X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=d88f3dde4925ead551a23a0362749fe68a4b7037;hp=067670ced9f380b7aff2623755bde19f72704293;hb=f7d46a20e8946e9a7a81208339596434dbcb745c;hpb=075877089921f3555f040cd953a6ce75d0d544d8

diff --git a/ginac/normal.cpp b/ginac/normal.cpp
index 067670ce..d88f3dde 100644
--- a/ginac/normal.cpp
+++ b/ginac/normal.cpp
@@ -23,7 +23,6 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
-#include <stdexcept>
 #include <algorithm>
 #include <map>
 
@@ -34,21 +33,18 @@
 #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 "matrix.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
@@ -78,10 +74,10 @@ static int heur_gcd_failed = 0;
 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";
+		std::cout << "gcd() called " << gcd_called << " times\n";
+		std::cout << "sr_gcd() called " << sr_gcd_called << " times\n";
+		std::cout << "heur_gcd() called " << heur_gcd_called << " times\n";
+		std::cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
 	}
 } stat_print;
 #endif
@@ -97,7 +93,7 @@ static struct _stat_print {
 static bool get_first_symbol(const ex &e, const symbol *&x)
 {
 	if (is_ex_exactly_of_type(e, symbol)) {
-		x = static_cast<symbol *>(e.bp);
+		x = &ex_to<symbol>(e);
 		return true;
 	} else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
 		for (unsigned i=0; i<e.nops(); i++)
@@ -140,8 +136,17 @@ struct sym_desc {
 	/** Maximum of deg_a and deg_b (Used for sorting) */
 	int max_deg;
 
+	/** Maximum number of terms of leading coefficient of symbol in both polynomials */
+	int max_lcnops;
+
 	/** Commparison operator for sorting */
-	bool operator<(const sym_desc &x) const {return max_deg < x.max_deg;}
+	bool operator<(const sym_desc &x) const
+	{
+		if (max_deg == x.max_deg)
+			return max_lcnops < x.max_lcnops;
+		else
+			return max_deg < x.max_deg;
+	}
 };
 
 // Vector of sym_desc structures
@@ -150,11 +155,11 @@ typedef std::vector<sym_desc> sym_desc_vec;
 // Add symbol the sym_desc_vec (used internally by get_symbol_stats())
 static void add_symbol(const symbol *s, sym_desc_vec &v)
 {
-	sym_desc_vec::iterator it = v.begin(), itend = v.end();
+	sym_desc_vec::const_iterator it = v.begin(), itend = v.end();
 	while (it != itend) {
 		if (it->sym->compare(*s) == 0)  // If it's already in there, don't add it a second time
 			return;
-		it++;
+		++it;
 	}
 	sym_desc d;
 	d.sym = s;
@@ -165,7 +170,7 @@ static void add_symbol(const symbol *s, sym_desc_vec &v)
 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);
+		add_symbol(&ex_to<symbol>(e), v);
 	} else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
 		for (unsigned i=0; i<e.nops(); i++)
 			collect_symbols(e.op(i), v);
@@ -196,19 +201,20 @@ 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->max_deg = std::max(deg_a,deg_b);
+		it->max_deg = std::max(deg_a, deg_b);
+		it->max_lcnops = std::max(a.lcoeff(*(it->sym)).nops(), b.lcoeff(*(it->sym)).nops());
 		it->ldeg_a = a.ldegree(*(it->sym));
 		it->ldeg_b = b.ldegree(*(it->sym));
-		it++;
+		++it;
 	}
-	sort(v.begin(), v.end());
+	std::sort(v.begin(), v.end());
 #if 0
 	std::clog << "Symbols:\n";
 	it = v.begin(); itend = v.end();
 	while (it != itend) {
-		std::clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << endl;
+		std::clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << ", max_lcnops=" << it->max_lcnops << endl;
 		std::clog << "  lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
-		it++;
+		++it;
 	}
 #endif
 }
@@ -223,19 +229,23 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
 static numeric lcmcoeff(const ex &e, const numeric &l)
 {
 	if (e.info(info_flags::rational))
-		return lcm(ex_to_numeric(e).denom(), l);
+		return lcm(ex_to<numeric>(e).denom(), l);
 	else if (is_ex_exactly_of_type(e, add)) {
-		numeric c = _num1();
+		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();
+		numeric c = _num1;
 		for (unsigned i=0; i<e.nops(); i++)
-			c *= lcmcoeff(e.op(i), _num1());
+			c *= lcmcoeff(e.op(i), _num1);
 		return lcm(c, l);
-	} else if (is_ex_exactly_of_type(e, power))
-		return pow(lcmcoeff(e.op(0), l), ex_to_numeric(e.op(1)));
+	} else if (is_ex_exactly_of_type(e, power)) {
+		if (is_ex_exactly_of_type(e.op(0), symbol))
+			return l;
+		else
+			return pow(lcmcoeff(e.op(0), l), ex_to<numeric>(e.op(1)));
+	}
 	return l;
 }
 
@@ -248,7 +258,7 @@ static numeric lcmcoeff(const ex &e, const numeric &l)
  *  @return LCM of denominators of coefficients */
 static numeric lcm_of_coefficients_denominators(const ex &e)
 {
-	return lcmcoeff(e, _num1());
+	return lcmcoeff(e, _num1);
 }
 
 /** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
@@ -259,22 +269,27 @@ static numeric lcm_of_coefficients_denominators(const ex &e)
 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();
+		unsigned num = e.nops();
+		exvector v; v.reserve(num + 1);
+		numeric lcm_accum = _num1;
 		for (unsigned i=0; i<e.nops(); i++) {
-			numeric op_lcm = lcmcoeff(e.op(i), _num1());
-			c *= multiply_lcm(e.op(i), op_lcm);
+			numeric op_lcm = lcmcoeff(e.op(i), _num1);
+			v.push_back(multiply_lcm(e.op(i), op_lcm));
 			lcm_accum *= op_lcm;
 		}
-		c *= lcm / lcm_accum;
-		return c;
+		v.push_back(lcm / lcm_accum);
+		return (new mul(v))->setflag(status_flags::dynallocated);
 	} 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;
+		unsigned num = e.nops();
+		exvector v; v.reserve(num);
+		for (unsigned i=0; i<num; i++)
+			v.push_back(multiply_lcm(e.op(i), lcm));
+		return (new add(v))->setflag(status_flags::dynallocated);
 	} else if (is_ex_exactly_of_type(e, power)) {
-		return pow(multiply_lcm(e.op(0), lcm.power(ex_to_numeric(e.op(1)).inverse())), e.op(1));
+		if (is_ex_exactly_of_type(e.op(0), symbol))
+			return e * lcm;
+		else
+			return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
 	} else
 		return e * lcm;
 }
@@ -293,7 +308,7 @@ numeric ex::integer_content(void) const
 
 numeric basic::integer_content(void) const
 {
-	return _num1();
+	return _num1;
 }
 
 numeric numeric::integer_content(void) const
@@ -305,15 +320,15 @@ numeric add::integer_content(void) const
 {
 	epvector::const_iterator it = seq.begin();
 	epvector::const_iterator itend = seq.end();
-	numeric c = _num0();
+	numeric c = _num0;
 	while (it != itend) {
-		GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-		GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
-		c = gcd(ex_to_numeric(it->coeff), c);
+		GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+		GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
+		c = gcd(ex_to<numeric>(it->coeff), c);
 		it++;
 	}
-	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-	c = gcd(ex_to_numeric(overall_coeff),c);
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	c = gcd(ex_to<numeric>(overall_coeff),c);
 	return c;
 }
 
@@ -323,12 +338,12 @@ numeric mul::integer_content(void) const
 	epvector::const_iterator it = seq.begin();
 	epvector::const_iterator itend = seq.end();
 	while (it != itend) {
-		GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
+		GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
 		++it;
 	}
 #endif // def DO_GINAC_ASSERT
-	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-	return abs(ex_to_numeric(overall_coeff));
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	return abs(ex_to<numeric>(overall_coeff));
 }
 
 
@@ -353,13 +368,12 @@ 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 _ex1();
+		return _ex1;
 #endif
 	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
 		throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));
 
 	// Polynomial long division
-	ex q = _ex0();
 	ex r = a.expand();
 	if (r.is_zero())
 		return r;
@@ -367,22 +381,23 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
 	int rdeg = r.degree(x);
 	ex blcoeff = b.expand().coeff(x, bdeg);
 	bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
+	exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
 	while (rdeg >= bdeg) {
 		ex term, rcoeff = r.coeff(x, rdeg);
 		if (blcoeff_is_numeric)
 			term = rcoeff / blcoeff;
 		else {
 			if (!divide(rcoeff, blcoeff, term, false))
-				return *new ex(fail());
+				return (new fail())->setflag(status_flags::dynallocated);
 		}
 		term *= power(x, rdeg - bdeg);
-		q += term;
+		v.push_back(term);
 		r -= (term * b).expand();
 		if (r.is_zero())
 			break;
 		rdeg = r.degree(x);
 	}
-	return q;
+	return (new add(v))->setflag(status_flags::dynallocated);
 }
 
 
@@ -401,13 +416,13 @@ 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 _ex0();
+			return _ex0;
 		else
-			return b;
+			return a;
 	}
 #if FAST_COMPARE
 	if (a.is_equal(b))
-		return _ex0();
+		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"));
@@ -426,7 +441,7 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
 			term = rcoeff / blcoeff;
 		else {
 			if (!divide(rcoeff, blcoeff, term, false))
-				return *new ex(fail());
+				return (new fail())->setflag(status_flags::dynallocated);
 		}
 		term *= power(x, rdeg - bdeg);
 		r -= (term * b).expand();
@@ -438,6 +453,24 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
 }
 
 
+/** Decompose rational function a(x)=N(x)/D(x) into P(x)+n(x)/D(x)
+ *  with degree(n, x) < degree(D, x).
+ *
+ *  @param a rational function in x
+ *  @param x a is a function of x
+ *  @return decomposed function. */
+ex decomp_rational(const ex &a, const symbol &x)
+{
+	ex nd = numer_denom(a);
+	ex numer = nd.op(0), denom = nd.op(1);
+	ex q = quo(numer, denom, x);
+	if (is_ex_exactly_of_type(q, fail))
+		return a;
+	else
+		return q + rem(numer, denom, x) / denom;
+}
+
+
 /** Pseudo-remainder of polynomials a(x) and b(x) in Z[x].
  *
  *  @param a  first polynomial in x (dividend)
@@ -452,7 +485,7 @@ 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 _ex0();
+			return _ex0;
 		else
 			return b;
 	}
@@ -468,18 +501,18 @@ 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 = _ex0();
+			eb = _ex0;
 		else
 			eb -= blcoeff * power(x, bdeg);
 	} else
-		blcoeff = _ex1();
+		blcoeff = _ex1;
 
 	int delta = rdeg - bdeg + 1, i = 0;
 	while (rdeg >= bdeg && !r.is_zero()) {
 		ex rlcoeff = r.coeff(x, rdeg);
 		ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
 		if (rdeg == 0)
-			r = _ex0();
+			r = _ex0;
 		else
 			r -= rlcoeff * power(x, rdeg);
 		r = (blcoeff * r).expand() - term;
@@ -498,14 +531,13 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @param check_args  check whether a and b are polynomials with rational
  *         coefficients (defaults to "true")
  *  @return sparse pseudo-remainder of a(x) and b(x) in Z[x] */
-
 ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
 {
 	if (b.is_zero())
 		throw(std::overflow_error("prem: division by zero"));
 	if (is_ex_exactly_of_type(a, numeric)) {
 		if (is_ex_exactly_of_type(b, numeric))
-			return _ex0();
+			return _ex0;
 		else
 			return b;
 	}
@@ -521,17 +553,17 @@ ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
 	if (bdeg <= rdeg) {
 		blcoeff = eb.coeff(x, bdeg);
 		if (bdeg == 0)
-			eb = _ex0();
+			eb = _ex0;
 		else
 			eb -= blcoeff * power(x, bdeg);
 	} else
-		blcoeff = _ex1();
+		blcoeff = _ex1;
 
 	while (rdeg >= bdeg && !r.is_zero()) {
 		ex rlcoeff = r.coeff(x, rdeg);
 		ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
 		if (rdeg == 0)
-			r = _ex0();
+			r = _ex0;
 		else
 			r -= rlcoeff * power(x, rdeg);
 		r = (blcoeff * r).expand() - term;
@@ -549,14 +581,15 @@ ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
  *  @param check_args  check whether a and b are polynomials with rational
  *         coefficients (defaults to "true")
  *  @return "true" when exact division succeeds (quotient returned in q),
- *          "false" otherwise */
+ *          "false" otherwise (q left untouched) */
 bool divide(const ex &a, const ex &b, ex &q, bool check_args)
 {
-	q = _ex0();
 	if (b.is_zero())
 		throw(std::overflow_error("divide: division by zero"));
-	if (a.is_zero())
+	if (a.is_zero()) {
+		q = _ex0;
 		return true;
+	}
 	if (is_ex_exactly_of_type(b, numeric)) {
 		q = a / b;
 		return true;
@@ -564,7 +597,7 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args)
 		return false;
 #if FAST_COMPARE
 	if (a.is_equal(b)) {
-		q = _ex1();
+		q = _ex1;
 		return true;
 	}
 #endif
@@ -579,12 +612,15 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args)
 
 	// Polynomial long division (recursive)
 	ex r = a.expand();
-	if (r.is_zero())
+	if (r.is_zero()) {
+		q = _ex0;
 		return true;
+	}
 	int bdeg = b.degree(*x);
 	int rdeg = r.degree(*x);
 	ex blcoeff = b.expand().coeff(*x, bdeg);
 	bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
+	exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
 	while (rdeg >= bdeg) {
 		ex term, rcoeff = r.coeff(*x, rdeg);
 		if (blcoeff_is_numeric)
@@ -593,10 +629,12 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args)
 			if (!divide(rcoeff, blcoeff, term, false))
 				return false;
 		term *= power(*x, rdeg - bdeg);
-		q += term;
+		v.push_back(term);
 		r -= (term * b).expand();
-		if (r.is_zero())
+		if (r.is_zero()) {
+			q = (new add(v))->setflag(status_flags::dynallocated);
 			return true;
+		}
 		rdeg = r.degree(*x);
 	}
 	return false;
@@ -612,9 +650,10 @@ typedef std::pair<ex, ex> ex2;
 typedef std::pair<ex, bool> exbool;
 
 struct ex2_less {
-	bool operator() (const ex2 p, const ex2 q) const 
+	bool operator() (const ex2 &p, const ex2 &q) const 
 	{
-		return p.first.compare(q.first) < 0 || (!(q.first.compare(p.first) < 0) && p.second.compare(q.second) < 0);        
+		int cmp = p.first.compare(q.first);
+		return ((cmp<0) || (!(cmp>0) && p.second.compare(q.second)<0));
 	}
 };
 
@@ -640,10 +679,10 @@ typedef std::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 = _ex0();
+	q = _ex0;
 	if (b.is_zero())
 		throw(std::overflow_error("divide_in_z: division by zero"));
-	if (b.is_equal(_ex1())) {
+	if (b.is_equal(_ex1)) {
 		q = a;
 		return true;
 	}
@@ -656,7 +695,7 @@ 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 = _ex1();
+		q = _ex1;
 		return true;
 	}
 #endif
@@ -687,24 +726,24 @@ 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 = _num0();
+	numeric point = _num0;
 	ex c;
 	for (i=0; i<=adeg; i++) {
 		ex bs = b.subs(*x == point);
 		while (bs.is_zero()) {
-			point += _num1();
+			point += _num1;
 			bs = b.subs(*x == point);
 		}
 		if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
 			return false;
 		alpha.push_back(point);
 		u.push_back(c);
-		point += _num1();
+		point += _num1;
 	}
 
 	// Compute inverses
 	vector<numeric> rcp; rcp.reserve(adeg + 1);
-	rcp.push_back(_num0());
+	rcp.push_back(_num0);
 	for (k=1; k<=adeg; k++) {
 		numeric product = alpha[k] - alpha[0];
 		for (i=1; i<k; i++)
@@ -742,14 +781,16 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
 	int rdeg = adeg;
 	ex eb = b.expand();
 	ex blcoeff = eb.coeff(*x, bdeg);
+	exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
 	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;
+		v.push_back(term);
 		r -= (term * eb).expand();
 		if (r.is_zero()) {
+			q = (new add(v))->setflag(status_flags::dynallocated);
 #if USE_REMEMBER
 			dr_remember[ex2(a, b)] = exbool(q, true);
 #endif
@@ -781,7 +822,7 @@ ex ex::unit(const symbol &x) const
 {
 	ex c = expand().lcoeff(x);
 	if (is_ex_exactly_of_type(c, numeric))
-		return c < _ex0() ? _ex_1() : _ex1();
+		return c < _ex0 ? _ex_1 : _ex1;
 	else {
 		const symbol *y;
 		if (get_first_symbol(c, y))
@@ -802,12 +843,12 @@ ex ex::unit(const symbol &x) const
 ex ex::content(const symbol &x) const
 {
 	if (is_zero())
-		return _ex0();
+		return _ex0;
 	if (is_ex_exactly_of_type(*this, numeric))
 		return info(info_flags::negative) ? -*this : *this;
 	ex e = expand();
 	if (e.is_zero())
-		return _ex0();
+		return _ex0;
 
 	// First, try the integer content
 	ex c = e.integer_content();
@@ -821,7 +862,7 @@ ex ex::content(const symbol &x) const
 	int ldeg = e.ldegree(x);
 	if (deg == ldeg)
 		return e.lcoeff(x) / e.unit(x);
-	c = _ex0();
+	c = _ex0;
 	for (int i=ldeg; i<=deg; i++)
 		c = gcd(e.coeff(x, i), c, NULL, NULL, false);
 	return c;
@@ -838,13 +879,13 @@ ex ex::content(const symbol &x) const
 ex ex::primpart(const symbol &x) const
 {
 	if (is_zero())
-		return _ex0();
+		return _ex0;
 	if (is_ex_exactly_of_type(*this, numeric))
-		return _ex1();
+		return _ex1;
 
 	ex c = content(x);
 	if (c.is_zero())
-		return _ex0();
+		return _ex0;
 	ex u = unit(x);
 	if (is_ex_exactly_of_type(c, numeric))
 		return *this / (c * u);
@@ -863,11 +904,11 @@ ex ex::primpart(const symbol &x) const
 ex ex::primpart(const symbol &x, const ex &c) const
 {
 	if (is_zero())
-		return _ex0();
+		return _ex0;
 	if (c.is_zero())
-		return _ex0();
+		return _ex0;
 	if (is_ex_exactly_of_type(*this, numeric))
-		return _ex1();
+		return _ex1;
 
 	ex u = unit(x);
 	if (is_ex_exactly_of_type(c, numeric))
@@ -1053,7 +1094,7 @@ static ex red_gcd(const ex &a, const ex &b, const symbol *x)
 	d = d.primpart(*x, cont_d);
 
 	// First element of divisor sequence
-	ex r, ri = _ex1();
+	ex r, ri = _ex1;
 	int delta = cdeg - ddeg;
 
 	for (;;) {
@@ -1127,7 +1168,7 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
 //std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
 
 	// First element of subresultant sequence
-	ex r = _ex0(), ri = _ex1(), psi = _ex1();
+	ex r = _ex0, ri = _ex1, psi = _ex1;
 	int delta = cdeg - ddeg;
 
 	for (;;) {
@@ -1174,9 +1215,11 @@ numeric ex::max_coefficient(void) const
 	return bp->max_coefficient();
 }
 
+/** Implementation ex::max_coefficient().
+ *  @see heur_gcd */
 numeric basic::max_coefficient(void) const
 {
-	return _num1();
+	return _num1;
 }
 
 numeric numeric::max_coefficient(void) const
@@ -1188,12 +1231,12 @@ numeric add::max_coefficient(void) const
 {
 	epvector::const_iterator it = seq.begin();
 	epvector::const_iterator itend = seq.end();
-	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-	numeric cur_max = abs(ex_to_numeric(overall_coeff));
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	numeric cur_max = abs(ex_to<numeric>(overall_coeff));
 	while (it != itend) {
 		numeric a;
-		GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-		a = abs(ex_to_numeric(it->coeff));
+		GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+		a = abs(ex_to<numeric>(it->coeff));
 		if (a > cur_max)
 			cur_max = a;
 		it++;
@@ -1207,28 +1250,21 @@ numeric mul::max_coefficient(void) const
 	epvector::const_iterator it = seq.begin();
 	epvector::const_iterator itend = seq.end();
 	while (it != itend) {
-		GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
+		GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
 		it++;
 	}
 #endif // def DO_GINAC_ASSERT
-	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-	return abs(ex_to_numeric(overall_coeff));
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	return abs(ex_to<numeric>(overall_coeff));
 }
 
 
-/** Apply symmetric modular homomorphism to a multivariate polynomial.
- *  This function is used internally by heur_gcd().
+/** Apply symmetric modular homomorphism to an expanded multivariate
+ *  polynomial.  This function is usually used internally by heur_gcd().
  *
- *  @param e  expanded multivariate polynomial
  *  @param xi  modulus
  *  @return mapped polynomial
  *  @see heur_gcd */
-ex ex::smod(const numeric &xi) const
-{
-	GINAC_ASSERT(bp!=0);
-	return bp->smod(xi);
-}
-
 ex basic::smod(const numeric &xi) const
 {
 	return *this;
@@ -1236,11 +1272,7 @@ 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
@@ -1250,22 +1282,14 @@ ex add::smod(const numeric &xi) const
 	epvector::const_iterator it = seq.begin();
 	epvector::const_iterator itend = seq.end();
 	while (it != itend) {
-		GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-#ifndef NO_NAMESPACE_GINAC
-		numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
-#else // ndef NO_NAMESPACE_GINAC
-		numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
-#endif // ndef NO_NAMESPACE_GINAC
+		GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+		numeric coeff = GiNaC::smod(ex_to<numeric>(it->coeff), xi);
 		if (!coeff.is_zero())
 			newseq.push_back(expair(it->rest, coeff));
 		it++;
 	}
-	GINAC_ASSERT(is_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
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
 	return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
 }
 
@@ -1275,17 +1299,13 @@ ex mul::smod(const numeric &xi) const
 	epvector::const_iterator it = seq.begin();
 	epvector::const_iterator itend = seq.end();
 	while (it != itend) {
-		GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
+		GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
 		it++;
 	}
 #endif // def DO_GINAC_ASSERT
-	mul * mulcopyp=new mul(*this);
-	GINAC_ASSERT(is_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
+	mul * mulcopyp = new mul(*this);
+	GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+	mulcopyp->overall_coeff = GiNaC::smod(ex_to<numeric>(overall_coeff),xi);
 	mulcopyp->clearflag(status_flags::evaluated);
 	mulcopyp->clearflag(status_flags::hash_calculated);
 	return mulcopyp->setflag(status_flags::dynallocated);
@@ -1293,17 +1313,17 @@ ex mul::smod(const numeric &xi) const
 
 
 /** xi-adic polynomial interpolation */
-static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x)
+static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x, int degree_hint = 1)
 {
-	ex g = _ex0();
+	exvector g; g.reserve(degree_hint);
 	ex e = gamma;
 	numeric rxi = xi.inverse();
 	for (int i=0; !e.is_zero(); i++) {
 		ex gi = e.smod(xi);
-		g += gi * power(x, i);
+		g.push_back(gi * power(x, i));
 		e = (e - gi) * rxi;
 	}
-	return g;
+	return (new add(g))->setflag(status_flags::dynallocated);
 }
 
 /** Exception thrown by heur_gcd() to signal failure. */
@@ -1331,17 +1351,17 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 	heur_gcd_called++;
 #endif
 
-	// Algorithms only works for non-vanishing input polynomials
+	// Algorithm only works for non-vanishing input polynomials
 	if (a.is_zero() || b.is_zero())
-		return *new ex(fail());
+		return (new fail())->setflag(status_flags::dynallocated);
 
 	// GCD of two numeric values -> CLN
 	if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
-		numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
+		numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
 		if (ca)
-			*ca = ex_to_numeric(a) / g;
+			*ca = ex_to<numeric>(a) / g;
 		if (cb)
-			*cb = ex_to_numeric(b) / g;
+			*cb = ex_to<numeric>(b) / g;
 		return g;
 	}
 
@@ -1353,21 +1373,21 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 	numeric rgc = gc.inverse();
 	ex p = a * rgc;
 	ex q = b * rgc;
-	int maxdeg =  std::max(p.degree(x),q.degree(x));
+	int maxdeg =  std::max(p.degree(x), q.degree(x));
 	
 	// Find evaluation point
 	numeric mp = p.max_coefficient();
 	numeric mq = q.max_coefficient();
 	numeric xi;
 	if (mp > mq)
-		xi = mq * _num2() + _num2();
+		xi = mq * _num2 + _num2;
 	else
-		xi = mp * _num2() + _num2();
+		xi = mp * _num2 + _num2;
 
 	// 6 tries maximum
 	for (int t=0; t<6; t++) {
 		if (xi.int_length() * maxdeg > 100000) {
-//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
+//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl;
 			throw gcdheu_failed();
 		}
 
@@ -1377,7 +1397,7 @@ 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 = interpolate(gamma, xi, x);
+			ex g = interpolate(gamma, xi, x, maxdeg);
 
 			// Remove integer content
 			g /= g.integer_content();
@@ -1387,7 +1407,7 @@ 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) && ex_to_numeric(lc).is_negative())
+				if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
 					return -g;
 				else
 					return g;
@@ -1400,7 +1420,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 					if (ca)
 						*ca = cp;
 					ex lc = g.lcoeff(x);
-					if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
+					if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
 						return -g;
 					else
 						return g;
@@ -1413,7 +1433,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 					if (cb)
 						*cb = cq;
 					ex lc = g.lcoeff(x);
-					if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
+					if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
 						return -g;
 					else
 						return g;
@@ -1425,7 +1445,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 		// Next evaluation point
 		xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
 	}
-	return *new ex(fail());
+	return (new fail())->setflag(status_flags::dynallocated);
 }
 
 
@@ -1446,25 +1466,25 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 
 	// 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));
+		numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
 		if (ca || cb) {
 			if (g.is_zero()) {
 				if (ca)
-					*ca = _ex0();
+					*ca = _ex0;
 				if (cb)
-					*cb = _ex0();
+					*cb = _ex0;
 			} else {
 				if (ca)
-					*ca = ex_to_numeric(a) / g;
+					*ca = ex_to<numeric>(a) / g;
 				if (cb)
-					*cb = ex_to_numeric(b) / g;
+					*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)) {
+	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"));
 	}
 
@@ -1473,38 +1493,40 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 		if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
 			goto factored_b;
 factored_a:
-		ex g = _ex1();
-		ex acc_ca = _ex1();
+		unsigned num = a.nops();
+		exvector g; g.reserve(num);
+		exvector acc_ca; acc_ca.reserve(num);
 		ex part_b = b;
-		for (unsigned i=0; i<a.nops(); i++) {
+		for (unsigned i=0; i<num; i++) {
 			ex part_ca, part_cb;
-			g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
-			acc_ca *= part_ca;
+			g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, check_args));
+			acc_ca.push_back(part_ca);
 			part_b = part_cb;
 		}
 		if (ca)
-			*ca = acc_ca;
+			*ca = (new mul(acc_ca))->setflag(status_flags::dynallocated);
 		if (cb)
 			*cb = part_b;
-		return g;
+		return (new mul(g))->setflag(status_flags::dynallocated);
 	} else if (is_ex_exactly_of_type(b, mul)) {
 		if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
 			goto factored_a;
 factored_b:
-		ex g = _ex1();
-		ex acc_cb = _ex1();
+		unsigned num = b.nops();
+		exvector g; g.reserve(num);
+		exvector acc_cb; acc_cb.reserve(num);
 		ex part_a = a;
-		for (unsigned i=0; i<b.nops(); i++) {
+		for (unsigned i=0; i<num; i++) {
 			ex part_ca, part_cb;
-			g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
-			acc_cb *= part_cb;
+			g.push_back(gcd(part_a, b.op(i), &part_ca, &part_cb, check_args));
+			acc_cb.push_back(part_cb);
 			part_a = part_ca;
 		}
 		if (ca)
 			*ca = part_a;
 		if (cb)
-			*cb = acc_cb;
-		return g;
+			*cb = (new mul(acc_cb))->setflag(status_flags::dynallocated);
+		return (new mul(g))->setflag(status_flags::dynallocated);
 	}
 
 #if FAST_COMPARE
@@ -1517,7 +1539,7 @@ factored_b:
 				ex exp_a = a.op(1), exp_b = b.op(1);
 				if (exp_a < exp_b) {
 					if (ca)
-						*ca = _ex1();
+						*ca = _ex1;
 					if (cb)
 						*cb = power(p, exp_b - exp_a);
 					return power(p, exp_a);
@@ -1525,7 +1547,7 @@ factored_b:
 					if (ca)
 						*ca = power(p, exp_a - exp_b);
 					if (cb)
-						*cb = _ex1();
+						*cb = _ex1;
 					return power(p, exp_b);
 				}
 			}
@@ -1535,7 +1557,7 @@ factored_b:
 				if (ca)
 					*ca = power(p, a.op(1) - 1);
 				if (cb)
-					*cb = _ex1();
+					*cb = _ex1;
 				return p;
 			}
 		}
@@ -1544,7 +1566,7 @@ factored_b:
 		if (p.is_equal(a)) {
 			// a = p, b = p^n, gcd = p
 			if (ca)
-				*ca = _ex1();
+				*ca = _ex1;
 			if (cb)
 				*cb = power(p, b.op(1) - 1);
 			return p;
@@ -1556,31 +1578,31 @@ factored_b:
 	ex aex = a.expand(), bex = b.expand();
 	if (aex.is_zero()) {
 		if (ca)
-			*ca = _ex0();
+			*ca = _ex0;
 		if (cb)
-			*cb = _ex1();
+			*cb = _ex1;
 		return b;
 	}
 	if (bex.is_zero()) {
 		if (ca)
-			*ca = _ex1();
+			*ca = _ex1;
 		if (cb)
-			*cb = _ex0();
+			*cb = _ex0;
 		return a;
 	}
-	if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) {
+	if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) {
 		if (ca)
 			*ca = a;
 		if (cb)
 			*cb = b;
-		return _ex1();
+		return _ex1;
 	}
 #if FAST_COMPARE
 	if (a.is_equal(b)) {
 		if (ca)
-			*ca = _ex1();
+			*ca = _ex1;
 		if (cb)
-			*cb = _ex1();
+			*cb = _ex1;
 		return a;
 	}
 #endif
@@ -1599,20 +1621,20 @@ factored_b:
 	int min_ldeg = std::min(ldeg_a,ldeg_b);
 	if (min_ldeg > 0) {
 		ex common = power(x, min_ldeg);
-//std::clog << "trivial common factor " << common << endl;
+//std::clog << "trivial common factor " << common << std::endl;
 		return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
 	}
 
 	// Try to eliminate variables
 	if (var->deg_a == 0) {
-//std::clog << "eliminating variable " << x << " from b" << endl;
+//std::clog << "eliminating variable " << x << " from b" << std::endl;
 		ex c = bex.content(x);
 		ex g = gcd(aex, c, ca, cb, false);
 		if (cb)
 			*cb *= bex.unit(x) * bex.primpart(x, c);
 		return g;
 	} else if (var->deg_b == 0) {
-//std::clog << "eliminating variable " << x << " from a" << endl;
+//std::clog << "eliminating variable " << x << " from a" << std::endl;
 		ex c = aex.content(x);
 		ex g = gcd(c, bex, ca, cb, false);
 		if (ca)
@@ -1626,10 +1648,10 @@ factored_b:
 	try {
 		g = heur_gcd(aex, bex, ca, cb, var);
 	} catch (gcdheu_failed) {
-		g = *new ex(fail());
+		g = fail();
 	}
 	if (is_ex_exactly_of_type(g, fail)) {
-//std::clog << "heuristics failed" << endl;
+//std::clog << "heuristics failed" << std::endl;
 #if STATISTICS
 		heur_gcd_failed++;
 #endif
@@ -1640,7 +1662,7 @@ factored_b:
 //		g = peu_gcd(aex, bex, &x);
 //		g = red_gcd(aex, bex, &x);
 		g = sr_gcd(aex, bex, var);
-		if (g.is_equal(_ex1())) {
+		if (g.is_equal(_ex1)) {
 			// Keep cofactors factored if possible
 			if (ca)
 				*ca = a;
@@ -1654,7 +1676,7 @@ factored_b:
 		}
 #if 1
 	} else {
-		if (g.is_equal(_ex1())) {
+		if (g.is_equal(_ex1)) {
 			// Keep cofactors factored if possible
 			if (ca)
 				*ca = a;
@@ -1677,8 +1699,8 @@ factored_b:
 ex lcm(const ex &a, const ex &b, bool check_args)
 {
 	if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
-		return lcm(ex_to_numeric(a), ex_to_numeric(b));
-	if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
+		return lcm(ex_to<numeric>(a), ex_to<numeric>(b));
+	if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
 		throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
 	
 	ex ca, cb;
@@ -1691,70 +1713,202 @@ ex lcm(const ex &a, const ex &b, bool check_args)
  *  Square-free factorization
  */
 
-// Univariate GCD of polynomials in Q[x] (used internally by sqrfree()).
-// a and b can be multivariate polynomials but they are treated as univariate polynomials in x.
-static ex univariate_gcd(const ex &a, const ex &b, const symbol &x)
+/** Compute square-free factorization of multivariate polynomial a(x) using
+ *  Yun´s algorithm.  Used internally by sqrfree().
+ *
+ *  @param a  multivariate polynomial over Z[X], treated here as univariate
+ *            polynomial in x.
+ *  @param x  variable to factor in
+ *  @return   vector of factors sorted in ascending degree */
+static exvector sqrfree_yun(const ex &a, const symbol &x)
 {
-	if (a.is_zero())
-		return b;
-	if (b.is_zero())
+	exvector res;
+	ex w = a;
+	ex z = w.diff(x);
+	ex g = gcd(w, z);
+	if (g.is_equal(_ex1)) {
+		res.push_back(a);
+		return res;
+	}
+	ex y;
+	do {
+		w = quo(w, g, x);
+		y = quo(z, g, x);
+		z = y - w.diff(x);
+		g = gcd(w, z);
+		res.push_back(g);
+	} while (!z.is_zero());
+	return res;
+}
+
+/** Compute a square-free factorization of a multivariate polynomial in Q[X].
+ *
+ *  @param a  multivariate polynomial over Q[X]
+ *  @param x  lst of variables to factor in, may be left empty for autodetection
+ *  @return   a square-free factorization of \p a.
+ *
+ * \note
+ * A polynomial \f$p(X) \in C[X]\f$ is said <EM>square-free</EM>
+ * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$
+ * are such that
+ * \f[
+ *     p(X) = q(X)^2 r(X),
+ * \f]
+ * we have \f$q(X) \in C\f$.
+ * This means that \f$p(X)\f$ has no repeated factors, apart
+ * eventually from constants.
+ * Given a polynomial \f$p(X) \in C[X]\f$, we say that the
+ * decomposition
+ * \f[
+ *   p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r}
+ * \f]
+ * is a <EM>square-free factorization</EM> of \f$p(X)\f$ if the
+ * following conditions hold:
+ * -#  \f$b \in C\f$ and \f$b \neq 0\f$;
+ * -#  \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$;
+ * -#  the degree of the polynomial \f$p_i\f$ is strictly positive
+ *     for \f$i = 1, \ldots, r\f$;
+ * -#  the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free.
+ *
+ * Square-free factorizations need not be unique.  For example, if
+ * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$
+ * into \f$-p_i(X)\f$.
+ * Observe also that the factors \f$p_i(X)\f$ need not be irreducible
+ * polynomials.
+ */
+ex sqrfree(const ex &a, const lst &l)
+{
+	if (is_a<numeric>(a) ||     // algorithm does not trap a==0
+	    is_a<symbol>(a))        // shortcut
 		return a;
-	if (a.is_equal(_ex1()) || b.is_equal(_ex1()))
-		return _ex1();
-	if (is_ex_of_type(a, numeric) && is_ex_of_type(b, numeric))
-		return gcd(ex_to_numeric(a), ex_to_numeric(b));
-	if (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
-		throw(std::invalid_argument("univariate_gcd: arguments must be polynomials over the rationals"));
 
-	// Euclidean algorithm
-	ex c, d, r;
-	if (a.degree(x) >= b.degree(x)) {
-		c = a;
-		d = b;
+	// If no lst of variables to factorize in was specified we have to
+	// invent one now.  Maybe one can optimize here by reversing the order
+	// or so, I don't know.
+	lst args;
+	if (l.nops()==0) {
+		sym_desc_vec sdv;
+		get_symbol_stats(a, _ex0, sdv);
+		sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end();
+		while (it != itend) {
+			args.append(*it->sym);
+			++it;
+		}
 	} else {
-		c = b;
-		d = a;
+		args = l;
 	}
-	for (;;) {
-		r = rem(c, d, x, false);
-		if (r.is_zero())
-			break;
-		c = d;
-		d = r;
+
+	// Find the symbol to factor in at this stage
+	if (!is_ex_of_type(args.op(0), symbol))
+		throw (std::runtime_error("sqrfree(): invalid factorization variable"));
+	const symbol &x = ex_to<symbol>(args.op(0));
+
+	// convert the argument from something in Q[X] to something in Z[X]
+	const numeric lcm = lcm_of_coefficients_denominators(a);
+	const ex tmp = multiply_lcm(a,lcm);
+
+	// find the factors
+	exvector factors = sqrfree_yun(tmp,x);
+
+	// construct the next list of symbols with the first element popped
+	lst newargs = args;
+	newargs.remove_first();
+
+	// recurse down the factors in remaining variables
+	if (newargs.nops()>0) {
+		exvector::iterator i = factors.begin();
+		while (i != factors.end()) {
+			*i = sqrfree(*i, newargs);
+			++i;
+		}
 	}
-	return d / d.lcoeff(x);
-}
 
+	// Done with recursion, now construct the final result
+	ex result = _ex1;
+	exvector::const_iterator it = factors.begin(), itend = factors.end();
+	for (int p = 1; it!=itend; ++it, ++p)
+		result *= power(*it, p);
 
-/** Compute square-free factorization of multivariate polynomial a(x) using
- *  Yun´s algorithm.
+	// Yun's algorithm does not account for constant factors.  (For univariate
+	// polynomials it works only in the monic case.)  We can correct this by
+	// inserting what has been lost back into the result.  For completeness
+	// we'll also have to recurse down that factor in the remaining variables.
+	if (newargs.nops()>0)
+		result *= sqrfree(quo(tmp, result, x), newargs);
+	else
+		result *= quo(tmp, result, x);
+
+	// Put in the reational overall factor again and return
+	return result *	lcm.inverse();
+}
+
+/** Compute square-free partial fraction decomposition of rational function
+ *  a(x).
  *
- * @param a  multivariate polynomial
- * @param x  variable to factor in
- * @return factored polynomial */
-ex sqrfree(const ex &a, const symbol &x)
+ *  @param a rational function over Z[x], treated as univariate polynomial
+ *           in x
+ *  @param x variable to factor in
+ *  @return decomposed rational function */
+ex sqrfree_parfrac(const ex & a, const symbol & x)
 {
-	int i = 1;
-	ex res = _ex1();
-	ex b = a.diff(x);
-	ex c = univariate_gcd(a, b, x);
-	ex w;
-	if (c.is_equal(_ex1())) {
-		w = a;
-	} else {
-		w = quo(a, c, x);
-		ex y = quo(b, c, x);
-		ex z = y - w.diff(x);
-		while (!z.is_zero()) {
-			ex g = univariate_gcd(w, z, x);
-			res *= power(g, i);
-			w = quo(w, g, x);
-			y = quo(z, g, x);
-			z = y - w.diff(x);
-			i++;
+	// Find numerator and denominator
+	ex nd = numer_denom(a);
+	ex numer = nd.op(0), denom = nd.op(1);
+//clog << "numer = " << numer << ", denom = " << denom << endl;
+
+	// Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D)
+	ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand();
+//clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl;
+
+	// Factorize denominator and compute cofactors
+	exvector yun = sqrfree_yun(denom, x);
+//clog << "yun factors: " << exprseq(yun) << endl;
+	unsigned num_yun = yun.size();
+	exvector factor; factor.reserve(num_yun);
+	exvector cofac; cofac.reserve(num_yun);
+	for (unsigned i=0; i<num_yun; i++) {
+		if (!yun[i].is_equal(_ex1)) {
+			for (unsigned j=0; j<=i; j++) {
+				factor.push_back(pow(yun[i], j+1));
+				ex prod = _ex1;
+				for (unsigned k=0; k<num_yun; k++) {
+					if (k == i)
+						prod *= pow(yun[k], i-j);
+					else
+						prod *= pow(yun[k], k+1);
+				}
+				cofac.push_back(prod.expand());
+			}
 		}
 	}
-	return res * power(w, i);
+	unsigned num_factors = factor.size();
+//clog << "factors  : " << exprseq(factor) << endl;
+//clog << "cofactors: " << exprseq(cofac) << endl;
+
+	// Construct coefficient matrix for decomposition
+	int max_denom_deg = denom.degree(x);
+	matrix sys(max_denom_deg + 1, num_factors);
+	matrix rhs(max_denom_deg + 1, 1);
+	for (int i=0; i<=max_denom_deg; i++) {
+		for (unsigned j=0; j<num_factors; j++)
+			sys(i, j) = cofac[j].coeff(x, i);
+		rhs(i, 0) = red_numer.coeff(x, i);
+	}
+//clog << "coeffs: " << sys << endl;
+//clog << "rhs   : " << rhs << endl;
+
+	// Solve resulting linear system
+	matrix vars(num_factors, 1);
+	for (unsigned i=0; i<num_factors; i++)
+		vars(i, 0) = symbol();
+	matrix sol = sys.solve(vars, rhs);
+
+	// Sum up decomposed fractions
+	ex sum = 0;
+	for (unsigned i=0; i<num_factors; i++)
+		sum += sol(i, 0) / factor[i];
+
+	return red_poly + sum;
 }
 
 
@@ -1770,6 +1924,7 @@ ex sqrfree(const ex &a, const symbol &x)
  *  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.
@@ -1795,7 +1950,7 @@ static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
 /** 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 */
+ *  @see basic::to_rational */
 static ex replace_with_symbol(const ex &e, lst &repl_lst)
 {
 	// Expression already in repl_lst? Then return the assigned symbol
@@ -1813,12 +1968,31 @@ static ex replace_with_symbol(const ex &e, lst &repl_lst)
 	return es;
 }
 
-/** Default implementation of ex::normal(). It replaces the object with a
- *  temporary symbol.
+
+/** Function object to be applied by basic::normal(). */
+struct normal_map_function : public map_function {
+	int level;
+	normal_map_function(int l) : level(l) {}
+	ex operator()(const ex & e) { return normal(e, level); }
+};
+
+/** Default implementation of ex::normal(). It normalizes the children and
+ *  replaces the object with a temporary symbol.
  *  @see ex::normal */
 ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-	return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+	if (nops() == 0)
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+	else {
+		if (level == 1)
+			return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+		else if (level == -max_recursion_level)
+			throw(std::runtime_error("max recursion level reached"));
+		else {
+			normal_map_function map_normal(level - 1);
+			return (new lst(replace_with_symbol(map(map_normal), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+		}
+	}
 }
 
 
@@ -1826,7 +2000,7 @@ ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal */
 ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-	return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+	return (new lst(*this, _ex1))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1862,13 +2036,17 @@ static ex frac_cancel(const ex &n, const ex &d)
 {
 	ex num = n;
 	ex den = d;
-	numeric pre_factor = _num1();
+	numeric pre_factor = _num1;
+
+//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
 
-//std::clog << "frac_cancel num = " << num << ", den = " << den << endl;
+	// Handle trivial case where denominator is 1
+	if (den.is_equal(_ex1))
+		return (new lst(num, den))->setflag(status_flags::dynallocated);
 
 	// Handle special cases where numerator or denominator is 0
 	if (num.is_zero())
-		return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
+		return (new lst(num, _ex1))->setflag(status_flags::dynallocated);
 	if (den.expand().is_zero())
 		throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
 
@@ -1882,7 +2060,7 @@ static ex frac_cancel(const ex &n, const ex &d)
 
 	// Cancel GCD from numerator and denominator
 	ex cnum, cden;
-	if (gcd(num, den, &cnum, &cden, false) != _ex1()) {
+	if (gcd(num, den, &cnum, &cden, false) != _ex1) {
 		num = cnum;
 		den = cden;
 	}
@@ -1891,15 +2069,15 @@ 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)) {
-		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();
+		GINAC_ASSERT(is_exactly_a<numeric>(den.unit(*x)));
+		if (ex_to<numeric>(den.unit(*x)).is_negative()) {
+			num *= _ex_1;
+			den *= _ex_1;
 		}
 	}
 
 	// Return result as list
-//std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl;
+//std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << std::endl;
 	return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
 }
 
@@ -1910,65 +2088,53 @@ static ex frac_cancel(const ex &n, const ex &d)
 ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
 	if (level == 1)
-		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 	else if (level == -max_recursion_level)
 		throw(std::runtime_error("max recursion level reached"));
 
-	// Normalize and expand children, chop into summands and split each
-	// one into numerator and denominator
+	// Normalize children and split each one into numerator and denominator
 	exvector nums, dens;
 	nums.reserve(seq.size()+1);
 	dens.reserve(seq.size()+1);
 	epvector::const_iterator it = seq.begin(), itend = seq.end();
 	while (it != itend) {
-
-		// Normalize and expand child
-		ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
-
-		// If numerator is a sum, chop into summands
-		if (is_ex_exactly_of_type(n.op(0), add)) {
-			epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
-			while (bit != bitend) {
-				nums.push_back(recombine_pair_to_ex(*bit));
-				dens.push_back(n.op(1));
-				bit++;
-			}
-
-			// The overall_coeff is already normalized (== rational), we just
-			// split it into numerator and denominator
-			GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
-			numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
-			nums.push_back(overall.numer());
-			dens.push_back(overall.denom() * n.op(1));
-		} else {
-			nums.push_back(n.op(0));
-			dens.push_back(n.op(1));
-		}
+		ex n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+		nums.push_back(n.op(0));
+		dens.push_back(n.op(1));
 		it++;
 	}
-	ex n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
+	ex n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
 	nums.push_back(n.op(0));
 	dens.push_back(n.op(1));
 	GINAC_ASSERT(nums.size() == dens.size());
 
 	// Now, nums is a vector of all numerators and dens is a vector of
 	// all denominators
+//std::clog << "add::normal uses " << nums.size() << " summands:\n";
 
 	// Add fractions sequentially
 	exvector::const_iterator num_it = nums.begin(), num_itend = nums.end();
 	exvector::const_iterator den_it = dens.begin(), den_itend = dens.end();
-//std::clog << "add::normal uses the following summands:\n";
-//std::clog << " num = " << *num_it << ", den = " << *den_it << endl;
+//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
 	ex num = *num_it++, den = *den_it++;
 	while (num_it != num_itend) {
-//std::clog << " num = " << *num_it << ", den = " << *den_it << endl;
+//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
+		ex next_num = *num_it++, next_den = *den_it++;
+
+		// Trivially add sequences of fractions with identical denominators
+		while ((den_it != den_itend) && next_den.is_equal(*den_it)) {
+			next_num += *num_it;
+			num_it++; den_it++;
+		}
+
+		// Additiion of two fractions, taking advantage of the fact that
+		// the heuristic GCD algorithm computes the cofactors at no extra cost
 		ex co_den1, co_den2;
-		ex g = gcd(den, *den_it, &co_den1, &co_den2, false);
-		num = (num * co_den2) + (*num_it * co_den1);
-		den *= co_den2;		// this is the lcm(den, *den_it)
-		num_it++; den_it++;
+		ex g = gcd(den, next_den, &co_den1, &co_den2, false);
+		num = ((num * co_den2) + (next_num * co_den1)).expand();
+		den *= co_den2;		// this is the lcm(den, next_den)
 	}
-//std::clog << " common denominator = " << den << endl;
+//std::clog << " common denominator = " << den << std::endl;
 
 	// Cancel common factors from num/den
 	return frac_cancel(num, den);
@@ -1981,27 +2147,28 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
 ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
 	if (level == 1)
-		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 	else if (level == -max_recursion_level)
 		throw(std::runtime_error("max recursion level reached"));
 
 	// Normalize children, separate into numerator and denominator
-	ex num = _ex1();
-	ex den = _ex1(); 
+	exvector num; num.reserve(seq.size());
+	exvector den; den.reserve(seq.size());
 	ex n;
 	epvector::const_iterator it = seq.begin(), itend = seq.end();
 	while (it != itend) {
-		n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
-		num *= n.op(0);
-		den *= n.op(1);
+		n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+		num.push_back(n.op(0));
+		den.push_back(n.op(1));
 		it++;
 	}
-	n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
-	num *= n.op(0);
-	den *= n.op(1);
+	n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+	num.push_back(n.op(0));
+	den.push_back(n.op(1));
 
 	// Perform fraction cancellation
-	return frac_cancel(num, den);
+	return frac_cancel((new mul(num))->setflag(status_flags::dynallocated),
+	                   (new mul(den))->setflag(status_flags::dynallocated));
 }
 
 
@@ -2012,78 +2179,72 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
 	if (level == 1)
-		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+		return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 	else if (level == -max_recursion_level)
 		throw(std::runtime_error("max recursion level reached"));
 
-	// Normalize basis
-	ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+	// Normalize basis and exponent (exponent gets reassembled)
+	ex n_basis = ex_to<basic>(basis).normal(sym_lst, repl_lst, level-1);
+	ex n_exponent = ex_to<basic>(exponent).normal(sym_lst, repl_lst, level-1);
+	n_exponent = n_exponent.op(0) / n_exponent.op(1);
 
-	if (exponent.info(info_flags::integer)) {
+	if (n_exponent.info(info_flags::integer)) {
 
-		if (exponent.info(info_flags::positive)) {
+		if (n_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);
+			return (new lst(power(n_basis.op(0), n_exponent), power(n_basis.op(1), n_exponent)))->setflag(status_flags::dynallocated);
 
-		} else if (exponent.info(info_flags::negative)) {
+		} else if (n_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);
+			return (new lst(power(n_basis.op(1), -n_exponent), power(n_basis.op(0), -n_exponent)))->setflag(status_flags::dynallocated);
 		}
 
 	} else {
 
-		if (exponent.info(info_flags::positive)) {
+		if (n_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);
+			return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 
-		} else if (exponent.info(info_flags::negative)) {
+		} else if (n_exponent.info(info_flags::negative)) {
 
-			if (n.op(1).is_equal(_ex1())) {
+			if (n_basis.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);
+				return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
 
 			} else {
 
 				// (a/b)^-x -> {sym((b/a)^x), 1}
-				return (new lst(replace_with_symbol(power(n.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+				return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 			}
 
-		} else {	// exponent not numeric
+		} else {	// n_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);
+			return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 		}
 	}
 }
 
 
-/** Implementation of ex::normal() for pseries. It normalizes each coefficient and
- *  replaces the series by a temporary symbol.
+/** Implementation of ex::normal() for pseries. It normalizes each coefficient
+ *  and replaces the series by a temporary symbol.
  *  @see ex::normal */
 ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-	epvector new_seq;
-	new_seq.reserve(seq.size());
-
-	epvector::const_iterator it = seq.begin(), itend = seq.end();
-	while (it != itend) {
-		new_seq.push_back(expair(it->rest.normal(), it->coeff));
-		it++;
-	}
-	ex n = pseries(relational(var,point), new_seq);
-	return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
-}
-
-
-/** Implementation of ex::normal() for relationals. It normalizes both sides.
- *  @see ex::normal */
-ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const
-{
-	return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated);
+	epvector newseq;
+	epvector::const_iterator i = seq.begin(), end = seq.end();
+	while (i != end) {
+		ex restexp = i->rest.normal();
+		if (!restexp.is_zero())
+			newseq.push_back(expair(restexp, i->coeff));
+		++i;
+	}
+	ex n = pseries(relational(var,point), newseq);
+	return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 }
 
 
@@ -2104,7 +2265,7 @@ 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));
+	GINAC_ASSERT(is_a<lst>(e));
 
 	// Re-insert replaced symbols
 	if (sym_lst.nops() > 0)
@@ -2114,9 +2275,9 @@ ex ex::normal(int level) const
 	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.
+/** Get numerator of an expression. If the expression is not of the normal
+ *  form "numerator/denominator", it is first converted to this form and
+ *  then the numerator is returned.
  *
  *  @see ex::normal
  *  @return numerator */
@@ -2125,7 +2286,7 @@ 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));
+	GINAC_ASSERT(is_a<lst>(e));
 
 	// Re-insert replaced symbols
 	if (sym_lst.nops() > 0)
@@ -2134,9 +2295,9 @@ ex ex::numer(void) const
 		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.
+/** Get denominator of an expression. If the expression is not of the normal
+ *  form "numerator/denominator", it is first converted to this form and
+ *  then the denominator is returned.
  *
  *  @see ex::normal
  *  @return denominator */
@@ -2145,7 +2306,7 @@ 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));
+	GINAC_ASSERT(is_a<lst>(e));
 
 	// Re-insert replaced symbols
 	if (sym_lst.nops() > 0)
@@ -2154,10 +2315,41 @@ ex ex::denom(void) const
 		return e.op(1);
 }
 
+/** Get numerator and denominator of an expression. If the expresison is not
+ *  of the normal form "numerator/denominator", it is first converted to this
+ *  form and then a list [numerator, denominator] is returned.
+ *
+ *  @see ex::normal
+ *  @return a list [numerator, denominator] */
+ex ex::numer_denom(void) const
+{
+	lst sym_lst, repl_lst;
+
+	ex e = bp->normal(sym_lst, repl_lst, 0);
+	GINAC_ASSERT(is_a<lst>(e));
+
+	// Re-insert replaced symbols
+	if (sym_lst.nops() > 0)
+		return e.subs(sym_lst, repl_lst);
+	else
+		return e;
+}
+
 
-/** Default implementation of ex::to_rational(). It replaces the object with a
- *  temporary symbol.
- *  @see ex::to_rational */
+/** Rationalization of non-rational functions.
+ *  This function converts a general expression to a rational polynomial
+ *  by replacing all non-rational subexpressions (like non-rational numbers,
+ *  non-integer powers or functions like sin(), cos() etc.) to temporary
+ *  symbols. This makes it possible to use functions like gcd() and divide()
+ *  on non-rational functions by applying to_rational() on the arguments,
+ *  calling the desired function and re-substituting the temporary symbols
+ *  in the result. To make the last step possible, all temporary symbols and
+ *  their associated expressions are collected in the list specified by the
+ *  repl_lst parameter in the form {symbol == expression}, ready to be passed
+ *  as an argument to ex::subs().
+ *
+ *  @param repl_lst collects a list of all temporary symbols and their replacements
+ *  @return rationalized expression */
 ex basic::to_rational(lst &repl_lst) const
 {
 	return replace_with_symbol(*this, repl_lst);
@@ -2165,8 +2357,7 @@ ex basic::to_rational(lst &repl_lst) const
 
 
 /** Implementation of ex::to_rational() for symbols. This returns the
- *  unmodified symbol.
- *  @see ex::to_rational */
+ *  unmodified symbol. */
 ex symbol::to_rational(lst &repl_lst) const
 {
 	return *this;
@@ -2175,8 +2366,7 @@ ex symbol::to_rational(lst &repl_lst) const
 
 /** 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 */
+ *  temporary symbol. */
 ex numeric::to_rational(lst &repl_lst) const
 {
 	if (is_real()) {
@@ -2194,8 +2384,7 @@ ex numeric::to_rational(lst &repl_lst) const
 
 
 /** Implementation of ex::to_rational() for powers. It replaces non-integer
- *  powers by temporary symbols.
- *  @see ex::to_rational */
+ *  powers by temporary symbols. */
 ex power::to_rational(lst &repl_lst) const
 {
 	if (exponent.info(info_flags::integer))
@@ -2205,44 +2394,23 @@ ex power::to_rational(lst &repl_lst) const
 }
 
 
-/** Implementation of ex::to_rational() for expairseqs.
- *  @see ex::to_rational */
+/** Implementation of ex::to_rational() for expairseqs. */
 ex expairseq::to_rational(lst &repl_lst) const
 {
 	epvector s;
 	s.reserve(seq.size());
-	for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) {
-		s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst)));
-		// s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst),
+	epvector::const_iterator i = seq.begin(), end = seq.end();
+	while (i != end) {
+		s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl_lst)));
+		++i;
 	}
 	ex oc = overall_coeff.to_rational(repl_lst);
 	if (oc.info(info_flags::numeric))
 		return thisexpairseq(s, overall_coeff);
-	else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1()));
+	else
+		s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1));
 	return thisexpairseq(s, default_overall_coeff());
 }
 
 
-/** Rationalization of non-rational functions.
- *  This function converts a general expression to a rational polynomial
- *  by replacing all non-rational subexpressions (like non-rational numbers,
- *  non-integer powers or functions like sin(), cos() etc.) to temporary
- *  symbols. This makes it possible to use functions like gcd() and divide()
- *  on non-rational functions by applying to_rational() on the arguments,
- *  calling the desired function and re-substituting the temporary symbols
- *  in the result. To make the last step possible, all temporary symbols and
- *  their associated expressions are collected in the list specified by the
- *  repl_lst parameter in the form {symbol == expression}, ready to be passed
- *  as an argument to ex::subs().
- *
- *  @param repl_lst collects a list of all temporary symbols and their replacements
- *  @return rationalized expression */
-ex ex::to_rational(lst &repl_lst) const
-{
-	return bp->to_rational(repl_lst);
-}
-
-
-#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC