Remove 'level' argument of normal().
[ginac.git] / ginac / inifcns.cpp
index 8103cd86bf3ec12ad02a2ebec48aff1b122b0c49..6141b8652d0d7faa9230b6476f8ac91d32703df3 100644 (file)
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's initially known functions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2011 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2016 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
@@ -24,6 +24,7 @@
 #include "ex.h"
 #include "constant.h"
 #include "lst.h"
+#include "fderivative.h"
 #include "matrix.h"
 #include "mul.h"
 #include "power.h"
@@ -66,6 +67,19 @@ static ex conjugate_conjugate(const ex & arg)
        return arg;
 }
 
+// If x is real then U.diff(x)-I*V.diff(x) represents both conjugate(U+I*V).diff(x) 
+// and conjugate((U+I*V).diff(x))
+static ex conjugate_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return conjugate(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(conjugate(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
 static ex conjugate_real_part(const ex & arg)
 {
        return arg.real_part();
@@ -76,8 +90,46 @@ static ex conjugate_imag_part(const ex & arg)
        return -arg.imag_part();
 }
 
+static bool func_arg_info(const ex & arg, unsigned inf)
+{
+       // for some functions we can return the info() of its argument
+       // (think of conjugate())
+       switch (inf) {
+               case info_flags::polynomial:
+               case info_flags::integer_polynomial:
+               case info_flags::cinteger_polynomial:
+               case info_flags::rational_polynomial:
+               case info_flags::real:
+               case info_flags::rational:
+               case info_flags::integer:
+               case info_flags::crational:
+               case info_flags::cinteger:
+               case info_flags::even:
+               case info_flags::odd:
+               case info_flags::prime:
+               case info_flags::crational_polynomial:
+               case info_flags::rational_function:
+               case info_flags::positive:
+               case info_flags::negative:
+               case info_flags::nonnegative:
+               case info_flags::posint:
+               case info_flags::negint:
+               case info_flags::nonnegint:
+               case info_flags::has_indices:
+                       return arg.info(inf);
+       }
+       return false;
+}
+
+static bool conjugate_info(const ex & arg, unsigned inf)
+{
+       return func_arg_info(arg, inf);
+}
+
 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
                                       evalf_func(conjugate_evalf).
+                                      expl_derivative_func(conjugate_expl_derivative).
+                                      info_func(conjugate_info).
                                       print_func<print_latex>(conjugate_print_latex).
                                       conjugate_func(conjugate_conjugate).
                                       real_part_func(conjugate_real_part).
@@ -121,8 +173,21 @@ static ex real_part_imag_part(const ex & arg)
        return 0;
 }
 
+// If x is real then Re(e).diff(x) is equal to Re(e.diff(x)) 
+static ex real_part_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return real_part_function(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(real_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
                                       evalf_func(real_part_evalf).
+                                      expl_derivative_func(real_part_expl_derivative).
                                       print_func<print_latex>(real_part_print_latex).
                                       conjugate_func(real_part_conjugate).
                                       real_part_func(real_part_real_part).
@@ -166,8 +231,21 @@ static ex imag_part_imag_part(const ex & arg)
        return 0;
 }
 
+// If x is real then Im(e).diff(x) is equal to Im(e.diff(x)) 
+static ex imag_part_expl_derivative(const ex & arg, const symbol & s)
+{
+       if (s.info(info_flags::real))
+               return imag_part_function(arg.diff(s));
+       else {
+               exvector vec_arg;
+               vec_arg.push_back(arg);
+               return fderivative(ex_to<function>(imag_part(arg)).get_serial(),0,vec_arg).hold()*arg.diff(s);
+       }
+}
+
 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
                                       evalf_func(imag_part_evalf).
+                                      expl_derivative_func(imag_part_expl_derivative).
                                       print_func<print_latex>(imag_part_print_latex).
                                       conjugate_func(imag_part_conjugate).
                                       real_part_func(imag_part_real_part).
@@ -194,12 +272,58 @@ static ex abs_eval(const ex & arg)
        if (arg.info(info_flags::nonnegative))
                return arg;
 
+       if (arg.info(info_flags::negative) || (-arg).info(info_flags::nonnegative))
+               return -arg;
+
        if (is_ex_the_function(arg, abs))
                return arg;
 
+       if (is_ex_the_function(arg, exp))
+               return exp(arg.op(0).real_part());
+
+       if (is_exactly_a<power>(arg)) {
+               const ex& base = arg.op(0);
+               const ex& exponent = arg.op(1);
+               if (base.info(info_flags::positive) || exponent.info(info_flags::real))
+                       return pow(abs(base), exponent.real_part());
+       }
+
+       if (is_ex_the_function(arg, conjugate_function))
+               return abs(arg.op(0));
+
+       if (is_ex_the_function(arg, step))
+               return arg;
+
        return abs(arg).hold();
 }
 
+static ex abs_expand(const ex & arg, unsigned options)
+{
+       if ((options & expand_options::expand_transcendental)
+               && is_exactly_a<mul>(arg)) {
+               exvector prodseq;
+               prodseq.reserve(arg.nops());
+               for (const_iterator i = arg.begin(); i != arg.end(); ++i) {
+                       if (options & expand_options::expand_function_args)
+                               prodseq.push_back(abs(i->expand(options)));
+                       else
+                               prodseq.push_back(abs(*i));
+               }
+               return dynallocate<mul>(prodseq).setflag(status_flags::expanded);
+       }
+
+       if (options & expand_options::expand_function_args)
+               return abs(arg.expand(options)).hold();
+       else
+               return abs(arg).hold();
+}
+
+static ex abs_expl_derivative(const ex & arg, const symbol & s)
+{
+       ex diff_arg = arg.diff(s);
+       return (diff_arg*arg.conjugate()+arg*diff_arg.conjugate())/2/abs(arg);
+}
+
 static void abs_print_latex(const ex & arg, const print_context & c)
 {
        c.s << "{|"; arg.print(c); c.s << "|}";
@@ -227,15 +351,47 @@ static ex abs_imag_part(const ex& arg)
 
 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
+       if ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even()) || exp.info(info_flags::even)) {
+               if (arg.info(info_flags::real) || arg.is_equal(arg.conjugate()))
+                       return pow(arg, exp);
+               else
+                       return pow(arg, exp/2) * pow(arg.conjugate(), exp/2);
+       } else
                return power(abs(arg), exp).hold();
 }
 
+bool abs_info(const ex & arg, unsigned inf)
+{
+       switch (inf) {
+               case info_flags::integer:
+               case info_flags::even:
+               case info_flags::odd:
+               case info_flags::prime:
+                       return arg.info(inf);
+               case info_flags::nonnegint:
+                       return arg.info(info_flags::integer);
+               case info_flags::nonnegative:
+               case info_flags::real:
+                       return true;
+               case info_flags::negative:
+                       return false;
+               case info_flags::positive:
+                       return arg.info(info_flags::positive) || arg.info(info_flags::negative);
+               case info_flags::has_indices: {
+                       if (arg.info(info_flags::has_indices))
+                               return true;
+                       else
+                               return false;
+               }
+       }
+       return false;
+}
+
 REGISTER_FUNCTION(abs, eval_func(abs_eval).
                        evalf_func(abs_evalf).
+                       expand_func(abs_expand).
+                       expl_derivative_func(abs_expl_derivative).
+                       info_func(abs_info).
                        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).
@@ -296,9 +452,8 @@ static ex step_series(const ex & arg,
            && !(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);
+       epvector seq { expair(step(arg_pt), _ex0) };
+       return pseries(rel, std::move(seq));
 }
 
 static ex step_conjugate(const ex& arg)
@@ -375,9 +530,8 @@ static ex csgn_series(const ex & arg,
            && !(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);
+       epvector seq { expair(csgn(arg_pt), _ex0) };
+       return pseries(rel, std::move(seq));
 }
 
 static ex csgn_conjugate(const ex& arg)
@@ -483,9 +637,8 @@ static ex eta_series(const ex & x, const ex & y,
            (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);
+       epvector seq { expair(eta(x_pt,y_pt), _ex0) };
+       return pseries(rel, std::move(seq));
 }
 
 static ex eta_conjugate(const ex & x, const ex & y)
@@ -588,9 +741,8 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                        // 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);
+                       epvector nseq { expair(Order(_ex1), order) };
+                       ser += pseries(rel, std::move(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
@@ -613,9 +765,8 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                        // 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);
+                       epvector nseq { expair(Order(_ex1), order) };
+                       ser += pseries(rel, std::move(nseq));
                        // reexpanding it will collapse the series again
                        return ser.series(rel, order);
                }
@@ -637,7 +788,7 @@ static ex Li2_series(const ex &x, const relational &rel, int order, unsigned opt
                                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);
+                       return pseries(rel, std::move(seq));
                }
        }
        // all other cases should be safe, by now:
@@ -799,7 +950,7 @@ static ex binomial_eval(const ex & x, const ex &y)
                return binomial(x, y).hold();
 }
 
-// At the moment the numeric evaluation of a binomail function always
+// At the moment the numeric evaluation of a binomial 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)
@@ -847,11 +998,10 @@ static ex Order_eval(const ex & x)
 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_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);
+       epvector new_seq { expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))) };
+       return pseries(r, std::move(new_seq));
 }
 
 static ex Order_conjugate(const ex & x)
@@ -871,11 +1021,15 @@ static ex Order_imag_part(const ex & x)
        return Order(x).hold();
 }
 
-// Differentiation is handled in function::derivative because of its special requirements
+static ex Order_expl_derivative(const ex & arg, const symbol & s)
+{
+       return Order(arg.diff(s));
+}
 
 REGISTER_FUNCTION(Order, eval_func(Order_eval).
                          series_func(Order_series).
                          latex_name("\\mathcal{O}").
+                         expl_derivative_func(Order_expl_derivative).
                          conjugate_func(Order_conjugate).
                          real_part_func(Order_real_part).
                          imag_part_func(Order_imag_part));
@@ -890,7 +1044,7 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
        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));
+               const ex sol = lsolve(lst{eqns}, lst{symbols});
                
                GINAC_ASSERT(sol.nops()==1);
                GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
@@ -948,12 +1102,12 @@ ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
        } catch (const std::runtime_error & e) {
                // Probably singular matrix or otherwise overdetermined system:
                // It is consistent to return an empty list
-               return lst();
+               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,...)
+       // 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));
@@ -1063,7 +1217,7 @@ fsolve(const ex& f_in, 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 = 0.984375;  // == 63/64 < 1
+                       constexpr 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();