X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=a161f82133a7a448ede06ff831868e9bb31e91d2;hp=82fcaf848dd250ea14106dab40a9b0d87b40edf2;hb=7bb797bc41e31fe098c94f0b5e57ddbba6cd26b6;hpb=9eab44408b9213d8909b7a9e525f404ad06064dd

diff --git a/ginac/normal.cpp b/ginac/normal.cpp
index 82fcaf84..a161f821 100644
--- a/ginac/normal.cpp
+++ b/ginac/normal.cpp
@@ -3,11 +3,10 @@
  *  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.
- */
+ *  computation, square-free factorization and rational function normalization. */
 
 /*
- *  GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2001 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
@@ -24,7 +23,6 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
-#include <stdexcept>
 #include <algorithm>
 #include <map>
 
@@ -35,16 +33,16 @@
 #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 "series.h"
+#include "matrix.h"
+#include "pseries.h"
 #include "symbol.h"
+#include "utils.h"
 
 namespace GiNaC {
 
@@ -54,7 +52,35 @@ namespace GiNaC {
 #define FAST_COMPARE 1
 
 // Set this if you want divide_in_z() to use remembering
-#define USE_REMEMBER 1
+#define USE_REMEMBER 0
+
+// Set this if you want divide_in_z() to use trial division followed by
+// polynomial interpolation (always slower except for completely dense
+// polynomials)
+#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() {
+		std::cout << "gcd() called " << gcd_called << " times\n";
+		std::cout << "sr_gcd() called " << sr_gcd_called << " times\n";
+		std::cout << "heur_gcd() called " << heur_gcd_called << " times\n";
+		std::cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
+	}
+} stat_print;
+#endif
 
 
 /** Return pointer to first symbol found in expression.  Due to GiNaC´s
@@ -64,21 +90,20 @@ namespace GiNaC {
  *  @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 (int 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;
+	if (is_ex_exactly_of_type(e, symbol)) {
+		x = &ex_to<symbol>(e);
+		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;
 }
 
 
@@ -93,56 +118,65 @@ static bool get_first_symbol(const ex &e, const symbol *&x)
  *
  *  @see get_symbol_stats */
 struct sym_desc {
-    /** Pointer to symbol */
-    const symbol *sym;
+	/** Pointer to symbol */
+	const symbol *sym;
+
+	/** Highest degree of symbol in polynomial "a" */
+	int deg_a;
 
-    /** Highest degree of symbol in polynomial "a" */
-    int deg_a;
+	/** Highest degree of symbol in polynomial "b" */
+	int deg_b;
 
-    /** 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 "a" */
-    int ldeg_a;
+	/** Lowest degree of symbol in polynomial "b" */
+	int ldeg_b;
 
-    /** Lowest degree of symbol in polynomial "b" */
-    int ldeg_b;
+	/** Maximum of deg_a and deg_b (Used for sorting) */
+	int max_deg;
 
-    /** Minimum of ldeg_a and ldeg_b (Used for sorting) */
-    int min_deg;
+	/** Maximum number of terms of leading coefficient of symbol in both polynomials */
+	int max_lcnops;
 
-    /** Commparison operator for sorting */
-    bool operator<(const sym_desc &x) const {return min_deg < x.min_deg;}
+	/** Commparison operator for sorting */
+	bool operator<(const sym_desc &x) const
+	{
+		if (max_deg == x.max_deg)
+			return max_lcnops < x.max_lcnops;
+		else
+			return max_deg < x.max_deg;
+	}
 };
 
 // Vector of sym_desc structures
-typedef vector<sym_desc> sym_desc_vec;
+typedef std::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);
+	sym_desc_vec::const_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 (int 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);
-    }
+	if (is_ex_exactly_of_type(e, symbol)) {
+		add_symbol(&ex_to<symbol>(e), 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.
@@ -157,23 +191,32 @@ static void collect_symbols(const ex &e, sym_desc_vec &v)
  *  @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->min_deg = min(deg_a, deg_b);
-        it->ldeg_a = a.ldegree(*(it->sym));
-        it->ldeg_b = b.ldegree(*(it->sym));
-        it++;
-    }
-    sort(v.begin(), v.end());
+	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 = std::max(deg_a, deg_b);
+		it->max_lcnops = std::max(a.lcoeff(*(it->sym)).nops(), b.lcoeff(*(it->sym)).nops());
+		it->ldeg_a = a.ldegree(*(it->sym));
+		it->ldeg_b = b.ldegree(*(it->sym));
+		++it;
+	}
+	std::sort(v.begin(), v.end());
+#if 0
+	std::clog << "Symbols:\n";
+	it = v.begin(); itend = v.end();
+	while (it != itend) {
+		std::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 << ", max_lcnops=" << it->max_lcnops << endl;
+		std::clog << "  lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
+		++it;
+	}
+#endif
 }
 
 
@@ -185,30 +228,70 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
 // 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) || is_ex_exactly_of_type(e, mul)) {
-        numeric c = numONE();
-        for (int i=0; i<e.nops(); i++) {
-            c = lcmcoeff(e.op(i), c);
-        }
-        return lcm(c, l);
-    } else if (is_ex_exactly_of_type(e, power))
-        return lcmcoeff(e.op(0), l);
-    return 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)) {
+		if (is_ex_exactly_of_type(e.op(0), symbol))
+			return l;
+		else
+			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].
+ *  to bring a polynomial from Q[X] to Z[X].
  *
- *  @param e  multivariate polynomial
+ *  @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.expand(), numONE());
+	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)) {
+		unsigned num = e.nops();
+		exvector v; v.reserve(num + 1);
+		numeric lcm_accum = _num1;
+		for (unsigned i=0; i<e.nops(); i++) {
+			numeric op_lcm = lcmcoeff(e.op(i), _num1);
+			v.push_back(multiply_lcm(e.op(i), op_lcm));
+			lcm_accum *= op_lcm;
+		}
+		v.push_back(lcm / lcm_accum);
+		return (new mul(v))->setflag(status_flags::dynallocated);
+	} else if (is_ex_exactly_of_type(e, add)) {
+		unsigned num = e.nops();
+		exvector v; v.reserve(num);
+		for (unsigned i=0; i<num; i++)
+			v.push_back(multiply_lcm(e.op(i), lcm));
+		return (new add(v))->setflag(status_flags::dynallocated);
+	} else if (is_ex_exactly_of_type(e, power)) {
+		if (is_ex_exactly_of_type(e.op(0), symbol))
+			return e * lcm;
+		else
+			return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
+	} else
+		return e * lcm;
 }
 
 
@@ -217,51 +300,50 @@ static numeric lcm_of_coefficients_denominators(const ex &e)
  *
  *  @param e  expanded polynomial
  *  @return integer content */
-
 numeric ex::integer_content(void) const
 {
-    ASSERT(bp!=0);
-    return bp->integer_content();
+	GINAC_ASSERT(bp!=0);
+	return bp->integer_content();
 }
 
 numeric basic::integer_content(void) const
 {
-    return numONE();
+	return _num1;
 }
 
 numeric numeric::integer_content(void) const
 {
-    return abs(*this);
+	return abs(*this);
 }
 
 numeric add::integer_content(void) const
 {
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    numeric c = numZERO();
-    while (it != itend) {
-        ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-        ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
-        c = gcd(ex_to_numeric(it->coeff), c);
-        it++;
-    }
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    c = gcd(ex_to_numeric(overall_coeff),c);
-    return c;
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	numeric c = _num0;
+	while (it != itend) {
+		GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+		GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
+		c = gcd(ex_to<numeric>(it->coeff), c);
+		it++;
+	}
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	c = gcd(ex_to<numeric>(overall_coeff),c);
+	return c;
 }
 
 numeric mul::integer_content(void) const
 {
-#ifdef DOASSERT
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    while (it != itend) {
-        ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
-        ++it;
-    }
-#endif // def DOASSERT
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    return abs(ex_to_numeric(overall_coeff));
+#ifdef DO_GINAC_ASSERT
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	while (it != itend) {
+		GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+		++it;
+	}
+#endif // def DO_GINAC_ASSERT
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	return abs(ex_to<numeric>(overall_coeff));
 }
 
 
@@ -278,45 +360,44 @@ numeric mul::integer_content(void) const
  *  @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 (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 exONE();
+	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 = exZERO();
-    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;
+	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 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);
+	exvector v; v.reserve(rdeg - bdeg + 1);
+	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 fail())->setflag(status_flags::dynallocated);
+		}
+		term *= power(x, rdeg - bdeg);
+		v.push_back(term);
+		r -= (term * b).expand();
+		if (r.is_zero())
+			break;
+		rdeg = r.degree(x);
+	}
+	return (new add(v))->setflag(status_flags::dynallocated);
 }
 
 
@@ -329,47 +410,64 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @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 exZERO();
-        else
-            return b;
-    }
+	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 a;
+	}
 #if FAST_COMPARE
-    if (a.is_equal(b))
-        return exZERO();
+	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;
+	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 fail())->setflag(status_flags::dynallocated);
+		}
+		term *= power(x, rdeg - bdeg);
+		r -= (term * b).expand();
+		if (r.is_zero())
+			break;
+		rdeg = r.degree(x);
+	}
+	return r;
+}
+
+
+/** Decompose rational function a(x)=N(x)/D(x) into P(x)+n(x)/D(x)
+ *  with degree(n, x) < degree(D, x).
+ *
+ *  @param a rational function in x
+ *  @param x a is a function of x
+ *  @return decomposed function. */
+ex decomp_rational(const ex &a, const symbol &x)
+{
+	ex nd = numer_denom(a);
+	ex numer = nd.op(0), denom = nd.op(1);
+	ex q = quo(numer, denom, x);
+	if (is_ex_exactly_of_type(q, fail))
+		return a;
+	else
+		return q + rem(numer, denom, x) / denom;
 }
 
 
@@ -381,48 +479,97 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @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 exZERO();
-        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 = exZERO();
-        else
-            eb -= blcoeff * power(x, bdeg);
-    } else
-        blcoeff = exONE();
-
-    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 = exZERO();
-        else
-            r -= rlcoeff * power(x, rdeg);
-        r = (blcoeff * r).expand() - term;
-        rdeg = r.degree(x);
-        i++;
-    }
-    return power(blcoeff, delta - i) * r;
+	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;
+}
+
+
+/** Sparse 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 sparse pseudo-remainder of a(x) and b(x) in Z[x] */
+ex sprem(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;
+
+	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);
+	}
+	return r;
 }
 
 
@@ -434,55 +581,63 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @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 */
-
+ *          "false" otherwise (q left untouched) */
 bool divide(const ex &a, const ex &b, ex &q, bool check_args)
 {
-    q = exZERO();
-    if (b.is_zero())
-        throw(std::overflow_error("divide: division by zero"));
-    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 (b.is_zero())
+		throw(std::overflow_error("divide: division by zero"));
+	if (a.is_zero()) {
+		q = _ex0;
+		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 = exONE();
-        return true;
-    }
+	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 (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()) {
+		q = _ex0;
+		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);
+	exvector v; v.reserve(rdeg - bdeg + 1);
+	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);
+		v.push_back(term);
+		r -= (term * b).expand();
+		if (r.is_zero()) {
+			q = (new add(v))->setflag(status_flags::dynallocated);
+			return true;
+		}
+		rdeg = r.degree(*x);
+	}
+	return false;
 }
 
 
@@ -491,17 +646,18 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args)
  *  Remembering
  */
 
-typedef pair<ex, ex> ex2;
-typedef pair<ex, bool> exbool;
+typedef std::pair<ex, ex> ex2;
+typedef std::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);        
-    }
+	bool operator() (const ex2 &p, const ex2 &q) const 
+	{
+		int cmp = p.first.compare(q.first);
+		return ((cmp<0) || (!(cmp>0) && p.second.compare(q.second)<0));
+	}
 };
 
-typedef map<ex2, exbool, ex2_less> ex2_exbool_remember;
+typedef std::map<ex2, exbool, ex2_less> ex2_exbool_remember;
 #endif
 
 
@@ -523,127 +679,130 @@ typedef map<ex2, exbool, ex2_less> ex2_exbool_remember;
  *  @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 = exZERO();
-    if (b.is_zero())
-        throw(std::overflow_error("divide_in_z: division by zero"));
-    if (b.is_equal(exONE())) {
-        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;
-    }
+	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 = exONE();
-        return true;
-    }
+	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;
-    }
+	// 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;
+	// 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;
+	}
 
-    // Compare degrees
-    int adeg = a.degree(*x), bdeg = b.degree(*x);
-    if (bdeg > adeg)
-        return false;
+	// 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());
+	}
 
-#if 1
+	// 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];
 
-    // 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 (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);
+	exvector v; v.reserve(rdeg - bdeg + 1);
+	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();
+		v.push_back(term);
+		r -= (term * eb).expand();
+		if (r.is_zero()) {
+			q = (new add(v))->setflag(status_flags::dynallocated);
 #if USE_REMEMBER
-            dr_remember[ex2(a, b)] = exbool(q, true);
+			dr_remember[ex2(a, b)] = exbool(q, true);
 #endif
-            return true;
-        }
-        rdeg = r.degree(*x);
-    }
+			return true;
+		}
+		rdeg = r.degree(*x);
+	}
 #if USE_REMEMBER
-    dr_remember[ex2(a, b)] = exbool(q, false);
+	dr_remember[ex2(a, b)] = exbool(q, false);
 #endif
-    return false;
+	return false;
 
-#else
-
-    // Trial division using 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 = numZERO();
-    ex c;
-    for (i=0; i<=adeg; i++) {
-        ex bs = b.subs(*x == point);
-        while (bs.is_zero()) {
-            point += numONE();
-            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 += numONE();
-    }
-
-    // Compute inverses
-    vector<numeric> rcp; rcp.reserve(adeg + 1);
-    rcp.push_back(0);
-    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;
 #endif
 }
 
@@ -661,16 +820,16 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
  *  @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 < exZERO() ? exMINUSONE() : exONE();
-    else {
-        const symbol *y;
-        if (get_first_symbol(c, y))
-            return c.unit(*y);
-        else
-            throw(std::invalid_argument("invalid expression in unit()"));
-    }
+	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()"));
+	}
 }
 
 
@@ -683,30 +842,30 @@ ex ex::unit(const symbol &x) const
  *  @see ex::unit, ex::primpart */
 ex ex::content(const symbol &x) const
 {
-    if (is_zero())
-        return exZERO();
-    if (is_ex_exactly_of_type(*this, numeric))
-        return info(info_flags::negative) ? -*this : *this;
-    ex e = expand();
-    if (e.is_zero())
-        return exZERO();
-
-    // 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 = exZERO();
-    for (int i=ldeg; i<=deg; i++)
-        c = gcd(e.coeff(x, i), c, NULL, NULL, false);
-    return c;
+	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;
 }
 
 
@@ -719,19 +878,19 @@ ex ex::content(const symbol &x) const
  *  @see ex::unit, ex::content */
 ex ex::primpart(const symbol &x) const
 {
-    if (is_zero())
-        return exZERO();
-    if (is_ex_exactly_of_type(*this, numeric))
-        return exONE();
-
-    ex c = content(x);
-    if (c.is_zero())
-        return exZERO();
-    ex u = unit(x);
-    if (is_ex_exactly_of_type(c, numeric))
-        return *this / (c * u);
-    else
-        return quo(*this, c * u, x, false);
+	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);
 }
 
 
@@ -742,21 +901,20 @@ ex ex::primpart(const symbol &x) const
  *  @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 exZERO();
-    if (c.is_zero())
-        return exZERO();
-    if (is_ex_exactly_of_type(*this, numeric))
-        return exONE();
-
-    ex u = unit(x);
-    if (is_ex_exactly_of_type(c, numeric))
-        return *this / (c * u);
-    else
-        return quo(*this, c * u, x, false);
+	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);
 }
 
 
@@ -764,8 +922,141 @@ ex ex::primpart(const symbol &x, const ex &c) const
  *  GCD of multivariate polynomials
  */
 
-/** Compute GCD of multivariate polynomials using the subresultant PRS
- *  algorithm. This function is used internally gy gcd().
+/** Compute GCD of polynomials in Q[X] using the Euclidean 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)
+{
+//std::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;
+	}
+
+	// Normalize in Q[x]
+	c = c / c.lcoeff(*x);
+	d = d / d.lcoeff(*x);
+
+	// Euclidean algorithm
+	ex r;
+	for (;;) {
+//std::clog << " d = " << d << endl;
+		r = rem(c, d, *x, false);
+		if (r.is_zero())
+			return d / d.lcoeff(*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)
+{
+//std::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;
+	}
+
+	// Calculate GCD of contents
+	ex gamma = gcd(c.content(*x), d.content(*x), NULL, NULL, false);
+
+	// Euclidean algorithm with pseudo-remainders
+	ex r;
+	for (;;) {
+//std::clog << " d = " << d << endl;
+		r = prem(c, d, *x, false);
+		if (r.is_zero())
+			return d.primpart(*x) * gamma;
+		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)
+{
+//std::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 (;;) {
+//std::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
@@ -773,62 +1064,142 @@ ex ex::primpart(const symbol &x, const ex &c) const
  *  @return the GCD as a new expression
  *  @see gcd */
 
-static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
+static ex red_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//std::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 divisor sequence
+	ex r, ri = _ex1;
+	int delta = cdeg - ddeg;
+
+	for (;;) {
+		// Calculate polynomial pseudo-remainder
+//std::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 var iterator to first element of vector of sym_desc structs
+ *  @return the GCD as a new expression
+ *  @see gcd */
+
+static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
 {
-    // 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 = exZERO(), ri = exONE(), psi = exONE();
-    int delta = cdeg - ddeg;
-
-    for (;;) {
-        // Calculate polynomial pseudo-remainder
-        r = prem(c, d, *x, false);
-        if (r.is_zero())
-            return gamma * d.primpart(*x);
-        c = d;
-        cdeg = ddeg;
-        if (!divide(r, ri * power(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
-        ri = c.expand().lcoeff(*x);
-        if (delta == 1)
-            psi = ri;
-        else if (delta)
-            divide(power(ri, delta), power(psi, delta-1), psi, false);
-        delta = cdeg - ddeg;
-    }
+//std::clog << "sr_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+	sr_gcd_called++;
+#endif
+
+	// The first symbol is our main variable
+	const symbol &x = *(var->sym);
+
+	// 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);
+//std::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
+//std::clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
+//std::clog << " d = " << d << endl;
+		r = prem(c, d, x, false);
+		if (r.is_zero())
+			return gamma * d.primpart(x);
+		c = d;
+		cdeg = ddeg;
+//std::clog << " dividing...\n";
+		if (!divide_in_z(r, ri * pow(psi, delta), d, var))
+			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
+//std::clog << " calculating next subresultant...\n";
+		ri = c.expand().lcoeff(x);
+		if (delta == 1)
+			psi = ri;
+		else if (delta)
+			divide_in_z(pow(ri, delta), pow(psi, delta-1), psi, var+1);
+		delta = cdeg - ddeg;
+	}
 }
 
 
@@ -838,117 +1209,124 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
  *  @param e  expanded multivariate polynomial
  *  @return maximum coefficient
  *  @see heur_gcd */
-
 numeric ex::max_coefficient(void) const
 {
-    ASSERT(bp!=0);
-    return bp->max_coefficient();
+	GINAC_ASSERT(bp!=0);
+	return bp->max_coefficient();
 }
 
+/** Implementation ex::max_coefficient().
+ *  @see heur_gcd */
 numeric basic::max_coefficient(void) const
 {
-    return numONE();
+	return _num1;
 }
 
 numeric numeric::max_coefficient(void) const
 {
-    return abs(*this);
+	return abs(*this);
 }
 
 numeric add::max_coefficient(void) const
 {
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    numeric cur_max = abs(ex_to_numeric(overall_coeff));
-    while (it != itend) {
-        numeric a;
-        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;
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	numeric cur_max = abs(ex_to<numeric>(overall_coeff));
+	while (it != itend) {
+		numeric a;
+		GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+		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 DOASSERT
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    while (it != itend) {
-        ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
-        it++;
-    }
-#endif // def DOASSERT
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    return abs(ex_to_numeric(overall_coeff));
+#ifdef DO_GINAC_ASSERT
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	while (it != itend) {
+		GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+		it++;
+	}
+#endif // def DO_GINAC_ASSERT
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	return abs(ex_to<numeric>(overall_coeff));
 }
 
 
-/** Apply symmetric modular homomorphism to a multivariate polynomial.
- *  This function is used internally by heur_gcd().
+/** Apply symmetric modular homomorphism to an expanded multivariate
+ *  polynomial.  This function is usually 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
-{
-    ASSERT(bp!=0);
-    return bp->smod(xi);
-}
-
 ex basic::smod(const numeric &xi) const
 {
-    return *this;
+	return *this;
 }
 
 ex numeric::smod(const numeric &xi) const
 {
-    return GiNaC::smod(*this, xi);
+	return GiNaC::smod(*this, xi);
 }
 
 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) {
-        ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-        numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
-        if (!coeff.is_zero())
-            newseq.push_back(expair(it->rest, coeff));
-        it++;
-    }
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
-    return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
+	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_exactly_a<numeric>(it->rest));
+		numeric coeff = GiNaC::smod(ex_to<numeric>(it->coeff), xi);
+		if (!coeff.is_zero())
+			newseq.push_back(expair(it->rest, coeff));
+		it++;
+	}
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
+	return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
 }
 
 ex mul::smod(const numeric &xi) const
 {
-#ifdef DOASSERT
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    while (it != itend) {
-        ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
-        it++;
-    }
-#endif // def DOASSERT
-    mul * mulcopyp=new mul(*this);
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-    mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
-    mulcopyp->clearflag(status_flags::evaluated);
-    mulcopyp->clearflag(status_flags::hash_calculated);
-    return mulcopyp->setflag(status_flags::dynallocated);
+#ifdef DO_GINAC_ASSERT
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	while (it != itend) {
+		GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+		it++;
+	}
+#endif // def DO_GINAC_ASSERT
+	mul * mulcopyp = new mul(*this);
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	mulcopyp->overall_coeff = GiNaC::smod(ex_to<numeric>(overall_coeff),xi);
+	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 */
+/** xi-adic polynomial interpolation */
+static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x, int degree_hint = 1)
+{
+	exvector g; g.reserve(degree_hint);
+	ex e = gamma;
+	numeric rxi = xi.inverse();
+	for (int i=0; !e.is_zero(); i++) {
+		ex gi = e.smod(xi);
+		g.push_back(gi * power(x, i));
+		e = (e - gi) * rxi;
+	}
+	return (new add(g))->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.
@@ -966,75 +1344,108 @@ class gcdheu_failed {};
  *  @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)
 {
-    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 * numTWO() + numTWO();
-    else
-        xi = mp * numTWO() + numTWO();
-
-    // 6 tries maximum
-    for (int t=0; t<6; t++) {
-        if (xi.int_length() * maxdeg > 50000)
-            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 = exZERO();
-            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) && lc.compare(exZERO()) < 0)
-                    return -g;
-                else
-                    return g;
-            }
-        }
-
-        // Next evaluation point
-        xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
-    }
-    return *new ex(fail());
+//std::clog << "heur_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+	heur_gcd_called++;
+#endif
+
+	// Algorithm only works for non-vanishing input polynomials
+	if (a.is_zero() || b.is_zero())
+		return (new fail())->setflag(status_flags::dynallocated);
+
+	// 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));
+		if (ca)
+			*ca = ex_to<numeric>(a) / g;
+		if (cb)
+			*cb = ex_to<numeric>(b) / g;
+		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 =  std::max(p.degree(x), q.degree(x));
+	
+	// Find evaluation point
+	numeric mp = p.max_coefficient();
+	numeric 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) {
+//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl;
+			throw gcdheu_failed();
+		}
+
+		// Apply evaluation homomorphism and calculate GCD
+		ex cp, cq;
+		ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), &cp, &cq, var+1).expand();
+		if (!is_ex_exactly_of_type(gamma, fail)) {
+
+			// Reconstruct polynomial from GCD of mapped polynomials
+			ex g = interpolate(gamma, xi, x, maxdeg);
+
+			// Remove integer content
+			g /= g.integer_content();
+
+			// If the calculated polynomial divides both p and q, 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;
+			}
+#if 0
+			cp = interpolate(cp, xi, x);
+			if (divide_in_z(cp, p, g, var)) {
+				if (divide_in_z(g, q, cb ? *cb : dummy, var)) {
+					g *= gc;
+					if (ca)
+						*ca = cp;
+					ex lc = g.lcoeff(x);
+					if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+						return -g;
+					else
+						return g;
+				}
+			}
+			cq = interpolate(cq, xi, x);
+			if (divide_in_z(cq, q, g, var)) {
+				if (divide_in_z(g, p, ca ? *ca : dummy, var)) {
+					g *= gc;
+					if (cb)
+						*cb = cq;
+					ex lc = g.lcoeff(x);
+					if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+						return -g;
+					else
+						return g;
+				}
+			}
+#endif
+		}
+
+		// Next evaluation point
+		xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
+	}
+	return (new fail())->setflag(status_flags::dynallocated);
 }
 
 
@@ -1046,105 +1457,235 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
  *  @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)
 {
-    // Some trivial cases
-    if (a.is_zero()) {
-        if (ca)
-            *ca = exZERO();
-        if (cb)
-            *cb = exONE();
-        return b;
-    }
-    if (b.is_zero()) {
-        if (ca)
-            *ca = exONE();
-        if (cb)
-            *cb = exZERO();
-        return a;
-    }
-    if (a.is_equal(exONE()) || b.is_equal(exONE())) {
-        if (ca)
-            *ca = a;
-        if (cb)
-            *cb = b;
-        return exONE();
-    }
+//std::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 || cb) {
+			if (g.is_zero()) {
+				if (ca)
+					*ca = _ex0;
+				if (cb)
+					*cb = _ex0;
+			} else {
+				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:
+		unsigned num = a.nops();
+		exvector g; g.reserve(num);
+		exvector acc_ca; acc_ca.reserve(num);
+		ex part_b = b;
+		for (unsigned i=0; i<num; i++) {
+			ex part_ca, part_cb;
+			g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, check_args));
+			acc_ca.push_back(part_ca);
+			part_b = part_cb;
+		}
+		if (ca)
+			*ca = (new mul(acc_ca))->setflag(status_flags::dynallocated);
+		if (cb)
+			*cb = part_b;
+		return (new mul(g))->setflag(status_flags::dynallocated);
+	} 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:
+		unsigned num = b.nops();
+		exvector g; g.reserve(num);
+		exvector acc_cb; acc_cb.reserve(num);
+		ex part_a = a;
+		for (unsigned i=0; i<num; i++) {
+			ex part_ca, part_cb;
+			g.push_back(gcd(part_a, b.op(i), &part_ca, &part_cb, check_args));
+			acc_cb.push_back(part_cb);
+			part_a = part_ca;
+		}
+		if (ca)
+			*ca = part_a;
+		if (cb)
+			*cb = (new mul(acc_cb))->setflag(status_flags::dynallocated);
+		return (new mul(g))->setflag(status_flags::dynallocated);
+	}
+
 #if FAST_COMPARE
-    if (a.is_equal(b)) {
-        if (ca)
-            *ca = exONE();
-        if (cb)
-            *cb = exONE();
-        return a;
-    }
+	// 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 = std::min(ldeg_a,ldeg_b);
+	if (min_ldeg > 0) {
+		ex common = power(x, min_ldeg);
+//std::clog << "trivial common factor " << common << std::endl;
+		return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
+	}
+
+	// Try to eliminate variables
+	if (var->deg_a == 0) {
+//std::clog << "eliminating variable " << x << " from b" << std::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) {
+//std::clog << "eliminating variable " << x << " from a" << std::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 = fail();
+	}
+	if (is_ex_exactly_of_type(g, fail)) {
+//std::clog << "heuristics failed" << std::endl;
+#if STATISTICS
+		heur_gcd_failed++;
 #endif
-    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;
-    }
-    if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
-        cerr << "a=" << a << endl;
-        cerr << "b=" << b << endl;
-        throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
-    }
-
-    // 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((a / common).expand(), (b / common).expand(), ca, cb, false) * common;
-    }
-
-    // Try to eliminate variables
-    if (var->deg_a == 0) {
-//clog << "eliminating variable " << *x << " from b" << endl;
-        ex c = b.content(*x);
-        ex g = gcd(a, c, ca, cb, false);
-        if (cb)
-            *cb *= b.unit(*x) * b.primpart(*x, c);
-        return g;
-    } else if (var->deg_b == 0) {
-//clog << "eliminating variable " << *x << " from a" << endl;
-        ex c = a.content(*x);
-        ex g = gcd(c, b, ca, cb, false);
-        if (ca)
-            *ca *= a.unit(*x) * a.primpart(*x, c);
-        return g;
-    }
-
-    // Try heuristic algorithm first, fall back to PRS if that failed
-    ex g;
-    try {
-        g = heur_gcd(a.expand(), b.expand(), ca, cb, var);
-    } catch (gcdheu_failed) {
-        g = *new ex(fail());
-    }
-    if (is_ex_exactly_of_type(g, fail)) {
-//clog << "heuristics failed\n";
-        g = sr_gcd(a, b, x);
-        if (ca)
-            divide(a, g, *ca, false);
-        if (cb)
-            divide(b, g, *cb, false);
-    }
-    return g;
+#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, var);
+		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;
 }
 
 
@@ -1157,14 +1698,14 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
  *  @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 gcd(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;
+	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;
 }
 
 
@@ -1172,70 +1713,172 @@ ex lcm(const ex &a, const ex &b, bool check_args)
  *  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)
+/** Compute square-free factorization of multivariate polynomial a(x) using
+ *  Yun´s algorithm.  Used internally by sqrfree().
+ *
+ *  @param a  multivariate polynomial over Z[X], treated here as univariate
+ *            polynomial in x.
+ *  @param x  variable to factor in
+ *  @return   vector of factors sorted in ascending degree */
+static exvector sqrfree_yun(const ex &a, const symbol &x)
 {
-    if (a.is_zero())
-        return b;
-    if (b.is_zero())
-        return a;
-    if (a.is_equal(exONE()) || b.is_equal(exONE()))
-        return exONE();
-    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);
+	exvector res;
+	ex w = a;
+	ex z = w.diff(x);
+	ex g = gcd(w, z);
+	if (g.is_equal(_ex1)) {
+		res.push_back(a);
+		return res;
+	}
+	ex y;
+	do {
+		w = quo(w, g, x);
+		y = quo(z, g, x);
+		z = y - w.diff(x);
+		g = gcd(w, z);
+		res.push_back(g);
+	} while (!z.is_zero());
+	return res;
 }
 
+/** Compute square-free factorization of multivariate polynomial in Q[X].
+ *
+ *  @param a  multivariate polynomial over Q[X]
+ *  @param x  lst of variables to factor in, may be left empty for autodetection
+ *  @return   polynomial a in square-free factored form. */
+ex sqrfree(const ex &a, const lst &l)
+{
+	if (is_a<numeric>(a) ||     // algorithm does not trap a==0
+	    is_a<symbol>(a))        // shortcut
+		return a;
+
+	// If no lst of variables to factorize in was specified we have to
+	// invent one now.  Maybe one can optimize here by reversing the order
+	// or so, I don't know.
+	lst args;
+	if (l.nops()==0) {
+		sym_desc_vec sdv;
+		get_symbol_stats(a, _ex0, sdv);
+		sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end();
+		while (it != itend) {
+			args.append(*it->sym);
+			++it;
+		}
+	} else {
+		args = l;
+	}
 
-/** Compute square-free factorization of multivariate polynomial a(x) using
- *  Yun´s algorithm.
+	// Find the symbol to factor in at this stage
+	if (!is_ex_of_type(args.op(0), symbol))
+		throw (std::runtime_error("sqrfree(): invalid factorization variable"));
+	const symbol &x = ex_to<symbol>(args.op(0));
+
+	// convert the argument from something in Q[X] to something in Z[X]
+	const numeric lcm = lcm_of_coefficients_denominators(a);
+	const ex tmp = multiply_lcm(a,lcm);
+
+	// find the factors
+	exvector factors = sqrfree_yun(tmp,x);
+
+	// construct the next list of symbols with the first element popped
+	lst newargs = args;
+	newargs.remove_first();
+
+	// recurse down the factors in remaining variables
+	if (newargs.nops()>0) {
+		exvector::iterator i = factors.begin();
+		while (i != factors.end()) {
+			*i = sqrfree(*i, newargs);
+			++i;
+		}
+	}
+
+	// Done with recursion, now construct the final result
+	ex result = _ex1;
+	exvector::const_iterator it = factors.begin(), itend = factors.end();
+	for (int p = 1; it!=itend; ++it, ++p)
+		result *= power(*it, p);
+
+	// Yun's algorithm does not account for constant factors.  (For univariate
+	// polynomials it works only in the monic case.)  We can correct this by
+	// inserting what has been lost back into the result.  For completeness
+	// we'll also have to recurse down that factor in the remaining variables.
+	if (newargs.nops()>0)
+		result *= sqrfree(quo(tmp, result, x), newargs);
+	else
+		result *= quo(tmp, result, x);
+
+	// Put in the reational overall factor again and return
+	return result *	lcm.inverse();
+}
+
+/** Compute square-free partial fraction decomposition of rational function
+ *  a(x).
  *
- * @param a  multivariate polynomial
- * @param x  variable to factor in
- * @return factored polynomial */
-ex sqrfree(const ex &a, const symbol &x)
+ *  @param a rational function over Z[x], treated as univariate polynomial
+ *           in x
+ *  @param x variable to factor in
+ *  @return decomposed rational function */
+ex sqrfree_parfrac(const ex & a, const symbol & x)
 {
-    int i = 1;
-    ex res = exONE();
-    ex b = a.diff(x);
-    ex c = univariate_gcd(a, b, x);
-    ex w;
-    if (c.is_equal(exONE())) {
-        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);
+	// Find numerator and denominator
+	ex nd = numer_denom(a);
+	ex numer = nd.op(0), denom = nd.op(1);
+//clog << "numer = " << numer << ", denom = " << denom << endl;
+
+	// Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D)
+	ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand();
+//clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl;
+
+	// Factorize denominator and compute cofactors
+	exvector yun = sqrfree_yun(denom, x);
+//clog << "yun factors: " << exprseq(yun) << endl;
+	unsigned num_yun = yun.size();
+	exvector factor; factor.reserve(num_yun);
+	exvector cofac; cofac.reserve(num_yun);
+	for (unsigned i=0; i<num_yun; i++) {
+		if (!yun[i].is_equal(_ex1)) {
+			for (unsigned j=0; j<=i; j++) {
+				factor.push_back(pow(yun[i], j+1));
+				ex prod = _ex1;
+				for (unsigned k=0; k<num_yun; k++) {
+					if (k == i)
+						prod *= pow(yun[k], i-j);
+					else
+						prod *= pow(yun[k], k+1);
+				}
+				cofac.push_back(prod.expand());
+			}
+		}
+	}
+	unsigned num_factors = factor.size();
+//clog << "factors  : " << exprseq(factor) << endl;
+//clog << "cofactors: " << exprseq(cofac) << endl;
+
+	// Construct coefficient matrix for decomposition
+	int max_denom_deg = denom.degree(x);
+	matrix sys(max_denom_deg + 1, num_factors);
+	matrix rhs(max_denom_deg + 1, 1);
+	for (int i=0; i<=max_denom_deg; i++) {
+		for (unsigned j=0; j<num_factors; j++)
+			sys(i, j) = cofac[j].coeff(x, i);
+		rhs(i, 0) = red_numer.coeff(x, i);
+	}
+//clog << "coeffs: " << sys << endl;
+//clog << "rhs   : " << rhs << endl;
+
+	// Solve resulting linear system
+	matrix vars(num_factors, 1);
+	for (unsigned i=0; i<num_factors; i++)
+		vars(i, 0) = symbol();
+	matrix sol = sys.solve(vars, rhs);
+
+	// Sum up decomposed fractions
+	ex sum = 0;
+	for (unsigned i=0; i<num_factors; i++)
+		sum += sol(i, 0) / factor[i];
+
+	return red_poly + sum;
 }
 
 
@@ -1243,42 +1886,91 @@ ex sqrfree(const ex &a, const symbol &x)
  *  Normal form of rational functions
  */
 
-// Create a symbol for replacing the expression "e" (or return a previously
-// assigned symbol). The symbol is appended to sym_list and returned, the
-// expression is appended to repl_list.
+/*
+ *  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 (int 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
+	// 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;
+	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 basic::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.
+/** Function object to be applied by basic::normal(). */
+struct normal_map_function : public map_function {
+	int level;
+	normal_map_function(int l) : level(l) {}
+	ex operator()(const ex & e) { return normal(e, level); }
+};
+
+/** Default implementation of ex::normal(). It normalizes the children and
+ *  replaces the object with a temporary symbol.
  *  @see ex::normal */
 ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    return replace_with_symbol(*this, sym_lst, repl_lst);
+	if (nops() == 0)
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+	else {
+		if (level == 1)
+			return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+		else if (level == -max_recursion_level)
+			throw(std::runtime_error("max recursion level reached"));
+		else {
+			normal_map_function map_normal(level - 1);
+			return (new lst(replace_with_symbol(map(map_normal), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+		}
+	}
 }
 
 
-/** Implementation of ex::normal() for symbols. This returns the unmodifies symbol.
+/** 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 *this;
+	return (new lst(*this, _ex1))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1288,53 +1980,57 @@ ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal */
 ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    if (is_real())
-        if (is_rational())
-            return *this;
-		else
-		    return replace_with_symbol(*this, sym_lst, repl_lst);
-    else { // complex
-        numeric re = real(), im = imag();
+	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);
-		return re_ex + im_ex * replace_with_symbol(I, 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);
+}
 
-/*
- *  Helper function for fraction cancellation (returns cancelled fraction n/d)
- */
 
+/** 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;
-    ex pre_factor = exONE();
-
-    // Handle special cases where numerator or denominator is 0
-    if (num.is_zero())
-        return exZERO();
-    if (den.expand().is_zero())
-        throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
-
-    // More special cases
-    if (is_ex_exactly_of_type(den, numeric))
-        return num / den;
-    if (num.is_zero())
-        return exZERO();
-
-    // Bring numerator and denominator to Z[X] by multiplying with
-    // LCM of all coefficients' denominators
-    ex num_lcm = lcm_of_coefficients_denominators(num);
-    ex den_lcm = lcm_of_coefficients_denominators(den);
-    num *= num_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) != exONE()) {
+	ex num = n;
+	ex den = d;
+	numeric pre_factor = _num1;
+
+//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
+
+	// Handle trivial case where denominator is 1
+	if (den.is_equal(_ex1))
+		return (new lst(num, den))->setflag(status_flags::dynallocated);
+
+	// Handle special cases where numerator or denominator is 0
+	if (num.is_zero())
+		return (new lst(num, _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;
 	}
@@ -1343,12 +2039,16 @@ static ex frac_cancel(const ex &n, const ex &d)
 	// as defined by get_first_symbol() is made positive)
 	const symbol *x;
 	if (get_first_symbol(den, x)) {
-		if (den.unit(*x).compare(exZERO()) < 0) {
-			num *= exMINUSONE();
-			den *= exMINUSONE();
+		GINAC_ASSERT(is_exactly_a<numeric>(den.unit(*x)));
+		if (ex_to<numeric>(den.unit(*x)).is_negative()) {
+			num *= _ex_1;
+			den *= _ex_1;
 		}
 	}
-    return pre_factor * num / den;
+
+	// Return result as list
+//std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << std::endl;
+	return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1357,51 +2057,57 @@ static ex frac_cancel(const ex &n, const ex &d)
  *  @see ex::normal */
 ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    // Normalize and expand children
-    exvector o;
-    o.reserve(seq.size()+1);
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-        ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
-        if (is_ex_exactly_of_type(n, add)) {
-            epvector::const_iterator bit = (static_cast<add *>(n.bp))->seq.begin(), bitend = (static_cast<add *>(n.bp))->seq.end();
-            while (bit != bitend) {
-                o.push_back(recombine_pair_to_ex(*bit));
-                bit++;
-            }
-            o.push_back((static_cast<add *>(n.bp))->overall_coeff);
-        } else
-            o.push_back(n);
-        it++;
-    }
-    o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
-
-    // Determine common denominator
-    ex den = exONE();
-    exvector::const_iterator ait = o.begin(), aitend = o.end();
-    while (ait != aitend) {
-        den = lcm((*ait).denom(false), den, false);
-        ait++;
-    }
-
-    // Add fractions
-    if (den.is_equal(exONE()))
-        return (new add(o))->setflag(status_flags::dynallocated);
-    else {
-        exvector num_seq;
-        for (ait=o.begin(); ait!=aitend; ait++) {
-            ex q;
-            if (!divide(den, (*ait).denom(false), q, false)) {
-                // should not happen
-                throw(std::runtime_error("invalid expression in add::normal, division failed"));
-            }
-            num_seq.push_back((*ait).numer(false) * q);
-        }
-        ex num = add(num_seq);
-
-        // Cancel common factors from num/den
-        return frac_cancel(num, den);
-    }
+	if (level == 1)
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+	else if (level == -max_recursion_level)
+		throw(std::runtime_error("max recursion level reached"));
+
+	// Normalize children and split each one into numerator and denominator
+	exvector nums, dens;
+	nums.reserve(seq.size()+1);
+	dens.reserve(seq.size()+1);
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		ex n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+		nums.push_back(n.op(0));
+		dens.push_back(n.op(1));
+		it++;
+	}
+	ex n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+	nums.push_back(n.op(0));
+	dens.push_back(n.op(1));
+	GINAC_ASSERT(nums.size() == dens.size());
+
+	// Now, nums is a vector of all numerators and dens is a vector of
+	// all denominators
+//std::clog << "add::normal uses " << nums.size() << " summands:\n";
+
+	// Add fractions sequentially
+	exvector::const_iterator num_it = nums.begin(), num_itend = nums.end();
+	exvector::const_iterator den_it = dens.begin(), den_itend = dens.end();
+//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
+	ex num = *num_it++, den = *den_it++;
+	while (num_it != num_itend) {
+//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
+		ex next_num = *num_it++, next_den = *den_it++;
+
+		// Trivially add sequences of fractions with identical denominators
+		while ((den_it != den_itend) && next_den.is_equal(*den_it)) {
+			next_num += *num_it;
+			num_it++; den_it++;
+		}
+
+		// Additiion of two fractions, taking advantage of the fact that
+		// the heuristic GCD algorithm computes the cofactors at no extra cost
+		ex co_den1, co_den2;
+		ex g = gcd(den, next_den, &co_den1, &co_den2, false);
+		num = ((num * co_den2) + (next_num * co_den1)).expand();
+		den *= co_den2;		// this is the lcm(den, next_den)
+	}
+//std::clog << " common denominator = " << den << std::endl;
+
+	// Cancel common factors from num/den
+	return frac_cancel(num, den);
 }
 
 
@@ -1410,17 +2116,29 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal() */
 ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    // Normalize children
-    exvector o;
-    o.reserve(seq.size()+1);
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-        o.push_back(recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1));
-        it++;
-    }
-    o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
-    ex n = (new mul(o))->setflag(status_flags::dynallocated);
-    return frac_cancel(n.numer(false), n.denom(false));
+	if (level == 1)
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+	else if (level == -max_recursion_level)
+		throw(std::runtime_error("max recursion level reached"));
+
+	// Normalize children, separate into numerator and denominator
+	exvector num; num.reserve(seq.size());
+	exvector den; den.reserve(seq.size());
+	ex n;
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+		num.push_back(n.op(0));
+		den.push_back(n.op(1));
+		it++;
+	}
+	n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+	num.push_back(n.op(0));
+	den.push_back(n.op(1));
+
+	// Perform fraction cancellation
+	return frac_cancel((new mul(num))->setflag(status_flags::dynallocated),
+	                   (new mul(den))->setflag(status_flags::dynallocated));
 }
 
 
@@ -1430,36 +2148,73 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal */
 ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    if (exponent.info(info_flags::integer)) {
-        // Integer powers are distributed
-        ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
-        ex num = n.numer(false);
-        ex den = n.denom(false);
-        return power(num, exponent) / power(den, exponent);
-    } else {
-        // Non-integer powers are replaced by temporary symbol (after normalizing basis)
-        ex n = power(basis.bp->normal(sym_lst, repl_lst, level-1), exponent);
-        return replace_with_symbol(n, sym_lst, repl_lst);
-    }
+	if (level == 1)
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+	else if (level == -max_recursion_level)
+		throw(std::runtime_error("max recursion level reached"));
+
+	// Normalize basis and exponent (exponent gets reassembled)
+	ex n_basis = ex_to<basic>(basis).normal(sym_lst, repl_lst, level-1);
+	ex n_exponent = ex_to<basic>(exponent).normal(sym_lst, repl_lst, level-1);
+	n_exponent = n_exponent.op(0) / n_exponent.op(1);
+
+	if (n_exponent.info(info_flags::integer)) {
+
+		if (n_exponent.info(info_flags::positive)) {
+
+			// (a/b)^n -> {a^n, b^n}
+			return (new lst(power(n_basis.op(0), n_exponent), power(n_basis.op(1), n_exponent)))->setflag(status_flags::dynallocated);
+
+		} else if (n_exponent.info(info_flags::negative)) {
+
+			// (a/b)^-n -> {b^n, a^n}
+			return (new lst(power(n_basis.op(1), -n_exponent), power(n_basis.op(0), -n_exponent)))->setflag(status_flags::dynallocated);
+		}
+
+	} else {
+
+		if (n_exponent.info(info_flags::positive)) {
+
+			// (a/b)^x -> {sym((a/b)^x), 1}
+			return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+
+		} else if (n_exponent.info(info_flags::negative)) {
+
+			if (n_basis.op(1).is_equal(_ex1)) {
+
+				// a^-x -> {1, sym(a^x)}
+				return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_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_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+			}
+
+		} else {	// n_exponent not numeric
+
+			// (a/b)^x -> {sym((a/b)^x, 1}
+			return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+		}
+	}
 }
 
 
-/** Implementation of ex::normal() for series. It normalizes each coefficient and
- *  replaces the series by a temporary symbol.
+/** Implementation of ex::normal() for pseries. It normalizes each coefficient
+ *  and replaces the series by a temporary symbol.
  *  @see ex::normal */
-ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
+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 = series(var, point, new_seq);
-    return replace_with_symbol(n, sym_lst, repl_lst);
+	epvector newseq;
+	epvector::const_iterator i = seq.begin(), end = seq.end();
+	while (i != end) {
+		ex restexp = i->rest.normal();
+		if (!restexp.is_zero())
+			newseq.push_back(expair(restexp, i->coeff));
+		++i;
+	}
+	ex n = pseries(relational(var,point), newseq);
+	return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1467,8 +2222,8 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  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
+ *  (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.
@@ -1477,12 +2232,155 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @return normalized expression */
 ex ex::normal(int level) const
 {
-    lst sym_lst, repl_lst;
-    ex e = bp->normal(sym_lst, repl_lst, level);
-    if (sym_lst.nops() > 0)
-        return e.subs(sym_lst, repl_lst);
-    else
-        return e;
+	lst sym_lst, repl_lst;
+
+	ex e = bp->normal(sym_lst, repl_lst, level);
+	GINAC_ASSERT(is_a<lst>(e));
+
+	// 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);
+}
+
+/** Get 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_a<lst>(e));
+
+	// Re-insert replaced symbols
+	if (sym_lst.nops() > 0)
+		return e.op(0).subs(sym_lst, repl_lst);
+	else
+		return e.op(0);
+}
+
+/** Get 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_a<lst>(e));
+
+	// Re-insert replaced symbols
+	if (sym_lst.nops() > 0)
+		return e.op(1).subs(sym_lst, repl_lst);
+	else
+		return e.op(1);
+}
+
+/** Get numerator and denominator of an expression. If the expresison is not
+ *  of the normal form "numerator/denominator", it is first converted to this
+ *  form and then a list [numerator, denominator] is returned.
+ *
+ *  @see ex::normal
+ *  @return a list [numerator, denominator] */
+ex ex::numer_denom(void) const
+{
+	lst sym_lst, repl_lst;
+
+	ex e = bp->normal(sym_lst, repl_lst, 0);
+	GINAC_ASSERT(is_a<lst>(e));
+
+	// Re-insert replaced symbols
+	if (sym_lst.nops() > 0)
+		return e.subs(sym_lst, repl_lst);
+	else
+		return e;
+}
+
+
+/** 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 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. */
+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. */
+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. */
+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. */
+ex expairseq::to_rational(lst &repl_lst) const
+{
+	epvector s;
+	s.reserve(seq.size());
+	epvector::const_iterator i = seq.begin(), end = seq.end();
+	while (i != end) {
+		s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl_lst)));
+		++i;
+	}
+	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());
+}
+
+
 } // namespace GiNaC