X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?a=blobdiff_plain;ds=sidebyside;f=ginac%2Fnormal.cpp;h=a5ebeb8b691ee4b6d95355fff45a17d68d87a414;hb=83c1a0ba18b918a02fea05272d6e9e514880b707;hp=d6661d99c611959367caeb8a19c488215d613898;hpb=955cb185a85535ab328ffedbfccdc508ce80fa91;p=ginac.git

diff --git a/ginac/normal.cpp b/ginac/normal.cpp
index d6661d99..a5ebeb8b 100644
--- a/ginac/normal.cpp
+++ b/ginac/normal.cpp
@@ -7,7 +7,7 @@
  */
 
 /*
- *  GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
@@ -43,13 +43,13 @@
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"
-#include "series.h"
+#include "pseries.h"
 #include "symbol.h"
 #include "utils.h"
 
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif // ndef NO_GINAC_NAMESPACE
+#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
@@ -74,7 +74,7 @@ static bool get_first_symbol(const ex &e, const symbol *&x)
         x = static_cast<symbol *>(e.bp);
         return true;
     } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
-        for (int i=0; i<e.nops(); i++)
+        for (unsigned i=0; i<e.nops(); i++)
             if (get_first_symbol(e.op(i), x))
                 return true;
     } else if (is_ex_exactly_of_type(e, power)) {
@@ -141,7 +141,7 @@ static void collect_symbols(const ex &e, sym_desc_vec &v)
     if (is_ex_exactly_of_type(e, symbol)) {
         add_symbol(static_cast<symbol *>(e.bp), v);
     } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
-        for (int i=0; i<e.nops(); i++)
+        for (unsigned i=0; i<e.nops(); i++)
             collect_symbols(e.op(i), v);
     } else if (is_ex_exactly_of_type(e, power)) {
         collect_symbols(e.op(0), v);
@@ -190,28 +190,61 @@ static numeric lcmcoeff(const ex &e, const numeric &l)
 {
     if (e.info(info_flags::rational))
         return lcm(ex_to_numeric(e).denom(), l);
-    else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
+    else if (is_ex_exactly_of_type(e, add)) {
         numeric c = _num1();
-        for (int i=0; i<e.nops(); i++) {
+        for (unsigned i=0; i<e.nops(); i++)
             c = lcmcoeff(e.op(i), c);
-        }
+        return lcm(c, l);
+    } else if (is_ex_exactly_of_type(e, mul)) {
+        numeric c = _num1();
+        for (unsigned i=0; i<e.nops(); i++)
+            c *= lcmcoeff(e.op(i), _num1());
         return lcm(c, l);
     } else if (is_ex_exactly_of_type(e, power))
-        return lcmcoeff(e.op(0), l);
+        return pow(lcmcoeff(e.op(0), l), ex_to_numeric(e.op(1)));
     return l;
 }
 
 /** Compute LCM of denominators of coefficients of a polynomial.
  *  Given a polynomial with rational coefficients, this function computes
  *  the LCM of the denominators of all coefficients. This can be used
- *  To bring a polynomial from Q[X] to Z[X].
+ *  to bring a polynomial from Q[X] to Z[X].
  *
- *  @param e  multivariate polynomial
+ *  @param e  multivariate polynomial (need not be expanded)
  *  @return LCM of denominators of coefficients */
 
 static numeric lcm_of_coefficients_denominators(const ex &e)
 {
-    return lcmcoeff(e.expand(), _num1());
+    return lcmcoeff(e, _num1());
+}
+
+/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
+ *  determined LCM of the coefficient's denominators.
+ *
+ *  @param e  multivariate polynomial (need not be expanded)
+ *  @param lcm  LCM to multiply in */
+
+static ex multiply_lcm(const ex &e, const numeric &lcm)
+{
+	if (is_ex_exactly_of_type(e, mul)) {
+		ex c = _ex1();
+		numeric lcm_accum = _num1();
+		for (unsigned i=0; i<e.nops(); i++) {
+			numeric op_lcm = lcmcoeff(e.op(i), _num1());
+			c *= multiply_lcm(e.op(i), op_lcm);
+			lcm_accum *= op_lcm;
+		}
+		c *= lcm / lcm_accum;
+		return c;
+	} else if (is_ex_exactly_of_type(e, add)) {
+		ex c = _ex0();
+		for (unsigned i=0; i<e.nops(); i++)
+			c += multiply_lcm(e.op(i), lcm);
+		return c;
+	} else if (is_ex_exactly_of_type(e, power)) {
+		return pow(multiply_lcm(e.op(0), lcm.power(ex_to_numeric(e.op(1)).inverse())), e.op(1));
+	} else
+		return e * lcm;
 }
 
 
@@ -911,11 +944,11 @@ ex basic::smod(const numeric &xi) const
 
 ex numeric::smod(const numeric &xi) const
 {
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
     return GiNaC::smod(*this, xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
     return ::smod(*this, xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
 }
 
 ex add::smod(const numeric &xi) const
@@ -926,21 +959,21 @@ ex add::smod(const numeric &xi) const
     epvector::const_iterator itend = seq.end();
     while (it != itend) {
         GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
         numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
         numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
         if (!coeff.is_zero())
             newseq.push_back(expair(it->rest, coeff));
         it++;
     }
     GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
     numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
     numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
     return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
 }
 
@@ -956,11 +989,11 @@ ex mul::smod(const numeric &xi) const
 #endif // def DO_GINAC_ASSERT
     mul * mulcopyp=new mul(*this);
     GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
     mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC
     mulcopyp->overall_coeff = ::smod(ex_to_numeric(overall_coeff),xi);
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
     mulcopyp->clearflag(status_flags::evaluated);
     mulcopyp->clearflag(status_flags::hash_calculated);
     return mulcopyp->setflag(status_flags::dynallocated);
@@ -1043,7 +1076,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
             if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
                 g *= gc;
                 ex lc = g.lcoeff(*x);
-                if (is_ex_exactly_of_type(lc, numeric) && lc.compare(_ex0()) < 0)
+                if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
                     return -g;
                 else
                     return g;
@@ -1068,6 +1101,47 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 
 ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 {
+//clog << "gcd(" << a << "," << b << ")\n";
+
+	// Partially factored cases (to avoid expanding large expressions)
+	if (is_ex_exactly_of_type(a, mul)) {
+		if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
+			goto factored_b;
+factored_a:
+		ex g = _ex1();
+		ex acc_ca = _ex1();
+		ex part_b = b;
+		for (unsigned i=0; i<a.nops(); i++) {
+			ex part_ca, part_cb;
+			g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
+			acc_ca *= part_ca;
+			part_b = part_cb;
+		}
+		if (ca)
+			*ca = acc_ca;
+		if (cb)
+			*cb = part_b;
+		return g;
+	} else if (is_ex_exactly_of_type(b, mul)) {
+		if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
+			goto factored_a;
+factored_b:
+		ex g = _ex1();
+		ex acc_cb = _ex1();
+		ex part_a = a;
+		for (unsigned i=0; i<b.nops(); i++) {
+			ex part_ca, part_cb;
+			g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
+			acc_cb *= part_cb;
+			part_a = part_ca;
+		}
+		if (ca)
+			*ca = part_a;
+		if (cb)
+			*cb = acc_cb;
+		return g;
+	}
+
     // Some trivial cases
 	ex aex = a.expand(), bex = b.expand();
     if (aex.is_zero()) {
@@ -1155,7 +1229,7 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
         g = *new ex(fail());
     }
     if (is_ex_exactly_of_type(g, fail)) {
-// clog << "heuristics failed" << endl;
+//clog << "heuristics failed" << endl;
         g = sr_gcd(aex, bex, x);
         if (ca)
             divide(aex, g, *ca, false);
@@ -1176,7 +1250,7 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 ex lcm(const ex &a, const ex &b, bool check_args)
 {
     if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
-        return gcd(ex_to_numeric(a), ex_to_numeric(b));
+        return lcm(ex_to_numeric(a), ex_to_numeric(b));
     if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
         throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
     
@@ -1261,13 +1335,22 @@ ex sqrfree(const ex &a, const symbol &x)
  *  Normal form of rational functions
  */
 
-// Create a symbol for replacing the expression "e" (or return a previously
-// assigned symbol). The symbol is appended to sym_list and returned, the
-// expression is appended to repl_list.
+/*
+ *  Note: The internal normal() functions (= basic::normal() and overloaded
+ *  functions) all return lists of the form {numerator, denominator}. This
+ *  is to get around mul::eval()'s automatic expansion of numeric coefficients.
+ *  E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
+ *  the information that (a+b) is the numerator and 3 is the denominator.
+ */
+
+/** Create a symbol for replacing the expression "e" (or return a previously
+ *  assigned symbol). The symbol is appended to sym_list and returned, the
+ *  expression is appended to repl_list.
+ *  @see ex::normal */
 static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
 {
     // Expression already in repl_lst? Then return the assigned symbol
-    for (int i=0; i<repl_lst.nops(); i++)
+    for (unsigned i=0; i<repl_lst.nops(); i++)
         if (repl_lst.op(i).is_equal(e))
             return sym_lst.op(i);
 
@@ -1288,15 +1371,15 @@ static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
  *  @see ex::normal */
 ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    return replace_with_symbol(*this, sym_lst, repl_lst);
+    return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
 }
 
 
-/** Implementation of ex::normal() for symbols. This returns the unmodifies symbol.
+/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
  *  @see ex::normal */
 ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    return *this;
+    return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1306,47 +1389,48 @@ ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal */
 ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    if (is_real())
-        if (is_rational())
-            return *this;
-        else
-            return replace_with_symbol(*this, sym_lst, repl_lst);
-    else { // complex
-        numeric re = real(), im = imag();
+	numeric num = numer();
+	ex numex = num;
+
+    if (num.is_real()) {
+        if (!num.is_integer())
+            numex = replace_with_symbol(numex, sym_lst, repl_lst);
+    } else { // complex
+        numeric re = num.real(), im = num.imag();
         ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
         ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
-        return re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
+        numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
     }
+
+	// Denominator is always a real integer (see numeric::denom())
+	return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
 }
 
 
-/*
- *  Helper function for fraction cancellation (returns cancelled fraction n/d)
- */
+/** Fraction cancellation.
+ *  @param n  numerator
+ *  @param d  denominator
+ *  @return cancelled fraction {n, d} as a list */
 static ex frac_cancel(const ex &n, const ex &d)
 {
     ex num = n;
     ex den = d;
-    ex pre_factor = _ex1();
+    numeric pre_factor = _num1();
+
+//clog << "frac_cancel num = " << num << ", den = " << den << endl;
 
     // Handle special cases where numerator or denominator is 0
     if (num.is_zero())
-        return _ex0();
+		return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
     if (den.expand().is_zero())
         throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
 
-    // More special cases
-    if (is_ex_exactly_of_type(den, numeric))
-        return num / den;
-    if (num.is_zero())
-        return _ex0();
-
     // Bring numerator and denominator to Z[X] by multiplying with
     // LCM of all coefficients' denominators
-    ex num_lcm = lcm_of_coefficients_denominators(num);
-    ex den_lcm = lcm_of_coefficients_denominators(den);
-    num *= num_lcm;
-    den *= den_lcm;
+    numeric num_lcm = lcm_of_coefficients_denominators(num);
+    numeric den_lcm = lcm_of_coefficients_denominators(den);
+	num = multiply_lcm(num, num_lcm);
+	den = multiply_lcm(den, den_lcm);
     pre_factor = den_lcm / num_lcm;
 
     // Cancel GCD from numerator and denominator
@@ -1360,12 +1444,16 @@ static ex frac_cancel(const ex &n, const ex &d)
 	// as defined by get_first_symbol() is made positive)
 	const symbol *x;
 	if (get_first_symbol(den, x)) {
-		if (den.unit(*x).compare(_ex0()) < 0) {
+                GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric));
+		if (ex_to_numeric(den.unit(*x)).is_negative()) {
 			num *= _ex_1();
 			den *= _ex_1();
 		}
 	}
-    return pre_factor * num / den;
+
+	// Return result as list
+//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl;
+    return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1374,47 +1462,70 @@ static ex frac_cancel(const ex &n, const ex &d)
  *  @see ex::normal */
 ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    // Normalize and expand children
+    // Normalize and expand children, chop into summands
     exvector o;
     o.reserve(seq.size()+1);
     epvector::const_iterator it = seq.begin(), itend = seq.end();
     while (it != itend) {
+
+		// Normalize and expand child
         ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
-        if (is_ex_exactly_of_type(n, add)) {
-            epvector::const_iterator bit = (static_cast<add *>(n.bp))->seq.begin(), bitend = (static_cast<add *>(n.bp))->seq.end();
+
+		// If numerator is a sum, chop into summands
+        if (is_ex_exactly_of_type(n.op(0), add)) {
+            epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
             while (bit != bitend) {
-                o.push_back(recombine_pair_to_ex(*bit));
+                o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated));
                 bit++;
             }
-            o.push_back((static_cast<add *>(n.bp))->overall_coeff);
+
+			// The overall_coeff is already normalized (== rational), we just
+			// split it into numerator and denominator
+			GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
+			numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
+            o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated));
         } else
             o.push_back(n);
         it++;
     }
     o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
 
+	// o is now a vector of {numerator, denominator} lists
+
     // Determine common denominator
     ex den = _ex1();
     exvector::const_iterator ait = o.begin(), aitend = o.end();
+//clog << "add::normal uses the following summands:\n";
     while (ait != aitend) {
-        den = lcm((*ait).denom(false), den, false);
+//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl;
+        den = lcm(ait->op(1), den, false);
         ait++;
     }
+//clog << " common denominator = " << den << endl;
 
     // Add fractions
-    if (den.is_equal(_ex1()))
-        return (new add(o))->setflag(status_flags::dynallocated);
-    else {
+    if (den.is_equal(_ex1())) {
+
+		// Common denominator is 1, simply add all numerators
+        exvector num_seq;
+		for (ait=o.begin(); ait!=aitend; ait++) {
+			num_seq.push_back(ait->op(0));
+		}
+		return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated);
+
+	} else {
+
+		// Perform fractional addition
         exvector num_seq;
         for (ait=o.begin(); ait!=aitend; ait++) {
             ex q;
-            if (!divide(den, (*ait).denom(false), q, false)) {
+            if (!divide(den, ait->op(1), q, false)) {
                 // should not happen
                 throw(std::runtime_error("invalid expression in add::normal, division failed"));
             }
-            num_seq.push_back((*ait).numer(false) * q);
+            num_seq.push_back((ait->op(0) * q).expand());
         }
-        ex num = add(num_seq);
+        ex num = (new add(num_seq))->setflag(status_flags::dynallocated);
 
         // Cancel common factors from num/den
         return frac_cancel(num, den);
@@ -1427,17 +1538,23 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal() */
 ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    // Normalize children
-    exvector o;
-    o.reserve(seq.size()+1);
+    // Normalize children, separate into numerator and denominator
+	ex num = _ex1();
+	ex den = _ex1(); 
+	ex n;
     epvector::const_iterator it = seq.begin(), itend = seq.end();
     while (it != itend) {
-        o.push_back(recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1));
+		n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
+		num *= n.op(0);
+		den *= n.op(1);
         it++;
     }
-    o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
-    ex n = (new mul(o))->setflag(status_flags::dynallocated);
-    return frac_cancel(n.numer(false), n.denom(false));
+	n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
+	num *= n.op(0);
+	den *= n.op(1);
+
+	// Perform fraction cancellation
+    return frac_cancel(num, den);
 }
 
 
@@ -1447,24 +1564,49 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal */
 ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
-    if (exponent.info(info_flags::integer)) {
-        // Integer powers are distributed
-        ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
-        ex num = n.numer(false);
-        ex den = n.denom(false);
-        return power(num, exponent) / power(den, exponent);
-    } else {
-        // Non-integer powers are replaced by temporary symbol (after normalizing basis)
-        ex n = power(basis.bp->normal(sym_lst, repl_lst, level-1), exponent);
-        return replace_with_symbol(n, sym_lst, repl_lst);
+	// Normalize basis
+    ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+
+	if (exponent.info(info_flags::integer)) {
+
+	    if (exponent.info(info_flags::positive)) {
+
+			// (a/b)^n -> {a^n, b^n}
+			return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
+
+		} else if (exponent.info(info_flags::negint)) {
+
+			// (a/b)^-n -> {b^n, a^n}
+			return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
+		}
+
+	} else {
+		if (exponent.info(info_flags::positive)) {
+
+			// (a/b)^z -> {sym((a/b)^z), 1}
+			return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+
+		} else {
+
+			if (n.op(1).is_equal(_ex1())) {
+
+				// a^-x -> {1, sym(a^x)}
+				return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+
+			} else {
+
+				// (a/b)^-x -> {(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);
+			}
+		}
     }
 }
 
 
-/** Implementation of ex::normal() for series. It normalizes each coefficient and
+/** Implementation of ex::normal() for pseries. It normalizes each coefficient and
  *  replaces the series by a temporary symbol.
  *  @see ex::normal */
-ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
     epvector new_seq;
     new_seq.reserve(seq.size());
@@ -1474,9 +1616,8 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
         new_seq.push_back(expair(it->rest.normal(), it->coeff));
         it++;
     }
-
-    ex n = series(var, point, new_seq);
-    return replace_with_symbol(n, sym_lst, repl_lst);
+    ex n = pseries(var, point, new_seq);
+	return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
 }
 
 
@@ -1495,13 +1636,58 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
 ex ex::normal(int level) const
 {
     lst sym_lst, repl_lst;
+
     ex e = bp->normal(sym_lst, repl_lst, level);
+	GINAC_ASSERT(is_ex_of_type(e, lst));
+
+	// Re-insert replaced symbols
     if (sym_lst.nops() > 0)
-        return e.subs(sym_lst, repl_lst);
-    else
-        return e;
+        e = e.subs(sym_lst, repl_lst);
+
+	// Convert {numerator, denominator} form back to fraction
+    return e.op(0) / e.op(1);
+}
+
+/** Numerator of an expression. If the expression is not of the normal form
+ *  "numerator/denominator", it is first converted to this form and then the
+ *  numerator is returned.
+ *
+ *  @see ex::normal
+ *  @return numerator */
+ex ex::numer(void) const
+{
+    lst sym_lst, repl_lst;
+
+    ex e = bp->normal(sym_lst, repl_lst, 0);
+	GINAC_ASSERT(is_ex_of_type(e, lst));
+
+	// Re-insert replaced symbols
+    if (sym_lst.nops() > 0)
+        return e.op(0).subs(sym_lst, repl_lst);
+	else
+		return e.op(0);
+}
+
+/** Denominator of an expression. If the expression is not of the normal form
+ *  "numerator/denominator", it is first converted to this form and then the
+ *  denominator is returned.
+ *
+ *  @see ex::normal
+ *  @return denominator */
+ex ex::denom(void) const
+{
+    lst sym_lst, repl_lst;
+
+    ex e = bp->normal(sym_lst, repl_lst, 0);
+	GINAC_ASSERT(is_ex_of_type(e, lst));
+
+	// Re-insert replaced symbols
+    if (sym_lst.nops() > 0)
+        return e.op(1).subs(sym_lst, repl_lst);
+	else
+		return e.op(1);
 }
 
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC