X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=a016a76bce4fb345dfea3bcd3aeccddd5fc8e68f;hp=f985573a5f6da669dba179ba91049f6b242c6cac;hb=b29bf684f84590c436c768a646b5c905ea530071;hpb=9961dfdd7a383f09b1040e195a8817a85d945e58

diff --git a/ginac/normal.cpp b/ginac/normal.cpp
index f985573a..a016a76b 100644
--- a/ginac/normal.cpp
+++ b/ginac/normal.cpp
@@ -39,6 +39,7 @@
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"
+#include "matrix.h"
 #include "pseries.h"
 #include "symbol.h"
 #include "utils.h"
@@ -228,7 +229,7 @@ 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();
 		for (unsigned i=0; i<e.nops(); i++)
@@ -243,7 +244,7 @@ static numeric lcmcoeff(const ex &e, const numeric &l)
 		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 pow(lcmcoeff(e.op(0), l), ex_to<numeric>(e.op(1)));
 	}
 	return l;
 }
@@ -268,25 +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();
+		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);
+			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)) {
 		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));
+			return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
 	} else
 		return e * lcm;
 }
@@ -321,11 +324,11 @@ numeric add::integer_content(void) const
 	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);
+		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);
+	c = gcd(ex_to<numeric>(overall_coeff),c);
 	return c;
 }
 
@@ -340,7 +343,7 @@ numeric mul::integer_content(void) const
 	}
 #endif // def DO_GINAC_ASSERT
 	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-	return abs(ex_to_numeric(overall_coeff));
+	return abs(ex_to<numeric>(overall_coeff));
 }
 
 
@@ -371,7 +374,6 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
 		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;
@@ -379,6 +381,7 @@ 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)
@@ -388,13 +391,13 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
 				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);
 }
 
 
@@ -415,7 +418,7 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
 		if  (is_ex_exactly_of_type(b, numeric))
 			return _ex0();
 		else
-			return b;
+			return a;
 	}
 #if FAST_COMPARE
 	if (a.is_equal(b))
@@ -450,6 +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)
@@ -510,7 +531,6 @@ 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())
@@ -597,6 +617,7 @@ bool divide(const ex &a, const ex &b, ex &q, 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)
@@ -605,10 +626,12 @@ 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;
@@ -755,14 +778,16 @@ 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
@@ -1204,11 +1229,11 @@ 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));
+	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));
+		a = abs(ex_to<numeric>(it->coeff));
 		if (a > cur_max)
 			cur_max = a;
 		it++;
@@ -1227,7 +1252,7 @@ numeric mul::max_coefficient(void) const
 	}
 #endif // def DO_GINAC_ASSERT
 	GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
-	return abs(ex_to_numeric(overall_coeff));
+	return abs(ex_to<numeric>(overall_coeff));
 }
 
 
@@ -1262,13 +1287,13 @@ 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));
-		numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
+		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));
-	numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
+	numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
 	return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
 }
 
@@ -1284,7 +1309,7 @@ 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));
-	mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
+	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);
@@ -1292,17 +1317,17 @@ 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. */
@@ -1330,17 +1355,17 @@ 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 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;
 	}
 
@@ -1352,7 +1377,7 @@ 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();
@@ -1376,7 +1401,7 @@ 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();
@@ -1386,7 +1411,7 @@ 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;
@@ -1399,7 +1424,7 @@ 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;
@@ -1412,7 +1437,7 @@ 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;
@@ -1445,7 +1470,7 @@ 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)
@@ -1454,9 +1479,9 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 					*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;
@@ -1472,38 +1497,40 @@ 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
@@ -1676,7 +1703,7 @@ 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"));
 	
@@ -1717,6 +1744,7 @@ static exvector sqrfree_yun(const ex &a, const symbol &x)
 	} while (!z.is_zero());
 	return res;
 }
+
 /** Compute square-free factorization of multivariate polynomial in Q[X].
  *
  *  @param a  multivariate polynomial over Q[X]
@@ -1742,7 +1770,7 @@ ex sqrfree(const ex &a, const lst &l)
 	// 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));
+	const symbol x = ex_to<symbol>(args.op(0));
 	// convert the argument from something in Q[X] to something in Z[X]
 	numeric lcm = lcm_of_coefficients_denominators(a);
 	ex tmp = multiply_lcm(a,lcm);
@@ -1769,6 +1797,75 @@ ex sqrfree(const ex &a, const lst &l)
 	return result *	lcm.inverse();
 }
 
+/** Compute square-free partial fraction decomposition of rational function
+ *  a(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)
+{
+	// 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;
+	int 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());
+			}
+		}
+	}
+	int 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 (unsigned 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;
+}
+
 
 /*
  *  Normal form of rational functions
@@ -1782,6 +1879,7 @@ ex sqrfree(const ex &a, const lst &l)
  *  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.
@@ -1825,12 +1923,31 @@ 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);
+		}
+	}
 }
 
 
@@ -1908,7 +2025,7 @@ static ex frac_cancel(const ex &n, const ex &d)
 	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()) {
+		if (ex_to<numeric>(den.unit(*x)).is_negative()) {
 			num *= _ex_1();
 			den *= _ex_1();
 		}
@@ -1990,22 +2107,23 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
 		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);
+		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);
+	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));
 }
 
 
@@ -2083,14 +2201,6 @@ ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
 }
 
 
-/** 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);
-}
-
-
 /** Normalization of rational functions.
  *  This function converts an expression to its normal form
  *  "numerator/denominator", where numerator and denominator are (relatively