X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns.cpp;h=f64597e5c3cb50031b57e577b7c221f060d4d999;hp=e0d1e91edcaa0520ce5f94c017dff93e3e67a8a2;hb=aa6281216091efd92dc5fcc3f96c7189114e80f1;hpb=fe9dbfb9947b24149b3ce7dd9285f27ab286cbd7

diff --git a/ginac/inifcns.cpp b/ginac/inifcns.cpp
index e0d1e91e..f64597e5 100644
--- a/ginac/inifcns.cpp
+++ b/ginac/inifcns.cpp
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's initially known functions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2001 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
@@ -29,37 +29,33 @@
 #include "lst.h"
 #include "matrix.h"
 #include "mul.h"
-#include "ncmul.h"
-#include "numeric.h"
 #include "power.h"
 #include "relational.h"
 #include "pseries.h"
 #include "symbol.h"
 #include "utils.h"
 
-#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
 
 //////////
 // absolute value
 //////////
 
-static ex abs_evalf(const ex & x)
+static ex abs_evalf(const ex & arg)
 {
-    BEGIN_TYPECHECK
-        TYPECHECK(x,numeric)
-    END_TYPECHECK(abs(x))
-    
-    return abs(ex_to_numeric(x));
+	BEGIN_TYPECHECK
+		TYPECHECK(arg,numeric)
+	END_TYPECHECK(abs(arg))
+	
+	return abs(ex_to<numeric>(arg));
 }
 
-static ex abs_eval(const ex & x)
+static ex abs_eval(const ex & arg)
 {
-    if (is_ex_exactly_of_type(x, numeric))
-        return abs(ex_to_numeric(x));
-    else
-        return abs(x).hold();
+	if (is_ex_exactly_of_type(arg, numeric))
+		return abs(ex_to<numeric>(arg));
+	else
+		return abs(arg).hold();
 }
 
 REGISTER_FUNCTION(abs, eval_func(abs_eval).
@@ -70,41 +66,42 @@ REGISTER_FUNCTION(abs, eval_func(abs_eval).
 // Complex sign
 //////////
 
-static ex csgn_evalf(const ex & x)
+static ex csgn_evalf(const ex & arg)
 {
-    BEGIN_TYPECHECK
-        TYPECHECK(x,numeric)
-    END_TYPECHECK(csgn(x))
-    
-    return csgn(ex_to_numeric(x));
+	BEGIN_TYPECHECK
+		TYPECHECK(arg,numeric)
+	END_TYPECHECK(csgn(arg))
+	
+	return csgn(ex_to<numeric>(arg));
 }
 
-static ex csgn_eval(const ex & x)
+static ex csgn_eval(const ex & arg)
 {
-    if (is_ex_exactly_of_type(x, numeric))
-        return csgn(ex_to_numeric(x));
-    
-    else if (is_ex_exactly_of_type(x, mul)) {
-        numeric oc = ex_to_numeric(x.op(x.nops()-1));
-        if (oc.is_real()) {
-            if (oc > 0)
-                // csgn(42*x) -> csgn(x)
-                return csgn(x/oc).hold();
-            else
-                // csgn(-42*x) -> -csgn(x)
-                return -csgn(x/oc).hold();
-        }
-        if (oc.real().is_zero()) {
-            if (oc.imag() > 0)
-                // csgn(42*I*x) -> csgn(I*x)
-                return csgn(I*x/oc).hold();
-            else
-                // csgn(-42*I*x) -> -csgn(I*x)
-                return -csgn(I*x/oc).hold();
-        }
+	if (is_ex_exactly_of_type(arg, numeric))
+		return csgn(ex_to<numeric>(arg));
+	
+	else if (is_ex_of_type(arg, mul) &&
+	         is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
+		numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
+		if (oc.is_real()) {
+			if (oc > 0)
+				// csgn(42*x) -> csgn(x)
+				return csgn(arg/oc).hold();
+			else
+				// csgn(-42*x) -> -csgn(x)
+				return -csgn(arg/oc).hold();
+		}
+		if (oc.real().is_zero()) {
+			if (oc.imag() > 0)
+				// csgn(42*I*x) -> csgn(I*x)
+				return csgn(I*arg/oc).hold();
+			else
+				// csgn(-42*I*x) -> -csgn(I*x)
+				return -csgn(I*arg/oc).hold();
+		}
 	}
-   
-    return csgn(x).hold();
+	
+	return csgn(arg).hold();
 }
 
 static ex csgn_series(const ex & arg,
@@ -112,159 +109,214 @@ static ex csgn_series(const ex & arg,
                       int order,
                       unsigned options)
 {
-    const ex arg_pt = arg.subs(rel);
-    if (arg_pt.info(info_flags::numeric)) {
-        if (ex_to_numeric(arg_pt).real().is_zero())
-            throw (std::domain_error("csgn_series(): on imaginary axis"));
-        epvector seq;
-        seq.push_back(expair(csgn(arg_pt), _ex0()));
-        return pseries(rel,seq);
-    }
-    epvector seq;
-    seq.push_back(expair(csgn(arg_pt), _ex0()));
-    return pseries(rel,seq);
+	const ex arg_pt = arg.subs(rel);
+	if (arg_pt.info(info_flags::numeric)
+	    && ex_to<numeric>(arg_pt).real().is_zero()
+	    && !(options & series_options::suppress_branchcut))
+		throw (std::domain_error("csgn_series(): on imaginary axis"));
+	
+	epvector seq;
+	seq.push_back(expair(csgn(arg_pt), _ex0()));
+	return pseries(rel,seq);
 }
 
 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
                         evalf_func(csgn_evalf).
                         series_func(csgn_series));
 
+
+//////////
+// Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
+//////////
+
+static ex eta_evalf(const ex & x, const ex & y)
+{
+	BEGIN_TYPECHECK
+		TYPECHECK(x,numeric)
+		TYPECHECK(y,numeric)
+	END_TYPECHECK(eta(x,y))
+		
+	numeric xim = imag(ex_to<numeric>(x));
+	numeric yim = imag(ex_to<numeric>(y));
+	numeric xyim = imag(ex_to<numeric>(x*y));
+	return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+}
+
+static ex eta_eval(const ex & x, const ex & y)
+{
+	if (is_ex_exactly_of_type(x, numeric) &&
+		is_ex_exactly_of_type(y, numeric)) {
+		// don't call eta_evalf here because it would call Pi.evalf()!
+		numeric xim = imag(ex_to<numeric>(x));
+		numeric yim = imag(ex_to<numeric>(y));
+		numeric xyim = imag(ex_to<numeric>(x*y));
+		return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+	}
+	
+	return eta(x,y).hold();
+}
+
+static ex eta_series(const ex & arg1,
+                     const ex & arg2,
+                     const relational & rel,
+                     int order,
+                     unsigned options)
+{
+	const ex arg1_pt = arg1.subs(rel);
+	const ex arg2_pt = arg2.subs(rel);
+	if (ex_to<numeric>(arg1_pt).imag().is_zero() ||
+		ex_to<numeric>(arg2_pt).imag().is_zero() ||
+		ex_to<numeric>(arg1_pt*arg2_pt).imag().is_zero()) {
+		throw (std::domain_error("eta_series(): on discontinuity"));
+	}
+	epvector seq;
+	seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
+	return pseries(rel,seq);
+}
+
+REGISTER_FUNCTION(eta, eval_func(eta_eval).
+                       evalf_func(eta_evalf).
+                       series_func(eta_series).
+                       latex_name("\\eta"));
+
+
 //////////
 // dilogarithm
 //////////
 
 static ex Li2_evalf(const ex & x)
 {
-    BEGIN_TYPECHECK
-        TYPECHECK(x,numeric)
-    END_TYPECHECK(Li2(x))
-    
-    return Li2(ex_to_numeric(x));  // -> numeric Li2(numeric)
+	BEGIN_TYPECHECK
+		TYPECHECK(x,numeric)
+	END_TYPECHECK(Li2(x))
+	
+	return Li2(ex_to<numeric>(x));  // -> numeric Li2(numeric)
 }
 
 static ex Li2_eval(const ex & x)
 {
-    if (x.info(info_flags::numeric)) {
-        // Li2(0) -> 0
-        if (x.is_zero())
-            return _ex0();
-        // Li2(1) -> Pi^2/6
-        if (x.is_equal(_ex1()))
-            return power(Pi,_ex2())/_ex6();
-        // Li2(1/2) -> Pi^2/12 - log(2)^2/2
-        if (x.is_equal(_ex1_2()))
-            return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
-        // Li2(-1) -> -Pi^2/12
-        if (x.is_equal(_ex_1()))
-            return -power(Pi,_ex2())/_ex12();
-        // Li2(I) -> -Pi^2/48+Catalan*I
-        if (x.is_equal(I))
-            return power(Pi,_ex2())/_ex_48() + Catalan*I;
-        // Li2(-I) -> -Pi^2/48-Catalan*I
-        if (x.is_equal(-I))
-            return power(Pi,_ex2())/_ex_48() - Catalan*I;
-        // Li2(float)
-        if (!x.info(info_flags::crational))
-            return Li2_evalf(x);
-    }
-    
-    return Li2(x).hold();
+	if (x.info(info_flags::numeric)) {
+		// Li2(0) -> 0
+		if (x.is_zero())
+			return _ex0();
+		// Li2(1) -> Pi^2/6
+		if (x.is_equal(_ex1()))
+			return power(Pi,_ex2())/_ex6();
+		// Li2(1/2) -> Pi^2/12 - log(2)^2/2
+		if (x.is_equal(_ex1_2()))
+			return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
+		// Li2(-1) -> -Pi^2/12
+		if (x.is_equal(_ex_1()))
+			return -power(Pi,_ex2())/_ex12();
+		// Li2(I) -> -Pi^2/48+Catalan*I
+		if (x.is_equal(I))
+			return power(Pi,_ex2())/_ex_48() + Catalan*I;
+		// Li2(-I) -> -Pi^2/48-Catalan*I
+		if (x.is_equal(-I))
+			return power(Pi,_ex2())/_ex_48() - Catalan*I;
+		// Li2(float)
+		if (!x.info(info_flags::crational))
+			return Li2_evalf(x);
+	}
+	
+	return Li2(x).hold();
 }
 
 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 {
-    GINAC_ASSERT(deriv_param==0);
-    
-    // d/dx Li2(x) -> -log(1-x)/x
-    return -log(1-x)/x;
+	GINAC_ASSERT(deriv_param==0);
+	
+	// d/dx Li2(x) -> -log(1-x)/x
+	return -log(1-x)/x;
 }
 
 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 {
-    const ex x_pt = x.subs(rel);
-    if (x_pt.info(info_flags::numeric)) {
-        // First special case: x==0 (derivatives have poles)
-        if (x_pt.is_zero()) {
-            // method:
-            // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
-            // simply substitute x==0.  The limit, however, exists: it is 1.
-            // We also know all higher derivatives' limits:
-            // (d/dx)^n Li2(x) == n!/n^2.
-            // So the primitive series expansion is
-            // Li2(x==0) == x + x^2/4 + x^3/9 + ...
-            // and so on.
-            // We first construct such a primitive series expansion manually in
-            // a dummy symbol s and then insert the argument's series expansion
-            // for s.  Reexpanding the resulting series returns the desired
-            // result.
-            const symbol s;
-            ex ser;
-            // manually construct the primitive expansion
-            for (int i=1; i<order; ++i)
-                ser += pow(s,i) / pow(numeric(i), _num2());
-            // substitute the argument's series expansion
-            ser = ser.subs(s==x.series(rel, order));
-            // maybe that was terminating, so add a proper order term
-            epvector nseq;
-            nseq.push_back(expair(Order(_ex1()), order));
-            ser += pseries(rel, nseq);
-            // reexpanding it will collapse the series again
-            return ser.series(rel, order);
-            // NB: Of course, this still does not allow us to compute anything
-            // like sin(Li2(x)).series(x==0,2), since then this code here is
-            // not reached and the derivative of sin(Li2(x)) doesn't allow the
-            // substitution x==0.  Probably limits *are* needed for the general
-            // cases.  In case L'Hospital's rule is implemented for limits and
-            // basic::series() takes care of this, this whole block is probably
-            // obsolete!
-        }
-        // second special case: x==1 (branch point)
-        if (x_pt == _ex1()) {
-            // method:
-            // construct series manually in a dummy symbol s
-            const symbol s;
-            ex ser = zeta(2);
-            // manually construct the primitive expansion
-            for (int i=1; i<order; ++i)
-                ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
-            // substitute the argument's series expansion
-            ser = ser.subs(s==x.series(rel, order));
-            // maybe that was terminating, so add a proper order term
-            epvector nseq;
-            nseq.push_back(expair(Order(_ex1()), order));
-            ser += pseries(rel, nseq);
-            // reexpanding it will collapse the series again
-            return ser.series(rel, order);
-        }
-        // third special case: x real, >=1 (branch cut)
-        if (!(options & series_options::suppress_branchcut) &&
-            ex_to_numeric(x_pt).is_real() && ex_to_numeric(x_pt)>1) {
-            // method:
-            // This is the branch cut: assemble the primitive series manually
-            // and then add the corresponding complex step function.
-            const symbol *s = static_cast<symbol *>(rel.lhs().bp);
-            const ex point = rel.rhs();
-            const symbol foo;
-            epvector seq;
-            // zeroth order term:
-            seq.push_back(expair(Li2(x_pt), _ex0()));
-            // compute the intermediate terms:
-            ex replarg = series(Li2(x), *s==foo, order);
-            for (unsigned i=1; i<replarg.nops()-1; ++i)
-                seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
-            // append an order term:
-            seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
-            return pseries(rel, seq);
-        }
-    }
-    // all other cases should be safe, by now:
-    throw do_taylor();  // caught by function::series()
+	const ex x_pt = x.subs(rel);
+	if (x_pt.info(info_flags::numeric)) {
+		// First special case: x==0 (derivatives have poles)
+		if (x_pt.is_zero()) {
+			// method:
+			// The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
+			// simply substitute x==0.  The limit, however, exists: it is 1.
+			// We also know all higher derivatives' limits:
+			// (d/dx)^n Li2(x) == n!/n^2.
+			// So the primitive series expansion is
+			// Li2(x==0) == x + x^2/4 + x^3/9 + ...
+			// and so on.
+			// We first construct such a primitive series expansion manually in
+			// a dummy symbol s and then insert the argument's series expansion
+			// for s.  Reexpanding the resulting series returns the desired
+			// result.
+			const symbol s;
+			ex ser;
+			// manually construct the primitive expansion
+			for (int i=1; i<order; ++i)
+				ser += pow(s,i) / pow(numeric(i), _num2());
+			// substitute the argument's series expansion
+			ser = ser.subs(s==x.series(rel, order));
+			// maybe that was terminating, so add a proper order term
+			epvector nseq;
+			nseq.push_back(expair(Order(_ex1()), order));
+			ser += pseries(rel, nseq);
+			// reexpanding it will collapse the series again
+			return ser.series(rel, order);
+			// NB: Of course, this still does not allow us to compute anything
+			// like sin(Li2(x)).series(x==0,2), since then this code here is
+			// not reached and the derivative of sin(Li2(x)) doesn't allow the
+			// substitution x==0.  Probably limits *are* needed for the general
+			// cases.  In case L'Hospital's rule is implemented for limits and
+			// basic::series() takes care of this, this whole block is probably
+			// obsolete!
+		}
+		// second special case: x==1 (branch point)
+		if (x_pt == _ex1()) {
+			// method:
+			// construct series manually in a dummy symbol s
+			const symbol s;
+			ex ser = zeta(_ex2());
+			// manually construct the primitive expansion
+			for (int i=1; i<order; ++i)
+				ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
+			// substitute the argument's series expansion
+			ser = ser.subs(s==x.series(rel, order));
+			// maybe that was terminating, so add a proper order term
+			epvector nseq;
+			nseq.push_back(expair(Order(_ex1()), order));
+			ser += pseries(rel, nseq);
+			// reexpanding it will collapse the series again
+			return ser.series(rel, order);
+		}
+		// third special case: x real, >=1 (branch cut)
+		if (!(options & series_options::suppress_branchcut) &&
+			ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
+			// method:
+			// This is the branch cut: assemble the primitive series manually
+			// and then add the corresponding complex step function.
+			const symbol *s = static_cast<symbol *>(rel.lhs().bp);
+			const ex point = rel.rhs();
+			const symbol foo;
+			epvector seq;
+			// zeroth order term:
+			seq.push_back(expair(Li2(x_pt), _ex0()));
+			// compute the intermediate terms:
+			ex replarg = series(Li2(x), *s==foo, order);
+			for (unsigned i=1; i<replarg.nops()-1; ++i)
+				seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
+			// append an order term:
+			seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
+			return pseries(rel, seq);
+		}
+	}
+	// all other cases should be safe, by now:
+	throw do_taylor();  // caught by function::series()
 }
 
 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
                        evalf_func(Li2_evalf).
                        derivative_func(Li2_deriv).
-                       series_func(Li2_series));
+                       series_func(Li2_series).
+                       latex_name("\\mbox{Li}_2"));
 
 //////////
 // trilogarithm
@@ -272,12 +324,13 @@ REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 
 static ex Li3_eval(const ex & x)
 {
-    if (x.is_zero())
-        return x;
-    return Li3(x).hold();
+	if (x.is_zero())
+		return x;
+	return Li3(x).hold();
 }
 
-REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
+REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
+                       latex_name("\\mbox{Li}_3"));
 
 //////////
 // factorial
@@ -285,15 +338,15 @@ REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
 
 static ex factorial_evalf(const ex & x)
 {
-    return factorial(x).hold();
+	return factorial(x).hold();
 }
 
 static ex factorial_eval(const ex & x)
 {
-    if (is_ex_exactly_of_type(x, numeric))
-        return factorial(ex_to_numeric(x));
-    else
-        return factorial(x).hold();
+	if (is_ex_exactly_of_type(x, numeric))
+		return factorial(ex_to<numeric>(x));
+	else
+		return factorial(x).hold();
 }
 
 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
@@ -305,15 +358,15 @@ REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 
 static ex binomial_evalf(const ex & x, const ex & y)
 {
-    return binomial(x, y).hold();
+	return binomial(x, y).hold();
 }
 
 static ex binomial_eval(const ex & x, const ex &y)
 {
-    if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
-        return binomial(ex_to_numeric(x), ex_to_numeric(y));
-    else
-        return binomial(x, y).hold();
+	if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
+		return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
+	else
+		return binomial(x, y).hold();
 }
 
 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
@@ -326,18 +379,16 @@ REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 static ex Order_eval(const ex & x)
 {
 	if (is_ex_exactly_of_type(x, numeric)) {
-
-		// O(c)=O(1)
-		return Order(_ex1()).hold();
-
+		// O(c) -> O(1) or 0
+		if (!x.is_zero())
+			return Order(_ex1()).hold();
+		else
+			return _ex0();
 	} else if (is_ex_exactly_of_type(x, mul)) {
-
-	 	mul *m = static_cast<mul *>(x.bp);
-		if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric)) {
-
-			// O(c*expr)=O(expr)
+		mul *m = static_cast<mul *>(x.bp);
+		// O(c*expr) -> O(expr)
+		if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
 			return Order(x / m->op(m->nops() - 1)).hold();
-		}
 	}
 	return Order(x).hold();
 }
@@ -346,8 +397,8 @@ static ex Order_series(const ex & x, const relational & r, int order, unsigned o
 {
 	// Just wrap the function into a pseries object
 	epvector new_seq;
-    GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
-    const symbol *s = static_cast<symbol *>(r.lhs().bp);
+	GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
+	const symbol *s = static_cast<symbol *>(r.lhs().bp);
 	new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
 	return pseries(r, new_seq);
 }
@@ -355,24 +406,8 @@ static ex Order_series(const ex & x, const relational & r, int order, unsigned o
 // Differentiation is handled in function::derivative because of its special requirements
 
 REGISTER_FUNCTION(Order, eval_func(Order_eval).
-                         series_func(Order_series));
-
-//////////
-// Inert partial differentiation operator
-//////////
-
-static ex Derivative_eval(const ex & f, const ex & l)
-{
-	if (!is_ex_exactly_of_type(f, function)) {
-        throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
-	}
-    if (!is_ex_exactly_of_type(l, lst)) {
-        throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
-    }
-	return Derivative(f, l).hold();
-}
-
-REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
+                         series_func(Order_series).
+                         latex_name("\\mathcal{O}"));
 
 //////////
 // Solve linear system
@@ -380,113 +415,84 @@ REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
 
 ex lsolve(const ex &eqns, const ex &symbols)
 {
-    // solve a system of linear equations
-    if (eqns.info(info_flags::relation_equal)) {
-        if (!symbols.info(info_flags::symbol))
-            throw(std::invalid_argument("lsolve: 2nd argument must be a symbol"));
-        ex sol=lsolve(lst(eqns),lst(symbols));
-        
-        GINAC_ASSERT(sol.nops()==1);
-        GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
-        
-        return sol.op(0).op(1); // return rhs of first solution
-    }
-    
-    // syntax checks
-    if (!eqns.info(info_flags::list)) {
-        throw(std::invalid_argument("lsolve: 1st argument must be a list"));
-    }
-    for (unsigned i=0; i<eqns.nops(); i++) {
-        if (!eqns.op(i).info(info_flags::relation_equal)) {
-            throw(std::invalid_argument("lsolve: 1st argument must be a list of equations"));
-        }
-    }
-    if (!symbols.info(info_flags::list)) {
-        throw(std::invalid_argument("lsolve: 2nd argument must be a list"));
-    }
-    for (unsigned i=0; i<symbols.nops(); i++) {
-        if (!symbols.op(i).info(info_flags::symbol)) {
-            throw(std::invalid_argument("lsolve: 2nd argument must be a list of symbols"));
-        }
-    }
-    
-    // build matrix from equation system
-    matrix sys(eqns.nops(),symbols.nops());
-    matrix rhs(eqns.nops(),1);
-    matrix vars(symbols.nops(),1);
-    
-    for (unsigned r=0; r<eqns.nops(); r++) {
-        ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
-        ex linpart = eq;
-        for (unsigned c=0; c<symbols.nops(); c++) {
-            ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
-            linpart -= co*symbols.op(c);
-            sys.set(r,c,co);
-        }
-        linpart=linpart.expand();
-        rhs.set(r,0,-linpart);
-    }
-    
-    // test if system is linear and fill vars matrix
-    for (unsigned i=0; i<symbols.nops(); i++) {
-        vars.set(i,0,symbols.op(i));
-        if (sys.has(symbols.op(i)))
-            throw(std::logic_error("lsolve: system is not linear"));
-        if (rhs.has(symbols.op(i)))
-            throw(std::logic_error("lsolve: system is not linear"));
-    }
-    
-    //matrix solution=sys.solve(rhs);
-    matrix solution;
-    try {
-        solution = sys.fraction_free_elim(vars,rhs);
-    } catch (const runtime_error & e) {
-        // probably singular matrix (or other error)
-        // return empty solution list
-        // cerr << e.what() << endl;
-        return lst();
-    }
-    
-    // return a list of equations
-    if (solution.cols()!=1) {
-        throw(std::runtime_error("lsolve: strange number of columns returned from matrix::solve"));
-    }
-    if (solution.rows()!=symbols.nops()) {
-        cout << "symbols.nops()=" << symbols.nops() << endl;
-        cout << "solution.rows()=" << solution.rows() << endl;
-        throw(std::runtime_error("lsolve: strange number of rows returned from matrix::solve"));
-    }
-    
-    // return list of the form lst(var1==sol1,var2==sol2,...)
-    lst sollist;
-    for (unsigned i=0; i<symbols.nops(); i++) {
-        sollist.append(symbols.op(i)==solution(i,0));
-    }
-    
-    return sollist;
-}
-
-/** non-commutative power. */
-ex ncpower(const ex &basis, unsigned exponent)
-{
-    if (exponent==0) {
-        return _ex1();
-    }
-
-    exvector v;
-    v.reserve(exponent);
-    for (unsigned i=0; i<exponent; ++i) {
-        v.push_back(basis);
-    }
-
-    return ncmul(v,1);
+	// solve a system of linear equations
+	if (eqns.info(info_flags::relation_equal)) {
+		if (!symbols.info(info_flags::symbol))
+			throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
+		ex sol=lsolve(lst(eqns),lst(symbols));
+		
+		GINAC_ASSERT(sol.nops()==1);
+		GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
+		
+		return sol.op(0).op(1); // return rhs of first solution
+	}
+	
+	// syntax checks
+	if (!eqns.info(info_flags::list)) {
+		throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
+	}
+	for (unsigned i=0; i<eqns.nops(); i++) {
+		if (!eqns.op(i).info(info_flags::relation_equal)) {
+			throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
+		}
+	}
+	if (!symbols.info(info_flags::list)) {
+		throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
+	}
+	for (unsigned i=0; i<symbols.nops(); i++) {
+		if (!symbols.op(i).info(info_flags::symbol)) {
+			throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
+		}
+	}
+	
+	// build matrix from equation system
+	matrix sys(eqns.nops(),symbols.nops());
+	matrix rhs(eqns.nops(),1);
+	matrix vars(symbols.nops(),1);
+	
+	for (unsigned r=0; r<eqns.nops(); r++) {
+		ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+		ex linpart = eq;
+		for (unsigned c=0; c<symbols.nops(); c++) {
+			ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
+			linpart -= co*symbols.op(c);
+			sys(r,c) = co;
+		}
+		linpart = linpart.expand();
+		rhs(r,0) = -linpart;
+	}
+	
+	// test if system is linear and fill vars matrix
+	for (unsigned i=0; i<symbols.nops(); i++) {
+		vars(i,0) = symbols.op(i);
+		if (sys.has(symbols.op(i)))
+			throw(std::logic_error("lsolve: system is not linear"));
+		if (rhs.has(symbols.op(i)))
+			throw(std::logic_error("lsolve: system is not linear"));
+	}
+	
+	matrix solution;
+	try {
+		solution = sys.solve(vars,rhs);
+	} catch (const std::runtime_error & e) {
+		// Probably singular matrix or otherwise overdetermined system:
+		// It is consistent to return an empty list
+		return lst();
+	}    
+	GINAC_ASSERT(solution.cols()==1);
+	GINAC_ASSERT(solution.rows()==symbols.nops());
+	
+	// return list of equations of the form lst(var1==sol1,var2==sol2,...)
+	lst sollist;
+	for (unsigned i=0; i<symbols.nops(); i++)
+		sollist.append(symbols.op(i)==solution(i,0));
+	
+	return sollist;
 }
 
-/** Force inclusion of functions from initcns_gamma and inifcns_zeta
- *  for static lib (so ginsh will see them). */
+/* Force inclusion of functions from inifcns_gamma and inifcns_zeta
+ * for static lib (so ginsh will see them). */
 unsigned force_include_tgamma = function_index_tgamma;
 unsigned force_include_zeta1 = function_index_zeta1;
 
-#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC