X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=9e24b99bc12e68bb2a5cdcbe263d4637bd9c64ea;hp=82fcaf848dd250ea14106dab40a9b0d87b40edf2;hb=fc80c36140aaf126150b2dea811177d8af35dac3;hpb=9eab44408b9213d8909b7a9e525f404ad06064dd

diff --git a/ginac/normal.cpp b/ginac/normal.cpp
index 82fcaf84..9e24b99b 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) 1999-2000 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
@@ -43,10 +42,13 @@
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"
-#include "series.h"
+#include "pseries.h"
 #include "symbol.h"
+#include "utils.h"
 
+#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
+#endif // ndef NO_NAMESPACE_GINAC
 
 // If comparing expressions (ex::compare()) is fast, you can set this to 1.
 // Some routines like quo(), rem() and gcd() will then return a quick answer
@@ -54,7 +56,34 @@ 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 (usually slower except for very large problems)
+#define USE_TRIAL_DIVISION 0
+
+// Set this to enable some statistical output for the GCD routines
+#define STATISTICS 0
+
+
+#if STATISTICS
+// Statistics variables
+static int gcd_called = 0;
+static int sr_gcd_called = 0;
+static int heur_gcd_called = 0;
+static int heur_gcd_failed = 0;
+
+// Print statistics at end of program
+static struct _stat_print {
+	_stat_print() {}
+	~_stat_print() {
+		cout << "gcd() called " << gcd_called << " times\n";
+		cout << "sr_gcd() called " << sr_gcd_called << " times\n";
+		cout << "heur_gcd() called " << heur_gcd_called << " times\n";
+		cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
+	}
+} stat_print;
+#endif
 
 
 /** Return pointer to first symbol found in expression.  Due to GiNaC´s
@@ -71,7 +100,7 @@ static bool get_first_symbol(const ex &e, const symbol *&x)
         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++)
+        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)) {
@@ -108,11 +137,11 @@ struct sym_desc {
     /** Lowest degree of symbol in polynomial "b" */
     int ldeg_b;
 
-    /** Minimum of ldeg_a and ldeg_b (Used for sorting) */
-    int min_deg;
+    /** Maximum of deg_a and deg_b (Used for sorting) */
+    int max_deg;
 
     /** Commparison operator for sorting */
-    bool operator<(const sym_desc &x) const {return min_deg < x.min_deg;}
+    bool operator<(const sym_desc &x) const {return max_deg < x.max_deg;}
 };
 
 // Vector of sym_desc structures
@@ -138,7 +167,7 @@ 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++)
+        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);
@@ -168,12 +197,21 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
         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->max_deg = max(deg_a, deg_b);
         it->ldeg_a = a.ldegree(*(it->sym));
         it->ldeg_b = b.ldegree(*(it->sym));
         it++;
     }
     sort(v.begin(), v.end());
+#if 0
+	clog << "Symbols:\n";
+	it = v.begin(); itend = v.end();
+	while (it != itend) {
+		clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << endl;
+		clog << "  lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
+		it++;
+	}
+#endif
 }
 
 
@@ -187,28 +225,61 @@ 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++) {
+    else if (is_ex_exactly_of_type(e, add)) {
+        numeric c = _num1();
+        for (unsigned i=0; i<e.nops(); i++)
             c = lcmcoeff(e.op(i), c);
-        }
+        return lcm(c, l);
+    } else if (is_ex_exactly_of_type(e, mul)) {
+        numeric c = _num1();
+        for (unsigned i=0; i<e.nops(); i++)
+            c *= lcmcoeff(e.op(i), _num1());
         return lcm(c, l);
     } else if (is_ex_exactly_of_type(e, power))
-        return lcmcoeff(e.op(0), l);
+        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)) {
+		ex c = _ex1();
+		numeric lcm_accum = _num1();
+		for (unsigned i=0; i<e.nops(); i++) {
+			numeric op_lcm = lcmcoeff(e.op(i), _num1());
+			c *= multiply_lcm(e.op(i), op_lcm);
+			lcm_accum *= op_lcm;
+		}
+		c *= lcm / lcm_accum;
+		return c;
+	} else if (is_ex_exactly_of_type(e, add)) {
+		ex c = _ex0();
+		for (unsigned i=0; i<e.nops(); i++)
+			c += multiply_lcm(e.op(i), lcm);
+		return c;
+	} else if (is_ex_exactly_of_type(e, power)) {
+		return pow(multiply_lcm(e.op(0), lcm.power(ex_to_numeric(e.op(1)).inverse())), e.op(1));
+	} else
+		return e * lcm;
 }
 
 
@@ -220,13 +291,13 @@ static numeric lcm_of_coefficients_denominators(const ex &e)
 
 numeric ex::integer_content(void) const
 {
-    ASSERT(bp!=0);
+    GINAC_ASSERT(bp!=0);
     return bp->integer_content();
 }
 
 numeric basic::integer_content(void) const
 {
-    return numONE();
+    return _num1();
 }
 
 numeric numeric::integer_content(void) const
@@ -238,29 +309,29 @@ numeric add::integer_content(void) const
 {
     epvector::const_iterator it = seq.begin();
     epvector::const_iterator itend = seq.end();
-    numeric c = numZERO();
+    numeric c = _num0();
     while (it != itend) {
-        ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-        ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
+        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
+        GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
         c = gcd(ex_to_numeric(it->coeff), c);
         it++;
     }
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
     c = gcd(ex_to_numeric(overall_coeff),c);
     return c;
 }
 
 numeric mul::integer_content(void) const
 {
-#ifdef DOASSERT
+#ifdef DO_GINAC_ASSERT
     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));
+        GINAC_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));
+#endif // def DO_GINAC_ASSERT
+    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
     return abs(ex_to_numeric(overall_coeff));
 }
 
@@ -287,13 +358,13 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
         return a / b;
 #if FAST_COMPARE
     if (a.is_equal(b))
-        return exONE();
+        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 q = _ex0();
     ex r = a.expand();
     if (r.is_zero())
         return r;
@@ -336,13 +407,13 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
         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();
+            return _ex0();
         else
             return b;
     }
 #if FAST_COMPARE
     if (a.is_equal(b))
-        return exZERO();
+        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"));
@@ -388,7 +459,7 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
         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();
+            return _ex0();
         else
             return b;
     }
@@ -404,18 +475,18 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
     if (bdeg <= rdeg) {
         blcoeff = eb.coeff(x, bdeg);
         if (bdeg == 0)
-            eb = exZERO();
+            eb = _ex0();
         else
             eb -= blcoeff * power(x, bdeg);
     } else
-        blcoeff = exONE();
+        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 = exZERO();
+            r = _ex0();
         else
             r -= rlcoeff * power(x, rdeg);
         r = (blcoeff * r).expand() - term;
@@ -438,9 +509,11 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
 
 bool divide(const ex &a, const ex &b, ex &q, bool check_args)
 {
-    q = exZERO();
+    q = _ex0();
     if (b.is_zero())
         throw(std::overflow_error("divide: division by zero"));
+	if (a.is_zero())
+		return true;
     if (is_ex_exactly_of_type(b, numeric)) {
         q = a / b;
         return true;
@@ -448,7 +521,7 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args)
         return false;
 #if FAST_COMPARE
     if (a.is_equal(b)) {
-        q = exONE();
+        q = _ex1();
         return true;
     }
 #endif
@@ -523,10 +596,10 @@ 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();
+    q = _ex0();
     if (b.is_zero())
         throw(std::overflow_error("divide_in_z: division by zero"));
-    if (b.is_equal(exONE())) {
+    if (b.is_equal(_ex1())) {
         q = a;
         return true;
     }
@@ -539,7 +612,7 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
     }
 #if FAST_COMPARE
     if (a.is_equal(b)) {
-        q = exONE();
+        q = _ex1();
         return true;
     }
 #endif
@@ -562,61 +635,32 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
     if (bdeg > adeg)
         return false;
 
-#if 1
+#if USE_TRIAL_DIVISION
 
-    // Polynomial long division (recursive)
-    ex r = a.expand();
-    if (r.is_zero())
-        return true;
-    int rdeg = adeg;
-    ex eb = b.expand();
-    ex blcoeff = eb.coeff(*x, bdeg);
-    while (rdeg >= bdeg) {
-        ex term, rcoeff = r.coeff(*x, rdeg);
-        if (!divide_in_z(rcoeff, blcoeff, term, var+1))
-            break;
-        term = (term * power(*x, rdeg - bdeg)).expand();
-        q += term;
-        r -= (term * eb).expand();
-        if (r.is_zero()) {
-#if USE_REMEMBER
-            dr_remember[ex2(a, b)] = exbool(q, true);
-#endif
-            return true;
-        }
-        rdeg = r.degree(*x);
-    }
-#if USE_REMEMBER
-    dr_remember[ex2(a, b)] = exbool(q, false);
-#endif
-    return false;
-
-#else
-
-    // Trial division using polynomial interpolation
+    // 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 = numZERO();
+    numeric point = _num0();
     ex c;
     for (i=0; i<=adeg; i++) {
         ex bs = b.subs(*x == point);
         while (bs.is_zero()) {
-            point += numONE();
+            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 += numONE();
+        point += _num1();
     }
 
     // Compute inverses
     vector<numeric> rcp; rcp.reserve(adeg + 1);
-    rcp.push_back(0);
+    rcp.push_back(_num0());
     for (k=1; k<=adeg; k++) {
         numeric product = alpha[k] - alpha[0];
         for (i=1; i<k; i++)
@@ -644,6 +688,36 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
         return true;
     } else
         return false;
+
+#else
+
+    // Polynomial long division (recursive)
+    ex r = a.expand();
+    if (r.is_zero())
+        return true;
+    int rdeg = adeg;
+    ex eb = b.expand();
+    ex blcoeff = eb.coeff(*x, bdeg);
+    while (rdeg >= bdeg) {
+        ex term, rcoeff = r.coeff(*x, rdeg);
+        if (!divide_in_z(rcoeff, blcoeff, term, var+1))
+            break;
+        term = (term * power(*x, rdeg - bdeg)).expand();
+        q += term;
+        r -= (term * eb).expand();
+        if (r.is_zero()) {
+#if USE_REMEMBER
+            dr_remember[ex2(a, b)] = exbool(q, true);
+#endif
+            return true;
+        }
+        rdeg = r.degree(*x);
+    }
+#if USE_REMEMBER
+    dr_remember[ex2(a, b)] = exbool(q, false);
+#endif
+    return false;
+
 #endif
 }
 
@@ -663,7 +737,7 @@ 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();
+        return c < _ex0() ? _ex_1() : _ex1();
     else {
         const symbol *y;
         if (get_first_symbol(c, y))
@@ -684,12 +758,12 @@ ex ex::unit(const symbol &x) const
 ex ex::content(const symbol &x) const
 {
     if (is_zero())
-        return exZERO();
+        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 exZERO();
+        return _ex0();
 
     // First, try the integer content
     ex c = e.integer_content();
@@ -703,7 +777,7 @@ ex ex::content(const symbol &x) const
     int ldeg = e.ldegree(x);
     if (deg == ldeg)
         return e.lcoeff(x) / e.unit(x);
-    c = exZERO();
+    c = _ex0();
     for (int i=ldeg; i<=deg; i++)
         c = gcd(e.coeff(x, i), c, NULL, NULL, false);
     return c;
@@ -720,13 +794,13 @@ ex ex::content(const symbol &x) const
 ex ex::primpart(const symbol &x) const
 {
     if (is_zero())
-        return exZERO();
+        return _ex0();
     if (is_ex_exactly_of_type(*this, numeric))
-        return exONE();
+        return _ex1();
 
     ex c = content(x);
     if (c.is_zero())
-        return exZERO();
+        return _ex0();
     ex u = unit(x);
     if (is_ex_exactly_of_type(c, numeric))
         return *this / (c * u);
@@ -746,11 +820,11 @@ ex ex::primpart(const symbol &x) const
 ex ex::primpart(const symbol &x, const ex &c) const
 {
     if (is_zero())
-        return exZERO();
+        return _ex0();
     if (c.is_zero())
-        return exZERO();
+        return _ex0();
     if (is_ex_exactly_of_type(*this, numeric))
-        return exONE();
+        return _ex1();
 
     ex u = unit(x);
     if (is_ex_exactly_of_type(c, numeric))
@@ -764,8 +838,201 @@ ex ex::primpart(const symbol &x, const ex &c) const
  *  GCD of multivariate polynomials
  */
 
+/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
+ *  (not really suited for multivariate GCDs). This function is only provided
+ *  for testing purposes.
+ *
+ *  @param a  first multivariate polynomial
+ *  @param b  second multivariate polynomial
+ *  @param x  pointer to symbol (main variable) in which to compute the GCD in
+ *  @return the GCD as a new expression
+ *  @see gcd */
+
+static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//clog << "eu_gcd(" << a << "," << b << ")\n";
+
+    // Sort c and d so that c has higher degree
+    ex c, d;
+    int adeg = a.degree(*x), bdeg = b.degree(*x);
+    if (adeg >= bdeg) {
+        c = a;
+        d = b;
+    } else {
+        c = b;
+        d = a;
+    }
+
+	// Euclidean algorithm
+    ex r;
+    for (;;) {
+//clog << " d = " << d << endl;
+        r = rem(c, d, *x, false);
+        if (r.is_zero())
+            return d.primpart(*x);
+        c = d;
+		d = r;
+    }
+}
+
+
+/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
+ *  with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]).
+ *  This function is only provided for testing purposes.
+ *
+ *  @param a  first multivariate polynomial
+ *  @param b  second multivariate polynomial
+ *  @param x  pointer to symbol (main variable) in which to compute the GCD in
+ *  @return the GCD as a new expression
+ *  @see gcd */
+
+static ex euprem_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//clog << "euprem_gcd(" << a << "," << b << ")\n";
+
+    // Sort c and d so that c has higher degree
+    ex c, d;
+    int adeg = a.degree(*x), bdeg = b.degree(*x);
+    if (adeg >= bdeg) {
+        c = a;
+        d = b;
+    } else {
+        c = b;
+        d = a;
+    }
+
+	// Euclidean algorithm with pseudo-remainders
+    ex r;
+    for (;;) {
+//clog << " d = " << d << endl;
+        r = prem(c, d, *x, false);
+        if (r.is_zero())
+            return d.primpart(*x);
+        c = d;
+		d = r;
+    }
+}
+
+
+/** Compute GCD of multivariate polynomials using the primitive Euclidean
+ *  PRS algorithm (complete content removal at each step). This function is
+ *  only provided for testing purposes.
+ *
+ *  @param a  first multivariate polynomial
+ *  @param b  second multivariate polynomial
+ *  @param x  pointer to symbol (main variable) in which to compute the GCD in
+ *  @return the GCD as a new expression
+ *  @see gcd */
+
+static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//clog << "peu_gcd(" << a << "," << b << ")\n";
+
+    // Sort c and d so that c has higher degree
+    ex c, d;
+    int adeg = a.degree(*x), bdeg = b.degree(*x);
+    int ddeg;
+    if (adeg >= bdeg) {
+        c = a;
+        d = b;
+        ddeg = bdeg;
+    } else {
+        c = b;
+        d = a;
+        ddeg = adeg;
+    }
+
+    // Remove content from c and d, to be attached to GCD later
+    ex cont_c = c.content(*x);
+    ex cont_d = d.content(*x);
+    ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
+    if (ddeg == 0)
+        return gamma;
+    c = c.primpart(*x, cont_c);
+    d = d.primpart(*x, cont_d);
+
+    // Euclidean algorithm with content removal
+	ex r;
+    for (;;) {
+//clog << " d = " << d << endl;
+        r = prem(c, d, *x, false);
+        if (r.is_zero())
+            return gamma * d;
+        c = d;
+		d = r.primpart(*x);
+    }
+}
+
+
+/** Compute GCD of multivariate polynomials using the reduced PRS algorithm.
+ *  This function is only provided for testing purposes.
+ *
+ *  @param a  first multivariate polynomial
+ *  @param b  second multivariate polynomial
+ *  @param x  pointer to symbol (main variable) in which to compute the GCD in
+ *  @return the GCD as a new expression
+ *  @see gcd */
+
+static ex red_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//clog << "red_gcd(" << a << "," << b << ")\n";
+
+    // Sort c and d so that c has higher degree
+    ex c, d;
+    int adeg = a.degree(*x), bdeg = b.degree(*x);
+    int cdeg, ddeg;
+    if (adeg >= bdeg) {
+        c = a;
+        d = b;
+        cdeg = adeg;
+        ddeg = bdeg;
+    } else {
+        c = b;
+        d = a;
+        cdeg = bdeg;
+        ddeg = adeg;
+    }
+
+    // Remove content from c and d, to be attached to GCD later
+    ex cont_c = c.content(*x);
+    ex cont_d = d.content(*x);
+    ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
+    if (ddeg == 0)
+        return gamma;
+    c = c.primpart(*x, cont_c);
+    d = d.primpart(*x, cont_d);
+
+    // First element of subresultant sequence
+    ex r, ri = _ex1();
+    int delta = cdeg - ddeg;
+
+    for (;;) {
+        // Calculate polynomial pseudo-remainder
+//clog << " d = " << d << endl;
+        r = prem(c, d, *x, false);
+        if (r.is_zero())
+            return gamma * d.primpart(*x);
+        c = d;
+        cdeg = ddeg;
+
+        if (!divide(r, pow(ri, delta), d, false))
+            throw(std::runtime_error("invalid expression in red_gcd(), division failed"));
+        ddeg = d.degree(*x);
+        if (ddeg == 0) {
+            if (is_ex_exactly_of_type(r, numeric))
+                return gamma;
+            else
+                return gamma * r.primpart(*x);
+        }
+
+        ri = c.expand().lcoeff(*x);
+        delta = cdeg - ddeg;
+    }
+}
+
+
 /** Compute GCD of multivariate polynomials using the subresultant PRS
- *  algorithm. This function is used internally gy gcd().
+ *  algorithm. This function is used internally by gcd().
  *
  *  @param a  first multivariate polynomial
  *  @param b  second multivariate polynomial
@@ -775,6 +1042,11 @@ ex ex::primpart(const symbol &x, const ex &c) const
 
 static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
 {
+//clog << "sr_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+	sr_gcd_called++;
+#endif
+
     // Sort c and d so that c has higher degree
     ex c, d;
     int adeg = a.degree(*x), bdeg = b.degree(*x);
@@ -799,19 +1071,23 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
         return gamma;
     c = c.primpart(*x, cont_c);
     d = d.primpart(*x, cont_d);
+//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
 
     // First element of subresultant sequence
-    ex r = exZERO(), ri = exONE(), psi = exONE();
+    ex r = _ex0(), ri = _ex1(), psi = _ex1();
     int delta = cdeg - ddeg;
 
     for (;;) {
         // Calculate polynomial pseudo-remainder
+//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
+//clog << " d = " << d << endl;
         r = prem(c, d, *x, false);
         if (r.is_zero())
             return gamma * d.primpart(*x);
         c = d;
         cdeg = ddeg;
-        if (!divide(r, ri * power(psi, delta), d, false))
+//clog << " dividing...\n";
+        if (!divide(r, ri * pow(psi, delta), d, false))
             throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
         ddeg = d.degree(*x);
         if (ddeg == 0) {
@@ -822,11 +1098,12 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
         }
 
         // Next element of subresultant sequence
+//clog << " calculating next subresultant...\n";
         ri = c.expand().lcoeff(*x);
         if (delta == 1)
             psi = ri;
         else if (delta)
-            divide(power(ri, delta), power(psi, delta-1), psi, false);
+            divide(pow(ri, delta), pow(psi, delta-1), psi, false);
         delta = cdeg - ddeg;
     }
 }
@@ -841,13 +1118,13 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
 
 numeric ex::max_coefficient(void) const
 {
-    ASSERT(bp!=0);
+    GINAC_ASSERT(bp!=0);
     return bp->max_coefficient();
 }
 
 numeric basic::max_coefficient(void) const
 {
-    return numONE();
+    return _num1();
 }
 
 numeric numeric::max_coefficient(void) const
@@ -859,11 +1136,11 @@ 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));
+    GINAC_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));
+        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
         a = abs(ex_to_numeric(it->coeff));
         if (a > cur_max)
             cur_max = a;
@@ -874,15 +1151,15 @@ numeric add::max_coefficient(void) const
 
 numeric mul::max_coefficient(void) const
 {
-#ifdef DOASSERT
+#ifdef DO_GINAC_ASSERT
     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));
+        GINAC_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));
+#endif // def DO_GINAC_ASSERT
+    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
     return abs(ex_to_numeric(overall_coeff));
 }
 
@@ -897,7 +1174,7 @@ numeric mul::max_coefficient(void) const
 
 ex ex::smod(const numeric &xi) const
 {
-    ASSERT(bp!=0);
+    GINAC_ASSERT(bp!=0);
     return bp->smod(xi);
 }
 
@@ -908,7 +1185,11 @@ ex basic::smod(const numeric &xi) const
 
 ex numeric::smod(const numeric &xi) const
 {
+#ifndef NO_NAMESPACE_GINAC
     return GiNaC::smod(*this, xi);
+#else // ndef NO_NAMESPACE_GINAC
+    return ::smod(*this, xi);
+#endif // ndef NO_NAMESPACE_GINAC
 }
 
 ex add::smod(const numeric &xi) const
@@ -918,37 +1199,49 @@ ex add::smod(const numeric &xi) const
     epvector::const_iterator it = seq.begin();
     epvector::const_iterator itend = seq.end();
     while (it != itend) {
-        ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
+        GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
+#ifndef NO_NAMESPACE_GINAC
         numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
+#else // ndef NO_NAMESPACE_GINAC
+        numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
+#endif // ndef NO_NAMESPACE_GINAC
         if (!coeff.is_zero())
             newseq.push_back(expair(it->rest, coeff));
         it++;
     }
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+#ifndef NO_NAMESPACE_GINAC
     numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
+#else // ndef NO_NAMESPACE_GINAC
+    numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi);
+#endif // ndef NO_NAMESPACE_GINAC
     return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
 }
 
 ex mul::smod(const numeric &xi) const
 {
-#ifdef DOASSERT
+#ifdef DO_GINAC_ASSERT
     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));
+        GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
         it++;
     }
-#endif // def DOASSERT
+#endif // def DO_GINAC_ASSERT
     mul * mulcopyp=new mul(*this);
-    ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+    GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+#ifndef NO_NAMESPACE_GINAC
     mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
+#else // ndef NO_NAMESPACE_GINAC
+    mulcopyp->overall_coeff = ::smod(ex_to_numeric(overall_coeff),xi);
+#endif // ndef NO_NAMESPACE_GINAC
     mulcopyp->clearflag(status_flags::evaluated);
     mulcopyp->clearflag(status_flags::hash_calculated);
     return mulcopyp->setflag(status_flags::dynallocated);
 }
 
 
-/** Exception thrown by heur_gcd() to signal failure */
+/** Exception thrown by heur_gcd() to signal failure. */
 class gcdheu_failed {};
 
 /** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
@@ -969,6 +1262,12 @@ class gcdheu_failed {};
 
 static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
 {
+//clog << "heur_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+	heur_gcd_called++;
+#endif
+
+	// GCD of two numeric values -> CLN
     if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
         numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
         numeric rg;
@@ -995,21 +1294,23 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
     numeric mp = p.max_coefficient(), mq = q.max_coefficient();
     numeric xi;
     if (mp > mq)
-        xi = mq * numTWO() + numTWO();
+        xi = mq * _num2() + _num2();
     else
-        xi = mp * numTWO() + numTWO();
+        xi = mp * _num2() + _num2();
 
     // 6 tries maximum
     for (int t=0; t<6; t++) {
-        if (xi.int_length() * maxdeg > 50000)
+        if (xi.int_length() * maxdeg > 100000) {
+//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
             throw gcdheu_failed();
+		}
 
         // Apply evaluation homomorphism and calculate GCD
         ex gamma = heur_gcd(p.subs(*x == xi), q.subs(*x == xi), NULL, NULL, var+1).expand();
         if (!is_ex_exactly_of_type(gamma, fail)) {
 
             // Reconstruct polynomial from GCD of mapped polynomials
-            ex g = exZERO();
+            ex g = _ex0();
             numeric rxi = xi.inverse();
             for (int i=0; !gamma.is_zero(); i++) {
                 ex gi = gamma.smod(xi);
@@ -1024,7 +1325,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
             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)
+                if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
                     return -g;
                 else
                     return g;
@@ -1049,50 +1350,142 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 
 ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 {
+//clog << "gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+	gcd_called++;
+#endif
+
+	// GCD of numerics -> CLN
+    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+        numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
+        if (ca)
+            *ca = ex_to_numeric(a) / g;
+        if (cb)
+            *cb = ex_to_numeric(b) / g;
+        return g;
+    }
+
+	// Check arguments
+    if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
+        throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
+    }
+
+	// Partially factored cases (to avoid expanding large expressions)
+	if (is_ex_exactly_of_type(a, mul)) {
+		if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
+			goto factored_b;
+factored_a:
+		ex g = _ex1();
+		ex acc_ca = _ex1();
+		ex part_b = b;
+		for (unsigned i=0; i<a.nops(); i++) {
+			ex part_ca, part_cb;
+			g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
+			acc_ca *= part_ca;
+			part_b = part_cb;
+		}
+		if (ca)
+			*ca = acc_ca;
+		if (cb)
+			*cb = part_b;
+		return g;
+	} else if (is_ex_exactly_of_type(b, mul)) {
+		if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
+			goto factored_a;
+factored_b:
+		ex g = _ex1();
+		ex acc_cb = _ex1();
+		ex part_a = a;
+		for (unsigned i=0; i<b.nops(); i++) {
+			ex part_ca, part_cb;
+			g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
+			acc_cb *= part_cb;
+			part_a = part_ca;
+		}
+		if (ca)
+			*ca = part_a;
+		if (cb)
+			*cb = acc_cb;
+		return g;
+	}
+
+#if FAST_COMPARE
+	// Input polynomials of the form poly^n are sometimes also trivial
+	if (is_ex_exactly_of_type(a, power)) {
+		ex p = a.op(0);
+		if (is_ex_exactly_of_type(b, power)) {
+			if (p.is_equal(b.op(0))) {
+				// a = p^n, b = p^m, gcd = p^min(n, m)
+				ex exp_a = a.op(1), exp_b = b.op(1);
+				if (exp_a < exp_b) {
+					if (ca)
+						*ca = _ex1();
+					if (cb)
+						*cb = power(p, exp_b - exp_a);
+					return power(p, exp_a);
+				} else {
+					if (ca)
+						*ca = power(p, exp_a - exp_b);
+					if (cb)
+						*cb = _ex1();
+					return power(p, exp_b);
+				}
+			}
+		} else {
+			if (p.is_equal(b)) {
+				// a = p^n, b = p, gcd = p
+				if (ca)
+					*ca = power(p, a.op(1) - 1);
+				if (cb)
+					*cb = _ex1();
+				return p;
+			}
+		}
+	} else if (is_ex_exactly_of_type(b, power)) {
+		ex p = b.op(0);
+		if (p.is_equal(a)) {
+			// a = p, b = p^n, gcd = p
+			if (ca)
+				*ca = _ex1();
+			if (cb)
+				*cb = power(p, b.op(1) - 1);
+			return p;
+		}
+	}
+#endif
+
     // Some trivial cases
-    if (a.is_zero()) {
+	ex aex = a.expand(), bex = b.expand();
+    if (aex.is_zero()) {
         if (ca)
-            *ca = exZERO();
+            *ca = _ex0();
         if (cb)
-            *cb = exONE();
+            *cb = _ex1();
         return b;
     }
-    if (b.is_zero()) {
+    if (bex.is_zero()) {
         if (ca)
-            *ca = exONE();
+            *ca = _ex1();
         if (cb)
-            *cb = exZERO();
+            *cb = _ex0();
         return a;
     }
-    if (a.is_equal(exONE()) || b.is_equal(exONE())) {
+    if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) {
         if (ca)
             *ca = a;
         if (cb)
             *cb = b;
-        return exONE();
+        return _ex1();
     }
 #if FAST_COMPARE
     if (a.is_equal(b)) {
         if (ca)
-            *ca = exONE();
+            *ca = _ex1();
         if (cb)
-            *cb = exONE();
+            *cb = _ex1();
         return a;
     }
 #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;
@@ -1109,41 +1502,69 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
     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;
+        return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
     }
 
     // Try to eliminate variables
     if (var->deg_a == 0) {
 //clog << "eliminating variable " << *x << " from b" << endl;
-        ex c = b.content(*x);
-        ex g = gcd(a, c, ca, cb, false);
+        ex c = bex.content(*x);
+        ex g = gcd(aex, c, ca, cb, false);
         if (cb)
-            *cb *= b.unit(*x) * b.primpart(*x, c);
+            *cb *= bex.unit(*x) * bex.primpart(*x, c);
         return g;
     } else if (var->deg_b == 0) {
 //clog << "eliminating variable " << *x << " from a" << endl;
-        ex c = a.content(*x);
-        ex g = gcd(c, b, ca, cb, false);
+        ex c = aex.content(*x);
+        ex g = gcd(c, bex, ca, cb, false);
         if (ca)
-            *ca *= a.unit(*x) * a.primpart(*x, c);
+            *ca *= aex.unit(*x) * aex.primpart(*x, c);
         return g;
     }
 
-    // Try heuristic algorithm first, fall back to PRS if that failed
     ex g;
+#if 1
+    // Try heuristic algorithm first, fall back to PRS if that failed
     try {
-        g = heur_gcd(a.expand(), b.expand(), ca, cb, var);
+        g = heur_gcd(aex, bex, ca, cb, var);
     } catch (gcdheu_failed) {
         g = *new ex(fail());
     }
     if (is_ex_exactly_of_type(g, fail)) {
-//clog << "heuristics failed\n";
-        g = sr_gcd(a, b, x);
-        if (ca)
-            divide(a, g, *ca, false);
-        if (cb)
-            divide(b, g, *cb, false);
-    }
+//clog << "heuristics failed" << endl;
+#if STATISTICS
+		heur_gcd_failed++;
+#endif
+#endif
+//		g = heur_gcd(aex, bex, ca, cb, var);
+//		g = eu_gcd(aex, bex, x);
+//		g = euprem_gcd(aex, bex, x);
+//		g = peu_gcd(aex, bex, x);
+//		g = red_gcd(aex, bex, x);
+		g = sr_gcd(aex, bex, x);
+		if (g.is_equal(_ex1())) {
+			// Keep cofactors factored if possible
+			if (ca)
+				*ca = a;
+			if (cb)
+				*cb = b;
+		} else {
+	        if (ca)
+	            divide(aex, g, *ca, false);
+	        if (cb)
+	            divide(bex, g, *cb, false);
+		}
+#if 1
+    } else {
+		if (g.is_equal(_ex1())) {
+			// Keep cofactors factored if possible
+			if (ca)
+				*ca = a;
+			if (cb)
+				*cb = b;
+		}
+	}
+#endif
     return g;
 }
 
@@ -1158,7 +1579,7 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 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));
+        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"));
     
@@ -1180,8 +1601,8 @@ static ex univariate_gcd(const ex &a, const ex &b, const symbol &x)
         return b;
     if (b.is_zero())
         return a;
-    if (a.is_equal(exONE()) || b.is_equal(exONE()))
-        return exONE();
+    if (a.is_equal(_ex1()) || b.is_equal(_ex1()))
+        return _ex1();
     if (is_ex_of_type(a, numeric) && is_ex_of_type(b, numeric))
         return gcd(ex_to_numeric(a), ex_to_numeric(b));
     if (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
@@ -1216,11 +1637,11 @@ static ex univariate_gcd(const ex &a, const ex &b, const symbol &x)
 ex sqrfree(const ex &a, const symbol &x)
 {
     int i = 1;
-    ex res = exONE();
+    ex res = _ex1();
     ex b = a.diff(x);
     ex c = univariate_gcd(a, b, x);
     ex w;
-    if (c.is_equal(exONE())) {
+    if (c.is_equal(_ex1())) {
         w = a;
     } else {
         w = quo(a, c, x);
@@ -1243,13 +1664,22 @@ 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++)
+    for (unsigned i=0; i<repl_lst.nops(); i++)
         if (repl_lst.op(i).is_equal(e))
             return sym_lst.op(i);
 
@@ -1264,21 +1694,41 @@ static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
     return es;
 }
 
+/** Create a symbol for replacing the expression "e" (or return a previously
+ *  assigned symbol). An expression of the form "symbol == expression" is added
+ *  to repl_lst and the symbol is returned.
+ *  @see ex::to_rational */
+static ex replace_with_symbol(const ex &e, lst &repl_lst)
+{
+    // Expression already in repl_lst? Then return the assigned symbol
+    for (unsigned i=0; i<repl_lst.nops(); i++)
+        if (repl_lst.op(i).op(1).is_equal(e))
+            return repl_lst.op(i).op(0);
+
+    // Otherwise create new symbol and add to list, taking care that the
+	// replacement expression doesn't contain symbols from the sym_lst
+	// because subs() is not recursive
+	symbol s;
+	ex es(s);
+	ex e_replaced = e.subs(repl_lst);
+    repl_lst.append(es == e_replaced);
+    return es;
+}
 
 /** Default implementation of ex::normal(). It replaces the object with a
  *  temporary symbol.
  *  @see ex::normal */
 ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    return replace_with_symbol(*this, sym_lst, repl_lst);
+    return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
 }
 
 
-/** Implementation of ex::normal() for symbols. This returns the 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 +1738,53 @@ 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();
-		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);
-	}
-}
+	numeric num = numer();
+	ex numex = num;
+
+    if (num.is_real()) {
+        if (!num.is_integer())
+            numex = replace_with_symbol(numex, sym_lst, repl_lst);
+    } else { // complex
+        numeric re = num.real(), im = num.imag();
+        ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
+        ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
+        numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
+    }
 
+	// Denominator is always a real integer (see numeric::denom())
+	return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
+}
 
-/*
- *  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();
+    numeric pre_factor = _num1();
+
+//clog << "frac_cancel num = " << num << ", den = " << den << endl;
 
     // Handle special cases where numerator or denominator is 0
     if (num.is_zero())
-        return exZERO();
+		return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
     if (den.expand().is_zero())
         throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
 
-    // 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;
+    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) != exONE()) {
+    if (gcd(num, den, &cnum, &cden, false) != _ex1()) {
 		num = cnum;
 		den = cden;
 	}
@@ -1343,12 +1793,16 @@ 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_ex_exactly_of_type(den.unit(*x),numeric));
+		if (ex_to_numeric(den.unit(*x)).is_negative()) {
+			num *= _ex_1();
+			den *= _ex_1();
 		}
 	}
-    return pre_factor * num / den;
+
+	// Return result as list
+//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl;
+    return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1357,47 +1811,70 @@ 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
+    // Normalize and expand children, chop into summands
     exvector o;
     o.reserve(seq.size()+1);
     epvector::const_iterator it = seq.begin(), itend = seq.end();
     while (it != itend) {
+
+		// Normalize and expand child
         ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
-        if (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();
+
+		// If numerator is a sum, chop into summands
+        if (is_ex_exactly_of_type(n.op(0), add)) {
+            epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
             while (bit != bitend) {
-                o.push_back(recombine_pair_to_ex(*bit));
+                o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated));
                 bit++;
             }
-            o.push_back((static_cast<add *>(n.bp))->overall_coeff);
+
+			// The overall_coeff is already normalized (== rational), we just
+			// split it into numerator and denominator
+			GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
+			numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
+            o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated));
         } else
             o.push_back(n);
         it++;
     }
     o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
 
+	// o is now a vector of {numerator, denominator} lists
+
     // Determine common denominator
-    ex den = exONE();
+    ex den = _ex1();
     exvector::const_iterator ait = o.begin(), aitend = o.end();
+//clog << "add::normal uses the following summands:\n";
     while (ait != aitend) {
-        den = lcm((*ait).denom(false), den, false);
+//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl;
+        den = lcm(ait->op(1), den, false);
         ait++;
     }
+//clog << " common denominator = " << den << endl;
 
     // Add fractions
-    if (den.is_equal(exONE()))
-        return (new add(o))->setflag(status_flags::dynallocated);
-    else {
+    if (den.is_equal(_ex1())) {
+
+		// Common denominator is 1, simply add all numerators
+        exvector num_seq;
+		for (ait=o.begin(); ait!=aitend; ait++) {
+			num_seq.push_back(ait->op(0));
+		}
+		return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated);
+
+	} else {
+
+		// Perform fractional addition
         exvector num_seq;
         for (ait=o.begin(); ait!=aitend; ait++) {
             ex q;
-            if (!divide(den, (*ait).denom(false), q, false)) {
+            if (!divide(den, ait->op(1), q, false)) {
                 // should not happen
                 throw(std::runtime_error("invalid expression in add::normal, division failed"));
             }
-            num_seq.push_back((*ait).numer(false) * q);
+            num_seq.push_back((ait->op(0) * q).expand());
         }
-        ex num = add(num_seq);
+        ex num = (new add(num_seq))->setflag(status_flags::dynallocated);
 
         // Cancel common factors from num/den
         return frac_cancel(num, den);
@@ -1410,17 +1887,23 @@ 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);
+    // Normalize children, separate into numerator and denominator
+	ex num = _ex1();
+	ex den = _ex1(); 
+	ex n;
     epvector::const_iterator it = seq.begin(), itend = seq.end();
     while (it != itend) {
-        o.push_back(recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1));
+		n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
+		num *= n.op(0);
+		den *= n.op(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));
+	n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
+	num *= n.op(0);
+	den *= n.op(1);
+
+	// Perform fraction cancellation
+    return frac_cancel(num, den);
 }
 
 
@@ -1430,24 +1913,55 @@ 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);
+	// Normalize basis
+    ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+
+	if (exponent.info(info_flags::integer)) {
+
+	    if (exponent.info(info_flags::positive)) {
+
+			// (a/b)^n -> {a^n, b^n}
+			return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
+
+		} else if (exponent.info(info_flags::negative)) {
+
+			// (a/b)^-n -> {b^n, a^n}
+			return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
+		}
+
+	} else {
+
+		if (exponent.info(info_flags::positive)) {
+
+			// (a/b)^x -> {sym((a/b)^x), 1}
+			return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+
+		} else if (exponent.info(info_flags::negative)) {
+
+			if (n.op(1).is_equal(_ex1())) {
+
+				// a^-x -> {1, sym(a^x)}
+				return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+
+			} else {
+
+				// (a/b)^-x -> {sym((b/a)^x), 1}
+				return (new lst(replace_with_symbol(power(n.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+			}
+
+		} else {	// exponent not numeric
+
+			// (a/b)^x -> {sym((a/b)^x, 1}
+			return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+		}
     }
 }
 
 
-/** Implementation of ex::normal() for series. It normalizes each coefficient and
+/** 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());
@@ -1457,9 +1971,16 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
         new_seq.push_back(expair(it->rest.normal(), it->coeff));
         it++;
     }
+    ex n = pseries(relational(var,point), new_seq);
+	return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+}
+
 
-    ex n = series(var, point, new_seq);
-    return replace_with_symbol(n, sym_lst, repl_lst);
+/** Implementation of ex::normal() for relationals. It normalizes both sides.
+ *  @see ex::normal */
+ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const
+{
+	return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1467,8 +1988,8 @@ 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.
@@ -1478,11 +1999,127 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
 ex ex::normal(int level) const
 {
     lst sym_lst, repl_lst;
+
     ex e = bp->normal(sym_lst, repl_lst, level);
+	GINAC_ASSERT(is_ex_of_type(e, lst));
+
+	// Re-insert replaced symbols
     if (sym_lst.nops() > 0)
-        return e.subs(sym_lst, repl_lst);
-    else
-        return e;
+        e = e.subs(sym_lst, repl_lst);
+
+	// Convert {numerator, denominator} form back to fraction
+    return e.op(0) / e.op(1);
+}
+
+/** Numerator of an expression. If the expression is not of the normal form
+ *  "numerator/denominator", it is first converted to this form and then the
+ *  numerator is returned.
+ *
+ *  @see ex::normal
+ *  @return numerator */
+ex ex::numer(void) const
+{
+    lst sym_lst, repl_lst;
+
+    ex e = bp->normal(sym_lst, repl_lst, 0);
+	GINAC_ASSERT(is_ex_of_type(e, lst));
+
+	// Re-insert replaced symbols
+    if (sym_lst.nops() > 0)
+        return e.op(0).subs(sym_lst, repl_lst);
+	else
+		return e.op(0);
 }
 
+/** Denominator of an expression. If the expression is not of the normal form
+ *  "numerator/denominator", it is first converted to this form and then the
+ *  denominator is returned.
+ *
+ *  @see ex::normal
+ *  @return denominator */
+ex ex::denom(void) const
+{
+    lst sym_lst, repl_lst;
+
+    ex e = bp->normal(sym_lst, repl_lst, 0);
+	GINAC_ASSERT(is_ex_of_type(e, lst));
+
+	// Re-insert replaced symbols
+    if (sym_lst.nops() > 0)
+        return e.op(1).subs(sym_lst, repl_lst);
+	else
+		return e.op(1);
+}
+
+
+/** Default implementation of ex::to_rational(). It replaces the object with a
+ *  temporary symbol.
+ *  @see ex::to_rational */
+ex basic::to_rational(lst &repl_lst) const
+{
+	return replace_with_symbol(*this, repl_lst);
+}
+
+
+/** Implementation of ex::to_rational() for symbols. This returns the unmodified symbol.
+ *  @see ex::to_rational */
+ex symbol::to_rational(lst &repl_lst) const
+{
+    return *this;
+}
+
+
+/** Implementation of ex::to_rational() for a numeric. It splits complex numbers
+ *  into re+I*im and replaces I and non-rational real numbers with a temporary
+ *  symbol.
+ *  @see ex::to_rational */
+ex numeric::to_rational(lst &repl_lst) const
+{
+    if (is_real()) {
+        if (!is_integer())
+            return replace_with_symbol(*this, repl_lst);
+    } else { // complex
+        numeric re = real(), im = imag();
+        ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst);
+        ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst);
+        return re_ex + im_ex * replace_with_symbol(I, repl_lst);
+    }
+	return *this;
+}
+
+
+/** Implementation of ex::to_rational() for powers. It replaces non-integer
+ *  powers by temporary symbols.
+ *  @see ex::to_rational */
+ex power::to_rational(lst &repl_lst) const
+{
+	if (exponent.info(info_flags::integer))
+		return power(basis.to_rational(repl_lst), exponent);
+	else
+		return replace_with_symbol(*this, repl_lst);
+}
+
+
+/** Rationalization of non-rational functions.
+ *  This function converts a general expression to a rational polynomial
+ *  by replacing all non-rational subexpressions (like non-rational numbers,
+ *  non-integer powers or functions like sin(), cos() etc.) to temporary
+ *  symbols. This makes it possible to use functions like gcd() and divide()
+ *  on non-rational functions by applying to_rational() on the arguments,
+ *  calling the desired function and re-substituting the temporary symbols
+ *  in the result. To make the last step possible, all temporary symbols and
+ *  their associated expressions are collected in the list specified by the
+ *  repl_lst parameter in the form {symbol == expression}, ready to be passed
+ *  as an argument to ex::subs().
+ *
+ *  @param repl_lst collects a list of all temporary symbols and their replacements
+ *  @return rationalized expression */
+ex ex::to_rational(lst &repl_lst) const
+{
+	return bp->to_rational(repl_lst);
+}
+
+
+#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
+#endif // ndef NO_NAMESPACE_GINAC