X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns.cpp;h=6141b8652d0d7faa9230b6476f8ac91d32703df3;hp=63c2d74f556d74c2c9cce124729107695a3bdf43;hb=ed914545e01d60ecf2544e6141d6c5142c01327f;hpb=67edef78ce992a8f6ad704bfac228b8dec6eacd2 diff --git a/ginac/inifcns.cpp b/ginac/inifcns.cpp index 63c2d74f..6141b865 100644 --- a/ginac/inifcns.cpp +++ b/ginac/inifcns.cpp @@ -3,7 +3,7 @@ * Implementation of GiNaC's initially known functions. */ /* - * GiNaC Copyright (C) 1999-2010 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(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(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(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(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(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(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(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(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(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 << "|}"; @@ -212,7 +336,7 @@ 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) @@ -227,14 +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(exp) && ex_to(exp).is_even()) - return power(arg, exp); - else + if ((is_a(exp) && ex_to(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(abs_print_latex). print_func(abs_print_csrc_float). print_func(abs_print_csrc_float). @@ -295,9 +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) @@ -319,8 +475,8 @@ 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)); + real_part_func(step_real_part). + imag_part_func(step_imag_part)); ////////// // Complex sign @@ -374,9 +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) @@ -398,7 +553,7 @@ static ex csgn_power(const ex & arg, const ex & exp) { if (is_a(exp) && exp.info(info_flags::positive) && ex_to(exp).is_integer()) { if (ex_to(exp).is_odd()) - return csgn(arg); + return csgn(arg).hold(); else return power(csgn(arg), _ex2).hold(); } else @@ -482,14 +637,13 @@ 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) { - return -eta(x, y); + return -eta(x, y).hold(); } static ex eta_real_part(const ex & x, const ex & y) @@ -587,9 +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 @@ -612,9 +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); } @@ -636,17 +788,32 @@ 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: 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(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). + conjugate_func(Li2_conjugate). latex_name("\\mathrm{Li}_2")); ////////// @@ -672,7 +839,7 @@ 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(); @@ -783,7 +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) @@ -831,11 +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(r.lhs())); const symbol &s = ex_to(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) @@ -855,11 +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)); @@ -874,7 +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(sol.op(0))); @@ -932,12 +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(ff.subs(x==xx[side]).evalf()); - fx[side] = ex_to(f.subs(x==xx[side]).evalf()); - if ((side==0 && xx[0]xxprev) || xx[0]>xx[1]) { + ex dx_ = ff.subs(x == xx[side]).evalf(); + if (!is_a(dx_)) + throw std::runtime_error("fsolve(): function derivative does not evaluate numerically"); + xx[side] += ex_to(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(f_x)) + throw std::runtime_error("fsolve(): function does not evaluate numerically"); + fx[side] = ex_to(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; @@ -1002,8 +1187,16 @@ fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2) side = !side; xxprev = xx[side]; fxprev = fx[side]; - xx[side] += ex_to(ff.subs(x==xx[side]).evalf()); - fx[side] = ex_to(f.subs(x==xx[side]).evalf()); + + ex dx_ = ff.subs(x == xx[side]).evalf(); + if (!is_a(dx_)) + throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]"); + xx[side] += ex_to(dx_); + + ex f_x = f.subs(x==xx[side]).evalf(); + if (!is_a(f_x)) + throw std::runtime_error("fsolve(): function does not evaluate numerically [2]"); + fx[side] = ex_to(f_x); } if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) { // Oops, the root isn't bracketed any more. @@ -1024,10 +1217,13 @@ 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])); - numeric fxmid = ex_to(f.subs(x==xxmid).evalf()); + ex fxmid_ = f.subs(x == xxmid).evalf(); + if (!is_a(fxmid_)) + throw std::runtime_error("fsolve(): function does not evaluate numerically [3]"); + numeric fxmid = ex_to(fxmid_); if (fxmid.is_zero()) { // Luck strikes... return xxmid;