/** @file inifcns.cpp
 *
 *  Implementation of GiNaC's initially known functions. */

/*
 *  GiNaC Copyright (C) 1999-2002 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
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  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
 */

#include <vector>
#include <stdexcept>

#include "inifcns.h"
#include "ex.h"
#include "constant.h"
#include "lst.h"
#include "matrix.h"
#include "mul.h"
#include "power.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
#include "symmetry.h"
#include "utils.h"

namespace GiNaC {

//////////
// absolute value
//////////

static ex abs_evalf(const ex & arg)
{
	if (is_exactly_a<numeric>(arg))
		return abs(ex_to<numeric>(arg));
	
	return abs(arg).hold();
}

static ex abs_eval(const ex & arg)
{
	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).
                       evalf_func(abs_evalf));


//////////
// 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_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(arg).hold();
}

static ex csgn_series(const ex & arg,
                      const relational & rel,
                      int order,
                      unsigned options)
{
	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: 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);
	const ex y_pt = y.subs(rel);
	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);
}

REGISTER_FUNCTION(eta, eval_func(eta_eval).
                       evalf_func(eta_evalf).
                       series_func(eta_series).
                       latex_name("\\eta").
                       set_symmetry(sy_symm(0, 1)));


//////////
// 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.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();
}

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);
	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.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));
			// 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 (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).
                       latex_name("\\mbox{Li}_2"));

//////////
// trilogarithm
//////////

static ex Li3_eval(const ex & x)
{
	if (x.is_zero())
		return x;
	return Li3(x).hold();
}

REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
                       latex_name("\\mbox{Li}_3"));

//////////
// factorial
//////////

static ex factorial_evalf(const ex & x)
{
	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();
}

REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
                             evalf_func(factorial_evalf));

//////////
// binomial
//////////

static ex binomial_evalf(const ex & x, const ex & y)
{
	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();
}

REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
                            evalf_func(binomial_evalf));

//////////
// Order term function (for truncated power series)
//////////

static ex Order_eval(const ex & x)
{
	if (is_ex_exactly_of_type(x, numeric)) {
		// 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)) {
		const mul &m = ex_to<mul>(x);
		// 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();
}

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;
	GINAC_ASSERT(is_exactly_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);
}

// Differentiation is handled in function::derivative because of its special requirements

REGISTER_FUNCTION(Order, eval_func(Order_eval).
                         series_func(Order_series).
                         latex_name("\\mathcal{O}"));

//////////
// Solve linear system
//////////

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 (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++) {
		const 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++) {
			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 (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,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 (unsigned i=0; i<symbols.nops(); i++)
		sollist.append(symbols.op(i)==solution(i,0));
	
	return sollist;
}

/* 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;

} // namespace GiNaC