X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=1c2dc823f7911b76fbd60a5d7fd1ebaa0bd87900;hp=c04596856f4e207a9ff5d762930f3278b303652a;hb=083b0f50275a536be807fa2a34c1e278098e12f5;hpb=a8507b8af1c08d9b27d98d57f95c7ca1a8671e27

diff --git a/ginac/normal.cpp b/ginac/normal.cpp
index c0459685..1c2dc823 100644
--- a/ginac/normal.cpp
+++ b/ginac/normal.cpp
@@ -4,9 +4,10 @@
  *  multivariate polynomials and rational functions.
  *  These functions include polynomial quotient and remainder, GCD and LCM
  *  computation, square-free factorization and rational function normalization.
+ */
 
- *
- *  GiNaC Copyright (C) 1999 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
@@ -24,8 +25,31 @@
  */
 
 #include <stdexcept>
+#include <algorithm>
+#include <map>
 
-#include "ginac.h"
+#include "normal.h"
+#include "basic.h"
+#include "ex.h"
+#include "add.h"
+#include "constant.h"
+#include "expairseq.h"
+#include "fail.h"
+#include "indexed.h"
+#include "inifcns.h"
+#include "lst.h"
+#include "mul.h"
+#include "ncmul.h"
+#include "numeric.h"
+#include "power.h"
+#include "relational.h"
+#include "pseries.h"
+#include "symbol.h"
+#include "utils.h"
+
+#ifndef NO_NAMESPACE_GINAC
+namespace GiNaC {
+#endif // ndef NO_NAMESPACE_GINAC
 
 // If comparing expressions (ex::compare()) is fast, you can set this to 1.
 // Some routines like quo(), rem() and gcd() will then return a quick answer
@@ -50,7 +74,7 @@ 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)) {
@@ -65,8 +89,6 @@ static bool get_first_symbol(const ex &e, const symbol *&x)
  *  Statistical information about symbols in polynomials
  */
 
-#include <algorithm>
-
 /** This structure holds information about the highest and lowest degrees
  *  in which a symbol appears in two multivariate polynomials "a" and "b".
  *  A vector of these structures with information about all symbols in
@@ -119,7 +141,7 @@ 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);
@@ -168,28 +190,61 @@ 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;
 }
 
 
@@ -201,13 +256,13 @@ 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
@@ -219,29 +274,29 @@ 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));
 }
 
@@ -268,13 +323,13 @@ 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;
@@ -317,13 +372,13 @@ 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"));
@@ -369,7 +424,7 @@ 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;
     }
@@ -385,18 +440,18 @@ 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;
@@ -419,7 +474,7 @@ 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 (is_ex_exactly_of_type(b, numeric)) {
@@ -429,7 +484,7 @@ 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
@@ -472,8 +527,6 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args)
  *  Remembering
  */
 
-#include <map>
-
 typedef pair<ex, ex> ex2;
 typedef pair<ex, bool> exbool;
 
@@ -506,10 +559,10 @@ 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;
     }
@@ -522,7 +575,7 @@ 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
@@ -582,19 +635,19 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
     // 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
@@ -646,7 +699,7 @@ 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))
@@ -667,12 +720,12 @@ 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();
@@ -686,7 +739,7 @@ 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;
@@ -703,13 +756,13 @@ 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);
@@ -729,11 +782,11 @@ 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))
@@ -784,7 +837,7 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
     d = d.primpart(*x, cont_d);
 
     // First element of subresultant sequence
-    ex r = exZERO(), ri = exONE(), psi = exONE();
+    ex r = _ex0(), ri = _ex1(), psi = _ex1();
     int delta = cdeg - ddeg;
 
     for (;;) {
@@ -824,13 +877,13 @@ 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
@@ -842,11 +895,11 @@ 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;
@@ -857,15 +910,15 @@ 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));
 }
 
@@ -880,7 +933,7 @@ numeric mul::max_coefficient(void) const
 
 ex ex::smod(const numeric &xi) const
 {
-    ASSERT(bp!=0);
+    GINAC_ASSERT(bp!=0);
     return bp->smod(xi);
 }
 
@@ -891,7 +944,11 @@ 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
@@ -901,37 +958,49 @@ 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));
-    mulcopyp->overall_coeff=::smod(ex_to_numeric(overall_coeff),xi);
+    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.
@@ -978,9 +1047,9 @@ 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++) {
@@ -992,7 +1061,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
         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);
@@ -1007,7 +1076,7 @@ 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) && lc.compare(_ex0()) < 0)
                     return -g;
                 else
                     return g;
@@ -1032,48 +1101,86 @@ 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)
 {
+	// 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;
+	}
+
     // 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 (is_ex_exactly_of_type(aex, numeric) && is_ex_exactly_of_type(bex, numeric)) {
+        numeric g = gcd(ex_to_numeric(aex), ex_to_numeric(bex));
         if (ca)
-            *ca = ex_to_numeric(a) / g;
+            *ca = ex_to_numeric(aex) / g;
         if (cb)
-            *cb = ex_to_numeric(b) / g;
+            *cb = ex_to_numeric(bex) / 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"));
     }
 
@@ -1092,40 +1199,40 @@ 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;
     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);
+//clog << "heuristics failed" << endl;
+        g = sr_gcd(aex, bex, x);
         if (ca)
-            divide(a, g, *ca, false);
+            divide(aex, g, *ca, false);
         if (cb)
-            divide(b, g, *cb, false);
+            divide(bex, g, *cb, false);
     }
     return g;
 }
@@ -1141,7 +1248,7 @@ 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"));
     
@@ -1163,8 +1270,8 @@ 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))
@@ -1199,11 +1306,11 @@ 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);
@@ -1226,13 +1333,22 @@ 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_list and returned, the
+ *  expression is appended to repl_list.
+ *  @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);
 
@@ -1253,15 +1369,15 @@ static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
  *  @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);
 }
 
 
@@ -1271,53 +1387,53 @@ 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;
 	}
@@ -1326,12 +1442,14 @@ 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();
+		if (den.unit(*x).compare(_ex0()) < 0) {
+			num *= _ex_1();
+			den *= _ex_1();
 		}
 	}
-    return pre_factor * num / den;
+
+	// Return result as list
+    return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1340,47 +1458,67 @@ 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()))->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();
     while (ait != aitend) {
-        den = lcm((*ait).denom(false), den, false);
+        den = lcm(ait->op(1), den, false);
         ait++;
     }
 
     // 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);
         }
-        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);
@@ -1393,17 +1531,23 @@ 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);
 }
 
 
@@ -1413,24 +1557,26 @@ 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)) {
+    if (exponent.info(info_flags::posint)) {
         // 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);
+		return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
+	} else if (exponent.info(info_flags::negint)) {
+        // Integer powers are distributed
+        ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+		return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
     } 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);
+		ex n = basis.bp->normal(sym_lst, repl_lst, level-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());
@@ -1440,9 +1586,8 @@ 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 = series(var, point, new_seq);
-    return replace_with_symbol(n, sym_lst, repl_lst);
+    ex n = pseries(var, point, new_seq);
+	return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1461,9 +1606,18 @@ 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);
 }
+
+#ifndef NO_NAMESPACE_GINAC
+} // namespace GiNaC
+#endif // ndef NO_NAMESPACE_GINAC