- normal() now internally keeps numerator and denominator separated and

[ginac.git] / ginac / normal.cpp

diff --git a/ginac/normal.cpp b/ginac/normal.cpp
index 0ee26ad5d0f86365742230fed8a9aa51d6b3b7cf..1c2dc823f7911b76fbd60a5d7fd1ebaa0bd87900 100644 (file)
--- a/ginac/normal.cpp
+++ b/ginac/normal.cpp
@@ -43,13 +43,13 @@
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"

-#include "series.h"
+#include "pseries.h"

 #include "symbol.h"
 #include "utils.h"
 
 #include "symbol.h"
 #include "utils.h"
 

-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC

 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
 
 // 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


@@ -190,27 +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);
 {
     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 (unsigned i=0; i<e.nops(); i++)
             c = lcmcoeff(e.op(i), c);
         return lcm(c, l);
         numeric c = _num1();
         for (unsigned i=0; i<e.nops(); i++)
             c = lcmcoeff(e.op(i), c);
         return lcm(c, l);

+    } else if (is_ex_exactly_of_type(e, mul)) {
+        numeric c = _num1();
+        for (unsigned i=0; i<e.nops(); i++)
+            c *= lcmcoeff(e.op(i), _num1());
+        return lcm(c, l);

     } else if (is_ex_exactly_of_type(e, power))
     } 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
     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 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;

 }
 
 
 }
 
 


@@ -910,11 +944,11 @@ ex basic::smod(const numeric &xi) const
 
 ex numeric::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);
     return GiNaC::smod(*this, xi);

-#else // ndef NO_GINAC_NAMESPACE
+#else // ndef NO_NAMESPACE_GINAC

     return ::smod(*this, xi);
     return ::smod(*this, xi);

-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC

 }
 
 ex add::smod(const numeric &xi) const
 }
 
 ex add::smod(const numeric &xi) const


@@ -925,21 +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));
     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);
         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);
         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));
         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);
     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);
     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);
 }
 
     return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
 }
 


@@ -955,11 +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));
 #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);
     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);
     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);
     mulcopyp->clearflag(status_flags::evaluated);
     mulcopyp->clearflag(status_flags::hash_calculated);
     return mulcopyp->setflag(status_flags::dynallocated);


@@ -1067,6 +1101,45 @@ 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)
 {
 
 ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 {

+       // Partially factored cases (to avoid expanding large expressions)
+       if (is_ex_exactly_of_type(a, mul)) {
+               if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
+                       goto factored_b;
+factored_a:
+               ex g = _ex1();
+               ex acc_ca = _ex1();
+               ex part_b = b;
+               for (unsigned i=0; i<a.nops(); i++) {
+                       ex part_ca, part_cb;
+                       g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
+                       acc_ca *= part_ca;
+                       part_b = part_cb;
+               }
+               if (ca)
+                       *ca = acc_ca;
+               if (cb)
+                       *cb = part_b;
+               return g;
+       } else if (is_ex_exactly_of_type(b, mul)) {
+               if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
+                       goto factored_a;
+factored_b:
+               ex g = _ex1();
+               ex acc_cb = _ex1();
+               ex part_a = a;
+               for (unsigned i=0; i<b.nops(); i++) {
+                       ex part_ca, part_cb;
+                       g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
+                       acc_cb *= part_cb;
+                       part_a = part_ca;
+               }
+               if (ca)
+                       *ca = part_a;
+               if (cb)
+                       *cb = acc_cb;
+               return g;
+       }
+

     // Some trivial cases
        ex aex = a.expand(), bex = b.expand();
     if (aex.is_zero()) {
     // Some trivial cases
        ex aex = a.expand(), bex = b.expand();
     if (aex.is_zero()) {


@@ -1154,7 +1227,7 @@ 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)) {
         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);
         g = sr_gcd(aex, bex, x);
         if (ca)
             divide(aex, g, *ca, false);


@@ -1175,7 +1248,7 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 ex lcm(const ex &a, const ex &b, bool check_args)
 {
     if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
 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"));
     
     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"));
     


@@ -1260,9 +1333,18 @@ ex sqrfree(const ex &a, const symbol &x)
  *  Normal form of rational functions
  */
 
  *  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
 static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
 {
     // Expression already in repl_lst? Then return the assigned symbol


@@ -1287,15 +1369,15 @@ static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
  *  @see ex::normal */
 ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
  *  @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
 {
  *  @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);

 }
 
 
 }
 
 


@@ -1305,47 +1387,48 @@ 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
 {
  *  @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);
         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;
 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())
 
     // 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"));
 
     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
     // 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
     pre_factor = den_lcm / num_lcm;
 
     // Cancel GCD from numerator and denominator


@@ -1364,7 +1447,9 @@ static ex frac_cancel(const ex &n, const ex &d)
                        den *= _ex_1();
                }
        }
                        den *= _ex_1();
                }
        }

-    return pre_factor * num / den;
+
+       // Return result as list
+    return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);

 }
 
 
 }
 
 


@@ -1373,47 +1458,67 @@ static ex frac_cancel(const ex &n, const ex &d)
  *  @see ex::normal */
 ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
  *  @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) {
     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();
         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) {
             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++;
             }
                 bit++;
             }

-            o.push_back((static_cast<add *>(n.bp))->overall_coeff);
+
+                       // The overall_coeff is already normalized (== rational), we just
+                       // split it into numerator and denominator
+                       GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
+                       numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
+            o.push_back((new lst(overall.numer(), overall.denom()))->setflag(status_flags::dynallocated));

         } else
             o.push_back(n);
         it++;
     }
     o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
 
         } 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();
     while (ait != aitend) {
     // Determine common denominator
     ex den = _ex1();
     exvector::const_iterator ait = o.begin(), aitend = o.end();
     while (ait != aitend) {

-        den = lcm((*ait).denom(false), den, false);
+        den = lcm(ait->op(1), den, false);

         ait++;
     }
 
     // Add fractions
         ait++;
     }
 
     // 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;
         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"));
             }
                 // should not happen
                 throw(std::runtime_error("invalid expression in add::normal, division failed"));
             }

-            num_seq.push_back((*ait).numer(false) * q);
+            num_seq.push_back(ait->op(0) * q);

         }
         }

-        ex num = add(num_seq);
+        ex num = (new add(num_seq))->setflag(status_flags::dynallocated);

 
         // Cancel common factors from num/den
         return frac_cancel(num, den);
 
         // Cancel common factors from num/den
         return frac_cancel(num, den);


@@ -1426,17 +1531,23 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal() */
 ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
  *  @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) {
     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++;
     }
         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);

 }
 
 
 }
 
 


@@ -1446,24 +1557,26 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
  *  @see ex::normal */
 ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
 {
  *  @see ex::normal */
 ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
 {

-    if (exponent.info(info_flags::integer)) {
+    if (exponent.info(info_flags::posint)) {
+        // Integer powers are distributed
+        ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+               return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
+       } else if (exponent.info(info_flags::negint)) {

         // Integer powers are distributed
         ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
         // Integer powers are distributed
         ex n = basis.bp->normal(sym_lst, repl_lst, level-1);

-        ex num = n.numer(false);
-        ex den = n.denom(false);
-        return power(num, exponent) / power(den, exponent);
+               return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);

     } else {
         // Non-integer powers are replaced by temporary symbol (after normalizing basis)
     } else {
         // Non-integer powers are replaced by temporary symbol (after normalizing basis)

-        ex n = power(basis.bp->normal(sym_lst, repl_lst, level-1), exponent);
-        return replace_with_symbol(n, sym_lst, repl_lst);
+               ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+               return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);

     }
 }
 
 
     }
 }
 
 

-/** Implementation of ex::normal() for series. It normalizes each coefficient and
+/** Implementation of ex::normal() for pseries. It normalizes each coefficient and

  *  replaces the series by a temporary symbol.
  *  @see ex::normal */
  *  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());
 {
     epvector new_seq;
     new_seq.reserve(seq.size());


@@ -1473,9 +1586,8 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
         new_seq.push_back(expair(it->rest.normal(), it->coeff));
         it++;
     }
         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);

 }
 
 
 }
 
 


@@ -1494,13 +1606,18 @@ ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
 ex ex::normal(int level) const
 {
     lst sym_lst, repl_lst;
 ex ex::normal(int level) const
 {
     lst sym_lst, repl_lst;

+

     ex e = bp->normal(sym_lst, repl_lst, level);
     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)
     if (sym_lst.nops() > 0)

-        return e.subs(sym_lst, repl_lst);
-    else
-        return e;
+        e = e.subs(sym_lst, repl_lst);
+
+       // Convert {numerator, denominator} form back to fraction
+    return e.op(0) / e.op(1);

 }
 
 }
 

-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC

 } // namespace GiNaC
 } // namespace GiNaC

-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC