* Implementation of GiNaC's initially known functions. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2014 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 "lst.h"
#include "matrix.h"
#include "mul.h"
-#include "ncmul.h"
-#include "numeric.h"
#include "power.h"
+#include "operators.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
+#include "symmetry.h"
#include "utils.h"
+#include <stdexcept>
+#include <vector>
+
namespace GiNaC {
+//////////
+// complex conjugate
+//////////
+
+static ex conjugate_evalf(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg)) {
+ return ex_to<numeric>(arg).conjugate();
+ }
+ return conjugate_function(arg).hold();
+}
+
+static ex conjugate_eval(const ex & arg)
+{
+ return arg.conjugate();
+}
+
+static void conjugate_print_latex(const ex & arg, const print_context & c)
+{
+ c.s << "\\bar{"; arg.print(c); c.s << "}";
+}
+
+static ex conjugate_conjugate(const ex & arg)
+{
+ 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();
+}
+
+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).
+ 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;
+}
+
+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
//////////
static ex abs_evalf(const ex & arg)
{
- BEGIN_TYPECHECK
- TYPECHECK(arg,numeric)
- END_TYPECHECK(abs(arg))
+ if (is_exactly_a<numeric>(arg))
+ return abs(ex_to<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));
+ if (is_exactly_a<numeric>(arg))
+ return abs(ex_to<numeric>(arg));
+
+ if (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 void abs_print_latex(const ex & arg, const print_context & c)
+{
+ c.s << "{|"; arg.print(c); c.s << "|}";
+}
+
+static void abs_print_csrc_float(const ex & arg, const print_context & c)
+{
+ c.s << "fabs("; arg.print(c); c.s << ")";
+}
+
+static ex abs_conjugate(const ex & 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 (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
+ 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));
+ evalf_func(abs_evalf).
+ expand_func(abs_expand).
+ 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).
+ 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_evalf(const ex & arg)
{
- BEGIN_TYPECHECK
- TYPECHECK(arg,numeric)
- END_TYPECHECK(csgn(arg))
+ if (is_exactly_a<numeric>(arg))
+ return csgn(ex_to<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));
+ if (is_exactly_a<numeric>(arg))
+ 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));
+ 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)
// csgn(42*x) -> csgn(x)
int order,
unsigned options)
{
- const ex arg_pt = arg.subs(rel);
+ 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()
+ && 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()));
+ seq.push_back(expair(csgn(arg_pt), _ex0));
return pseries(rel,seq);
}
+static ex csgn_conjugate(const ex& 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));
+ series_func(csgn_series).
+ conjugate_func(csgn_conjugate).
+ real_part_func(csgn_real_part).
+ imag_part_func(csgn_imag_part).
+ power_func(csgn_power));
//////////
-// Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
+// 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)
+static ex eta_evalf(const ex &x, const ex &y)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- TYPECHECK(y,numeric)
- END_TYPECHECK(eta(x,y))
-
- numeric xim = imag(ex_to_numeric(x));
- numeric yim = imag(ex_to_numeric(y));
- numeric xyim = imag(ex_to_numeric(x*y));
- return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+ // 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)
+static ex eta_eval(const ex &x, const ex &y)
{
- if (is_ex_exactly_of_type(x, numeric) &&
- is_ex_exactly_of_type(y, numeric)) {
+ // 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()!
- numeric xim = imag(ex_to_numeric(x));
- numeric yim = imag(ex_to_numeric(y));
- numeric xyim = imag(ex_to_numeric(x*y));
- return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+ 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 & arg1,
- const ex & arg2,
+static ex eta_series(const ex & x, const ex & y,
const relational & rel,
int order,
unsigned options)
{
- const ex arg1_pt = arg1.subs(rel);
- const ex arg2_pt = arg2.subs(rel);
- if (ex_to_numeric(arg1_pt).imag().is_zero() ||
- ex_to_numeric(arg2_pt).imag().is_zero() ||
- ex_to_numeric(arg1_pt*arg2_pt).imag().is_zero()) {
- throw (std::domain_error("eta_series(): on discontinuity"));
- }
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ const ex y_pt = y.subs(rel, subs_options::no_pattern);
+ 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(arg1_pt,arg2_pt), _ex0()));
+ seq.push_back(expair(eta(x_pt,y_pt), _ex0));
return pseries(rel,seq);
}
+static ex eta_conjugate(const ex & x, const ex & 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).
evalf_func(eta_evalf).
series_func(eta_series).
- latex_name("\\eta"));
+ latex_name("\\eta").
+ set_symmetry(sy_symm(0, 1)).
+ conjugate_func(eta_conjugate).
+ real_part_func(eta_real_part).
+ imag_part_func(eta_imag_part));
//////////
static ex Li2_evalf(const ex & x)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(Li2(x))
+ if (is_exactly_a<numeric>(x))
+ return Li2(ex_to<numeric>(x));
- return Li2(ex_to_numeric(x)); // -> numeric Li2(numeric)
+ 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();
+ return _ex0;
// Li2(1) -> Pi^2/6
- if (x.is_equal(_ex1()))
- return power(Pi,_ex2())/_ex6();
+ 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();
+ 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();
+ 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;
+ 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;
+ return power(Pi,_ex2)/_ex_48 - Catalan*I;
// Li2(float)
if (!x.info(info_flags::crational))
- return Li2_evalf(x);
+ return Li2(ex_to<numeric>(x));
}
return Li2(x).hold();
GINAC_ASSERT(deriv_param==0);
// d/dx Li2(x) -> -log(1-x)/x
- return -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);
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
if (x_pt.info(info_flags::numeric)) {
// First special case: x==0 (derivatives have poles)
if (x_pt.is_zero()) {
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));
+ 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));
+ nseq.push_back(expair(Order(_ex1), order));
ser += pseries(rel, nseq);
// reexpanding it will collapse the series again
return ser.series(rel, order);
// obsolete!
}
// second special case: x==1 (branch point)
- if (x_pt == _ex1()) {
+ if (x_pt.is_equal(_ex1)) {
// method:
// construct series manually in a dummy symbol s
const symbol s;
- ex ser = zeta(2);
+ 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));
+ 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));
+ 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) {
+ 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 = static_cast<symbol *>(rel.lhs().bp);
+ 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()));
+ 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));
+ ex replarg = series(Li2(x), s==foo, order);
+ for (size_t 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, subs_options::no_pattern),i));
// append an order term:
- seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
+ seq.push_back(expair(Order(_ex1), replarg.nops()-1));
return pseries(rel, seq);
}
}
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)
+//////////
+
+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).hold();
+ }
+
+ return zetaderiv(n, x).hold();
+}
+
+static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param<2);
+
+ if (deriv_param==0) {
+ // d/dn zeta(n,x)
+ throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
+ }
+ // d/dx psi(n,x)
+ return zetaderiv(n+1,x);
+}
+
+REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
+ derivative_func(zetaderiv_deriv).
+ latex_name("\\zeta^\\prime"));
//////////
// factorial
static ex factorial_eval(const ex & x)
{
- if (is_ex_exactly_of_type(x, numeric))
- return factorial(ex_to_numeric(x));
+ if (is_exactly_a<numeric>(x))
+ return factorial(ex_to<numeric>(x));
else
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).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));
+ evalf_func(factorial_evalf).
+ 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
return binomial(x, y).hold();
}
+static ex binomial_sym(const ex & x, const numeric & y)
+{
+ if (y.is_integer()) {
+ if (y.is_nonneg_integer()) {
+ const unsigned N = y.to_int();
+ 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 _ex0;
+ }
+
+ 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
+ if (is_exactly_a<numeric>(y)) {
+ if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
+ return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
+ else
+ return binomial_sym(x, ex_to<numeric>(y));
+ } else
return binomial(x, y).hold();
}
+// At the moment the numeric evaluation of a binomail 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)
+{
+ 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));
+ evalf_func(binomial_evalf).
+ 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_eval(const ex & x)
{
- if (is_ex_exactly_of_type(x, numeric)) {
+ if (is_exactly_a<numeric>(x)) {
// O(c) -> O(1) or 0
if (!x.is_zero())
- return Order(_ex1()).hold();
+ return Order(_ex1).hold();
else
- return _ex0();
- } else if (is_ex_exactly_of_type(x, mul)) {
- mul *m = static_cast<mul *>(x.bp);
+ return _ex0;
+ } else if (is_exactly_a<mul>(x)) {
+ 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();
+ if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
+ return Order(x / m.op(m.nops() - 1)).hold();
}
return Order(x).hold();
}
{
// Just wrap the function into a pseries object
epvector new_seq;
- GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
- const symbol *s = static_cast<symbol *>(r.lhs().bp);
- new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
+ 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);
}
-// 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}"));
+static ex Order_conjugate(const ex & x)
+{
+ return Order(x).hold();
+}
-//////////
-// Inert partial differentiation operator
-//////////
+static ex Order_real_part(const ex & x)
+{
+ return Order(x).hold();
+}
-static ex Derivative_eval(const ex & f, const ex & l)
+static ex Order_imag_part(const ex & x)
{
- if (!is_ex_exactly_of_type(f, function)) {
- throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
- }
- if (!is_ex_exactly_of_type(l, lst)) {
- throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
- }
- return Derivative(f, l).hold();
+ if(x.info(info_flags::real))
+ return 0;
+ return Order(x).hold();
}
-REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
+// 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).
+ real_part_func(Order_real_part).
+ imag_part_func(Order_imag_part));
//////////
// Solve linear system
//////////
-ex lsolve(const ex &eqns, const ex &symbols)
+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"));
- ex sol=lsolve(lst(eqns),lst(symbols));
+ const ex sol = lsolve(lst(eqns),lst(symbols));
GINAC_ASSERT(sol.nops()==1);
- GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
+ 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"));
+ throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
}
- for (unsigned i=0; i<eqns.nops(); i++) {
+ for (size_t 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"));
+ throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
}
- for (unsigned i=0; i<symbols.nops(); i++) {
+ for (size_t 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"));
}
matrix rhs(eqns.nops(),1);
matrix vars(symbols.nops(),1);
- for (unsigned r=0; r<eqns.nops(); r++) {
- ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+ for (size_t 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++) {
- ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
+ for (size_t 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;
}
}
// test if system is linear and fill vars matrix
- for (unsigned i=0; i<symbols.nops(); i++) {
+ for (size_t 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"));
matrix solution;
try {
- solution = sys.solve(vars,rhs);
+ 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++)
+ for (size_t i=0; i<symbols.nops(); i++)
sollist.append(symbols.op(i)==solution(i,0));
return sollist;
}
-// Symmetrize/antisymmetrize over a vector of objects
-static ex symm(const ex & e, exvector::const_iterator first, exvector::const_iterator last, bool asymmetric)
-{
- // Need at least 2 objects for this operation
- int num = last - first;
- if (num < 2)
- return e;
-
- // Sort object vector, transform it into a list, and make a copy so we
- // will know which objects get substituted for which
- exlist iv_lst;
- iv_lst.insert(iv_lst.begin(), first, last);
- shaker_sort(iv_lst.begin(), iv_lst.end(), ex_is_less());
- lst orig_lst(iv_lst);
-
- // Loop over all permutations (the first permutation, which is the
- // identity, is unrolled)
- ex sum = e;
- while (next_permutation(iv_lst.begin(), iv_lst.end(), ex_is_less())) {
- ex term = e.subs(orig_lst, lst(iv_lst));
- if (asymmetric) {
- exlist test_lst = iv_lst;
- term *= permutation_sign(test_lst.begin(), test_lst.end(), ex_is_less());
- }
- sum += term;
- }
- return sum / factorial(numeric(num));
-}
+//////////
+// Find real root of f(x) numerically
+//////////
-ex symmetrize(const ex & e, exvector::const_iterator first, exvector::const_iterator last)
+const numeric
+fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
{
- return symm(e, first, last, false);
-}
+ 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");
+ }
-ex antisymmetrize(const ex & e, exvector::const_iterator first, exvector::const_iterator last)
-{
- return symm(e, first, last, true);
-}
+ // 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_);
+ // 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;
+ fx[side] = fxprev;
+ side = !side;
+ xxprev = xx[side];
+ fxprev = fx[side];
-/** Symmetrize expression over a list of objects (symbols, indices). */
-ex ex::symmetrize(const lst & l) const
-{
- exvector v;
- v.reserve(l.nops());
- for (unsigned i=0; i<l.nops(); i++)
- v.push_back(l.op(i));
- return symm(*this, v.begin(), v.end(), false);
-}
+ 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_);
-/** Antisymmetrize expression over a list of objects (symbols, indices). */
-ex ex::antisymmetrize(const lst & l) const
-{
- exvector v;
- v.reserve(l.nops());
- for (unsigned i=0; i<l.nops(); i++)
- v.push_back(l.op(i));
- return symm(*this, v.begin(), v.end(), true);
+ 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 initcns_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;
+
+/* 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;
+unsigned force_include_zeta1 = zeta1_SERIAL::serial;
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff