/** @file numeric.cpp
*
* This file contains the interface to the underlying bignum package.
* Its most important design principle is to completely hide the inner
* working of that other package from the user of GiNaC. It must either
* provide implementation of arithmetic operators and numerical evaluation
* of special functions or implement the interface to the bignum package. */
/*
* GiNaC Copyright (C) 1999-2007 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
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include "config.h"
#include <vector>
#include <stdexcept>
#include <string>
#include <sstream>
#include <limits>
#include "numeric.h"
#include "ex.h"
#include "operators.h"
#include "archive.h"
#include "tostring.h"
#include "utils.h"
// CLN should pollute the global namespace as little as possible. Hence, we
// include most of it here and include only the part needed for properly
// declaring cln::cl_number in numeric.h. This can only be safely done in
// namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
// subset of CLN, so we don't include the complete <cln/cln.h> but only the
// essential stuff:
#include <cln/output.h>
#include <cln/integer_io.h>
#include <cln/integer_ring.h>
#include <cln/rational_io.h>
#include <cln/rational_ring.h>
#include <cln/lfloat_class.h>
#include <cln/lfloat_io.h>
#include <cln/real_io.h>
#include <cln/real_ring.h>
#include <cln/complex_io.h>
#include <cln/complex_ring.h>
#include <cln/numtheory.h>
namespace GiNaC {
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(numeric, basic,
print_func<print_context>(&numeric::do_print).
print_func<print_latex>(&numeric::do_print_latex).
print_func<print_csrc>(&numeric::do_print_csrc).
print_func<print_csrc_cl_N>(&numeric::do_print_csrc_cl_N).
print_func<print_tree>(&numeric::do_print_tree).
print_func<print_python_repr>(&numeric::do_print_python_repr))
//////////
// default constructor
//////////
/** default ctor. Numerically it initializes to an integer zero. */
numeric::numeric() : basic(TINFO_numeric)
{
value = cln::cl_I(0);
setflag(status_flags::evaluated | status_flags::expanded);
}
//////////
// other constructors
//////////
// public
numeric::numeric(int i) : basic(TINFO_numeric)
{
// Not the whole int-range is available if we don't cast to long
// first. This is due to the behaviour of the cl_I-ctor, which
// emphasizes efficiency. However, if the integer is small enough
// we save space and dereferences by using an immediate type.
// (C.f. <cln/object.h>)
// The #if clause prevents compiler warnings on 64bit machines where the
// comparision is always true.
#if cl_value_len >= 32
value = cln::cl_I(i);
#else
if (i < (1L << (cl_value_len-1)) && i >= -(1L << (cl_value_len-1)))
value = cln::cl_I(i);
else
value = cln::cl_I(static_cast<long>(i));
#endif
setflag(status_flags::evaluated | status_flags::expanded);
}
numeric::numeric(unsigned int i) : basic(TINFO_numeric)
{
// Not the whole uint-range is available if we don't cast to ulong
// first. This is due to the behaviour of the cl_I-ctor, which
// emphasizes efficiency. However, if the integer is small enough
// we save space and dereferences by using an immediate type.
// (C.f. <cln/object.h>)
// The #if clause prevents compiler warnings on 64bit machines where the
// comparision is always true.
#if cl_value_len >= 32
value = cln::cl_I(i);
#else
if (i < (1UL << (cl_value_len-1)))
value = cln::cl_I(i);
else
value = cln::cl_I(static_cast<unsigned long>(i));
#endif
setflag(status_flags::evaluated | status_flags::expanded);
}
numeric::numeric(long i) : basic(TINFO_numeric)
{
value = cln::cl_I(i);
setflag(status_flags::evaluated | status_flags::expanded);
}
numeric::numeric(unsigned long i) : basic(TINFO_numeric)
{
value = cln::cl_I(i);
setflag(status_flags::evaluated | status_flags::expanded);
}
/** Constructor for rational numerics a/b.
*
* @exception overflow_error (division by zero) */
numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
{
if (!denom)
throw std::overflow_error("division by zero");
value = cln::cl_I(numer) / cln::cl_I(denom);
setflag(status_flags::evaluated | status_flags::expanded);
}
numeric::numeric(double d) : basic(TINFO_numeric)
{
// We really want to explicitly use the type cl_LF instead of the
// more general cl_F, since that would give us a cl_DF only which
// will not be promoted to cl_LF if overflow occurs:
value = cln::cl_float(d, cln::default_float_format);
setflag(status_flags::evaluated | status_flags::expanded);
}
/** ctor from C-style string. It also accepts complex numbers in GiNaC
* notation like "2+5*I". */
numeric::numeric(const char *s) : basic(TINFO_numeric)
{
cln::cl_N ctorval = 0;
// parse complex numbers (functional but not completely safe, unfortunately
// std::string does not understand regexpese):
// ss should represent a simple sum like 2+5*I
std::string ss = s;
std::string::size_type delim;
// make this implementation safe by adding explicit sign
if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
ss = '+' + ss;
// We use 'E' as exponent marker in the output, but some people insist on
// writing 'e' at input, so let's substitute them right at the beginning:
while ((delim = ss.find("e"))!=std::string::npos)
ss.replace(delim,1,"E");
// main parser loop:
do {
// chop ss into terms from left to right
std::string term;
bool imaginary = false;
delim = ss.find_first_of(std::string("+-"),1);
// Do we have an exponent marker like "31.415E-1"? If so, hop on!
if (delim!=std::string::npos && ss.at(delim-1)=='E')
delim = ss.find_first_of(std::string("+-"),delim+1);
term = ss.substr(0,delim);
if (delim!=std::string::npos)
ss = ss.substr(delim);
// is the term imaginary?
if (term.find("I")!=std::string::npos) {
// erase 'I':
term.erase(term.find("I"),1);
// erase '*':
if (term.find("*")!=std::string::npos)
term.erase(term.find("*"),1);
// correct for trivial +/-I without explicit factor on I:
if (term.size()==1)
term += '1';
imaginary = true;
}
if (term.find('.')!=std::string::npos || term.find('E')!=std::string::npos) {
// CLN's short type cl_SF is not very useful within the GiNaC
// framework where we are mainly interested in the arbitrary
// precision type cl_LF. Hence we go straight to the construction
// of generic floats. In order to create them we have to convert
// our own floating point notation used for output and construction
// from char * to CLN's generic notation:
// 3.14 --> 3.14e0_<Digits>
// 31.4E-1 --> 31.4e-1_<Digits>
// and s on.
// No exponent marker? Let's add a trivial one.
if (term.find("E")==std::string::npos)
term += "E0";
// E to lower case
term = term.replace(term.find("E"),1,"e");
// append _<Digits> to term
term += "_" + ToString((unsigned)Digits);
// construct float using cln::cl_F(const char *) ctor.
if (imaginary)
ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
else
ctorval = ctorval + cln::cl_F(term.c_str());
} else {
// this is not a floating point number...
if (imaginary)
ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
else
ctorval = ctorval + cln::cl_R(term.c_str());
}
} while (delim != std::string::npos);
value = ctorval;
setflag(status_flags::evaluated | status_flags::expanded);
}
/** Ctor from CLN types. This is for the initiated user or internal use
* only. */
numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
{
value = z;
setflag(status_flags::evaluated | status_flags::expanded);
}
//////////
// archiving
//////////
numeric::numeric(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
{
cln::cl_N ctorval = 0;
// Read number as string
std::string str;
if (n.find_string("number", str)) {
std::istringstream s(str.c_str());
cln::cl_idecoded_float re, im;
char c;
s.get(c);
switch (c) {
case 'R': // Integer-decoded real number
s >> re.sign >> re.mantissa >> re.exponent;
ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
break;
case 'C': // Integer-decoded complex number
s >> re.sign >> re.mantissa >> re.exponent;
s >> im.sign >> im.mantissa >> im.exponent;
ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
break;
default: // Ordinary number
s.putback(c);
s >> ctorval;
break;
}
}
value = ctorval;
setflag(status_flags::evaluated | status_flags::expanded);
}
void numeric::archive(archive_node &n) const
{
inherited::archive(n);
// Write number as string
std::ostringstream s;
if (this->is_crational())
s << value;
else {
// Non-rational numbers are written in an integer-decoded format
// to preserve the precision
if (this->is_real()) {
cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
s << "R";
s << re.sign << " " << re.mantissa << " " << re.exponent;
} else {
cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
s << "C";
s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
s << im.sign << " " << im.mantissa << " " << im.exponent;
}
}
n.add_string("number", s.str());
}
DEFAULT_UNARCHIVE(numeric)
//////////
// functions overriding virtual functions from base classes
//////////
/** Helper function to print a real number in a nicer way than is CLN's
* default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
* and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
* long as it only uses cl_LF and no other floating point types that we might
* want to visibly distinguish from cl_LF.
*
* @see numeric::print() */
static void print_real_number(const print_context & c, const cln::cl_R & x)
{
cln::cl_print_flags ourflags;
if (cln::instanceof(x, cln::cl_RA_ring)) {
// case 1: integer or rational
if (cln::instanceof(x, cln::cl_I_ring) ||
!is_a<print_latex>(c)) {
cln::print_real(c.s, ourflags, x);
} else { // rational output in LaTeX context
if (x < 0)
c.s << "-";
c.s << "\\frac{";
cln::print_real(c.s, ourflags, cln::abs(cln::numerator(cln::the<cln::cl_RA>(x))));
c.s << "}{";
cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
c.s << '}';
}
} else {
// case 2: float
// make CLN believe this number has default_float_format, so it prints
// 'E' as exponent marker instead of 'L':
ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
cln::print_real(c.s, ourflags, x);
}
}
/** Helper function to print integer number in C++ source format.
*
* @see numeric::print() */
static void print_integer_csrc(const print_context & c, const cln::cl_I & x)
{
// Print small numbers in compact float format, but larger numbers in
// scientific format
const int max_cln_int = 536870911; // 2^29-1
if (x >= cln::cl_I(-max_cln_int) && x <= cln::cl_I(max_cln_int))
c.s << cln::cl_I_to_int(x) << ".0";
else
c.s << cln::double_approx(x);
}
/** Helper function to print real number in C++ source format.
*
* @see numeric::print() */
static void print_real_csrc(const print_context & c, const cln::cl_R & x)
{
if (cln::instanceof(x, cln::cl_I_ring)) {
// Integer number
print_integer_csrc(c, cln::the<cln::cl_I>(x));
} else if (cln::instanceof(x, cln::cl_RA_ring)) {
// Rational number
const cln::cl_I numer = cln::numerator(cln::the<cln::cl_RA>(x));
const cln::cl_I denom = cln::denominator(cln::the<cln::cl_RA>(x));
if (cln::plusp(x) > 0) {
c.s << "(";
print_integer_csrc(c, numer);
} else {
c.s << "-(";
print_integer_csrc(c, -numer);
}
c.s << "/";
print_integer_csrc(c, denom);
c.s << ")";
} else {
// Anything else
c.s << cln::double_approx(x);
}
}
/** Helper function to print real number in C++ source format using cl_N types.
*
* @see numeric::print() */
static void print_real_cl_N(const print_context & c, const cln::cl_R & x)
{
if (cln::instanceof(x, cln::cl_I_ring)) {
// Integer number
c.s << "cln::cl_I(\"";
print_real_number(c, x);
c.s << "\")";
} else if (cln::instanceof(x, cln::cl_RA_ring)) {
// Rational number
cln::cl_print_flags ourflags;
c.s << "cln::cl_RA(\"";
cln::print_rational(c.s, ourflags, cln::the<cln::cl_RA>(x));
c.s << "\")";
} else {
// Anything else
c.s << "cln::cl_F(\"";
print_real_number(c, cln::cl_float(1.0, cln::default_float_format) * x);
c.s << "_" << Digits << "\")";
}
}
void numeric::print_numeric(const print_context & c, const char *par_open, const char *par_close, const char *imag_sym, const char *mul_sym, unsigned level) const
{
const cln::cl_R r = cln::realpart(value);
const cln::cl_R i = cln::imagpart(value);
if (cln::zerop(i)) {
// case 1, real: x or -x
if ((precedence() <= level) && (!this->is_nonneg_integer())) {
c.s << par_open;
print_real_number(c, r);
c.s << par_close;
} else {
print_real_number(c, r);
}
} else {
if (cln::zerop(r)) {
// case 2, imaginary: y*I or -y*I
if (i == 1)
c.s << imag_sym;
else {
if (precedence()<=level)
c.s << par_open;
if (i == -1)
c.s << "-" << imag_sym;
else {
print_real_number(c, i);
c.s << mul_sym << imag_sym;
}
if (precedence()<=level)
c.s << par_close;
}
} else {
// case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
if (precedence() <= level)
c.s << par_open;
print_real_number(c, r);
if (i < 0) {
if (i == -1) {
c.s << "-" << imag_sym;
} else {
print_real_number(c, i);
c.s << mul_sym << imag_sym;
}
} else {
if (i == 1) {
c.s << "+" << imag_sym;
} else {
c.s << "+";
print_real_number(c, i);
c.s << mul_sym << imag_sym;
}
}
if (precedence() <= level)
c.s << par_close;
}
}
}
void numeric::do_print(const print_context & c, unsigned level) const
{
print_numeric(c, "(", ")", "I", "*", level);
}
void numeric::do_print_latex(const print_latex & c, unsigned level) const
{
print_numeric(c, "{(", ")}", "i", " ", level);
}
void numeric::do_print_csrc(const print_csrc & c, unsigned level) const
{
std::ios::fmtflags oldflags = c.s.flags();
c.s.setf(std::ios::scientific);
int oldprec = c.s.precision();
// Set precision
if (is_a<print_csrc_double>(c))
c.s.precision(std::numeric_limits<double>::digits10 + 1);
else
c.s.precision(std::numeric_limits<float>::digits10 + 1);
if (this->is_real()) {
// Real number
print_real_csrc(c, cln::the<cln::cl_R>(value));
} else {
// Complex number
c.s << "std::complex<";
if (is_a<print_csrc_double>(c))
c.s << "double>(";
else
c.s << "float>(";
print_real_csrc(c, cln::realpart(value));
c.s << ",";
print_real_csrc(c, cln::imagpart(value));
c.s << ")";
}
c.s.flags(oldflags);
c.s.precision(oldprec);
}
void numeric::do_print_csrc_cl_N(const print_csrc_cl_N & c, unsigned level) const
{
if (this->is_real()) {
// Real number
print_real_cl_N(c, cln::the<cln::cl_R>(value));
} else {
// Complex number
c.s << "cln::complex(";
print_real_cl_N(c, cln::realpart(value));
c.s << ",";
print_real_cl_N(c, cln::imagpart(value));
c.s << ")";
}
}
void numeric::do_print_tree(const print_tree & c, unsigned level) const
{
c.s << std::string(level, ' ') << value
<< " (" << class_name() << ")" << " @" << this
<< std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
<< std::endl;
}
void numeric::do_print_python_repr(const print_python_repr & c, unsigned level) const
{
c.s << class_name() << "('";
print_numeric(c, "(", ")", "I", "*", level);
c.s << "')";
}
bool numeric::info(unsigned inf) const
{
switch (inf) {
case info_flags::numeric:
case info_flags::polynomial:
case info_flags::rational_function:
case info_flags::expanded:
return true;
case info_flags::real:
return is_real();
case info_flags::rational:
case info_flags::rational_polynomial:
return is_rational();
case info_flags::crational:
case info_flags::crational_polynomial:
return is_crational();
case info_flags::integer:
case info_flags::integer_polynomial:
return is_integer();
case info_flags::cinteger:
case info_flags::cinteger_polynomial:
return is_cinteger();
case info_flags::positive:
return is_positive();
case info_flags::negative:
return is_negative();
case info_flags::nonnegative:
return !is_negative();
case info_flags::posint:
return is_pos_integer();
case info_flags::negint:
return is_integer() && is_negative();
case info_flags::nonnegint:
return is_nonneg_integer();
case info_flags::even:
return is_even();
case info_flags::odd:
return is_odd();
case info_flags::prime:
return is_prime();
case info_flags::algebraic:
return !is_real();
}
return false;
}
int numeric::degree(const ex & s) const
{
return 0;
}
int numeric::ldegree(const ex & s) const
{
return 0;
}
ex numeric::coeff(const ex & s, int n) const
{
return n==0 ? *this : _ex0;
}
/** Disassemble real part and imaginary part to scan for the occurrence of a
* single number. Also handles the imaginary unit. It ignores the sign on
* both this and the argument, which may lead to what might appear as funny
* results: (2+I).has(-2) -> true. But this is consistent, since we also
* would like to have (-2+I).has(2) -> true and we want to think about the
* sign as a multiplicative factor. */
bool numeric::has(const ex &other) const
{
if (!is_exactly_a<numeric>(other))
return false;
const numeric &o = ex_to<numeric>(other);
if (this->is_equal(o) || this->is_equal(-o))
return true;
if (o.imag().is_zero()) { // e.g. scan for 3 in -3*I
if (!this->real().is_equal(*_num0_p))
if (this->real().is_equal(o) || this->real().is_equal(-o))
return true;
if (!this->imag().is_equal(*_num0_p))
if (this->imag().is_equal(o) || this->imag().is_equal(-o))
return true;
return false;
}
else {
if (o.is_equal(I)) // e.g scan for I in 42*I
return !this->is_real();
if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
if (!this->imag().is_equal(*_num0_p))
if (this->imag().is_equal(o*I) || this->imag().is_equal(-o*I))
return true;
}
return false;
}
/** Evaluation of numbers doesn't do anything at all. */
ex numeric::eval(int level) const
{
// Warning: if this is ever gonna do something, the ex ctors from all kinds
// of numbers should be checking for status_flags::evaluated.
return this->hold();
}
/** Cast numeric into a floating-point object. For example exact numeric(1) is
* returned as a 1.0000000000000000000000 and so on according to how Digits is
* currently set. In case the object already was a floating point number the
* precision is trimmed to match the currently set default.
*
* @param level ignored, only needed for overriding basic::evalf.
* @return an ex-handle to a numeric. */
ex numeric::evalf(int level) const
{
// level can safely be discarded for numeric objects.
return numeric(cln::cl_float(1.0, cln::default_float_format) * value);
}
ex numeric::conjugate() const
{
if (is_real()) {
return *this;
}
return numeric(cln::conjugate(this->value));
}
// protected
int numeric::compare_same_type(const basic &other) const
{
GINAC_ASSERT(is_exactly_a<numeric>(other));
const numeric &o = static_cast<const numeric &>(other);
return this->compare(o);
}
bool numeric::is_equal_same_type(const basic &other) const
{
GINAC_ASSERT(is_exactly_a<numeric>(other));
const numeric &o = static_cast<const numeric &>(other);
return this->is_equal(o);
}
unsigned numeric::calchash() const
{
// Base computation of hashvalue on CLN's hashcode. Note: That depends
// only on the number's value, not its type or precision (i.e. a true
// equivalence relation on numbers). As a consequence, 3 and 3.0 share
// the same hashvalue. That shouldn't really matter, though.
setflag(status_flags::hash_calculated);
hashvalue = golden_ratio_hash(cln::equal_hashcode(value));
return hashvalue;
}
//////////
// new virtual functions which can be overridden by derived classes
//////////
// none
//////////
// non-virtual functions in this class
//////////
// public
/** Numerical addition method. Adds argument to *this and returns result as
* a numeric object. */
const numeric numeric::add(const numeric &other) const
{
return numeric(value + other.value);
}
/** Numerical subtraction method. Subtracts argument from *this and returns
* result as a numeric object. */
const numeric numeric::sub(const numeric &other) const
{
return numeric(value - other.value);
}
/** Numerical multiplication method. Multiplies *this and argument and returns
* result as a numeric object. */
const numeric numeric::mul(const numeric &other) const
{
return numeric(value * other.value);
}
/** Numerical division method. Divides *this by argument and returns result as
* a numeric object.
*
* @exception overflow_error (division by zero) */
const numeric numeric::div(const numeric &other) const
{
if (cln::zerop(other.value))
throw std::overflow_error("numeric::div(): division by zero");
return numeric(value / other.value);
}
/** Numerical exponentiation. Raises *this to the power given as argument and
* returns result as a numeric object. */
const numeric numeric::power(const numeric &other) const
{
// Shortcut for efficiency and numeric stability (as in 1.0 exponent):
// trap the neutral exponent.
if (&other==_num1_p || cln::equal(other.value,_num1_p->value))
return *this;
if (cln::zerop(value)) {
if (cln::zerop(other.value))
throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
else if (cln::zerop(cln::realpart(other.value)))
throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
else if (cln::minusp(cln::realpart(other.value)))
throw std::overflow_error("numeric::eval(): division by zero");
else
return *_num0_p;
}
return numeric(cln::expt(value, other.value));
}
/** Numerical addition method. Adds argument to *this and returns result as
* a numeric object on the heap. Use internally only for direct wrapping into
* an ex object, where the result would end up on the heap anyways. */
const numeric &numeric::add_dyn(const numeric &other) const
{
// Efficiency shortcut: trap the neutral element by pointer. This hack
// is supposed to keep the number of distinct numeric objects low.
if (this==_num0_p)
return other;
else if (&other==_num0_p)
return *this;
return static_cast<const numeric &>((new numeric(value + other.value))->
setflag(status_flags::dynallocated));
}
/** Numerical subtraction method. Subtracts argument from *this and returns
* result as a numeric object on the heap. Use internally only for direct
* wrapping into an ex object, where the result would end up on the heap
* anyways. */
const numeric &numeric::sub_dyn(const numeric &other) const
{
// Efficiency shortcut: trap the neutral exponent (first by pointer). This
// hack is supposed to keep the number of distinct numeric objects low.
if (&other==_num0_p || cln::zerop(other.value))
return *this;
return static_cast<const numeric &>((new numeric(value - other.value))->
setflag(status_flags::dynallocated));
}
/** Numerical multiplication method. Multiplies *this and argument and returns
* result as a numeric object on the heap. Use internally only for direct
* wrapping into an ex object, where the result would end up on the heap
* anyways. */
const numeric &numeric::mul_dyn(const numeric &other) const
{
// Efficiency shortcut: trap the neutral element by pointer. This hack
// is supposed to keep the number of distinct numeric objects low.
if (this==_num1_p)
return other;
else if (&other==_num1_p)
return *this;
return static_cast<const numeric &>((new numeric(value * other.value))->
setflag(status_flags::dynallocated));
}
/** Numerical division method. Divides *this by argument and returns result as
* a numeric object on the heap. Use internally only for direct wrapping
* into an ex object, where the result would end up on the heap
* anyways.
*
* @exception overflow_error (division by zero) */
const numeric &numeric::div_dyn(const numeric &other) const
{
// Efficiency shortcut: trap the neutral element by pointer. This hack
// is supposed to keep the number of distinct numeric objects low.
if (&other==_num1_p)
return *this;
if (cln::zerop(cln::the<cln::cl_N>(other.value)))
throw std::overflow_error("division by zero");
return static_cast<const numeric &>((new numeric(value / other.value))->
setflag(status_flags::dynallocated));
}
/** Numerical exponentiation. Raises *this to the power given as argument and
* returns result as a numeric object on the heap. Use internally only for
* direct wrapping into an ex object, where the result would end up on the
* heap anyways. */
const numeric &numeric::power_dyn(const numeric &other) const
{
// Efficiency shortcut: trap the neutral exponent (first try by pointer, then
// try harder, since calls to cln::expt() below may return amazing results for
// floating point exponent 1.0).
if (&other==_num1_p || cln::equal(other.value, _num1_p->value))
return *this;
if (cln::zerop(value)) {
if (cln::zerop(other.value))
throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
else if (cln::zerop(cln::realpart(other.value)))
throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
else if (cln::minusp(cln::realpart(other.value)))
throw std::overflow_error("numeric::eval(): division by zero");
else
return *_num0_p;
}
return static_cast<const numeric &>((new numeric(cln::expt(value, other.value)))->
setflag(status_flags::dynallocated));
}
const numeric &numeric::operator=(int i)
{
return operator=(numeric(i));
}
const numeric &numeric::operator=(unsigned int i)
{
return operator=(numeric(i));
}
const numeric &numeric::operator=(long i)
{
return operator=(numeric(i));
}
const numeric &numeric::operator=(unsigned long i)
{
return operator=(numeric(i));
}
const numeric &numeric::operator=(double d)
{
return operator=(numeric(d));
}
const numeric &numeric::operator=(const char * s)
{
return operator=(numeric(s));
}
/** Inverse of a number. */
const numeric numeric::inverse() const
{
if (cln::zerop(value))
throw std::overflow_error("numeric::inverse(): division by zero");
return numeric(cln::recip(value));
}
/** Return the complex half-plane (left or right) in which the number lies.
* csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
* csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
*
* @see numeric::compare(const numeric &other) */
int numeric::csgn() const
{
if (cln::zerop(value))
return 0;
cln::cl_R r = cln::realpart(value);
if (!cln::zerop(r)) {
if (cln::plusp(r))
return 1;
else
return -1;
} else {
if (cln::plusp(cln::imagpart(value)))
return 1;
else
return -1;
}
}
/** This method establishes a canonical order on all numbers. For complex
* numbers this is not possible in a mathematically consistent way but we need
* to establish some order and it ought to be fast. So we simply define it
* to be compatible with our method csgn.
*
* @return csgn(*this-other)
* @see numeric::csgn() */
int numeric::compare(const numeric &other) const
{
// Comparing two real numbers?
if (cln::instanceof(value, cln::cl_R_ring) &&
cln::instanceof(other.value, cln::cl_R_ring))
// Yes, so just cln::compare them
return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
else {
// No, first cln::compare real parts...
cl_signean real_cmp = cln::compare(cln::realpart(value), cln::realpart(other.value));
if (real_cmp)
return real_cmp;
// ...and then the imaginary parts.
return cln::compare(cln::imagpart(value), cln::imagpart(other.value));
}
}
bool numeric::is_equal(const numeric &other) const
{
return cln::equal(value, other.value);
}
/** True if object is zero. */
bool numeric::is_zero() const
{
return cln::zerop(value);
}
/** True if object is not complex and greater than zero. */
bool numeric::is_positive() const
{
if (cln::instanceof(value, cln::cl_R_ring)) // real?
return cln::plusp(cln::the<cln::cl_R>(value));
return false;
}
/** True if object is not complex and less than zero. */
bool numeric::is_negative() const
{
if (cln::instanceof(value, cln::cl_R_ring)) // real?
return cln::minusp(cln::the<cln::cl_R>(value));
return false;
}
/** True if object is a non-complex integer. */
bool numeric::is_integer() const
{
return cln::instanceof(value, cln::cl_I_ring);
}
/** True if object is an exact integer greater than zero. */
bool numeric::is_pos_integer() const
{
return (cln::instanceof(value, cln::cl_I_ring) && cln::plusp(cln::the<cln::cl_I>(value)));
}
/** True if object is an exact integer greater or equal zero. */
bool numeric::is_nonneg_integer() const
{
return (cln::instanceof(value, cln::cl_I_ring) && !cln::minusp(cln::the<cln::cl_I>(value)));
}
/** True if object is an exact even integer. */
bool numeric::is_even() const
{
return (cln::instanceof(value, cln::cl_I_ring) && cln::evenp(cln::the<cln::cl_I>(value)));
}
/** True if object is an exact odd integer. */
bool numeric::is_odd() const
{
return (cln::instanceof(value, cln::cl_I_ring) && cln::oddp(cln::the<cln::cl_I>(value)));
}
/** Probabilistic primality test.
*
* @return true if object is exact integer and prime. */
bool numeric::is_prime() const
{
return (cln::instanceof(value, cln::cl_I_ring) // integer?
&& cln::plusp(cln::the<cln::cl_I>(value)) // positive?
&& cln::isprobprime(cln::the<cln::cl_I>(value)));
}
/** True if object is an exact rational number, may even be complex
* (denominator may be unity). */
bool numeric::is_rational() const
{
return cln::instanceof(value, cln::cl_RA_ring);
}
/** True if object is a real integer, rational or float (but not complex). */
bool numeric::is_real() const
{
return cln::instanceof(value, cln::cl_R_ring);
}
bool numeric::operator==(const numeric &other) const
{
return cln::equal(value, other.value);
}
bool numeric::operator!=(const numeric &other) const
{
return !cln::equal(value, other.value);
}
/** True if object is element of the domain of integers extended by I, i.e. is
* of the form a+b*I, where a and b are integers. */
bool numeric::is_cinteger() const
{
if (cln::instanceof(value, cln::cl_I_ring))
return true;
else if (!this->is_real()) { // complex case, handle n+m*I
if (cln::instanceof(cln::realpart(value), cln::cl_I_ring) &&
cln::instanceof(cln::imagpart(value), cln::cl_I_ring))
return true;
}
return false;
}
/** True if object is an exact rational number, may even be complex
* (denominator may be unity). */
bool numeric::is_crational() const
{
if (cln::instanceof(value, cln::cl_RA_ring))
return true;
else if (!this->is_real()) { // complex case, handle Q(i):
if (cln::instanceof(cln::realpart(value), cln::cl_RA_ring) &&
cln::instanceof(cln::imagpart(value), cln::cl_RA_ring))
return true;
}
return false;
}
/** Numerical comparison: less.
*
* @exception invalid_argument (complex inequality) */
bool numeric::operator<(const numeric &other) const
{
if (this->is_real() && other.is_real())
return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
throw std::invalid_argument("numeric::operator<(): complex inequality");
}
/** Numerical comparison: less or equal.
*
* @exception invalid_argument (complex inequality) */
bool numeric::operator<=(const numeric &other) const
{
if (this->is_real() && other.is_real())
return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
throw std::invalid_argument("numeric::operator<=(): complex inequality");
}
/** Numerical comparison: greater.
*
* @exception invalid_argument (complex inequality) */
bool numeric::operator>(const numeric &other) const
{
if (this->is_real() && other.is_real())
return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
throw std::invalid_argument("numeric::operator>(): complex inequality");
}
/** Numerical comparison: greater or equal.
*
* @exception invalid_argument (complex inequality) */
bool numeric::operator>=(const numeric &other) const
{
if (this->is_real() && other.is_real())
return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
throw std::invalid_argument("numeric::operator>=(): complex inequality");
}
/** Converts numeric types to machine's int. You should check with
* is_integer() if the number is really an integer before calling this method.
* You may also consider checking the range first. */
int numeric::to_int() const
{
GINAC_ASSERT(this->is_integer());
return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
}
/** Converts numeric types to machine's long. You should check with
* is_integer() if the number is really an integer before calling this method.
* You may also consider checking the range first. */
long numeric::to_long() const
{
GINAC_ASSERT(this->is_integer());
return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
}
/** Converts numeric types to machine's double. You should check with is_real()
* if the number is really not complex before calling this method. */
double numeric::to_double() const
{
GINAC_ASSERT(this->is_real());
return cln::double_approx(cln::realpart(value));
}
/** Returns a new CLN object of type cl_N, representing the value of *this.
* This method may be used when mixing GiNaC and CLN in one project.
*/
cln::cl_N numeric::to_cl_N() const
{
return value;
}
/** Real part of a number. */
const numeric numeric::real() const
{
return numeric(cln::realpart(value));
}
/** Imaginary part of a number. */
const numeric numeric::imag() const
{
return numeric(cln::imagpart(value));
}
/** Numerator. Computes the numerator of rational numbers, rationalized
* numerator of complex if real and imaginary part are both rational numbers
* (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
* cases. */
const numeric numeric::numer() const
{
if (cln::instanceof(value, cln::cl_I_ring))
return numeric(*this); // integer case
else if (cln::instanceof(value, cln::cl_RA_ring))
return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
else if (!this->is_real()) { // complex case, handle Q(i):
const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(value));
const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(value));
if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
return numeric(*this);
if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
}
}
// at least one float encountered
return numeric(*this);
}
/** Denominator. Computes the denominator of rational numbers, common integer
* denominator of complex if real and imaginary part are both rational numbers
* (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
const numeric numeric::denom() const
{
if (cln::instanceof(value, cln::cl_I_ring))
return *_num1_p; // integer case
if (cln::instanceof(value, cln::cl_RA_ring))
return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
if (!this->is_real()) { // complex case, handle Q(i):
const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(value));
const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(value));
if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
return *_num1_p;
if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
return numeric(cln::denominator(i));
if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
return numeric(cln::denominator(r));
if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
}
// at least one float encountered
return *_num1_p;
}
/** Size in binary notation. For integers, this is the smallest n >= 0 such
* that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
* 2^(n-1) <= x < 2^n.
*
* @return number of bits (excluding sign) needed to represent that number
* in two's complement if it is an integer, 0 otherwise. */
int numeric::int_length() const
{
if (cln::instanceof(value, cln::cl_I_ring))
return cln::integer_length(cln::the<cln::cl_I>(value));
else
return 0;
}
//////////
// global constants
//////////
/** Imaginary unit. This is not a constant but a numeric since we are
* natively handing complex numbers anyways, so in each expression containing
* an I it is automatically eval'ed away anyhow. */
const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
/** Exponential function.
*
* @return arbitrary precision numerical exp(x). */
const numeric exp(const numeric &x)
{
return cln::exp(x.to_cl_N());
}
/** Natural logarithm.
*
* @param x complex number
* @return arbitrary precision numerical log(x).
* @exception pole_error("log(): logarithmic pole",0) */
const numeric log(const numeric &x)
{
if (x.is_zero())
throw pole_error("log(): logarithmic pole",0);
return cln::log(x.to_cl_N());
}
/** Numeric sine (trigonometric function).
*
* @return arbitrary precision numerical sin(x). */
const numeric sin(const numeric &x)
{
return cln::sin(x.to_cl_N());
}
/** Numeric cosine (trigonometric function).
*
* @return arbitrary precision numerical cos(x). */
const numeric cos(const numeric &x)
{
return cln::cos(x.to_cl_N());
}
/** Numeric tangent (trigonometric function).
*
* @return arbitrary precision numerical tan(x). */
const numeric tan(const numeric &x)
{
return cln::tan(x.to_cl_N());
}
/** Numeric inverse sine (trigonometric function).
*
* @return arbitrary precision numerical asin(x). */
const numeric asin(const numeric &x)
{
return cln::asin(x.to_cl_N());
}
/** Numeric inverse cosine (trigonometric function).
*
* @return arbitrary precision numerical acos(x). */
const numeric acos(const numeric &x)
{
return cln::acos(x.to_cl_N());
}
/** Numeric arcustangent.
*
* @param x complex number
* @return atan(x)
* @exception pole_error("atan(): logarithmic pole",0) if x==I or x==-I. */
const numeric atan(const numeric &x)
{
if (!x.is_real() &&
x.real().is_zero() &&
abs(x.imag()).is_equal(*_num1_p))
throw pole_error("atan(): logarithmic pole",0);
return cln::atan(x.to_cl_N());
}
/** Numeric arcustangent of two arguments, analytically continued in a suitable way.
*
* @param y complex number
* @param x complex number
* @return -I*log((x+I*y)/sqrt(x^2+y^2)), which is equal to atan(y/x) if y and
* x are both real.
* @exception pole_error("atan(): logarithmic pole",0) if y/x==+I or y/x==-I. */
const numeric atan(const numeric &y, const numeric &x)
{
if (x.is_zero() && y.is_zero())
return *_num0_p;
if (x.is_real() && y.is_real())
return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
cln::the<cln::cl_R>(y.to_cl_N()));
// Compute -I*log((x+I*y)/sqrt(x^2+y^2))
// == -I*log((x+I*y)/sqrt((x+I*y)*(x-I*y)))
// Do not "simplify" this to -I/2*log((x+I*y)/(x-I*y))) or likewise.
// The branch cuts are easily messed up.
const cln::cl_N aux_p = x.to_cl_N()+cln::complex(0,1)*y.to_cl_N();
if (cln::zerop(aux_p)) {
// x+I*y==0 => y/x==I, so this is a pole (we have x!=0).
throw pole_error("atan(): logarithmic pole",0);
}
const cln::cl_N aux_m = x.to_cl_N()-cln::complex(0,1)*y.to_cl_N();
if (cln::zerop(aux_m)) {
// x-I*y==0 => y/x==-I, so this is a pole (we have x!=0).
throw pole_error("atan(): logarithmic pole",0);
}
return cln::complex(0,-1)*cln::log(aux_p/cln::sqrt(aux_p*aux_m));
}
/** Numeric hyperbolic sine (trigonometric function).
*
* @return arbitrary precision numerical sinh(x). */
const numeric sinh(const numeric &x)
{
return cln::sinh(x.to_cl_N());
}
/** Numeric hyperbolic cosine (trigonometric function).
*
* @return arbitrary precision numerical cosh(x). */
const numeric cosh(const numeric &x)
{
return cln::cosh(x.to_cl_N());
}
/** Numeric hyperbolic tangent (trigonometric function).
*
* @return arbitrary precision numerical tanh(x). */
const numeric tanh(const numeric &x)
{
return cln::tanh(x.to_cl_N());
}
/** Numeric inverse hyperbolic sine (trigonometric function).
*
* @return arbitrary precision numerical asinh(x). */
const numeric asinh(const numeric &x)
{
return cln::asinh(x.to_cl_N());
}
/** Numeric inverse hyperbolic cosine (trigonometric function).
*
* @return arbitrary precision numerical acosh(x). */
const numeric acosh(const numeric &x)
{
return cln::acosh(x.to_cl_N());
}
/** Numeric inverse hyperbolic tangent (trigonometric function).
*
* @return arbitrary precision numerical atanh(x). */
const numeric atanh(const numeric &x)
{
return cln::atanh(x.to_cl_N());
}
/*static cln::cl_N Li2_series(const ::cl_N &x,
const ::float_format_t &prec)
{
// Note: argument must be in the unit circle
// This is very inefficient unless we have fast floating point Bernoulli
// numbers implemented!
cln::cl_N c1 = -cln::log(1-x);
cln::cl_N c2 = c1;
// hard-wire the first two Bernoulli numbers
cln::cl_N acc = c1 - cln::square(c1)/4;
cln::cl_N aug;
cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
unsigned i = 1;
c1 = cln::square(c1);
do {
c2 = c1 * c2;
piac = piac * pisq;
aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
// aug = c2 * cln::cl_I(i%2 ? 1 : -1) / cln::cl_I(2*i+1) * cln::cl_zeta(2*i, prec) / piac / (cln::cl_I(1)<<(2*i-1));
acc = acc + aug;
++i;
} while (acc != acc+aug);
return acc;
}*/
/** Numeric evaluation of Dilogarithm within circle of convergence (unit
* circle) using a power series. */
static cln::cl_N Li2_series(const cln::cl_N &x,
const cln::float_format_t &prec)
{
// Note: argument must be in the unit circle
cln::cl_N aug, acc;
cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
cln::cl_I den = 0;
unsigned i = 1;
do {
num = num * x;
den = den + i; // 1, 4, 9, 16, ...
i += 2;
aug = num / den;
acc = acc + aug;
} while (acc != acc+aug);
return acc;
}
/** Folds Li2's argument inside a small rectangle to enhance convergence. */
static cln::cl_N Li2_projection(const cln::cl_N &x,
const cln::float_format_t &prec)
{
const cln::cl_R re = cln::realpart(x);
const cln::cl_R im = cln::imagpart(x);
if (re > cln::cl_F(".5"))
// zeta(2) - Li2(1-x) - log(x)*log(1-x)
return(cln::zeta(2)
- Li2_series(1-x, prec)
- cln::log(x)*cln::log(1-x));
if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
// -log(1-x)^2 / 2 - Li2(x/(x-1))
return(- cln::square(cln::log(1-x))/2
- Li2_series(x/(x-1), prec));
if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
// Li2(x^2)/2 - Li2(-x)
return(Li2_projection(cln::square(x), prec)/2
- Li2_projection(-x, prec));
return Li2_series(x, prec);
}
/** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
* the branch cut lies along the positive real axis, starting at 1 and
* continuous with quadrant IV.
*
* @return arbitrary precision numerical Li2(x). */
const numeric Li2(const numeric &x)
{
if (x.is_zero())
return *_num0_p;
// what is the desired float format?
// first guess: default format
cln::float_format_t prec = cln::default_float_format;
const cln::cl_N value = x.to_cl_N();
// second guess: the argument's format
if (!x.real().is_rational())
prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
else if (!x.imag().is_rational())
prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
if (value==1) // may cause trouble with log(1-x)
return cln::zeta(2, prec);
if (cln::abs(value) > 1)
// -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
return(- cln::square(cln::log(-value))/2
- cln::zeta(2, prec)
- Li2_projection(cln::recip(value), prec));
else
return Li2_projection(x.to_cl_N(), prec);
}
/** Numeric evaluation of Riemann's Zeta function. Currently works only for
* integer arguments. */
const numeric zeta(const numeric &x)
{
// A dirty hack to allow for things like zeta(3.0), since CLN currently
// only knows about integer arguments and zeta(3).evalf() automatically
// cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
// being an exact zero for CLN, which can be tested and then we can just
// pass the number casted to an int:
if (x.is_real()) {
const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
if (cln::zerop(x.to_cl_N()-aux))
return cln::zeta(aux);
}
throw dunno();
}
/** The Gamma function.
* This is only a stub! */
const numeric lgamma(const numeric &x)
{
throw dunno();
}
const numeric tgamma(const numeric &x)
{
throw dunno();
}
/** The psi function (aka polygamma function).
* This is only a stub! */
const numeric psi(const numeric &x)
{
throw dunno();
}
/** The psi functions (aka polygamma functions).
* This is only a stub! */
const numeric psi(const numeric &n, const numeric &x)
{
throw dunno();
}
/** Factorial combinatorial function.
*
* @param n integer argument >= 0
* @exception range_error (argument must be integer >= 0) */
const numeric factorial(const numeric &n)
{
if (!n.is_nonneg_integer())
throw std::range_error("numeric::factorial(): argument must be integer >= 0");
return numeric(cln::factorial(n.to_int()));
}
/** The double factorial combinatorial function. (Scarcely used, but still
* useful in cases, like for exact results of tgamma(n+1/2) for instance.)
*
* @param n integer argument >= -1
* @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
* @exception range_error (argument must be integer >= -1) */
const numeric doublefactorial(const numeric &n)
{
if (n.is_equal(*_num_1_p))
return *_num1_p;
if (!n.is_nonneg_integer())
throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
return numeric(cln::doublefactorial(n.to_int()));
}
/** The Binomial coefficients. It computes the binomial coefficients. For
* integer n and k and positive n this is the number of ways of choosing k
* objects from n distinct objects. If n is negative, the formula
* binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
const numeric binomial(const numeric &n, const numeric &k)
{
if (n.is_integer() && k.is_integer()) {
if (n.is_nonneg_integer()) {
if (k.compare(n)!=1 && k.compare(*_num0_p)!=-1)
return numeric(cln::binomial(n.to_int(),k.to_int()));
else
return *_num0_p;
} else {
return _num_1_p->power(k)*binomial(k-n-(*_num1_p),k);
}
}
// should really be gamma(n+1)/gamma(k+1)/gamma(n-k+1) or a suitable limit
throw std::range_error("numeric::binomial(): don't know how to evaluate that.");
}
/** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
* in the expansion of the function x/(e^x-1).
*
* @return the nth Bernoulli number (a rational number).
* @exception range_error (argument must be integer >= 0) */
const numeric bernoulli(const numeric &nn)
{
if (!nn.is_integer() || nn.is_negative())
throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
// Method:
//
// The Bernoulli numbers are rational numbers that may be computed using
// the relation
//
// B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
//
// with B(0) = 1. Since the n'th Bernoulli number depends on all the
// previous ones, the computation is necessarily very expensive. There are
// several other ways of computing them, a particularly good one being
// cl_I s = 1;
// cl_I c = n+1;
// cl_RA Bern = 0;
// for (unsigned i=0; i<n; i++) {
// c = exquo(c*(i-n),(i+2));
// Bern = Bern + c*s/(i+2);
// s = s + expt_pos(cl_I(i+2),n);
// }
// return Bern;
//
// But if somebody works with the n'th Bernoulli number she is likely to
// also need all previous Bernoulli numbers. So we need a complete remember
// table and above divide and conquer algorithm is not suited to build one
// up. The formula below accomplishes this. It is a modification of the
// defining formula above but the computation of the binomial coefficients
// is carried along in an inline fashion. It also honors the fact that
// B_n is zero when n is odd and greater than 1.
//
// (There is an interesting relation with the tangent polynomials described
// in `Concrete Mathematics', which leads to a program a little faster as
// our implementation below, but it requires storing one such polynomial in
// addition to the remember table. This doubles the memory footprint so
// we don't use it.)
const unsigned n = nn.to_int();
// the special cases not covered by the algorithm below
if (n & 1)
return (n==1) ? (*_num_1_2_p) : (*_num0_p);
if (!n)
return *_num1_p;
// store nonvanishing Bernoulli numbers here
static std::vector< cln::cl_RA > results;
static unsigned next_r = 0;
// algorithm not applicable to B(2), so just store it
if (!next_r) {
results.push_back(cln::recip(cln::cl_RA(6)));
next_r = 4;
}
if (n<next_r)
return results[n/2-1];
results.reserve(n/2);
for (unsigned p=next_r; p<=n; p+=2) {
cln::cl_I c = 1; // seed for binonmial coefficients
cln::cl_RA b = cln::cl_RA(p-1)/-2;
const unsigned p3 = p+3;
const unsigned pm = p-2;
unsigned i, k, p_2;
// test if intermediate unsigned int can be represented by immediate
// objects by CLN (i.e. < 2^29 for 32 Bit machines, see <cln/object.h>)
if (p < (1UL<<cl_value_len/2)) {
for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
c = cln::exquo(c * ((p3-i) * p_2), (i-1)*k);
b = b + c*results[k-1];
}
} else {
for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
c = cln::exquo((c * (p3-i)) * p_2, cln::cl_I(i-1)*k);
b = b + c*results[k-1];
}
}
results.push_back(-b/(p+1));
}
next_r = n+2;
return results[n/2-1];
}
/** Fibonacci number. The nth Fibonacci number F(n) is defined by the
* recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
*
* @param n an integer
* @return the nth Fibonacci number F(n) (an integer number)
* @exception range_error (argument must be an integer) */
const numeric fibonacci(const numeric &n)
{
if (!n.is_integer())
throw std::range_error("numeric::fibonacci(): argument must be integer");
// Method:
//
// The following addition formula holds:
//
// F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
//
// (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
// w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
// agree.)
// Replace m by m+1:
// F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
// Now put in m = n, to get
// F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
// F(2n+1) = F(n)^2 + F(n+1)^2
// hence
// F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
if (n.is_zero())
return *_num0_p;
if (n.is_negative())
if (n.is_even())
return -fibonacci(-n);
else
return fibonacci(-n);
cln::cl_I u(0);
cln::cl_I v(1);
cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
for (uintL bit=cln::integer_length(m); bit>0; --bit) {
// Since a squaring is cheaper than a multiplication, better use
// three squarings instead of one multiplication and two squarings.
cln::cl_I u2 = cln::square(u);
cln::cl_I v2 = cln::square(v);
if (cln::logbitp(bit-1, m)) {
v = cln::square(u + v) - u2;
u = u2 + v2;
} else {
u = v2 - cln::square(v - u);
v = u2 + v2;
}
}
if (n.is_even())
// Here we don't use the squaring formula because one multiplication
// is cheaper than two squarings.
return u * ((v << 1) - u);
else
return cln::square(u) + cln::square(v);
}
/** Absolute value. */
const numeric abs(const numeric& x)
{
return cln::abs(x.to_cl_N());
}
/** Modulus (in positive representation).
* In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
* sign of a or is zero. This is different from Maple's modp, where the sign
* of b is ignored. It is in agreement with Mathematica's Mod.
*
* @return a mod b in the range [0,abs(b)-1] with sign of b if both are
* integer, 0 otherwise. */
const numeric mod(const numeric &a, const numeric &b)
{
if (a.is_integer() && b.is_integer())
return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
cln::the<cln::cl_I>(b.to_cl_N()));
else
return *_num0_p;
}
/** Modulus (in symmetric representation).
* Equivalent to Maple's mods.
*
* @return a mod b in the range [-iquo(abs(b)-1,2), iquo(abs(b),2)]. */
const numeric smod(const numeric &a, const numeric &b)
{
if (a.is_integer() && b.is_integer()) {
const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
cln::the<cln::cl_I>(b.to_cl_N())) - b2;
} else
return *_num0_p;
}
/** Numeric integer remainder.
* Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
* In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
* sign of a or is zero.
*
* @return remainder of a/b if both are integer, 0 otherwise.
* @exception overflow_error (division by zero) if b is zero. */
const numeric irem(const numeric &a, const numeric &b)
{
if (b.is_zero())
throw std::overflow_error("numeric::irem(): division by zero");
if (a.is_integer() && b.is_integer())
return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
cln::the<cln::cl_I>(b.to_cl_N()));
else
return *_num0_p;
}
/** Numeric integer remainder.
* Equivalent to Maple's irem(a,b,'q') it obeyes the relation
* irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
* and irem(a,b) has the sign of a or is zero.
*
* @return remainder of a/b and quotient stored in q if both are integer,
* 0 otherwise.
* @exception overflow_error (division by zero) if b is zero. */
const numeric irem(const numeric &a, const numeric &b, numeric &q)
{
if (b.is_zero())
throw std::overflow_error("numeric::irem(): division by zero");
if (a.is_integer() && b.is_integer()) {
const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
cln::the<cln::cl_I>(b.to_cl_N()));
q = rem_quo.quotient;
return rem_quo.remainder;
} else {
q = *_num0_p;
return *_num0_p;
}
}
/** Numeric integer quotient.
* Equivalent to Maple's iquo as far as sign conventions are concerned.
*
* @return truncated quotient of a/b if both are integer, 0 otherwise.
* @exception overflow_error (division by zero) if b is zero. */
const numeric iquo(const numeric &a, const numeric &b)
{
if (b.is_zero())
throw std::overflow_error("numeric::iquo(): division by zero");
if (a.is_integer() && b.is_integer())
return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
cln::the<cln::cl_I>(b.to_cl_N()));
else
return *_num0_p;
}
/** Numeric integer quotient.
* Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
* r == a - iquo(a,b,r)*b.
*
* @return truncated quotient of a/b and remainder stored in r if both are
* integer, 0 otherwise.
* @exception overflow_error (division by zero) if b is zero. */
const numeric iquo(const numeric &a, const numeric &b, numeric &r)
{
if (b.is_zero())
throw std::overflow_error("numeric::iquo(): division by zero");
if (a.is_integer() && b.is_integer()) {
const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
cln::the<cln::cl_I>(b.to_cl_N()));
r = rem_quo.remainder;
return rem_quo.quotient;
} else {
r = *_num0_p;
return *_num0_p;
}
}
/** Greatest Common Divisor.
*
* @return The GCD of two numbers if both are integer, a numerical 1
* if they are not. */
const numeric gcd(const numeric &a, const numeric &b)
{
if (a.is_integer() && b.is_integer())
return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
cln::the<cln::cl_I>(b.to_cl_N()));
else
return *_num1_p;
}
/** Least Common Multiple.
*
* @return The LCM of two numbers if both are integer, the product of those
* two numbers if they are not. */
const numeric lcm(const numeric &a, const numeric &b)
{
if (a.is_integer() && b.is_integer())
return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
cln::the<cln::cl_I>(b.to_cl_N()));
else
return a.mul(b);
}
/** Numeric square root.
* If possible, sqrt(x) should respect squares of exact numbers, i.e. sqrt(4)
* should return integer 2.
*
* @param x numeric argument
* @return square root of x. Branch cut along negative real axis, the negative
* real axis itself where imag(x)==0 and real(x)<0 belongs to the upper part
* where imag(x)>0. */
const numeric sqrt(const numeric &x)
{
return cln::sqrt(x.to_cl_N());
}
/** Integer numeric square root. */
const numeric isqrt(const numeric &x)
{
if (x.is_integer()) {
cln::cl_I root;
cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
return root;
} else
return *_num0_p;
}
/** Floating point evaluation of Archimedes' constant Pi. */
ex PiEvalf()
{
return numeric(cln::pi(cln::default_float_format));
}
/** Floating point evaluation of Euler's constant gamma. */
ex EulerEvalf()
{
return numeric(cln::eulerconst(cln::default_float_format));
}
/** Floating point evaluation of Catalan's constant. */
ex CatalanEvalf()
{
return numeric(cln::catalanconst(cln::default_float_format));
}
/** _numeric_digits default ctor, checking for singleton invariance. */
_numeric_digits::_numeric_digits()
: digits(17)
{
// It initializes to 17 digits, because in CLN float_format(17) turns out
// to be 61 (<64) while float_format(18)=65. The reason is we want to
// have a cl_LF instead of cl_SF, cl_FF or cl_DF.
if (too_late)
throw(std::runtime_error("I told you not to do instantiate me!"));
too_late = true;
cln::default_float_format = cln::float_format(17);
}
/** Assign a native long to global Digits object. */
_numeric_digits& _numeric_digits::operator=(long prec)
{
digits = prec;
cln::default_float_format = cln::float_format(prec);
return *this;
}
/** Convert global Digits object to native type long. */
_numeric_digits::operator long()
{
// BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
return (long)digits;
}
/** Append global Digits object to ostream. */
void _numeric_digits::print(std::ostream &os) const
{
os << digits;
}
std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
{
e.print(os);
return os;
}
//////////
// static member variables
//////////
// private
bool _numeric_digits::too_late = false;
/** Accuracy in decimal digits. Only object of this type! Can be set using
* assignment from C++ unsigned ints and evaluated like any built-in type. */
_numeric_digits Digits;
} // namespace GiNaC
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