X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns.cpp;h=f366e779e70fbdb18a64e0447b8b21263f3704ee;hp=ad4e035aaf6d793904e63d641654962508af2b78;hb=9993a7aac97abf383624fc5dae4beecb29531fbd;hpb=f1dedbe2e079cc7aa0bf34445aa920e9a9046ee5

diff --git a/ginac/inifcns.cpp b/ginac/inifcns.cpp
index ad4e035a..f366e779 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) 1999-2010 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
@@ -17,12 +17,9 @@
  *
  *  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>
-#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)
@@ -107,13 +215,112 @@ static ex abs_conjugate(const ex & arg)
 	return abs(arg);
 }
 
+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())
+		return power(arg, exp);
+	else
+		return power(abs(arg), exp).hold();
+}
+
 REGISTER_FUNCTION(abs, eval_func(abs_eval).
                        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));
+                       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 +381,38 @@ 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);
+		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 +489,17 @@ 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);
+}
+
+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 +507,9 @@ 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));
 
 
 //////////
@@ -339,7 +583,7 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
 			ex ser;
 			// manually construct the primitive expansion
 			for (int i=1; i<order; ++i)
-				ser += pow(s,i) / pow(numeric(i), _num2);
+				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
@@ -399,11 +643,26 @@ 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);
+	}
+	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 +676,7 @@ 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)
@@ -480,14 +739,26 @@ 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
@@ -503,14 +774,14 @@ 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 _num0;
+			if (N == 0) return _ex1;
 			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 _num0;
+			return _ex0;
 	}
 
 	return binomial(x, y).hold();
@@ -532,12 +803,24 @@ 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 +855,19 @@ 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 +875,9 @@ 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 +899,7 @@ 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 +907,7 @@ 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)) {
@@ -663,6 +960,125 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 	return sollist;
 }
 
+//////////
+// Find real root of f(x) numerically
+//////////
+
+const numeric
+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");
+	}
+	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 };
+	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])) {
+		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];
+		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_);
+
+		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 ((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];
+
+			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.
+			// 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 = 0.984375;  // == 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]));
+			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;
+			}
+			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;