* Implementation of GiNaC's initially known functions. */
/*
- * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2015 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
* 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"
#include "lst.h"
+#include "fderivative.h"
#include "matrix.h"
#include "mul.h"
#include "power.h"
#include "symmetry.h"
#include "utils.h"
+#include <stdexcept>
+#include <vector>
+
namespace GiNaC {
//////////
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();
+}
+
+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::algebraic:
+ 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).
+ 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;
+}
+
+// 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).
+ 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;
+}
+
+// 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).
+ imag_part_func(imag_part_imag_part).
+ set_name("imag_part","imag_part"));
+
//////////
// absolute value
//////////
{
if (is_exactly_a<numeric>(arg))
return abs(ex_to<numeric>(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 (new mul(prodseq))->setflag(status_flags::dynallocated | 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 << "|}";
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 ((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 power(arg, exp);
+ else
+ return power(arg, exp/2)*power(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).
- 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
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));
//////////
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).
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));
//////////
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
}
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)
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();
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
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)
// 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)
static ex Order_conjugate(const ex & x)
{
- return Order(x);
+ return Order(x).hold();
}
-// Differentiation is handled in function::derivative because of its special requirements
+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();
+}
+
+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}").
- conjugate_func(Order_conjugate));
+ expl_derivative_func(Order_expl_derivative).
+ conjugate_func(Order_conjugate).
+ real_part_func(Order_real_part).
+ imag_part_func(Order_imag_part));
//////////
// Solve linear system
// 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)) {
}
}
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)) {
//////////
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");
// 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])) {
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;
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.
// 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;
git://www.ginac.de / ginac.git / blobdiff