- input parser recognizes "sqrt()", which is also used in the output
[ginac.git] / ginac / normal.cpp
index 6d0bdbd5ae67e3288fc99d1cbe222012bafd9f54..a161f82133a7a448ede06ff831868e9bb31e91d2 100644 (file)
@@ -23,7 +23,6 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
-#include <stdexcept>
 #include <algorithm>
 #include <map>
 
 #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(rdeg - bdeg + 1);
        while (rdeg >= bdeg) {
                ex term, rcoeff = r.coeff(x, rdeg);
                if (blcoeff_is_numeric)
                        term = rcoeff / blcoeff;
                else {
                        if (!divide(rcoeff, blcoeff, term, false))
-                               return *new 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(rdeg - bdeg + 1);
        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(rdeg - bdeg + 1);
        while (rdeg >= bdeg) {
                ex term, rcoeff = r.coeff(*x, rdeg);
                if (!divide_in_z(rcoeff, blcoeff, term, var+1))
                        break;
                term = (term * power(*x, rdeg - bdeg)).expand();
-               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);
 }
 
 
@@ -1446,18 +1466,18 @@ 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;
@@ -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;
@@ -1677,7 +1699,7 @@ 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));
+               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"));
        
@@ -1691,70 +1713,172 @@ ex lcm(const ex &a, const ex &b, bool check_args)
  *  Square-free factorization
  */
 
-// Univariate GCD of polynomials in Q[x] (used internally by sqrfree()).
-// a and b can be multivariate polynomials but they are treated as univariate polynomials in x.
-static ex univariate_gcd(const ex &a, const ex &b, const symbol &x)
+/** Compute square-free factorization of multivariate polynomial a(x) using
+ *  Yun´s algorithm.  Used internally by sqrfree().
+ *
+ *  @param a  multivariate polynomial over Z[X], treated here as univariate
+ *            polynomial in x.
+ *  @param x  variable to factor in
+ *  @return   vector of factors sorted in ascending degree */
+static exvector sqrfree_yun(const ex &a, const symbol &x)
+{
+       exvector res;
+       ex w = a;
+       ex z = w.diff(x);
+       ex g = gcd(w, z);
+       if (g.is_equal(_ex1)) {
+               res.push_back(a);
+               return res;
+       }
+       ex y;
+       do {
+               w = quo(w, g, x);
+               y = quo(z, g, x);
+               z = y - w.diff(x);
+               g = gcd(w, z);
+               res.push_back(g);
+       } while (!z.is_zero());
+       return res;
+}
+
+/** Compute square-free factorization of multivariate polynomial in Q[X].
+ *
+ *  @param a  multivariate polynomial over Q[X]
+ *  @param x  lst of variables to factor in, may be left empty for autodetection
+ *  @return   polynomial a in square-free factored form. */
+ex sqrfree(const ex &a, const lst &l)
 {
-       if (a.is_zero())
-               return b;
-       if (b.is_zero())
+       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 +1894,7 @@ 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 +1920,7 @@ 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 +1938,31 @@ 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 +1970,7 @@ 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);
 }
 
 
@@ -1862,17 +2006,17 @@ 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 << endl;
+//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
 
        // Handle trivial case where denominator is 1
-       if (den.is_equal(_ex1()))
+       if (den.is_equal(_ex1))
                return (new lst(num, den))->setflag(status_flags::dynallocated);
 
        // Handle special cases where numerator or denominator is 0
        if (num.is_zero())
-               return (new lst(num, _ex1()))->setflag(status_flags::dynallocated);
+               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"));
 
@@ -1886,7 +2030,7 @@ 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;
        }
@@ -1895,15 +2039,15 @@ 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);
 }
 
@@ -1914,7 +2058,7 @@ 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"));
 
@@ -1924,12 +2068,12 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
        dens.reserve(seq.size()+1);
        epvector::const_iterator it = seq.begin(), itend = seq.end();
        while (it != itend) {
-               ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
+               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());
@@ -1941,10 +2085,10 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
        // Add fractions sequentially
        exvector::const_iterator num_it = nums.begin(), num_itend = nums.end();
        exvector::const_iterator den_it = dens.begin(), den_itend = dens.end();
-//std::clog << " num = " << *num_it << ", den = " << *den_it << 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
@@ -1960,7 +2104,7 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
                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);
@@ -1973,27 +2117,28 @@ 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));
 }
 
 
@@ -2004,13 +2149,13 @@ 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 and exponent (exponent gets reassembled)
-       ex n_basis = basis.bp->normal(sym_lst, repl_lst, level-1);
-       ex n_exponent = exponent.bp->normal(sym_lst, repl_lst, level-1);
+       ex n_basis = ex_to<basic>(basis).normal(sym_lst, repl_lst, level-1);
+       ex n_exponent = ex_to<basic>(exponent).normal(sym_lst, repl_lst, level-1);
        n_exponent = n_exponent.op(0) / n_exponent.op(1);
 
        if (n_exponent.info(info_flags::integer)) {
@@ -2031,53 +2176,45 @@ ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
                if (n_exponent.info(info_flags::positive)) {
 
                        // (a/b)^x -> {sym((a/b)^x), 1}
-                       return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+                       return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
 
                } else if (n_exponent.info(info_flags::negative)) {
 
-                       if (n_basis.op(1).is_equal(_ex1())) {
+                       if (n_basis.op(1).is_equal(_ex1)) {
 
                                // a^-x -> {1, sym(a^x)}
-                               return (new lst(_ex1(), replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+                               return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
 
                        } else {
 
                                // (a/b)^-x -> {sym((b/a)^x), 1}
-                               return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+                               return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
                        }
 
                } else {        // n_exponent not numeric
 
                        // (a/b)^x -> {sym((a/b)^x, 1}
-                       return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+                       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);
 }
 
 
@@ -2098,7 +2235,7 @@ 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)
@@ -2108,9 +2245,9 @@ 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 */
@@ -2119,7 +2256,7 @@ 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)
@@ -2128,9 +2265,9 @@ 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 */
@@ -2139,7 +2276,7 @@ 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)
@@ -2148,10 +2285,41 @@ 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));
 
-/** Default implementation of ex::to_rational(). It replaces the object with a
- *  temporary symbol.
- *  @see ex::to_rational */
+       // Re-insert replaced symbols
+       if (sym_lst.nops() > 0)
+               return e.subs(sym_lst, repl_lst);
+       else
+               return e;
+}
+
+
+/** Rationalization of non-rational functions.
+ *  This function converts a general expression to a rational polynomial
+ *  by replacing all non-rational subexpressions (like non-rational numbers,
+ *  non-integer powers or functions like sin(), cos() etc.) to temporary
+ *  symbols. This makes it possible to use functions like gcd() and divide()
+ *  on non-rational functions by applying to_rational() on the arguments,
+ *  calling the desired function and re-substituting the temporary symbols
+ *  in the result. To make the last step possible, all temporary symbols and
+ *  their associated expressions are collected in the list specified by the
+ *  repl_lst parameter in the form {symbol == expression}, ready to be passed
+ *  as an argument to ex::subs().
+ *
+ *  @param repl_lst collects a list of all temporary symbols and their replacements
+ *  @return rationalized expression */
 ex basic::to_rational(lst &repl_lst) const
 {
        return replace_with_symbol(*this, repl_lst);
@@ -2159,8 +2327,7 @@ 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;
@@ -2169,8 +2336,7 @@ 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()) {
@@ -2188,8 +2354,7 @@ 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))
@@ -2199,44 +2364,23 @@ 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