbol_stats */
struct sym_desc {
    /** Pointer to symbol */
    const symbol *sym;

    /** Highest degree of symbol in polynomial "a" */
    int deg_a;

    /** Highest degree of symbol in polynomial "b" */
    int deg_b;

    /** Lowest degree of symbol in polynomial "a" */
    int ldeg_a;

    /** Lowest degree of symbol in polynomial "b" */
    int ldeg_b;

    /** Maximum of deg_a and deg_b (Used for sorting) */
    int max_deg;

    /** Commparison operator for sorting */
    bool operator<(const sym_desc &x) const {return max_deg < x.max_deg;}
};

// Vector of sym_desc structures
typedef vector<sym_desc> sym_desc_vec;

// Add symbol the sym_desc_vec (used internally by get_symbol_stats())
static void add_symbol(const symbol *s, sym_desc_vec &v)
{
    sym_desc_vec::iterator it = v.begin(), itend = v.end();
    while (it != itend) {
        if (it->sym->compare(*s) == 0)  // If it's already in there, don't add it a second time
            return;
        it++;
    }
    sym_desc d;
    d.sym = s;
    v.push_back(d);
}

// Collect all symbols of an expression (used internally by get_symbol_stats())
static void collect_symbols(const ex &e, sym_desc_vec &v)
{
    if (is_ex_exactly_of_type(e, symbol)) {
        add_symbol(static_cast<symbol *>(e.bp), v);
    } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
        for (unsigned i=0; i<e.nops(); i++)
            collect_symbols(e.op(i), v);
    } else if (is_ex_exactly_of_type(e, power)) {
        collect_symbols(e.op(0), v);
    }
}

/** Collect statistical information about symbols in polynomials.
 *  This function fills in a vector of "sym_desc" structs which contain
 *  information about the highest and lowest degrees of all symbols that
 *  appear in two polynomials. The vector is then sorted by minimum
 *  degree (lowest to highest). The information gathered by this
 *  function is used by the GCD routines to identify trivial factors
 *  and to determine which variable to choose as the main variable
 *  for GCD computation.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param v  vector of sym_desc structs (filled in) */
static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
{
    collect_symbols(a.eval(), v);   // eval() to expand assigned symbols
    collect_symbols(b.eval(), v);
    sym_desc_vec::iterator it = v.begin(), itend = v.end();
    while (it != itend) {
        int deg_a = a.degree(*(it->sym));
        int deg_b = b.degree(*(it->sym));
        it->deg_a = deg_a;
        it->deg_b = deg_b;
        it->max_deg = max(deg_a, deg_b);
        it->ldeg_a = a.ldegree(*(it->sym));
        it->ldeg_b = b.ldegree(*(it->sym));
        it++;
    }
    sort(v.begin(), v.end());
#if 0
        clog << "Symbols:\n";
        it = v.begin(); itend = v.end();
        while (it != itend) {
                clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << endl;
                clog << "  lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
                it++;
        }
#endif
}


/*
 *  Computation of LCM of denominators of coefficients of a polynomial
 */

// Compute LCM of denominators of coefficients by going through the
// expression recursively (used internally by lcm_of_coefficients_denominators())
static numeric lcmcoeff(const ex &e, const numeric &l)
{
    if (e.info(info_flags::rational))
        return lcm(ex_to_numeric(e).denom(), l);
    else if (is_ex_exactly_of_type(e, add)) {
        numeric c = _num1();
        for (unsigned i=0; i<e.nops(); i++)
            c = lcmcoeff(e.op(i), c);
        return lcm(c, l);
    } else if (is_ex_exactly_of_type(e, mul)) {
        numeric c = _num1();
        for (unsigned i=0; i<e.nops(); i++)
            c *= lcmcoeff(e.op(i), _num1());
        return lcm(c, l);
    } else if (is_ex_exactly_of_type(e, power))
        return pow(lcmcoeff(e.op(0), l), ex_to_numeric(e.op(1)));
    return l;
}

/** Compute LCM of denominators of coefficients of a polynomial.
 *  Given a polynomial with rational coefficients, this function computes
 *  the LCM of the denominators of all coefficients. This can be used
 *  to bring a polynomial from Q[X] to Z[X].
 *
 *  @param e  multivariate polynomial (need not be expanded)
 *  @return LCM of denominators of coefficients */
static numeric lcm_of_coefficients_denominators(const ex &e)
{
    return lcmcoeff(e, _num1());
}

/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
 *  determined LCM of the coefficient's denominators.
 *
 *  @param e  multivariate polynomial (need not be expanded)
 *  @param lcm  LCM to multiply in */
static ex multiply_lcm(const ex &e, const numeric &lcm)
{
        if (is_ex_exactly_of_type(e, mul)) {
                ex c = _ex1();
                numeric lcm_accum = _num1();
                for (unsigned i=0; i<e.nops(); i++) {
                        numeric op_lcm = lcmcoeff(e.op(i), _num1());
                        c *= multiply_lcm(e.op(i), op_lcm);
                        lcm_accum *= op_lcm;
                }
                c *= lcm / lcm_accum;
                return c;
        } else if (is_ex_exactly_of_type(e, add)) {
                ex c = _ex0();
                for (unsigned i=0; i<e.nops(); i++)
                        c += multiply_lcm(e.op(i), lcm);
                return c;
        } else if (is_ex_exactly_of_type(e, power)) {
                return pow(multiply_lcm(e.op(0), lcm.power(ex_to_numeric(e.op(1)).inverse())), e.op(1));
        } else
                return e * lcm;
}


/** Compute the integer content (= GCD of all numeric coefficients) of an
 *  expanded polynomial.
 *
 *  @param e  expanded polynomial
 *  @return integer content */
numeric ex::integer_content(void) const
{
    GINAC_ASSERT(bp!=0);
    return bp->integer_content();
}

numeric basic::integer_content(void) const
{
    return _num1();
}

numeric numeric::integer_content(void) const
{
    return abs(*this);
}

numeric add::integer_content(void) const
{
    epvector::const_iterator it = seq.begin();
    epvector::const_iterator itend = seq.end();
    numeric c = _num0();
    while (it != itend) {
        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
        GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
        c = gcd(ex_to_numeric(it->coeff), c);
        it++;
    }
    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
    c = gcd(ex_to_numeric(overall_coeff),c);
    return c;
}

numeric mul::integer_content(void) const
{
#ifdef DO_GINAC_ASSERT
    epvector::const_iterator it = seq.begin();
    epvector::const_iterator itend = seq.end();
    while (it != itend) {
        GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
        ++it;
    }
#endif // def DO_GINAC_ASSERT
    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
    return abs(ex_to_numeric(overall_coeff));
}


/*
 *  Polynomial quotients and remainders
 */

/** Quotient q(x) of polynomials a(x) and b(x) in Q[x].
 *  It satisfies a(x)=b(x)*q(x)+r(x).
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return quotient of a and b in Q[x] */
ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
{
    if (b.is_zero())
        throw(std::overflow_error("quo: division by zero"));
    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
        return a / b;
#if FAST_COMPARE
    if (a.is_equal(b))
        return _ex1();
#endif
    if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
        throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));

    // Polynomial long division
    ex q = _ex0();
    ex r = a.expand();
    if (r.is_zero())
        return r;
    int bdeg = b.degree(x);
    int rdeg = r.degree(x);
    ex blcoeff = b.expand().coeff(x, bdeg);
    bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
    while (rdeg >= bdeg) {
        ex term, rcoeff = r.coeff(x, rdeg);
        if (blcoeff_is_numeric)
            term = rcoeff / blcoeff;
        else {
            if (!divide(rcoeff, blcoeff, term, false))
                return *new ex(fail());
        }
        term *= power(x, rdeg - bdeg);
        q += term;
        r -= (term * b).expand();
        if (r.is_zero())
            break;
        rdeg = r.degree(x);
    }
    return q;
}


/** Remainder r(x) of polynomials a(x) and b(x) in Q[x].
 *  It satisfies a(x)=b(x)*q(x)+r(x).
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return remainder of a(x) and b(x) in Q[x] */
ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
    if (b.is_zero())
        throw(std::overflow_error("rem: division by zero"));
    if (is_ex_exactly_of_type(a, numeric)) {
        if  (is_ex_exactly_of_type(b, numeric))
            return _ex0();
        else
            return b;
    }
#if FAST_COMPARE
    if (a.is_equal(b))
        return _ex0();
#endif
    if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
        throw(std::invalid_argument("rem: arguments must be polynomials over the rationals"));

    // Polynomial long division
    ex r = a.expand();
    if (r.is_zero())
        return r;
    int bdeg = b.degree(x);
    int rdeg = r.degree(x);
    ex blcoeff = b.expand().coeff(x, bdeg);
    bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
    while (rdeg >= bdeg) {
        ex term, rcoeff = r.coeff(x, rdeg);
        if (blcoeff_is_numeric)
            term = rcoeff / blcoeff;
        else {
            if (!divide(rcoeff, blcoeff, term, false))
                return *new ex(fail());
        }
        term *= power(x, rdeg - bdeg);
        r -= (term * b).expand();
        if (r.is_zero())
            break;
        rdeg = r.degree(x);
    }
    return r;
}


/** Pseudo-remainder of polynomials a(x) and b(x) in Z[x].
 *
 *  @param a  first polynomial in x (dividend)
 *  @param b  second polynomial in x (divisor)
 *  @param x  a and b are polynomials in x
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return pseudo-remainder of a(x) and b(x) in Z[x] */
ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
    if (b.is_zero())
        throw(std::overflow_error("prem: division by zero"));
    if (is_ex_exactly_of_type(a, numeric)) {
        if (is_ex_exactly_of_type(b, numeric))
            return _ex0();
        else
            return b;
    }
    if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
        throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));

    // Polynomial long division
    ex r = a.expand();
    ex eb = b.expand();
    int rdeg = r.degree(x);
    int bdeg = eb.degree(x);
    ex blcoeff;
    if (bdeg <= rdeg) {
        blcoeff = eb.coeff(x, bdeg);
        if (bdeg == 0)
            eb = _ex0();
        else
            eb -= blcoeff * power(x, bdeg);
    } else
        blcoeff = _ex1();

    int delta = rdeg - bdeg + 1, i = 0;
    while (rdeg >= bdeg && !r.is_zero()) {
        ex rlcoeff = r.coeff(x, rdeg);
        ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
        if (rdeg == 0)
            r = _ex0();
        else
            r -= rlcoeff * power(x, rdeg);
        r = (blcoeff * r).expand() - term;
        rdeg = r.degree(x);
        i++;
    }
    return power(blcoeff, delta - i) * r;
}


/** Exact polynomial division of a(X) by b(X) in Q[X].
 *  
 *  @param a  first multivariate polynomial (dividend)
 *  @param b  second multivariate polynomial (divisor)
 *  @param q  quotient (returned)
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return "true" when exact division succeeds (quotient returned in q),
 *          "false" otherwise */
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
    q = _ex0();
    if (b.is_zero())
        throw(std::overflow_error("divide: division by zero"));
    if (a.is_zero())
        return true;
    if (is_ex_exactly_of_type(b, numeric)) {
        q = a / b;
        return true;
    } else if (is_ex_exactly_of_type(a, numeric))
        return false;
#if FAST_COMPARE
    if (a.is_equal(b)) {
        q = _ex1();
        return true;
    }
#endif
    if (check_args && (!a.info(info_flags::rational_polynomial) ||
                       !b.info(info_flags::rational_polynomial)))
        throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));

    // Find first symbol
    const symbol *x;
    if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
        throw(std::invalid_argument("invalid expression in divide()"));

    // Polynomial long division (recursive)
    ex r = a.expand();
    if (r.is_zero())
        return true;
    int bdeg = b.degree(*x);
    int rdeg = r.degree(*x);
    ex blcoeff = b.expand().coeff(*x, bdeg);
    bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
    while (rdeg >= bdeg) {
        ex term, rcoeff = r.coeff(*x, rdeg);
        if (blcoeff_is_numeric)
            term = rcoeff / blcoeff;
        else
            if (!divide(rcoeff, blcoeff, term, false))
                return false;
        term *= power(*x, rdeg - bdeg);
        q += term;
        r -= (term * b).expand();
        if (r.is_zero())
            return true;
        rdeg = r.degree(*x);
    }
    return false;
}


#if USE_REMEMBER
/*
 *  Remembering
 */

typedef pair<ex, ex> ex2;
typedef pair<ex, bool> exbool;

struct ex2_less {
    bool operator() (const ex2 p, const ex2 q) const 
    {
        return p.first.compare(q.first) < 0 || (!(q.first.compare(p.first) < 0) && p.second.compare(q.second) < 0);        
    }
};

typedef map<ex2, exbool, ex2_less> ex2_exbool_remember;
#endif


/** Exact polynomial division of a(X) by b(X) in Z[X].
 *  This functions works like divide() but the input and output polynomials are
 *  in Z[X] instead of Q[X] (i.e. they have integer coefficients). Unlike
 *  divide(), it doesn´t check whether the input polynomials really are integer
 *  polynomials, so be careful of what you pass in. Also, you have to run
 *  get_symbol_stats() over the input polynomials before calling this function
 *  and pass an iterator to the first element of the sym_desc vector. This
 *  function is used internally by the heur_gcd().
 *  
 *  @param a  first multivariate polynomial (dividend)
 *  @param b  second multivariate polynomial (divisor)
 *  @param q  quotient (returned)
 *  @param var  iterator to first element of vector of sym_desc structs
 *  @return "true" when exact division succeeds (the quotient is returned in
 *          q), "false" otherwise.
 *  @see get_symbol_stats, heur_gcd */
static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_iterator var)
{
    q = _ex0();
    if (b.is_zero())
        throw(std::overflow_error("divide_in_z: division by zero"));
    if (b.is_equal(_ex1())) {
        q = a;
        return true;
    }
    if (is_ex_exactly_of_type(a, numeric)) {
        if (is_ex_exactly_of_type(b, numeric)) {
            q = a / b;
            return q.info(info_flags::integer);
        } else
            return false;
    }
#if FAST_COMPARE
    if (a.is_equal(b)) {
        q = _ex1();
        return true;
    }
#endif

#if USE_REMEMBER
    // Remembering
    static ex2_exbool_remember dr_remember;
    ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b));
    if (remembered != dr_remember.end()) {
        q = remembered->second.first;
        return remembered->second.second;
    }
#endif

    // Main symbol
    const symbol *x = var->sym;

    // Compare degrees
    int adeg = a.degree(*x), bdeg = b.degree(*x);
    if (bdeg > adeg)
        return false;

#if USE_TRIAL_DIVISION

    // Trial division with polynomial interpolation
    int i, k;

    // Compute values at evaluation points 0..adeg
    vector<numeric> alpha; alpha.reserve(adeg + 1);
    exvector u; u.reserve(adeg + 1);
    numeric point = _num0();
    ex c;
    for (i=0; i<=adeg; i++) {
        ex bs = b.subs(*x == point);
        while (bs.is_zero()) {
            point += _num1();
            bs = b.subs(*x == point);
        }
        if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
            return false;
        alpha.push_back(point);
        u.push_back(c);
        point += _num1();
    }

    // Compute inverses
    vector<numeric> rcp; rcp.reserve(adeg + 1);
    rcp.push_back(_num0());
    for (k=1; k<=adeg; k++) {
        numeric product = alpha[k] - alpha[0];
        for (i=1; i<k; i++)
            product *= alpha[k] - alpha[i];
        rcp.push_back(product.inverse());
    }

    // Compute Newton coefficients
    exvector v; v.reserve(adeg + 1);
    v.push_back(u[0]);
    for (k=1; k<=adeg; k++) {
        ex temp = v[k - 1];
        for (i=k-2; i>=0; i--)
            temp = temp * (alpha[k] - alpha[i]) + v[i];
        v.push_back((u[k] - temp) * rcp[k]);
    }

    // Convert from Newton form to standard form
    c = v[adeg];
    for (k=adeg-1; k>=0; k--)
        c = c * (*x - alpha[k]) + v[k];

    if (c.degree(*x) == (adeg - bdeg)) {
        q = c.expand();
        return true;
    } else
        return false;

#else

    // Polynomial long division (recursive)
    ex r = a.expand();
    if (r.is_zero())
        return true;
    int rdeg = adeg;
    ex eb = b.expand();
    ex blcoeff = eb.coeff(*x, bdeg);
    while (rdeg >= bdeg) {
        ex term, rcoeff = r.coeff(*x, rdeg);
        if (!divide_in_z(rcoeff, blcoeff, term, var+1))
            break;
        term = (term * power(*x, rdeg - bdeg)).expand();
        q += term;
        r -= (term * eb).expand();
        if (r.is_zero()) {
#if USE_REMEMBER
            dr_remember[ex2(a, b)] = exbool(q, true);
#endif
            return true;
        }
        rdeg = r.degree(*x);
    }
#if USE_REMEMBER
    dr_remember[ex2(a, b)] = exbool(q, false);
#endif
    return false;

#endif
}


/*
 *  Separation of unit part, content part and primitive part of polynomials
 */

/** Compute unit part (= sign of leading coefficient) of a multivariate
 *  polynomial in Z[x]. The product of unit part, content part, and primitive
 *  part is the polynomial itself.
 *
 *  @param x  variable in which to compute the unit part
 *  @return unit part
 *  @see ex::content, ex::primpart */
ex ex::unit(const symbol &x) const
{
    ex c = expand().lcoeff(x);
    if (is_ex_exactly_of_type(c, numeric))
        return c < _ex0() ? _ex_1() : _ex1();
    else {
        const symbol *y;
        if (get_first_symbol(c, y))
            return c.unit(*y);
        else
            throw(std::invalid_argument("invalid expression in unit()"));
    }
}


/** Compute content part (= unit normal GCD of all coefficients) of a
 *  multivariate polynomial in Z[x].  The product of unit part, content part,
 *  and primitive part is the polynomial itself.
 *
 *  @param x  variable in which to compute the content part
 *  @return content part
 *  @see ex::unit, ex::primpart */
ex ex::content(const symbol &x) const
{
    if (is_zero())
        return _ex0();
    if (is_ex_exactly_of_type(*this, numeric))
        return info(info_flags::negative) ? -*this : *this;
    ex e = expand();
    if (e.is_zero())
        return _ex0();

    // First, try the integer content
    ex c = e.integer_content();
    ex r = e / c;
    ex lcoeff = r.lcoeff(x);
    if (lcoeff.info(info_flags::integer))
        return c;

    // GCD of all coefficients
    int deg = e.degree(x);
    int ldeg = e.ldegree(x);
    if (deg == ldeg)
        return e.lcoeff(x) / e.unit(x);
    c = _ex0();
    for (int i=ldeg; i<=deg; i++)
        c = gcd(e.coeff(x, i), c, NULL, NULL, false);
    return c;
}


/** Compute primitive part of a multivariate polynomial in Z[x].
 *  The product of unit part, content part, and primitive part is the
 *  polynomial itself.
 *
 *  @param x  variable in which to compute the primitive part
 *  @return primitive part
 *  @see ex::unit, ex::content */
ex ex::primpart(const symbol &x) const
{
    if (is_zero())
        return _ex0();
    if (is_ex_exactly_of_type(*this, numeric))
        return _ex1();

    ex c = content(x);
    if (c.is_zero())
        return _ex0();
    ex u = unit(x);
    if (is_ex_exactly_of_type(c, numeric))
        return *this / (c * u);
    else
        return quo(*this, c * u, x, false);
}


/** Compute primitive part of a multivariate polynomial in Z[x] when the
 *  content part is already known. This function is faster in computing the
 *  primitive part than the previous function.
 *
 *  @param x  variable in which to compute the primitive part
 *  @param c  previously computed content part
 *  @return primitive part */
ex ex::primpart(const symbol &x, const ex &c) const
{
    if (is_zero())
        return _ex0();
    if (c.is_zero())
        return _ex0();
    if (is_ex_exactly_of_type(*this, numeric))
        return _ex1();

    ex u = unit(x);
    if (is_ex_exactly_of_type(c, numeric))
        return *this / (c * u);
    else
        return quo(*this, c * u, x, false);
}


/*
 *  GCD of multivariate polynomials
 */

/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
 *  (not really suited for multivariate GCDs). This function is only provided
 *  for testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
{
//clog << "eu_gcd(" << a << "," << b << ")\n";

    // Sort c and d so that c has higher degree
    ex c, d;
    int adeg = a.degree(*x), bdeg = b.degree(*x);
    if (adeg >= bdeg) {
        c = a;
        d = b;
    } else {
        c = b;
        d = a;
    }

        // Euclidean algorithm
    ex r;
    for (;;) {
//clog << " d = " << d << endl;
        r = rem(c, d, *x, false);
        if (r.is_zero())
            return d.primpart(*x);
        c = d;
                d = r;
    }
}


/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
 *  with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]).
 *  This function is only provided for testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex euprem_gcd(const ex &a, const ex &b, const symbol *x)
{
//clog << "euprem_gcd(" << a << "," << b << ")\n";

    // Sort c and d so that c has higher degree
    ex c, d;
    int adeg = a.degree(*x), bdeg = b.degree(*x);
    if (adeg >= bdeg) {
        c = a;
        d = b;
    } else {
        c = b;
        d = a;
    }

        // Euclidean algorithm with pseudo-remainders
    ex r;
    for (;;) {
//clog << " d = " << d << endl;
        r = prem(c, d, *x, false);
        if (r.is_zero())
            return d.primpart(*x);
        c = d;
                d = r;
    }
}


/** Compute GCD of multivariate polynomials using the primitive Euclidean
 *  PRS algorithm (complete content removal at each step). This function is
 *  only provided for testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
{
//clog << "peu_gcd(" << a << "," << b << ")\n";

    // Sort c and d so that c has higher degree
    ex c, d;
    int adeg = a.degree(*x), bdeg = b.degree(*x);
    int ddeg;
    if (adeg >= bdeg) {
        c = a;
        d = b;
        ddeg = bdeg;
    } else {
        c = b;
        d = a;
        ddeg = adeg;
    }

    // Remove content from c and d, to be attached to GCD later
    ex cont_c = c.content(*x);
    ex cont_d = d.content(*x);
    ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
    if (ddeg == 0)
        return gamma;
    c = c.primpart(*x, cont_c);
    d = d.primpart(*x, cont_d);

    // Euclidean algorithm with content removal
        ex r;
    for (;;) {
//clog << " d = " << d << endl;
        r = prem(c, d, *x, false);
        if (r.is_zero())
            return gamma * d;
        c = d;
                d = r.primpart(*x);
    }
}


/** Compute GCD of multivariate polynomials using the reduced PRS algorithm.
 *  This function is only provided for testing purposes.
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */

static ex red_gcd(const ex &a, const ex &b, const symbol *x)
{
//clog << "red_gcd(" << a << "," << b << ")\n";

    // Sort c and d so that c has higher degree
    ex c, d;
    int adeg = a.degree(*x), bdeg = b.degree(*x);
    int cdeg, ddeg;
    if (adeg >= bdeg) {
        c = a;
        d = b;
        cdeg = adeg;
        ddeg = bdeg;
    } else {
        c = b;
        d = a;
        cdeg = bdeg;
        ddeg = adeg;
    }

    // Remove content from c and d, to be attached to GCD later
    ex cont_c = c.content(*x);
    ex cont_d = d.content(*x);
    ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
    if (ddeg == 0)
        return gamma;
    c = c.primpart(*x, cont_c);
    d = d.primpart(*x, cont_d);

    // First element of subresultant sequence
    ex r, ri = _ex1();
    int delta = cdeg - ddeg;

    for (;;) {
        // Calculate polynomial pseudo-remainder
//clog << " d = " << d << endl;
        r = prem(c, d, *x, false);
        if (r.is_zero())
            return gamma * d.primpart(*x);
        c = d;
        cdeg = ddeg;

        if (!divide(r, pow(ri, delta), d, false))
            throw(std::runtime_error("invalid expression in red_gcd(), division failed"));
        ddeg = d.degree(*x);
        if (ddeg == 0) {
            if (is_ex_exactly_of_type(r, numeric))
                return gamma;
            else
                return gamma * r.primpart(*x);
        }

        ri = c.expand().lcoeff(*x);
        delta = cdeg - ddeg;
    }
}


/** Compute GCD of multivariate polynomials using the subresultant PRS
 *  algorithm. This function is used internally by gcd().
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param x  pointer to symbol (main variable) in which to compute the GCD in
 *  @return the GCD as a new expression
 *  @see gcd */
static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
{
//clog << "sr_gcd(" << a << "," << b << ")\n";
#if STATISTICS
        sr_gcd_called++;
#endif

    // Sort c and d so that c has higher degree
    ex c, d;
    int adeg = a.degree(*x), bdeg = b.degree(*x);
    int cdeg, ddeg;
    if (adeg >= bdeg) {
        c = a;
        d = b;
        cdeg = adeg;
        ddeg = bdeg;
    } else {
        c = b;
        d = a;
        cdeg = bdeg;
        ddeg = adeg;
    }

    // Remove content from c and d, to be attached to GCD later
    ex cont_c = c.content(*x);
    ex cont_d = d.content(*x);
    ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
    if (ddeg == 0)
        return gamma;
    c = c.primpart(*x, cont_c);
    d = d.primpart(*x, cont_d);
//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";

    // First element of subresultant sequence
    ex r = _ex0(), ri = _ex1(), psi = _ex1();
    int delta = cdeg - ddeg;

    for (;;) {
        // Calculate polynomial pseudo-remainder
//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
//clog << " d = " << d << endl;
        r = prem(c, d, *x, false);
        if (r.is_zero())
            return gamma * d.primpart(*x);
        c = d;
        cdeg = ddeg;
//clog << " dividing...\n";
        if (!divide(r, ri * pow(psi, delta), d, false))
            throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
        ddeg = d.degree(*x);
        if (ddeg == 0) {
            if (is_ex_exactly_of_type(r, numeric))
                return gamma;
            else
                return gamma * r.primpart(*x);
        }

        // Next element of subresultant sequence
//clog << " calculating next subresultant...\n";
        ri = c.expand().lcoeff(*x);
        if (delta == 1)
            psi = ri;
        else if (delta)
            divide(pow(ri, delta), pow(psi, delta-1), psi, false);
        delta = cdeg - ddeg;
    }
}


/** Return maximum (absolute value) coefficient of a polynomial.
 *  This function is used internally by heur_gcd().
 *
 *  @param e  expanded multivariate polynomial
 *  @return maximum coefficient
 *  @see heur_gcd */
numeric ex::max_coefficient(void) const
{
    GINAC_ASSERT(bp!=0);
    return bp->max_coefficient();
}

numeric basic::max_coefficient(void) const
{
    return _num1();
}

numeric numeric::max_coefficient(void) const
{
    return abs(*this);
}

numeric add::max_coefficient(void) const
{
    epvector::const_iterator it = seq.begin();
    epvector::const_iterator itend = seq.end();
    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
    numeric cur_max = abs(ex_to_numeric(overall_coeff));
    while (it != itend) {
        numeric a;
        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
        a = abs(ex_to_numeric(it->coeff));
        if (a > cur_max)
            cur_max = a;
        it++;
    }
    return cur_max;
}

numeric mul::max_coefficient(void) const
{
#ifdef DO_GINAC_ASSERT
    epvector::const_iterator it = seq.begin();
    epvector::const_iterator itend = seq.end();
    while (it != itend) {
        GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
        it++;
    }
#endif // def DO_GINAC_ASSERT
    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
    return abs(ex_to_numeric(overall_coeff));
}


/** Apply symmetric modular homomorphism to a multivariate polynomial.
 *  This function is used internally by heur_gcd().
 *
 *  @param e  expanded multivariate polynomial
 *  @param xi  modulus
 *  @return mapped polynomial
 *  @see heur_gcd */
ex ex::smod(const numeric &xi) const
{
    GINAC_ASSERT(bp!=0);
    return bp->smod(xi);
}

ex basic::smod(const numeric &xi) const
{
    return *this;
}

ex numeric::smod(const numeric &xi) const
{
#ifndef NO_NAMESPACE_GINAC
    return GiNaC::smod(*this, xi);
#else // ndef NO_NAMESPACE_GINAC
    return ::smod(*this, xi);
#endif // ndef NO_NAMESPACE_GINAC
}

ex add::smod(const numeric &xi) const
{
    epvector newseq;
    newseq.reserve(seq.size()+1);
    epvector::const_iterator it = seq.begin();
    epvector::const_iterator itend = seq.end();
    while (it != itend) {
        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
#ifndef NO_NAMESPACE_GINAC
        numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
#else // ndef NO_NAMESPACE_GINAC
        numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
#endif // ndef NO_NAMESPACE_GINAC
        if (!coeff.is_zero())
            newseq.push_back(expair(it->rest, coeff));
        it++;
    }
    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
#ifndef NO_NAMESPACE_GINAC
    numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
#else // ndef NO_NAMESPACE_GINAC
    numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi);
#endif // ndef NO_NAMESPACE_GINAC
    return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
}

ex mul::smod(const numeric &xi) const
{
#ifdef DO_GINAC_ASSERT
    epvector::const_iterator it = seq.begin();
    epvector::const_iterator itend = seq.end();
    while (it != itend) {
        GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
        it++;
    }
#endif // def DO_GINAC_ASSERT
    mul * mulcopyp=new mul(*this);
    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
#ifndef NO_NAMESPACE_GINAC
    mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
#else // ndef NO_NAMESPACE_GINAC
    mulcopyp->overall_coeff = ::smod(ex_to_numeric(overall_coeff),xi);
#endif // ndef NO_NAMESPACE_GINAC
    mulcopyp->clearflag(status_flags::evaluated);
    mulcopyp->clearflag(status_flags::hash_calculated);
    return mulcopyp->setflag(status_flags::dynallocated);
}


/** Exception thrown by heur_gcd() to signal failure. */
class gcdheu_failed {};

/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
 *  get_symbol_stats() must have been called previously with the input
 *  polynomials and an iterator to the first element of the sym_desc vector
 *  passed in. This function is used internally by gcd().
 *
 *  @param a  first multivariate polynomial (expanded)
 *  @param b  second multivariate polynomial (expanded)
 *  @param ca  cofactor of polynomial a (returned), NULL to suppress
 *             calculation of cofactor
 *  @param cb  cofactor of polynomial b (returned), NULL to suppress
 *             calculation of cofactor
 *  @param var iterator to first element of vector of sym_desc structs
 *  @return the GCD as a new expression
 *  @see gcd
 *  @exception gcdheu_failed() */
static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
{
//clog << "heur_gcd(" << a << "," << b << ")\n";
#if STATISTICS
        heur_gcd_called++;
#endif

        // GCD of two numeric values -> CLN
    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
        numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
        numeric rg;
        if (ca || cb)
            rg = g.inverse();
        if (ca)
            *ca = ex_to_numeric(a).mul(rg);
        if (cb)
            *cb = ex_to_numeric(b).mul(rg);
        return g;
    }

    // The first symbol is our main variable
    const symbol *x = var->sym;

    // Remove integer content
    numeric gc = gcd(a.integer_content(), b.integer_content());
    numeric rgc = gc.inverse();
    ex p = a * rgc;
    ex q = b * rgc;
    int maxdeg = max(p.degree(*x), q.degree(*x));

    // Find evaluation point
    numeric mp = p.max_coefficient(), mq = q.max_coefficient();
    numeric xi;
    if (mp > mq)
        xi = mq * _num2() + _num2();
    else
        xi = mp * _num2() + _num2();

    // 6 tries maximum
    for (int t=0; t<6; t++) {
        if (xi.int_length() * maxdeg > 100000) {
//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
            throw gcdheu_failed();
                }

        // Apply evaluation homomorphism and calculate GCD
        ex gamma = heur_gcd(p.subs(*x == xi), q.subs(*x == xi), NULL, NULL, var+1).expand();
        if (!is_ex_exactly_of_type(gamma, fail)) {

            // Reconstruct polynomial from GCD of mapped polynomials
            ex g = _ex0();
            numeric rxi = xi.inverse();
            for (int i=0; !gamma.is_zero(); i++) {
                ex gi = gamma.smod(xi);
                g += gi * power(*x, i);
                gamma = (gamma - gi) * rxi;
            }
            // Remove integer content
            g /= g.integer_content();

            // If the calculated polynomial divides both a and b, this is the GCD
            ex dummy;
            if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
                g *= gc;
                ex lc = g.lcoeff(*x);
                if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
                    return -g;
                else
                    return g;
            }
        }

        // Next evaluation point
        xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
    }
    return *new ex(fail());
}


/** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
 *  and b(X) in Z[X].
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return the GCD as a new expression */
ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
{
//clog << "gcd(" << a << "," << b << ")\n";
#if STATISTICS
        gcd_called++;
#endif

        // GCD of numerics -> CLN
    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
        numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
        if (ca)
            *ca = ex_to_numeric(a) / g;
        if (cb)
            *cb = ex_to_numeric(b) / g;
        return g;
    }

        // Check arguments
    if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
        throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
    }

        // Partially factored cases (to avoid expanding large expressions)
        if (is_ex_exactly_of_type(a, mul)) {
                if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
                        goto factored_b;
factored_a:
                ex g = _ex1();
                ex acc_ca = _ex1();
                ex part_b = b;
                for (unsigned i=0; i<a.nops(); i++) {
                        ex part_ca, part_cb;
                        g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
                        acc_ca *= part_ca;
                        part_b = part_cb;
                }
                if (ca)
                        *ca = acc_ca;
                if (cb)
                        *cb = part_b;
                return g;
        } else if (is_ex_exactly_of_type(b, mul)) {
                if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
                        goto factored_a;
factored_b:
                ex g = _ex1();
                ex acc_cb = _ex1();
                ex part_a = a;
                for (unsigned i=0; i<b.nops(); i++) {
                        ex part_ca, part_cb;
                        g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
                        acc_cb *= part_cb;
                        part_a = part_ca;
                }
                if (ca)
                        *ca = part_a;
                if (cb)
                        *cb = acc_cb;
                return g;
        }

#if FAST_COMPARE
        // Input polynomials of the form poly^n are sometimes also trivial
        if (is_ex_exactly_of_type(a, power)) {
                ex p = a.op(0);
                if (is_ex_exactly_of_type(b, power)) {
                        if (p.is_equal(b.op(0))) {
                                // a = p^n, b = p^m, gcd = p^min(n, m)
                                ex exp_a = a.op(1), exp_b = b.op(1);
                                if (exp_a < exp_b) {
                                        if (ca)
                                                *ca = _ex1();
                                        if (cb)
                                                *cb = power(p, exp_b - exp_a);
                                        return power(p, exp_a);
                                } else {
                                        if (ca)
                                                *ca = power(p, exp_a - exp_b);
                                        if (cb)
                                                *cb = _ex1();
                                        return power(p, exp_b);
                                }
                        }
                } else {
                        if (p.is_equal(b)) {
                                // a = p^n, b = p, gcd = p
                                if (ca)
                                        *ca = power(p, a.op(1) - 1);
                                if (cb)
                                        *cb = _ex1();
                                return p;
                        }
                }
        } else if (is_ex_exactly_of_type(b, power)) {
                ex p = b.op(0);
                if (p.is_equal(a)) {
                        // a = p, b = p^n, gcd = p
                        if (ca)
                                *ca = _ex1();
                        if (cb)
                                *cb = power(p, b.op(1) - 1);
                        return p;
                }
        }
#endif

    // Some trivial cases
        ex aex = a.expand(), bex = b.expand();
    if (aex.is_zero()) {
        if (ca)
            *ca = _ex0();
        if (cb)
            *cb = _ex1();
        return b;
    }
    if (bex.is_zero()) {
        if (ca)
            *ca = _ex1();
        if (cb)
            *cb = _ex0();
        return a;
    }
    if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) {
        if (ca)
            *ca = a;
        if (cb)
            *cb = b;
        return _ex1();
    }
#if FAST_COMPARE
    if (a.is_equal(b)) {
        if (ca)
            *ca = _ex1();
        if (cb)
            *cb = _ex1();
        return a;
    }
#endif

    // Gather symbol statistics
    sym_desc_vec sym_stats;
    get_symbol_stats(a, b, sym_stats);

    // The symbol with least degree is our main variable
    sym_desc_vec::const_iterator var = sym_stats.begin();
    const symbol *x = var->sym;

    // Cancel trivial common factor
    int ldeg_a = var->ldeg_a;
    int ldeg_b = var->ldeg_b;
    int min_ldeg = min(ldeg_a, ldeg_b);
    if (min_ldeg > 0) {
        ex common = power(*x, min_ldeg);
//clog << "trivial common factor " << common << endl;
        return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
    }

    // Try to eliminate variables
    if (var->deg_a == 0) {
//clog << "eliminating variable " << *x << " from b" << endl;
        ex c = bex.content(*x);
        ex g = gcd(aex, c, ca, cb, false);
        if (cb)
            *cb *= bex.unit(*x) * bex.primpart(*x, c);
        return g;
    } else if (var->deg_b == 0) {
//clog << "eliminating variable " << *x << " from a" << endl;
        ex c = aex.content(*x);
        ex g = gcd(c, bex, ca, cb, false);
        if (ca)
            *ca *= aex.unit(*x) * aex.primpart(*x, c);
        return g;
    }

    ex g;
#if 1
    // Try heuristic algorithm first, fall back to PRS if that failed
    try {
        g = heur_gcd(aex, bex, ca, cb, var);
    } catch (gcdheu_failed) {
        g = *new ex(fail());
    }
    if (is_ex_exactly_of_type(g, fail)) {
//clog << "heuristics failed" << endl;
#if STATISTICS
                heur_gcd_failed++;
#endif
#endif
//              g = heur_gcd(aex, bex, ca, cb, var);
//              g = eu_gcd(aex, bex, x);
//              g = euprem_gcd(aex, bex, x);
//              g = peu_gcd(aex, bex, x);
//              g = red_gcd(aex, bex, x);
                g = sr_gcd(aex, bex, x);
                if (g.is_equal(_ex1())) {
                        // Keep cofactors factored if possible
                        if (ca)
                                *ca = a;
                        if (cb)
                                *cb = b;
                } else {
                if (ca)
                    divide(aex, g, *ca, false);
                if (cb)
                    divide(bex, g, *cb, false);
                }
#if 1
    } else {
                if (g.is_equal(_ex1())) {
                        // Keep cofactors factored if possible
                        if (ca)
                                *ca = a;
                        if (cb)
                                *cb = b;
                }
        }
#endif
    return g;
}


/** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X].
 *
 *  @param a  first multivariate polynomial
 *  @param b  second multivariate polynomial
 *  @param check_args  check whether a and b are polynomials with rational
 *         coefficients (defaults to "true")
 *  @return the LCM as a new expression */
ex lcm(const ex &a, const ex &b, bool check_args)
{
    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
        return lcm(ex_to_numeric(a), ex_to_numeric(b));
    if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
        throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
    
    ex ca, cb;
    ex g = gcd(a, b, &ca, &cb, false);
    return ca * cb * g;
}


/*
 *  Square-free factorization
 */

// Univariate GCD of polynomials in Q[x] (used internally by sqrfree()).
// a and b can be multivariate polynomials but they are treated as univariate polynomials in x.
static ex univariate_gcd(const ex &a, const ex &b, const symbol &x)
{
    if (a.is_zero())
        return b;
    if (b.is_zero())
        return a;
    if (a.is_equal(_ex1()) || b.is_equal(_ex1()))
        return _ex1();
    if (is_ex_of_type(a, numeric) && is_ex_of_type(b, numeric))
        return gcd(ex_to_numeric(a), ex_to_numeric(b));
    if (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
        throw(std::invalid_argument("univariate_gcd: arguments must be polynomials over the rationals"));

    // Euclidean algorithm
    ex c, d, r;
    if (a.degree(x) >= b.degree(x)) {
        c = a;
        d = b;
    } else {
        c = b;
        d = a;
    }
    for (;;) {
        r = rem(c, d, x, false);
        if (r.is_zero())
            break;
        c = d;
        d = r;
    }
    return d / d.lcoeff(x);
}


/** Compute square-free factorization of multivariate polynomial a(x) using
 *  Yun´s algorithm.
 *
 * @param a  multivariate polynomial
 * @param x  variable to factor in
 * @return factored polynomial */
ex sqrfree(const ex &a, const symbol &x)
{
    int i = 1;
    ex res = _ex1();
    ex b = a.diff(x);
    ex c = univariate_gcd(a, b, x);
    ex w;
    if (c.is_equal(_ex1())) {
        w = a;
    } else {
        w = quo(a, c, x);
        ex y = quo(b, c, x);
        ex z = y - w.diff(x);
        while (!z.is_zero()) {
            ex g = univariate_gcd(w, z, x);
            res *= power(g, i);
            w = quo(w, g, x);
            y = quo(z, g, x);
            z = y - w.diff(x);
            i++;
        }
    }
    return res * power(w, i);
}


/*
 *  Normal form of rational functions
 */

/*
 *  Note: The internal normal() functions (= basic::normal() and overloaded
 *  functions) all return lists of the form {numerator, denominator}. This
 *  is to get around mul::eval()'s automatic expansion of numeric coefficients.
 *  E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
 *  the information that (a+b) is the numerator and 3 is the denominator.
 */

/** Create a symbol for replacing the expression "e" (or return a previously
 *  assigned symbol). The symbol is appended to sym_lst and returned, the
 *  expression is appended to repl_lst.
 *  @see ex::normal */
static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
{
    // Expression already in repl_lst? Then return the assigned symbol
    for (unsigned i=0; i<repl_lst.nops(); i++)
        if (repl_lst.op(i).is_equal(e))
            return sym_lst.op(i);
    
    // Otherwise create new symbol and add to list, taking care that the
        // replacement expression doesn't contain symbols from the sym_lst
        // because subs() is not recursive
        symbol s;
        ex es(s);
        ex e_replaced = e.subs(sym_lst, repl_lst);
    sym_lst.append(es);
    repl_lst.append(e_replaced);
    return es;
}

/** Create a symbol for replacing the expression "e" (or return a previously
 *  assigned symbol). An expression of the form "symbol == expression" is added
 *  to repl_lst and the symbol is returned.
 *  @see ex::to_rational */
static ex replace_with_symbol(const ex &e, lst &repl_lst)
{
    // Expression already in repl_lst? Then return the assigned symbol
    for (unsigned i=0; i<repl_lst.nops(); i++)
        if (repl_lst.op(i).op(1).is_equal(e))
            return repl_lst.op(i).op(0);
    
    // Otherwise create new symbol and add to list, taking care that the
        // replacement expression doesn't contain symbols from the sym_lst
        // because subs() is not recursive
        symbol s;
        ex es(s);
        ex e_replaced = e.subs(repl_lst);
    repl_lst.append(es == e_replaced);
    return es;
}

/** Default implementation of ex::normal(). It replaces the object with a
 *  temporary symbol.
 *  @see ex::normal */
ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
{
    return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
 *  @see ex::normal */
ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
{
    return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for a numeric. It splits complex numbers
 *  into re+I*im and replaces I and non-rational real numbers with a temporary
 *  symbol.
 *  @see ex::normal */
ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
{
        numeric num = numer();
        ex numex = num;

    if (num.is_real()) {
        if (!num.is_integer())
            numex = replace_with_symbol(numex, sym_lst, repl_lst);
    } else { // complex
        numeric re = num.real(), im = num.imag();
        ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
        ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
        numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
    }

        // Denominator is always a real integer (see numeric::denom())
        return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
}


/** Fraction cancellation.
 *  @param n  numerator
 *  @param d  denominator
 *  @return cancelled fraction {n, d} as a list */
static ex frac_cancel(const ex &n, const ex &d)
{
    ex num = n;
    ex den = d;
    numeric pre_factor = _num1();

//clog << "frac_cancel num = " << num << ", den = " << den << endl;

    // Handle special cases where numerator or denominator is 0
    if (num.is_zero())
                return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
    if (den.expand().is_zero())
        throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));

    // Bring numerator and denominator to Z[X] by multiplying with
    // LCM of all coefficients' denominators
    numeric num_lcm = lcm_of_coefficients_denominators(num);
    numeric den_lcm = lcm_of_coefficients_denominators(den);
        num = multiply_lcm(num, num_lcm);
        den = multiply_lcm(den, den_lcm);
    pre_factor = den_lcm / num_lcm;

    // Cancel GCD from numerator and denominator
    ex cnum, cden;
    if (gcd(num, den, &cnum, &cden, false) != _ex1()) {
                num = cnum;
                den = cden;
        }

        // Make denominator unit normal (i.e. coefficient of first symbol
        // as defined by get_first_symbol() is made positive)
        const symbol *x;
        if (get_first_symbol(den, x)) {
                GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric));
                if (ex_to_numeric(den.unit(*x)).is_negative()) {
                        num *= _ex_1();
                        den *= _ex_1();
                }
        }

        // Return result as list
//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl;
    return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for a sum. It expands terms and performs
 *  fractional addition.
 *  @see ex::normal */
ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
{
    // Normalize and expand children, chop into summands
    exvector o;
    o.reserve(seq.size()+1);
    epvector::const_iterator it = seq.begin(), itend = seq.end();
    while (it != itend) {

                // Normalize and expand child
        ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();

                // If numerator is a sum, chop into summands
        if (is_ex_exactly_of_type(n.op(0), add)) {
            epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
            while (bit != bitend) {
                o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated));
                bit++;
            }

                        // The overall_coeff is already normalized (== rational), we just
                        // split it into numerator and denominator
                        GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
                        numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
            o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated));
        } else
            o.push_back(n);
        it++;
    }
    o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));

        // o is now a vector of {numerator, denominator} lists

    // Determine common denominator
    ex den = _ex1();
    exvector::const_iterator ait = o.begin(), aitend = o.end();
//clog << "add::normal uses the following summands:\n";
    while (ait != aitend) {
//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl;
        den = lcm(ait->op(1), den, false);
        ait++;
    }
//clog << " common denominator = " << den << endl;

    // Add fractions
    if (den.is_equal(_ex1())) {

                // Common denominator is 1, simply add all numerators
        exvector num_seq;
                for (ait=o.begin(); ait!=aitend; ait++) {
                        num_seq.push_back(ait->op(0));
                }
                return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated);

        } else {

                // Perform fractional addition
        exvector num_seq;
        for (ait=o.begin(); ait!=aitend; ait++) {
            ex q;
            if (!divide(den, ait->op(1), q, false)) {
                // should not happen
                throw(std::runtime_error("invalid expression in add::normal, division failed"));
            }
            num_seq.push_back((ait->op(0) * q).expand());
        }
        ex num = (new add(num_seq))->setflag(status_flags::dynallocated);

        // Cancel common factors from num/den
        return frac_cancel(num, den);
    }
}


/** Implementation of ex::normal() for a product. It cancels common factors
 *  from fractions.
 *  @see ex::normal() */
ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
{
    // Normalize children, separate into numerator and denominator
        ex num = _ex1();
        ex den = _ex1(); 
        ex n;
    epvector::const_iterator it = seq.begin(), itend = seq.end();
    while (it != itend) {
                n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
                num *= n.op(0);
                den *= n.op(1);
        it++;
    }
        n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
        num *= n.op(0);
        den *= n.op(1);

        // Perform fraction cancellation
    return frac_cancel(num, den);
}


/** Implementation of ex::normal() for powers. It normalizes the basis,
 *  distributes integer exponents to numerator and denominator, and replaces
 *  non-integer powers by temporary symbols.
 *  @see ex::normal */
ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
{
        // Normalize basis
    ex n = basis.bp->normal(sym_lst, repl_lst, level-1);

        if (exponent.info(info_flags::integer)) {

            if (exponent.info(info_flags::positive)) {

                        // (a/b)^n -> {a^n, b^n}
                        return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);

                } else if (exponent.info(info_flags::negative)) {

                        // (a/b)^-n -> {b^n, a^n}
                        return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
                }

        } else {

                if (exponent.info(info_flags::positive)) {

                        // (a/b)^x -> {sym((a/b)^x), 1}
                        return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);

                } else if (exponent.info(info_flags::negative)) {

                        if (n.op(1).is_equal(_ex1())) {

                                // a^-x -> {1, sym(a^x)}
                                return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);

                        } else {

                                // (a/b)^-x -> {sym((b/a)^x), 1}
                                return (new lst(replace_with_symbol(power(n.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
                        }

                } else {        // exponent not numeric

                        // (a/b)^x -> {sym((a/b)^x, 1}
                        return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
                }
    }
}


/** Implementation of ex::normal() for pseries. It normalizes each coefficient and
 *  replaces the series by a temporary symbol.
 *  @see ex::normal */
ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
{
    epvector new_seq;
    new_seq.reserve(seq.size());

    epvector::const_iterator it = seq.begin(), itend = seq.end();
    while (it != itend) {
        new_seq.push_back(expair(it->rest.normal(), it->coeff));
        it++;
    }
    ex n = pseries(relational(var,point), new_seq);
        return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
}


/** Implementation of ex::normal() for relationals. It normalizes both sides.
 *  @see ex::normal */
ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const
{
        return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated);
}


/** Normalization of rational functions.
 *  This function converts an expression to its normal form
 *  "numerator/denominator", where numerator and denominator are (relatively
 *  prime) polynomials. Any subexpressions which are not rational functions
 *  (like non-rational numbers, non-integer powers or functions like sin(),
 *  cos() etc.) are replaced by temporary symbols which are re-substituted by
 *  the (normalized) subexpressions before normal() returns (this way, any
 *  expression can be treated as a rational function). normal() is applied
 *  recursively to arguments of functions etc.
 *
 *  @param level maximum depth of recursion
 *  @return normalized expression */
ex ex::normal(int level) const
{
    lst sym_lst, repl_lst;

    ex e = bp->normal(sym_lst, repl_lst, level);
        GINAC_ASSERT(is_ex_of_type(e, lst));

        // Re-insert replaced symbols
    if (sym_lst.nops() > 0)
        e = e.subs(sym_lst, repl_lst);

        // Convert {numerator, denominator} form back to fraction
    return e.op(0) / e.op(1);
}

/** Numerator of an expression. If the expression is not of the normal form
 *  "numerator/denominator", it is first converted to this form and then the
 *  numerator is returned.
 *
 *  @see ex::normal
 *  @return numerator */
ex ex::numer(void) const
{
    lst sym_lst, repl_lst;

    ex e = bp->normal(sym_lst, repl_lst, 0);
        GINAC_ASSERT(is_ex_of_type(e, lst));

        // Re-insert replaced symbols
    if (sym_lst.nops() > 0)
        return e.op(0).subs(sym_lst, repl_lst);
        else
                return e.op(0);
}

/** Denominator of an expression. If the expression is not of the normal form
 *  "numerator/denominator", it is first converted to this form and then the
 *  denominator is returned.
 *
 *  @see ex::normal
 *  @return denominator */
ex ex::denom(void) const
{
    lst sym_lst, repl_lst;

    ex e = bp->normal(sym_lst, repl_lst, 0);
        GINAC_ASSERT(is_ex_of_type(e, lst));

        // Re-insert replaced symbols
    if (sym_lst.nops() > 0)
        return e.op(1).subs(sym_lst, repl_lst);
        else
                return e.op(1);
}


/** Default implementation of ex::to_rational(). It replaces the object with a
 *  temporary symbol.
 *  @see ex::to_rational */
ex basic::to_rational(lst &repl_lst) const
{
        return replace_with_symbol(*this, repl_lst);
}


/** Implementation of ex::to_rational() for symbols. This returns the
 *  unmodified symbol.
 *  @see ex::to_rational */
ex symbol::to_rational(lst &repl_lst) const
{
    return *this;
}


/** Implementation of ex::to_rational() for a numeric. It splits complex
 *  numbers into re+I*im and replaces I and non-rational real numbers with a
 *  temporary symbol.
 *  @see ex::to_rational */
ex numeric::to_rational(lst &repl_lst) const
{
    if (is_real()) {
        if (!is_rational())
            return replace_with_symbol(*this, repl_lst);
    } else { // complex
        numeric re = real();
        numeric im = imag();
        ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst);
        ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst);
        return re_ex + im_ex * replace_with_symbol(I, repl_lst);
    }
        return *this;
}


/** Implementation of ex::to_rational() for powers. It replaces non-integer
 *  powers by temporary symbols.
 *  @see ex::to_rational */
ex power::to_rational(lst &repl_lst) const
{
        if (exponent.info(info_flags::integer))
                return power(basis.to_rational(repl_lst), exponent);
        else
                return replace_with_symbol(*this, repl_lst);
}


/** Implementation of ex::to_rational() for expairseqs.
 *  @see ex::to_rational */
ex expairseq::to_rational(lst &repl_lst) const
{
    epvector s;
    s.reserve(seq.size());
    for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) {
        s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst)));
        // s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst),
    }
    ex oc = overall_coeff.to_rational(repl_lst);
    if (oc.info(info_flags::numeric))
        return thisexpairseq(s, overall_coeff);
    else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1()));
    return thisexpairseq(s, default_overall_coeff());
}


/** Rationalization of non-rational functions.
 *  This function converts a general expression to a rational polynomial
 *  by replacing all non-rational subexpressions (like non-rational numbers,
 *  non-integer powers or functions like sin(), cos() etc.) to temporary
 *  symbols. This makes it possible to use functions like gcd() and divide()
 *  on non-rational functions by applying to_rational() on the arguments,
 *  calling the desired function and re-substituting the temporary symbols
 *  in the result. To make the last step possible, all temporary symbols and
 *  their associated expressions are collected in the list specified by the
 *  repl_lst parameter in the form {symbol == expression}, ready to be passed
 *  as an argument to ex::subs().
 *
 *  @param repl_lst collects a list of all temporary symbols and their replacements
 *  @return rationalized expression */
ex ex::to_rational(lst &repl_lst) const
{
        return bp->to_rational(repl_lst);
}


#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
#endif // ndef NO_NAMESPACE_GINAC