X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns.cpp;h=a9ad5ce5223ec84c2e0b0d4e55813d01a8d3a07a;hp=bc401f47266272789b371d9bdced87464ddc3ba7;hb=4d59c02d51fbf50ff24d616b00296aa4cbb1ea5e;hpb=ca04cff6c8e848782a4e123688a6edbb38821884

diff --git a/ginac/inifcns.cpp b/ginac/inifcns.cpp
index bc401f47..a9ad5ce5 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) 1999-2005 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
@@ -17,7 +17,7 @@
  *
  *  You should have received a copy of the GNU General Public License
  *  along with this program; if not, write to the Free Software
- *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+ *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
 #include <vector>
@@ -29,58 +29,381 @@
 #include "lst.h"
 #include "matrix.h"
 #include "mul.h"
-#include "ncmul.h"
-#include "numeric.h"
 #include "power.h"
+#include "operators.h"
 #include "relational.h"
 #include "pseries.h"
 #include "symbol.h"
+#include "symmetry.h"
 #include "utils.h"
 
-#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
+
+//////////
+// complex conjugate
+//////////
+
+static ex conjugate_evalf(const ex & arg)
+{
+	if (is_exactly_a<numeric>(arg)) {
+		return ex_to<numeric>(arg).conjugate();
+	}
+	return conjugate_function(arg).hold();
+}
+
+static ex conjugate_eval(const ex & arg)
+{
+	return arg.conjugate();
+}
+
+static void conjugate_print_latex(const ex & arg, const print_context & c)
+{
+	c.s << "\\bar{"; arg.print(c); c.s << "}";
+}
+
+static ex conjugate_conjugate(const ex & arg)
+{
+	return arg;
+}
+
+REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
+                                      evalf_func(conjugate_evalf).
+                                      print_func<print_latex>(conjugate_print_latex).
+                                      conjugate_func(conjugate_conjugate).
+                                      set_name("conjugate","conjugate"));
 
 //////////
 // 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));
+	if (is_exactly_a<numeric>(arg))
+		return abs(ex_to<numeric>(arg));
+	
+	return abs(arg).hold();
 }
 
-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_exactly_a<numeric>(arg))
+		return abs(ex_to<numeric>(arg));
+	else
+		return abs(arg).hold();
+}
+
+static void abs_print_latex(const ex & arg, const print_context & c)
+{
+	c.s << "{|"; arg.print(c); c.s << "|}";
+}
+
+static void abs_print_csrc_float(const ex & arg, const print_context & c)
+{
+	c.s << "fabs("; arg.print(c); c.s << ")";
+}
+
+static ex abs_conjugate(const ex & arg)
+{
+	return abs(arg);
 }
 
 REGISTER_FUNCTION(abs, eval_func(abs_eval).
-                       evalf_func(abs_evalf));
+                       evalf_func(abs_evalf).
+                       print_func<print_latex>(abs_print_latex).
+                       print_func<print_csrc_float>(abs_print_csrc_float).
+                       print_func<print_csrc_double>(abs_print_csrc_float).
+                       conjugate_func(abs_conjugate));
+
+
+//////////
+// Complex sign
+//////////
+
+static ex csgn_evalf(const ex & arg)
+{
+	if (is_exactly_a<numeric>(arg))
+		return csgn(ex_to<numeric>(arg));
+	
+	return csgn(arg).hold();
+}
+
+static ex csgn_eval(const ex & arg)
+{
+	if (is_exactly_a<numeric>(arg))
+		return csgn(ex_to<numeric>(arg));
+	
+	else if (is_exactly_a<mul>(arg) &&
+	         is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
+		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(arg).hold();
+}
+
+static ex csgn_series(const ex & arg,
+                      const relational & rel,
+                      int order,
+                      unsigned options)
+{
+	const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
+	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);
+}
+
+static ex csgn_conjugate(const ex& arg)
+{
+	return csgn(arg);
+}
+
+REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
+                        evalf_func(csgn_evalf).
+                        series_func(csgn_series).
+                        conjugate_func(csgn_conjugate));
+
+
+//////////
+// Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
+// This function is closely related to the unwinding number K, sometimes found
+// in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
+//////////
+
+static ex eta_evalf(const ex &x, const ex &y)
+{
+	// It seems like we basically have to replicate the eval function here,
+	// since the expression might not be fully evaluated yet.
+	if (x.info(info_flags::positive) || y.info(info_flags::positive))
+		return _ex0;
+
+	if (x.info(info_flags::numeric) &&	y.info(info_flags::numeric)) {
+		const numeric nx = ex_to<numeric>(x);
+		const numeric ny = ex_to<numeric>(y);
+		const numeric nxy = ex_to<numeric>(x*y);
+		int cut = 0;
+		if (nx.is_real() && nx.is_negative())
+			cut -= 4;
+		if (ny.is_real() && ny.is_negative())
+			cut -= 4;
+		if (nxy.is_real() && nxy.is_negative())
+			cut += 4;
+		return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
+		                      (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
+	}
+
+	return eta(x,y).hold();
+}
+
+static ex eta_eval(const ex &x, const ex &y)
+{
+	// trivial:  eta(x,c) -> 0  if c is real and positive
+	if (x.info(info_flags::positive) || y.info(info_flags::positive))
+		return _ex0;
+
+	if (x.info(info_flags::numeric) &&	y.info(info_flags::numeric)) {
+		// don't call eta_evalf here because it would call Pi.evalf()!
+		const numeric nx = ex_to<numeric>(x);
+		const numeric ny = ex_to<numeric>(y);
+		const numeric nxy = ex_to<numeric>(x*y);
+		int cut = 0;
+		if (nx.is_real() && nx.is_negative())
+			cut -= 4;
+		if (ny.is_real() && ny.is_negative())
+			cut -= 4;
+		if (nxy.is_real() && nxy.is_negative())
+			cut += 4;
+		return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
+		                 (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
+	}
+	
+	return eta(x,y).hold();
+}
+
+static ex eta_series(const ex & x, const ex & y,
+                     const relational & rel,
+                     int order,
+                     unsigned options)
+{
+	const ex x_pt = x.subs(rel, subs_options::no_pattern);
+	const ex y_pt = y.subs(rel, subs_options::no_pattern);
+	if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
+	    (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
+	    ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
+			throw (std::domain_error("eta_series(): on discontinuity"));
+	epvector seq;
+	seq.push_back(expair(eta(x_pt,y_pt), _ex0));
+	return pseries(rel,seq);
+}
+
+static ex eta_conjugate(const ex & x, const ex & y)
+{
+	return -eta(x,y);
+}
+
+REGISTER_FUNCTION(eta, eval_func(eta_eval).
+                       evalf_func(eta_evalf).
+                       series_func(eta_series).
+                       latex_name("\\eta").
+                       set_symmetry(sy_symm(0, 1)).
+                       conjugate_func(eta_conjugate));
+
 
 //////////
 // dilogarithm
 //////////
 
+static ex Li2_evalf(const ex & x)
+{
+	if (is_exactly_a<numeric>(x))
+		return Li2(ex_to<numeric>(x));
+	
+	return Li2(x).hold();
+}
+
 static ex Li2_eval(const ex & x)
 {
-    if (x.is_zero())
-        return x;
-    if (x.is_equal(_ex1()))
-        return power(Pi, _ex2()) / _ex6();
-    if (x.is_equal(_ex_1()))
-        return -power(Pi, _ex2()) / _ex12();
-    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(ex_to<numeric>(x));
+	}
+	
+	return Li2(x).hold();
 }
 
-REGISTER_FUNCTION(Li2, eval_func(Li2_eval));
+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(_ex1-x)/x;
+}
+
+static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
+{
+	const ex x_pt = x.subs(rel, subs_options::no_pattern);
+	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_p);
+			// substitute the argument's series expansion
+			ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+			// 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.is_equal(_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), subs_options::no_pattern);
+			// 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 = ex_to<symbol>(rel.lhs());
+			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 (size_t 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, subs_options::no_pattern),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).
+                       latex_name("\\mbox{Li}_2"));
 
 //////////
 // trilogarithm
@@ -88,12 +411,44 @@ 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).
+                       latex_name("\\mbox{Li}_3"));
+
+//////////
+// Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
+//////////
+
+static ex zetaderiv_eval(const ex & n, const ex & x)
+{
+	if (n.info(info_flags::numeric)) {
+		// zetaderiv(0,x) -> zeta(x)
+		if (n.is_zero())
+			return zeta(x);
+	}
+	
+	return zetaderiv(n, x).hold();
+}
+
+static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
+{
+	GINAC_ASSERT(deriv_param<2);
+	
+	if (deriv_param==0) {
+		// d/dn zeta(n,x)
+		throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
+	}
+	// d/dx psi(n,x)
+	return zetaderiv(n+1,x);
 }
 
-REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
+REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
+	                       	 derivative_func(zetaderiv_deriv).
+  	                         latex_name("\\zeta^\\prime"));
 
 //////////
 // factorial
@@ -101,19 +456,38 @@ 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_exactly_a<numeric>(x))
+		return factorial(ex_to<numeric>(x));
+	else
+		return factorial(x).hold();
+}
+
+static void factorial_print_dflt_latex(const ex & x, const print_context & c)
+{
+	if (is_exactly_a<symbol>(x) ||
+	    is_exactly_a<constant>(x) ||
+		is_exactly_a<function>(x)) {
+		x.print(c); c.s << "!";
+	} else {
+		c.s << "("; x.print(c); c.s << ")!";
+	}
+}
+
+static ex factorial_conjugate(const ex & x)
+{
+	return factorial(x);
 }
 
 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
-                             evalf_func(factorial_evalf));
+                             evalf_func(factorial_evalf).
+                             print_func<print_dflt>(factorial_print_dflt_latex).
+                             print_func<print_latex>(factorial_print_dflt_latex).
+                             conjugate_func(factorial_conjugate));
 
 //////////
 // binomial
@@ -121,19 +495,49 @@ 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_sym(const ex & x, const numeric & y)
+{
+	if (y.is_integer()) {
+		if (y.is_nonneg_integer()) {
+			const unsigned N = y.to_int();
+			if (N == 0) return _ex0;
+			if (N == 1) return x;
+			ex t = x.expand();
+			for (unsigned i = 2; i <= N; ++i)
+				t = (t * (x + i - y - 1)).expand() / i;
+			return t;
+		} else
+			return _ex0;
+	}
+
+	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_exactly_a<numeric>(y)) {
+		if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
+			return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
+		else
+			return binomial_sym(x, ex_to<numeric>(y));
+	} else
+		return binomial(x, y).hold();
+}
+
+// At the moment the numeric evaluation of a binomail function always
+// gives a real number, but if this would be implemented using the gamma
+// function, also complex conjugation should be changed (or rather, deleted).
+static ex binomial_conjugate(const ex & x, const ex & y)
+{
+	return binomial(x,y);
 }
 
 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
-                            evalf_func(binomial_evalf));
+                            evalf_func(binomial_evalf).
+                            conjugate_func(binomial_conjugate));
 
 //////////
 // Order term function (for truncated power series)
@@ -141,150 +545,221 @@ 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();
-
-	} 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)
-			return Order(x / m->op(m->nops() - 1)).hold();
-		}
+	if (is_exactly_a<numeric>(x)) {
+		// O(c) -> O(1) or 0
+		if (!x.is_zero())
+			return Order(_ex1).hold();
+		else
+			return _ex0;
+	} else if (is_exactly_a<mul>(x)) {
+		const mul &m = ex_to<mul>(x);
+		// O(c*expr) -> O(expr)
+		if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
+			return Order(x / m.op(m.nops() - 1)).hold();
 	}
 	return Order(x).hold();
 }
 
-static ex Order_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 {
 	// Just wrap the function into a pseries object
 	epvector new_seq;
-	new_seq.push_back(expair(Order(_ex1()), numeric(min(x.ldegree(s), order))));
-	return pseries(s, point, new_seq);
+	GINAC_ASSERT(is_a<symbol>(r.lhs()));
+	const symbol &s = ex_to<symbol>(r.lhs());
+	new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
+	return pseries(r, new_seq);
+}
+
+static ex Order_conjugate(const ex & x)
+{
+	return Order(x);
 }
 
+// Differentiation is handled in function::derivative because of its special requirements
+
 REGISTER_FUNCTION(Order, eval_func(Order_eval).
-                         series_func(Order_series));
+                         series_func(Order_series).
+                         latex_name("\\mathcal{O}").
+                         conjugate_func(Order_conjugate));
 
 //////////
 // Solve linear system
 //////////
 
-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);
-}
-
-/** Force inclusion of functions from initcns_gamma and inifcns_zeta
- *  for static lib (so ginsh will see them). */
-unsigned force_include_gamma = function_index_gamma;
-unsigned force_include_zeta1 = function_index_zeta1;
-
-#ifndef NO_NAMESPACE_GINAC
+ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
+{
+	// 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"));
+		const ex sol = lsolve(lst(eqns),lst(symbols));
+		
+		GINAC_ASSERT(sol.nops()==1);
+		GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
+		
+		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 (size_t 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 (size_t 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 (size_t r=0; r<eqns.nops(); r++) {
+		const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+		ex linpart = eq;
+		for (size_t c=0; c<symbols.nops(); c++) {
+			const 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 (size_t 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,options);
+	} 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 (size_t i=0; i<symbols.nops(); i++)
+		sollist.append(symbols.op(i)==solution(i,0));
+	
+	return sollist;
+}
+
+//////////
+// Find real root of f(x) numerically
+//////////
+
+const numeric
+fsolve(const ex& f, const symbol& x, const numeric& x1, const numeric& x2)
+{
+	if (!x1.is_real() || !x2.is_real()) {
+		throw std::runtime_error("fsolve(): interval not bounded by real numbers");
+	}
+	if (x1==x2) {
+		throw std::runtime_error("fsolve(): vanishing interval");
+	}
+	// xx[0] == left interval limit, xx[1] == right interval limit.
+	// fx[0] == f(xx[0]), fx[1] == f(xx[1]).
+	// We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
+	numeric xx[2] = { x1<x2 ? x1 : x2,
+	                  x1<x2 ? x2 : x1 };
+	const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
+	                    f.subs(x==xx[1]).evalf() };
+	if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
+		throw std::runtime_error("fsolve(): function does not evaluate numerically");
+	}
+	numeric fx[2] = { ex_to<numeric>(fx_[0]),
+	                  ex_to<numeric>(fx_[1]) };
+	if (!fx[0].is_real() || !fx[1].is_real()) {
+		throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
+	}
+	if (fx[0]*fx[1]>=0) {
+		throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
+	}
+
+	// The Newton-Raphson method has quadratic convergence!  Simply put, it
+	// replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
+	const ex ff = normal(-f/f.diff(x));
+	int side = 0;  // Start at left interval limit.
+	numeric xxprev;
+	numeric fxprev;
+	do {
+		xxprev = xx[side];
+		fxprev = fx[side];
+		xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
+		fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
+		if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
+			// Oops, Newton-Raphson method shot out of the interval.
+			// Restore, and try again with the other side instead!
+			xx[side] = xxprev;
+			fx[side] = fxprev;
+			side = !side;
+			xxprev = xx[side];
+			fxprev = fx[side];
+			xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
+			fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
+		}
+		if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
+			// Oops, the root isn't bracketed any more.
+			// Restore, and perform a bisection!
+			xx[side] = xxprev;
+			fx[side] = fxprev;
+
+			// Ah, the bisection! Bisections converge linearly. Unfortunately,
+			// they occur pretty often when Newton-Raphson arrives at an x too
+			// close to the result on one side of the interval and
+			// f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
+			// precision errors! Recall that this function does not have a
+			// precision goal as one of its arguments but instead relies on
+			// x converging to a fixed point. We speed up the (safe but slow)
+			// bisection method by mixing in a dash of the (unsafer but faster)
+			// secant method: Instead of splitting the interval at the
+			// arithmetic mean (bisection), we split it nearer to the root as
+			// determined by the secant between the values xx[0] and xx[1].
+			// Don't set the secant_weight to one because that could disturb
+			// the convergence in some corner cases!
+			static const double secant_weight = 0.96875;  // == 31/32 < 1
+			numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
+			    + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
+			numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
+			if (fxmid.is_zero()) {
+				// Luck strikes...
+				return xxmid;
+			}
+			if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
+				side = !side;
+			}
+			xxprev = xx[side];
+			fxprev = fx[side];
+			xx[side] = xxmid;
+			fx[side] = fxmid;
+		}
+	} while (xxprev!=xx[side]);
+	return xxprev;
+}
+
+
+/* Force inclusion of functions from inifcns_gamma and inifcns_zeta
+ * for static lib (so ginsh will see them). */
+unsigned force_include_tgamma = tgamma_SERIAL::serial;
+unsigned force_include_zeta1 = zeta1_SERIAL::serial;
+
 } // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC