X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns.cpp;h=8103cd86bf3ec12ad02a2ebec48aff1b122b0c49;hp=a9ad5ce5223ec84c2e0b0d4e55813d01a8d3a07a;hb=10365850aa3803337bfea1fc201b81b6752096d4;hpb=4d59c02d51fbf50ff24d616b00296aa4cbb1ea5e

diff --git a/ginac/inifcns.cpp b/ginac/inifcns.cpp
index a9ad5ce5..8103cd86 100644
--- a/ginac/inifcns.cpp
+++ b/ginac/inifcns.cpp
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's initially known functions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2011 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
@@ -20,9 +20,6 @@
  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
-#include <vector>
-#include <stdexcept>
-
 #include "inifcns.h"
 #include "ex.h"
 #include "constant.h"
@@ -37,6 +34,9 @@
 #include "symmetry.h"
 #include "utils.h"
 
+#include <stdexcept>
+#include <vector>
+
 namespace GiNaC {
 
 //////////
@@ -66,12 +66,114 @@ static ex conjugate_conjugate(const ex & arg)
 	return arg;
 }
 
+static ex conjugate_real_part(const ex & arg)
+{
+	return arg.real_part();
+}
+
+static ex conjugate_imag_part(const ex & arg)
+{
+	return -arg.imag_part();
+}
+
 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
                                       evalf_func(conjugate_evalf).
                                       print_func<print_latex>(conjugate_print_latex).
                                       conjugate_func(conjugate_conjugate).
+                                      real_part_func(conjugate_real_part).
+                                      imag_part_func(conjugate_imag_part).
                                       set_name("conjugate","conjugate"));
 
+//////////
+// real part
+//////////
+
+static ex real_part_evalf(const ex & arg)
+{
+	if (is_exactly_a<numeric>(arg)) {
+		return ex_to<numeric>(arg).real();
+	}
+	return real_part_function(arg).hold();
+}
+
+static ex real_part_eval(const ex & arg)
+{
+	return arg.real_part();
+}
+
+static void real_part_print_latex(const ex & arg, const print_context & c)
+{
+	c.s << "\\Re"; arg.print(c); c.s << "";
+}
+
+static ex real_part_conjugate(const ex & arg)
+{
+	return real_part_function(arg).hold();
+}
+
+static ex real_part_real_part(const ex & arg)
+{
+	return real_part_function(arg).hold();
+}
+
+static ex real_part_imag_part(const ex & arg)
+{
+	return 0;
+}
+
+REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
+                                      evalf_func(real_part_evalf).
+                                      print_func<print_latex>(real_part_print_latex).
+                                      conjugate_func(real_part_conjugate).
+                                      real_part_func(real_part_real_part).
+                                      imag_part_func(real_part_imag_part).
+                                      set_name("real_part","real_part"));
+
+//////////
+// imag part
+//////////
+
+static ex imag_part_evalf(const ex & arg)
+{
+	if (is_exactly_a<numeric>(arg)) {
+		return ex_to<numeric>(arg).imag();
+	}
+	return imag_part_function(arg).hold();
+}
+
+static ex imag_part_eval(const ex & arg)
+{
+	return arg.imag_part();
+}
+
+static void imag_part_print_latex(const ex & arg, const print_context & c)
+{
+	c.s << "\\Im"; arg.print(c); c.s << "";
+}
+
+static ex imag_part_conjugate(const ex & arg)
+{
+	return imag_part_function(arg).hold();
+}
+
+static ex imag_part_real_part(const ex & arg)
+{
+	return imag_part_function(arg).hold();
+}
+
+static ex imag_part_imag_part(const ex & arg)
+{
+	return 0;
+}
+
+REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
+                                      evalf_func(imag_part_evalf).
+                                      print_func<print_latex>(imag_part_print_latex).
+                                      conjugate_func(imag_part_conjugate).
+                                      real_part_func(imag_part_real_part).
+                                      imag_part_func(imag_part_imag_part).
+                                      set_name("imag_part","imag_part"));
+
 //////////
 // absolute value
 //////////
@@ -88,8 +190,14 @@ static ex abs_eval(const ex & arg)
 {
 	if (is_exactly_a<numeric>(arg))
 		return abs(ex_to<numeric>(arg));
-	else
-		return abs(arg).hold();
+
+	if (arg.info(info_flags::nonnegative))
+		return arg;
+
+	if (is_ex_the_function(arg, abs))
+		return arg;
+
+	return abs(arg).hold();
 }
 
 static void abs_print_latex(const ex & arg, const print_context & c)
@@ -104,7 +212,26 @@ static void abs_print_csrc_float(const ex & arg, const print_context & c)
 
 static ex abs_conjugate(const ex & arg)
 {
-	return abs(arg);
+	return abs(arg).hold();
+}
+
+static ex abs_real_part(const ex & arg)
+{
+	return abs(arg).hold();
+}
+
+static ex abs_imag_part(const ex& arg)
+{
+	return 0;
+}
+
+static ex abs_power(const ex & arg, const ex & exp)
+{
+	if (arg.is_equal(arg.conjugate()) && ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
+						|| exp.info(info_flags::even)))
+		return power(arg, exp);
+	else
+		return power(abs(arg), exp).hold();
 }
 
 REGISTER_FUNCTION(abs, eval_func(abs_eval).
@@ -112,8 +239,89 @@ REGISTER_FUNCTION(abs, eval_func(abs_eval).
                        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));
+                       conjugate_func(abs_conjugate).
+                       real_part_func(abs_real_part).
+                       imag_part_func(abs_imag_part).
+                       power_func(abs_power));
+
+//////////
+// Step function
+//////////
+
+static ex step_evalf(const ex & arg)
+{
+	if (is_exactly_a<numeric>(arg))
+		return step(ex_to<numeric>(arg));
+	
+	return step(arg).hold();
+}
+
+static ex step_eval(const ex & arg)
+{
+	if (is_exactly_a<numeric>(arg))
+		return step(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)
+				// step(42*x) -> step(x)
+				return step(arg/oc).hold();
+			else
+				// step(-42*x) -> step(-x)
+				return step(-arg/oc).hold();
+		}
+		if (oc.real().is_zero()) {
+			if (oc.imag() > 0)
+				// step(42*I*x) -> step(I*x)
+				return step(I*arg/oc).hold();
+			else
+				// step(-42*I*x) -> step(-I*x)
+				return step(-I*arg/oc).hold();
+		}
+	}
+	
+	return step(arg).hold();
+}
 
+static ex step_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("step_series(): on imaginary axis"));
+	
+	epvector seq;
+	seq.push_back(expair(step(arg_pt), _ex0));
+	return pseries(rel,seq);
+}
+
+static ex step_conjugate(const ex& arg)
+{
+	return step(arg).hold();
+}
+
+static ex step_real_part(const ex& arg)
+{
+	return step(arg).hold();
+}
+
+static ex step_imag_part(const ex& arg)
+{
+	return 0;
+}
+
+REGISTER_FUNCTION(step, eval_func(step_eval).
+                        evalf_func(step_evalf).
+                        series_func(step_series).
+                        conjugate_func(step_conjugate).
+                        real_part_func(step_real_part).
+                        imag_part_func(step_imag_part));
 
 //////////
 // Complex sign
@@ -174,13 +382,38 @@ static ex csgn_series(const ex & arg,
 
 static ex csgn_conjugate(const ex& arg)
 {
-	return csgn(arg);
+	return csgn(arg).hold();
 }
 
+static ex csgn_real_part(const ex& arg)
+{
+	return csgn(arg).hold();
+}
+
+static ex csgn_imag_part(const ex& arg)
+{
+	return 0;
+}
+
+static ex csgn_power(const ex & arg, const ex & exp)
+{
+	if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
+		if (ex_to<numeric>(exp).is_odd())
+			return csgn(arg).hold();
+		else
+			return power(csgn(arg), _ex2).hold();
+	} else
+		return power(csgn(arg), exp).hold();
+}
+
+
 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
                         evalf_func(csgn_evalf).
                         series_func(csgn_series).
-                        conjugate_func(csgn_conjugate));
+                        conjugate_func(csgn_conjugate).
+                        real_part_func(csgn_real_part).
+                        imag_part_func(csgn_imag_part).
+                        power_func(csgn_power));
 
 
 //////////
@@ -257,7 +490,17 @@ static ex eta_series(const ex & x, const ex & y,
 
 static ex eta_conjugate(const ex & x, const ex & y)
 {
-	return -eta(x,y);
+	return -eta(x, y).hold();
+}
+
+static ex eta_real_part(const ex & x, const ex & y)
+{
+	return 0;
+}
+
+static ex eta_imag_part(const ex & x, const ex & y)
+{
+	return -I*eta(x, y).hold();
 }
 
 REGISTER_FUNCTION(eta, eval_func(eta_eval).
@@ -265,7 +508,9 @@ REGISTER_FUNCTION(eta, eval_func(eta_eval).
                        series_func(eta_series).
                        latex_name("\\eta").
                        set_symmetry(sy_symm(0, 1)).
-                       conjugate_func(eta_conjugate));
+                       conjugate_func(eta_conjugate).
+                       real_part_func(eta_real_part).
+                       imag_part_func(eta_imag_part));
 
 
 //////////
@@ -399,11 +644,26 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
 	throw do_taylor();  // caught by function::series()
 }
 
+static ex Li2_conjugate(const ex & x)
+{
+	// conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
+	// run along the positive real axis beginning at 1.
+	if (x.info(info_flags::negative)) {
+		return Li2(x).hold();
+	}
+	if (is_exactly_a<numeric>(x) &&
+	    (!x.imag_part().is_zero() || x < *_num1_p)) {
+		return Li2(x.conjugate());
+	}
+	return conjugate_function(Li2(x)).hold();
+}
+
 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
                        evalf_func(Li2_evalf).
                        derivative_func(Li2_deriv).
                        series_func(Li2_series).
-                       latex_name("\\mbox{Li}_2"));
+                       conjugate_func(Li2_conjugate).
+                       latex_name("\\mathrm{Li}_2"));
 
 //////////
 // trilogarithm
@@ -417,7 +677,7 @@ static ex Li3_eval(const ex & x)
 }
 
 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
-                       latex_name("\\mbox{Li}_3"));
+                       latex_name("\\mathrm{Li}_3"));
 
 //////////
 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
@@ -428,7 +688,7 @@ 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 zeta(x).hold();
 	}
 	
 	return zetaderiv(n, x).hold();
@@ -480,14 +740,26 @@ static void factorial_print_dflt_latex(const ex & x, const print_context & c)
 
 static ex factorial_conjugate(const ex & x)
 {
-	return factorial(x);
+	return factorial(x).hold();
+}
+
+static ex factorial_real_part(const ex & x)
+{
+	return factorial(x).hold();
+}
+
+static ex factorial_imag_part(const ex & x)
+{
+	return 0;
 }
 
 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
                              evalf_func(factorial_evalf).
                              print_func<print_dflt>(factorial_print_dflt_latex).
                              print_func<print_latex>(factorial_print_dflt_latex).
-                             conjugate_func(factorial_conjugate));
+                             conjugate_func(factorial_conjugate).
+                             real_part_func(factorial_real_part).
+                             imag_part_func(factorial_imag_part));
 
 //////////
 // binomial
@@ -503,7 +775,7 @@ 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 == 0) return _ex1;
 			if (N == 1) return x;
 			ex t = x.expand();
 			for (unsigned i = 2; i <= N; ++i)
@@ -532,12 +804,24 @@ static ex binomial_eval(const ex & x, const ex &y)
 // function, also complex conjugation should be changed (or rather, deleted).
 static ex binomial_conjugate(const ex & x, const ex & y)
 {
-	return binomial(x,y);
+	return binomial(x,y).hold();
+}
+
+static ex binomial_real_part(const ex & x, const ex & y)
+{
+	return binomial(x,y).hold();
+}
+
+static ex binomial_imag_part(const ex & x, const ex & y)
+{
+	return 0;
 }
 
 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
                             evalf_func(binomial_evalf).
-                            conjugate_func(binomial_conjugate));
+                            conjugate_func(binomial_conjugate).
+                            real_part_func(binomial_real_part).
+                            imag_part_func(binomial_imag_part));
 
 //////////
 // Order term function (for truncated power series)
@@ -572,7 +856,19 @@ static ex Order_series(const ex & x, const relational & r, int order, unsigned o
 
 static ex Order_conjugate(const ex & x)
 {
-	return Order(x);
+	return Order(x).hold();
+}
+
+static ex Order_real_part(const ex & x)
+{
+	return Order(x).hold();
+}
+
+static ex Order_imag_part(const ex & x)
+{
+	if(x.info(info_flags::real))
+		return 0;
+	return Order(x).hold();
 }
 
 // Differentiation is handled in function::derivative because of its special requirements
@@ -580,7 +876,9 @@ static ex Order_conjugate(const ex & x)
 REGISTER_FUNCTION(Order, eval_func(Order_eval).
                          series_func(Order_series).
                          latex_name("\\mathcal{O}").
-                         conjugate_func(Order_conjugate));
+                         conjugate_func(Order_conjugate).
+                         real_part_func(Order_real_part).
+                         imag_part_func(Order_imag_part));
 
 //////////
 // Solve linear system
@@ -602,7 +900,7 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 	
 	// syntax checks
 	if (!eqns.info(info_flags::list)) {
-		throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
+		throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
 	}
 	for (size_t i=0; i<eqns.nops(); i++) {
 		if (!eqns.op(i).info(info_flags::relation_equal)) {
@@ -610,7 +908,7 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 		}
 	}
 	if (!symbols.info(info_flags::list)) {
-		throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
+		throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
 	}
 	for (size_t i=0; i<symbols.nops(); i++) {
 		if (!symbols.op(i).info(info_flags::symbol)) {
@@ -668,7 +966,7 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 //////////
 
 const numeric
-fsolve(const ex& f, const symbol& x, const numeric& x1, const numeric& x2)
+fsolve(const ex& f_in, 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");
@@ -681,6 +979,12 @@ fsolve(const ex& f, const symbol& x, const numeric& x1, const numeric& x2)
 	// 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 };
+	ex f;
+	if (is_a<relational>(f_in)) {
+		f = f_in.lhs()-f_in.rhs();
+	} else {
+		f = f_in;
+	}
 	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])) {
@@ -704,9 +1008,24 @@ fsolve(const ex& f, const symbol& x, const numeric& x1, const numeric& x2)
 	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]) {
+		ex dx_ = ff.subs(x == xx[side]).evalf();
+		if (!is_a<numeric>(dx_))
+			throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
+		xx[side] += ex_to<numeric>(dx_);
+		// Now check if Newton-Raphson method shot out of the interval 
+		bool bad_shot = (side == 0 && xx[0] < xxprev) || 
+				(side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
+		if (!bad_shot) {
+			// Compute f(x) only if new x is inside the interval.
+			// The function might be difficult to compute numerically
+			// or even ill defined outside the interval. Also it's
+			// a small optimization. 
+			ex f_x = f.subs(x == xx[side]).evalf();
+			if (!is_a<numeric>(f_x))
+				throw std::runtime_error("fsolve(): function does not evaluate numerically");
+			fx[side] = ex_to<numeric>(f_x);
+		}
+		if (bad_shot) {
 			// Oops, Newton-Raphson method shot out of the interval.
 			// Restore, and try again with the other side instead!
 			xx[side] = xxprev;
@@ -714,8 +1033,16 @@ fsolve(const ex& f, const symbol& x, const numeric& x1, const numeric& x2)
 			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());
+
+			ex dx_ = ff.subs(x == xx[side]).evalf();
+			if (!is_a<numeric>(dx_))
+				throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
+			xx[side] += ex_to<numeric>(dx_);
+
+			ex f_x = f.subs(x==xx[side]).evalf();
+			if (!is_a<numeric>(f_x))
+				throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
+			fx[side] = ex_to<numeric>(f_x);
 		}
 		if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
 			// Oops, the root isn't bracketed any more.
@@ -736,10 +1063,13 @@ fsolve(const ex& f, const symbol& x, const numeric& x1, const numeric& x2)
 			// 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
+			static const double secant_weight = 0.984375;  // == 63/64 < 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());
+			ex fxmid_ = f.subs(x == xxmid).evalf();
+			if (!is_a<numeric>(fxmid_))
+				throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
+			numeric fxmid = ex_to<numeric>(fxmid_);
 			if (fxmid.is_zero()) {
 				// Luck strikes...
 				return xxmid;