* functions. */
/*
- * GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2004 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
#include "constant.h"
#include "numeric.h"
#include "power.h"
+#include "operators.h"
#include "relational.h"
#include "symbol.h"
#include "pseries.h"
if (x.is_zero()) {
return _ex1;
}
+
// exp(n*Pi*I/2) -> {+1|+I|-1|-I}
const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
if (TwoExOverPiI.info(info_flags::integer)) {
if (z.is_equal(_num3))
return ex(-I);
}
+
// exp(log(x)) -> x
if (is_ex_the_function(x, log))
return x.op(0);
- // exp(float)
+ // exp(float) -> float
if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
return exp(ex_to<numeric>(x));
return (Pi*I*_num1_2);
if (x.is_equal(-I)) // log(-I) -> -Pi*I/2
return (Pi*I*_num_1_2);
- // log(float)
+
+ // log(float) -> float
if (!x.info(info_flags::crational))
return log(ex_to<numeric>(x));
}
+
// log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
if (is_ex_the_function(x, exp)) {
const ex &t = x.op(0);
+ if (is_a<symbol>(t) && t.info(info_flags::real)) {
+ return t;
+ }
if (t.info(info_flags::numeric)) {
const numeric &nt = ex_to<numeric>(t);
if (nt.is_real())
return power(x, _ex_1);
}
-// This is a strange workaround for a compiliation problem with the try statement
-// below. With -O1 the exception is not caucht properly as of GCC-2.95.2, at
-// least on i386. Version 2.95.4 seems to have fixed this silly problem, though.
-// Funnily, with a simple extern declaration here it mysteriously works again.
-#if defined(__GNUC__) && (__GNUC__==2)
-extern "C" int putchar(int);
-#endif
-
static ex log_series(const ex &arg,
const relational &rel,
int order,
bool must_expand_arg = false;
// maybe substitution of rel into arg fails because of a pole
try {
- arg_pt = arg.subs(rel);
+ arg_pt = arg.subs(rel, subs_options::no_pattern);
} catch (pole_error) {
must_expand_arg = true;
}
// in this case n more (or less) terms are needed
// (sadly, to generate them, we have to start from the beginning)
const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
+ if (n == 0 && coeff == 1) {
+ epvector epv;
+ ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
+ epv.reserve(2);
+ epv.push_back(expair(-1, _ex0));
+ epv.push_back(expair(Order(_ex1), order));
+ ex rest = pseries(rel, epv).add_series(argser);
+ for (int i = order-1; i>0; --i) {
+ epvector cterm;
+ cterm.reserve(1);
+ cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
+ acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
+ acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
+ }
+ return acc;
+ }
return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
} else // it was a monomial
return pseries(rel, seq);
const symbol &s = ex_to<symbol>(rel.lhs());
const ex &point = rel.rhs();
const symbol foo;
- const ex replarg = series(log(arg), s==foo, order).subs(foo==point);
+ const ex replarg = series(log(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
epvector seq;
seq.push_back(expair(-I*csgn(arg*I)*Pi, _ex0));
seq.push_back(expair(Order(_ex1), order));
if (z.is_equal(_num30)) // sin(Pi/2) -> 1
return sign;
}
-
+
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// sin(asin(x)) -> x
if (is_ex_the_function(x, asin))
return t;
+
// sin(acos(x)) -> sqrt(1-x^2)
if (is_ex_the_function(x, acos))
return sqrt(_ex1-power(t,_ex2));
+
// sin(atan(x)) -> x/sqrt(1+x^2)
if (is_ex_the_function(x, atan))
return t*power(_ex1+power(t,_ex2),_ex_1_2);
// sin(float) -> float
if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
return sin(ex_to<numeric>(x));
+
+ // sin() is odd
+ if (x.info(info_flags::negative))
+ return -sin(-x);
return sin(x).hold();
}
if (z.is_equal(_num30)) // cos(Pi/2) -> 0
return _ex0;
}
-
+
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// cos(acos(x)) -> x
if (is_ex_the_function(x, acos))
return t;
+
// cos(asin(x)) -> sqrt(1-x^2)
if (is_ex_the_function(x, asin))
return sqrt(_ex1-power(t,_ex2));
+
// cos(atan(x)) -> 1/sqrt(1+x^2)
if (is_ex_the_function(x, atan))
return power(_ex1+power(t,_ex2),_ex_1_2);
if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
return cos(ex_to<numeric>(x));
+ // cos() is even
+ if (x.info(info_flags::negative))
+ return cos(-x);
+
return cos(x).hold();
}
if (z.is_equal(_num30)) // tan(Pi/2) -> infinity
throw (pole_error("tan_eval(): simple pole",1));
}
-
+
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// tan(atan(x)) -> x
if (is_ex_the_function(x, atan))
return t;
+
// tan(asin(x)) -> x/sqrt(1+x^2)
if (is_ex_the_function(x, asin))
return t*power(_ex1-power(t,_ex2),_ex_1_2);
+
// tan(acos(x)) -> sqrt(1-x^2)/x
if (is_ex_the_function(x, acos))
return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
return tan(ex_to<numeric>(x));
}
+ // tan() is odd
+ if (x.info(info_flags::negative))
+ return -tan(-x);
+
return tan(x).hold();
}
// method:
// Taylor series where there is no pole falls back to tan_deriv.
// On a pole simply expand sin(x)/cos(x).
- const ex x_pt = x.subs(rel);
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
if (!(2*x_pt/Pi).info(info_flags::odd))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole
- return (sin(x)/cos(x)).series(rel, order+2, options);
+ return (sin(x)/cos(x)).series(rel, order, options);
}
REGISTER_FUNCTION(tan, eval_func(tan_eval).
static ex asin_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// asin(0) -> 0
if (x.is_zero())
return x;
+
// asin(1/2) -> Pi/6
if (x.is_equal(_ex1_2))
return numeric(1,6)*Pi;
+
// asin(1) -> Pi/2
if (x.is_equal(_ex1))
return _num1_2*Pi;
+
// asin(-1/2) -> -Pi/6
if (x.is_equal(_ex_1_2))
return numeric(-1,6)*Pi;
+
// asin(-1) -> -Pi/2
if (x.is_equal(_ex_1))
return _num_1_2*Pi;
+
// asin(float) -> float
if (!x.info(info_flags::crational))
return asin(ex_to<numeric>(x));
+
+ // asin() is odd
+ if (x.info(info_flags::negative))
+ return -asin(-x);
}
return asin(x).hold();
static ex acos_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// acos(1) -> 0
if (x.is_equal(_ex1))
return _ex0;
+
// acos(1/2) -> Pi/3
if (x.is_equal(_ex1_2))
return _ex1_3*Pi;
+
// acos(0) -> Pi/2
if (x.is_zero())
return _ex1_2*Pi;
+
// acos(-1/2) -> 2/3*Pi
if (x.is_equal(_ex_1_2))
return numeric(2,3)*Pi;
+
// acos(-1) -> Pi
if (x.is_equal(_ex_1))
return Pi;
+
// acos(float) -> float
if (!x.info(info_flags::crational))
return acos(ex_to<numeric>(x));
+
+ // acos(-x) -> Pi-acos(x)
+ if (x.info(info_flags::negative))
+ return Pi-acos(-x);
}
return acos(x).hold();
static ex atan_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// atan(0) -> 0
if (x.is_zero())
return _ex0;
+
// atan(1) -> Pi/4
if (x.is_equal(_ex1))
return _ex1_4*Pi;
+
// atan(-1) -> -Pi/4
if (x.is_equal(_ex_1))
return _ex_1_4*Pi;
+
if (x.is_equal(I) || x.is_equal(-I))
throw (pole_error("atan_eval(): logarithmic pole",0));
+
// atan(float) -> float
if (!x.info(info_flags::crational))
return atan(ex_to<numeric>(x));
+
+ // atan() is odd
+ if (x.info(info_flags::negative))
+ return -atan(-x);
}
return atan(x).hold();
// On the branch cuts and the poles series expand
// (log(1+I*x)-log(1-I*x))/(2*I)
// instead.
- const ex arg_pt = arg.subs(rel);
+ const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
if (!(I*arg_pt).info(info_flags::real))
throw do_taylor(); // Re(x) != 0
if ((I*arg_pt).info(info_flags::real) && abs(I*arg_pt)<_ex1)
const symbol &s = ex_to<symbol>(rel.lhs());
const ex &point = rel.rhs();
const symbol foo;
- const ex replarg = series(atan(arg), s==foo, order).subs(foo==point);
+ const ex replarg = series(atan(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
ex Order0correction = replarg.op(0)+csgn(arg)*Pi*_ex_1_2;
if ((I*arg_pt)<_ex0)
Order0correction += log((I*arg_pt+_ex_1)/(I*arg_pt+_ex1))*I*_ex_1_2;
static ex atan2_evalf(const ex &y, const ex &x)
{
if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
- return atan2(ex_to<numeric>(y), ex_to<numeric>(x));
+ return atan(ex_to<numeric>(y), ex_to<numeric>(x));
return atan2(y, x).hold();
}
static ex atan2_eval(const ex & y, const ex & x)
{
- if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
- x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
- return atan2_evalf(y,x);
+ if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) {
+
+ if (y.is_zero()) {
+
+ // atan(0, 0) -> 0
+ if (x.is_zero())
+ return _ex0;
+
+ // atan(0, x), x real and positive -> 0
+ if (x.info(info_flags::positive))
+ return _ex0;
+
+ // atan(0, x), x real and negative -> -Pi
+ if (x.info(info_flags::negative))
+ return _ex_1*Pi;
+ }
+
+ if (x.is_zero()) {
+
+ // atan(y, 0), y real and positive -> Pi/2
+ if (y.info(info_flags::positive))
+ return _ex1_2*Pi;
+
+ // atan(y, 0), y real and negative -> -Pi/2
+ if (y.info(info_flags::negative))
+ return _ex_1_2*Pi;
+ }
+
+ if (y.is_equal(x)) {
+
+ // atan(y, y), y real and positive -> Pi/4
+ if (y.info(info_flags::positive))
+ return _ex1_4*Pi;
+
+ // atan(y, y), y real and negative -> -3/4*Pi
+ if (y.info(info_flags::negative))
+ return numeric(-3, 4)*Pi;
+ }
+
+ if (y.is_equal(-x)) {
+
+ // atan(y, -y), y real and positive -> 3*Pi/4
+ if (y.info(info_flags::positive))
+ return numeric(3, 4)*Pi;
+
+ // atan(y, -y), y real and negative -> -Pi/4
+ if (y.info(info_flags::negative))
+ return _ex_1_4*Pi;
+ }
+
+ // atan(float, float) -> float
+ if (!y.info(info_flags::crational) && !x.info(info_flags::crational))
+ return atan(ex_to<numeric>(y), ex_to<numeric>(x));
+
+ // atan(real, real) -> atan(y/x) +/- Pi
+ if (y.info(info_flags::real) && x.info(info_flags::real)) {
+ if (x.info(info_flags::positive))
+ return atan(y/x);
+ else if(y.info(info_flags::positive))
+ return atan(y/x)+Pi;
+ else
+ return atan(y/x)-Pi;
+ }
}
-
- return atan2(y,x).hold();
+
+ return atan2(y, x).hold();
}
static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
static ex sinh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- if (x.is_zero()) // sinh(0) -> 0
+
+ // sinh(0) -> 0
+ if (x.is_zero())
return _ex0;
- if (!x.info(info_flags::crational)) // sinh(float) -> float
+
+ // sinh(float) -> float
+ if (!x.info(info_flags::crational))
return sinh(ex_to<numeric>(x));
+
+ // sinh() is odd
+ if (x.info(info_flags::negative))
+ return -sinh(-x);
}
if ((x/Pi).info(info_flags::numeric) &&
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// sinh(asinh(x)) -> x
if (is_ex_the_function(x, asinh))
return t;
+
// sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
if (is_ex_the_function(x, acosh))
return sqrt(t-_ex1)*sqrt(t+_ex1);
+
// sinh(atanh(x)) -> x/sqrt(1-x^2)
if (is_ex_the_function(x, atanh))
return t*power(_ex1-power(t,_ex2),_ex_1_2);
static ex cosh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- if (x.is_zero()) // cosh(0) -> 1
+
+ // cosh(0) -> 1
+ if (x.is_zero())
return _ex1;
- if (!x.info(info_flags::crational)) // cosh(float) -> float
+
+ // cosh(float) -> float
+ if (!x.info(info_flags::crational))
return cosh(ex_to<numeric>(x));
+
+ // cosh() is even
+ if (x.info(info_flags::negative))
+ return cosh(-x);
}
if ((x/Pi).info(info_flags::numeric) &&
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// cosh(acosh(x)) -> x
if (is_ex_the_function(x, acosh))
return t;
+
// cosh(asinh(x)) -> sqrt(1+x^2)
if (is_ex_the_function(x, asinh))
return sqrt(_ex1+power(t,_ex2));
+
// cosh(atanh(x)) -> 1/sqrt(1-x^2)
if (is_ex_the_function(x, atanh))
return power(_ex1-power(t,_ex2),_ex_1_2);
static ex tanh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
- if (x.is_zero()) // tanh(0) -> 0
+
+ // tanh(0) -> 0
+ if (x.is_zero())
return _ex0;
- if (!x.info(info_flags::crational)) // tanh(float) -> float
+
+ // tanh(float) -> float
+ if (!x.info(info_flags::crational))
return tanh(ex_to<numeric>(x));
+
+ // tanh() is odd
+ if (x.info(info_flags::negative))
+ return -tanh(-x);
}
if ((x/Pi).info(info_flags::numeric) &&
if (is_exactly_a<function>(x)) {
const ex &t = x.op(0);
+
// tanh(atanh(x)) -> x
if (is_ex_the_function(x, atanh))
return t;
+
// tanh(asinh(x)) -> x/sqrt(1+x^2)
if (is_ex_the_function(x, asinh))
return t*power(_ex1+power(t,_ex2),_ex_1_2);
+
// tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
if (is_ex_the_function(x, acosh))
return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
// method:
// Taylor series where there is no pole falls back to tanh_deriv.
// On a pole simply expand sinh(x)/cosh(x).
- const ex x_pt = x.subs(rel);
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
if (!(2*I*x_pt/Pi).info(info_flags::odd))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole
- return (sinh(x)/cosh(x)).series(rel, order+2, options);
+ return (sinh(x)/cosh(x)).series(rel, order, options);
}
REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
static ex asinh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// asinh(0) -> 0
if (x.is_zero())
return _ex0;
+
// asinh(float) -> float
if (!x.info(info_flags::crational))
return asinh(ex_to<numeric>(x));
+
+ // asinh() is odd
+ if (x.info(info_flags::negative))
+ return -asinh(-x);
}
return asinh(x).hold();
static ex acosh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// acosh(0) -> Pi*I/2
if (x.is_zero())
return Pi*I*numeric(1,2);
+
// acosh(1) -> 0
if (x.is_equal(_ex1))
return _ex0;
+
// acosh(-1) -> Pi*I
if (x.is_equal(_ex_1))
return Pi*I;
+
// acosh(float) -> float
if (!x.info(info_flags::crational))
return acosh(ex_to<numeric>(x));
+
+ // acosh(-x) -> Pi*I-acosh(x)
+ if (x.info(info_flags::negative))
+ return Pi*I-acosh(-x);
}
return acosh(x).hold();
static ex atanh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
+
// atanh(0) -> 0
if (x.is_zero())
return _ex0;
+
// atanh({+|-}1) -> throw
if (x.is_equal(_ex1) || x.is_equal(_ex_1))
throw (pole_error("atanh_eval(): logarithmic pole",0));
+
// atanh(float) -> float
if (!x.info(info_flags::crational))
return atanh(ex_to<numeric>(x));
+
+ // atanh() is odd
+ if (x.info(info_flags::negative))
+ return -atanh(-x);
}
return atanh(x).hold();
// On the branch cuts and the poles series expand
// (log(1+x)-log(1-x))/2
// instead.
- const ex arg_pt = arg.subs(rel);
+ const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
if (!(arg_pt).info(info_flags::real))
throw do_taylor(); // Im(x) != 0
if ((arg_pt).info(info_flags::real) && abs(arg_pt)<_ex1)
const symbol &s = ex_to<symbol>(rel.lhs());
const ex &point = rel.rhs();
const symbol foo;
- const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point);
+ const ex replarg = series(atanh(arg), s==foo, order).subs(foo==point, subs_options::no_pattern);
ex Order0correction = replarg.op(0)+csgn(I*arg)*Pi*I*_ex1_2;
if (arg_pt<_ex0)
Order0correction += log((arg_pt+_ex_1)/(arg_pt+_ex1))*_ex1_2;
git://www.ginac.de / ginac.git / blobdiff