* 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
*
* 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"
#include "symmetry.h"
#include "utils.h"
+#include <stdexcept>
+#include <vector>
+
namespace GiNaC {
//////////
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).
- set_name("conjugate","conjugate"));
+ 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
{
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)
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
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));
//////////
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).
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));
//////////
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
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
}
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)
return factorial(x).hold();
}
+static void factorial_print_dflt_latex(const ex & x, const print_context & c)
+{
+ if (is_exactly_a<symbol>(x) ||
+ is_exactly_a<constant>(x) ||
+ is_exactly_a<function>(x)) {
+ x.print(c); c.s << "!";
+ } else {
+ c.s << "("; x.print(c); c.s << ")!";
+ }
+}
+
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).
- conjugate_func(factorial_conjugate));
+ print_func<print_dflt>(factorial_print_dflt_latex).
+ print_func<print_latex>(factorial_print_dflt_latex).
+ 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 _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();
// 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();
+}
+
+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
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
// 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)) {
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;
git://www.ginac.de / ginac.git / blobdiff