* of special functions or implement the interface to the bignum package. */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * 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
*
* 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 "config.h"
#include <vector>
#include <stdexcept>
#include <string>
-
-#if defined(HAVE_SSTREAM)
#include <sstream>
-#elif defined(HAVE_STRSTREAM)
-#include <strstream>
-#else
-#error Need either sstream or strstream
-#endif
+#include <limits>
#include "numeric.h"
#include "ex.h"
+#include "operators.h"
#include "archive.h"
-#include "debugmsg.h"
+#include "tostring.h"
#include "utils.h"
-// CLN should not pollute the global namespace, hence we include it here
-// instead of in some header file where it would propagate to other parts.
-// Also, we only need a subset of CLN, so we don't include the complete cln.h:
-#ifdef HAVE_CLN_CLN_H
-#include <cln/cl_integer_io.h>
-#include <cln/cl_integer_ring.h>
-#include <cln/cl_rational_io.h>
-#include <cln/cl_rational_ring.h>
-#include <cln/cl_lfloat_class.h>
-#include <cln/cl_lfloat_io.h>
-#include <cln/cl_real_io.h>
-#include <cln/cl_real_ring.h>
-#include <cln/cl_complex_io.h>
-#include <cln/cl_complex_ring.h>
-#include <cln/cl_numtheory.h>
-#else // def HAVE_CLN_CLN_H
-#include <cl_integer_io.h>
-#include <cl_integer_ring.h>
-#include <cl_rational_io.h>
-#include <cl_rational_ring.h>
-#include <cl_lfloat_class.h>
-#include <cl_lfloat_io.h>
-#include <cl_real_io.h>
-#include <cl_real_ring.h>
-#include <cl_complex_io.h>
-#include <cl_complex_ring.h>
-#include <cl_numtheory.h>
-#endif // def HAVE_CLN_CLN_H
-
-#ifndef NO_NAMESPACE_GINAC
-namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
+// 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>
-// linker has no problems finding text symbols for numerator or denominator
-//#define SANE_LINKER
+namespace GiNaC {
-GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
+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, destructor, copy constructor assignment
-// operator and helpers
+// default constructor
//////////
-// public
-
/** default ctor. Numerically it initializes to an integer zero. */
-numeric::numeric() : basic(TINFO_numeric)
+numeric::numeric() : basic(&numeric::tinfo_static)
{
- debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
- value = new cl_N;
- *value = cl_I(0);
- calchash();
- setflag(status_flags::evaluated |
- status_flags::expanded |
- status_flags::hash_calculated);
-}
-
-numeric::~numeric()
-{
- debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
- destroy(0);
-}
-
-numeric::numeric(const numeric & other)
-{
- debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
- copy(other);
-}
-
-const numeric & numeric::operator=(const numeric & other)
-{
- debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
- if (this != &other) {
- destroy(1);
- copy(other);
- }
- return *this;
-}
-
-// protected
-
-void numeric::copy(const numeric & other)
-{
- basic::copy(other);
- value = new cl_N(*other.value);
-}
-
-void numeric::destroy(bool call_parent)
-{
- delete value;
- if (call_parent) basic::destroy(call_parent);
+ value = cln::cl_I(0);
+ setflag(status_flags::evaluated | status_flags::expanded);
}
//////////
// public
-numeric::numeric(int i) : basic(TINFO_numeric)
-{
- debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
- // 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:
- value = new cl_I((long) i);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+numeric::numeric(int i) : basic(&numeric::tinfo_static)
+{
+ // 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)
+numeric::numeric(unsigned int i) : basic(&numeric::tinfo_static)
{
- debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
- // 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:
- value = new cl_I((unsigned long)i);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ // 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)
+numeric::numeric(long i) : basic(&numeric::tinfo_static)
{
- debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
- value = new cl_I(i);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ value = cln::cl_I(i);
+ setflag(status_flags::evaluated | status_flags::expanded);
}
-numeric::numeric(unsigned long i) : basic(TINFO_numeric)
+numeric::numeric(unsigned long i) : basic(&numeric::tinfo_static)
{
- debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
- value = new cl_I(i);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ value = cln::cl_I(i);
+ setflag(status_flags::evaluated | status_flags::expanded);
}
-/** Ctor for rational numerics a/b.
+
+/** Constructor for rational numerics a/b.
*
* @exception overflow_error (division by zero) */
-numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
-{
- debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
- if (!denom)
- throw (std::overflow_error("division by zero"));
- value = new cl_I(numer);
- *value = *value / cl_I(denom);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
-}
-
-
-numeric::numeric(double d) : basic(TINFO_numeric)
-{
- debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
- // 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 = new cl_N;
- *value = cl_float(d, cl_default_float_format);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+numeric::numeric(long numer, long denom) : basic(&numeric::tinfo_static)
+{
+ 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(&numeric::tinfo_static)
+{
+ // 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(&numeric::tinfo_static)
+{
+ 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);
}
-numeric::numeric(const char *s) : basic(TINFO_numeric)
-{ // MISSING: treatment of complex and ints and rationals.
- debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
- if (strchr(s, '.'))
- value = new cl_LF(s);
- else
- value = new cl_I(s);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
-}
-
/** Ctor from CLN types. This is for the initiated user or internal use
* only. */
-numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
+numeric::numeric(const cln::cl_N &z) : basic(&numeric::tinfo_static)
{
- debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
- value = new cl_N(z);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ value = z;
+ setflag(status_flags::evaluated | status_flags::expanded);
}
+
//////////
// archiving
//////////
-/** Construct object from archive_node. */
-numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
-{
- debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
- value = new cl_N;
-#ifdef HAVE_SSTREAM
- // Read number as string
- string str;
- if (n.find_string("number", str)) {
- istringstream s(str);
- cl_idecoded_float re, im;
- char c;
- s.get(c);
- switch (c) {
- case 'N': // Ordinary number
- case 'R': // Integer-decoded real number
- s >> re.sign >> re.mantissa >> re.exponent;
- *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_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;
- *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
- im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
- break;
- default: // Ordinary number
+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);
+ 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 >> *value;
- break;
- }
- }
-#else
- // Read number as string
- string str;
- if (n.find_string("number", str)) {
- istrstream f(str.c_str(), str.size() + 1);
- cl_idecoded_float re, im;
- char c;
- f.get(c);
- switch (c) {
- case 'R': // Integer-decoded real number
- f >> re.sign >> re.mantissa >> re.exponent;
- *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
- break;
- case 'C': // Integer-decoded complex number
- f >> re.sign >> re.mantissa >> re.exponent;
- f >> im.sign >> im.mantissa >> im.exponent;
- *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
- im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
- break;
- default: // Ordinary number
- f.putback(c);
- f >> *value;
+ s >> ctorval;
break;
- }
- }
-#endif
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
-}
-
-/** Unarchive the object. */
-ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
-{
- return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
+ }
+ }
+ value = ctorval;
+ setflag(status_flags::evaluated | status_flags::expanded);
}
-/** Archive the object. */
void numeric::archive(archive_node &n) const
{
- inherited::archive(n);
-#ifdef HAVE_SSTREAM
- // Write number as string
- 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()) {
- cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
- s << "R";
- s << re.sign << " " << re.mantissa << " " << re.exponent;
- } else {
- cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
- cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
- s << "C";
- s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
- s << im.sign << " " << im.mantissa << " " << im.exponent;
- }
- }
- n.add_string("number", s.str());
-#else
- // Write number as string
- char buf[1024];
- ostrstream f(buf, 1024);
- if (this->is_crational())
- f << *value << ends;
- else {
- // Non-rational numbers are written in an integer-decoded format
- // to preserve the precision
- if (this->is_real()) {
- cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
- f << "R";
- f << re.sign << " " << re.mantissa << " " << re.exponent << ends;
- } else {
- cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
- cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
- f << "C";
- f << re.sign << " " << re.mantissa << " " << re.exponent << " ";
- f << im.sign << " " << im.mantissa << " " << im.exponent << ends;
- }
- }
- string str(buf);
- n.add_string("number", str);
-#endif
-}
+ 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 bases classes
+// functions overriding virtual functions from base classes
//////////
-// public
+/** 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);
+ }
+}
+
+template<typename T1, typename T2>
+static inline bool coerce(T1& dst, const T2& arg);
-basic * numeric::duplicate() const
+/**
+ * @brief Check if CLN integer can be converted into int
+ *
+ * @sa http://www.ginac.de/pipermail/cln-list/2006-October/000248.html
+ */
+template<>
+static inline bool coerce<int, cln::cl_I>(int& dst, const cln::cl_I& arg)
{
- debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
- return new numeric(*this);
+ static const cln::cl_I cl_max_int =
+ (cln::cl_I)(long)(std::numeric_limits<int>::max());
+ static const cln::cl_I cl_min_int =
+ (cln::cl_I)(long)(std::numeric_limits<int>::min());
+ if ((arg >= cl_min_int) && (arg <= cl_max_int)) {
+ dst = cl_I_to_int(arg);
+ return true;
+ }
+ return false;
}
-void numeric::print(ostream & os, unsigned upper_precedence) const
+template<>
+static inline bool coerce<unsigned int, cln::cl_I>(unsigned int& dst, const cln::cl_I& arg)
{
- // The method print adds to the output so it blends more consistently
- // together with the other routines and produces something compatible to
- // ginsh input.
- debugmsg("numeric print", LOGLEVEL_PRINT);
- if (this->is_real()) {
- // case 1, real: x or -x
- if ((precedence<=upper_precedence) && (!this->is_pos_integer())) {
- os << "(" << *value << ")";
- } else {
- os << *value;
- }
+ static const cln::cl_I cl_max_uint =
+ (cln::cl_I)(unsigned long)(std::numeric_limits<unsigned int>::max());
+ if ((! minusp(arg)) && (arg <= cl_max_uint)) {
+ dst = cl_I_to_uint(arg);
+ return true;
+ }
+ return false;
+}
+
+/** 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)) {
+
+ int dst;
+ // fixnum
+ if (coerce(dst, cln::the<cln::cl_I>(x))) {
+ // can be converted to native int
+ if (dst < 0)
+ c.s << "(-" << dst << ")";
+ else
+ c.s << dst;
} else {
- // case 2, imaginary: y*I or -y*I
- if (::realpart(*value) == 0) {
- if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
- if (::imagpart(*value) == -1) {
- os << "(-I)";
- } else {
- os << "(" << ::imagpart(*value) << "*I)";
- }
- } else {
- if (::imagpart(*value) == 1) {
- os << "I";
- } else {
- if (::imagpart (*value) == -1) {
- os << "-I";
- } else {
- os << ::imagpart(*value) << "*I";
- }
- }
- }
- } else {
- // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
- if (precedence <= upper_precedence) os << "(";
- os << ::realpart(*value);
- if (::imagpart(*value) < 0) {
- if (::imagpart(*value) == -1) {
- os << "-I";
- } else {
- os << ::imagpart(*value) << "*I";
- }
- } else {
- if (::imagpart(*value) == 1) {
- os << "+I";
- } else {
- os << "+" << ::imagpart(*value) << "*I";
- }
- }
- if (precedence <= upper_precedence) os << ")";
- }
+ // bignum
+ 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;
+ }
-void numeric::printraw(ostream & os) const
+ } 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
{
- // The method printraw doesn't do much, it simply uses CLN's operator<<()
- // for output, which is ugly but reliable. e.g: 2+2i
- debugmsg("numeric printraw", LOGLEVEL_PRINT);
- os << "numeric(" << *value << ")";
+ print_numeric(c, "(", ")", "I", "*", level);
}
+void numeric::do_print_latex(const print_latex & c, unsigned level) const
+{
+ print_numeric(c, "{(", ")}", "i", " ", level);
+}
-void numeric::printtree(ostream & os, unsigned indent) const
+void numeric::do_print_csrc(const print_csrc & c, unsigned level) const
{
- debugmsg("numeric printtree", LOGLEVEL_PRINT);
- os << string(indent,' ') << *value
- << " (numeric): "
- << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
- << ", flags=" << flags << endl;
+ 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::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
+void numeric::do_print_tree(const print_tree & c, unsigned level) const
{
- debugmsg("numeric print csrc", LOGLEVEL_PRINT);
- ios::fmtflags oldflags = os.flags();
- os.setf(ios::scientific);
- if (this->is_rational() && !this->is_integer()) {
- if (compare(_num0()) > 0) {
- os << "(";
- if (type == csrc_types::ctype_cl_N)
- os << "cl_F(\"" << numer().evalf() << "\")";
- else
- os << numer().to_double();
- } else {
- os << "-(";
- if (type == csrc_types::ctype_cl_N)
- os << "cl_F(\"" << -numer().evalf() << "\")";
- else
- os << -numer().to_double();
- }
- os << "/";
- if (type == csrc_types::ctype_cl_N)
- os << "cl_F(\"" << denom().evalf() << "\")";
- else
- os << denom().to_double();
- os << ")";
- } else {
- if (type == csrc_types::ctype_cl_N)
- os << "cl_F(\"" << evalf() << "\")";
- else
- os << to_double();
- }
- os.flags(oldflags);
+ 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:
- 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();
- }
- return false;
+ 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;
+}
+
+bool numeric::is_polynomial(const ex & var) const
+{
+ return true;
+}
+
+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
* 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_of_type(*other.bp, numeric))
- return false;
- const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
- if (this->is_equal(o) || this->is_equal(-o))
- return true;
- if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
- return (this->real().is_equal(o) || this->imag().is_equal(o) ||
- this->real().is_equal(-o) || this->imag().is_equal(-o));
- 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
- return (this->real().has(o*I) || this->imag().has(o*I) ||
- this->real().has(-o*I) || this->imag().has(-o*I));
- }
- return false;
+bool numeric::has(const ex &other, unsigned options) 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();
+ // 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.
+ * 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, but needed for overriding basic::evalf.
+ * @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(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
+ // level can safely be discarded for numeric objects.
+ return numeric(cln::cl_float(1.0, cln::default_float_format) * value);
}
-// protected
-
-/** Implementation of ex::diff() for a numeric. It always returns 0.
- *
- * @see ex::diff */
-ex numeric::derivative(const symbol & s) const
+ex numeric::conjugate() const
{
- return _ex0();
+ if (is_real()) {
+ return *this;
+ }
+ return numeric(cln::conjugate(this->value));
}
-
-int numeric::compare_same_type(const basic & other) const
+ex numeric::real_part() const
{
- GINAC_ASSERT(is_exactly_of_type(other, numeric));
- const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
-
- if (*value == *o.value) {
- return 0;
- }
+ return numeric(cln::realpart(value));
+}
- return compare(o);
+ex numeric::imag_part() const
+{
+ return numeric(cln::imagpart(value));
}
+// protected
-bool numeric::is_equal_same_type(const basic & other) const
+int numeric::compare_same_type(const basic &other) const
{
- GINAC_ASSERT(is_exactly_of_type(other,numeric));
- const numeric *o = static_cast<const numeric *>(&other);
-
- return this->is_equal(*o);
+ GINAC_ASSERT(is_exactly_a<numeric>(other));
+ const numeric &o = static_cast<const numeric &>(other);
+
+ return this->compare(o);
}
-unsigned numeric::calchash(void) const
+
+bool numeric::is_equal_same_type(const basic &other) const
{
- return (hashvalue=cl_equal_hashcode(*value) | 0x80000000U);
- /*
- cout << *value << "->" << hashvalue << endl;
- hashvalue=HASHVALUE_NUMERIC+1000U;
- return HASHVALUE_NUMERIC+1000U;
- */
+ GINAC_ASSERT(is_exactly_a<numeric>(other));
+ const numeric &o = static_cast<const numeric &>(other);
+
+ return this->is_equal(o);
}
-/*
-unsigned numeric::calchash(void) const
+
+unsigned numeric::calchash() const
{
- double d=to_double();
- int s=d>0 ? 1 : -1;
- d=fabs(d);
- if (d>0x07FF0000) {
- d=0x07FF0000;
- }
- return 0x88000000U+s*unsigned(d/0x07FF0000);
+ // 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;
}
-*/
//////////
// public
/** Numerical addition method. Adds argument to *this and returns result as
- * a new numeric object. */
-numeric numeric::add(const numeric & other) const
+ * a numeric object. */
+const numeric numeric::add(const numeric &other) const
{
- return numeric((*value)+(*other.value));
+ return numeric(value + other.value);
}
+
/** Numerical subtraction method. Subtracts argument from *this and returns
- * result as a new numeric object. */
-numeric numeric::sub(const numeric & other) const
+ * result as a numeric object. */
+const numeric numeric::sub(const numeric &other) const
{
- return numeric((*value)-(*other.value));
+ return numeric(value - other.value);
}
+
/** Numerical multiplication method. Multiplies *this and argument and returns
- * result as a new numeric object. */
-numeric numeric::mul(const numeric & other) const
-{
- static const numeric * _num1p=&_num1();
- if (this==_num1p) {
- return other;
- } else if (&other==_num1p) {
- return *this;
- }
- return numeric((*value)*(*other.value));
+ * 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 new numeric object.
+ * a numeric object.
*
* @exception overflow_error (division by zero) */
-numeric numeric::div(const numeric & other) const
-{
- if (::zerop(*other.value))
- throw (std::overflow_error("division by zero"));
- return numeric((*value)/(*other.value));
-}
-
-numeric numeric::power(const numeric & other) const
-{
- static const numeric * _num1p = &_num1();
- if (&other==_num1p)
- return *this;
- if (::zerop(*value)) {
- if (::zerop(*other.value))
- throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
- else if (::zerop(::realpart(*other.value)))
- throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
- else if (::minusp(::realpart(*other.value)))
- throw (std::overflow_error("numeric::eval(): division by zero"));
- else
- return _num0();
- }
- return numeric(::expt(*value,*other.value));
+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);
}
-/** Inverse of a number. */
-numeric numeric::inverse(void) const
+
+/** 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
{
- return numeric(::recip(*value)); // -> CLN
+ // 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));
}
-const numeric & numeric::add_dyn(const numeric & other) const
+
+
+/** 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
{
- return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
- setflag(status_flags::dynallocated));
+ // 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));
}
-const numeric & numeric::sub_dyn(const numeric & other) const
+
+/** 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
{
- return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
- setflag(status_flags::dynallocated));
+ // 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));
}
-const numeric & numeric::mul_dyn(const numeric & other) const
+
+/** 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
{
- static const numeric * _num1p=&_num1();
- if (this==_num1p) {
- return other;
- } else if (&other==_num1p) {
- return *this;
- }
- return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
- setflag(status_flags::dynallocated));
+ // 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));
}
-const numeric & numeric::div_dyn(const numeric & other) const
+
+/** 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
{
- if (::zerop(*other.value))
- throw (std::overflow_error("division by zero"));
- return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
- setflag(status_flags::dynallocated));
+ // 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::power_dyn(const numeric & other) const
+
+const numeric &numeric::operator=(int i)
{
- static const numeric * _num1p=&_num1();
- if (&other==_num1p)
- return *this;
- if (::zerop(*value)) {
- if (::zerop(*other.value))
- throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
- else if (::zerop(::realpart(*other.value)))
- throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
- else if (::minusp(::realpart(*other.value)))
- throw (std::overflow_error("numeric::eval(): division by zero"));
- else
- return _num0();
- }
- return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
- setflag(status_flags::dynallocated));
+ return operator=(numeric(i));
}
-const numeric & numeric::operator=(int i)
+
+const numeric &numeric::operator=(unsigned int i)
{
- return operator=(numeric(i));
+ return operator=(numeric(i));
}
-const numeric & numeric::operator=(unsigned int i)
+
+const numeric &numeric::operator=(long i)
{
- return operator=(numeric(i));
+ return operator=(numeric(i));
}
-const numeric & numeric::operator=(long i)
+
+const numeric &numeric::operator=(unsigned long i)
{
- return operator=(numeric(i));
+ return operator=(numeric(i));
}
-const numeric & numeric::operator=(unsigned long i)
+
+const numeric &numeric::operator=(double d)
{
- return operator=(numeric(i));
+ return operator=(numeric(d));
}
-const numeric & numeric::operator=(double d)
+
+const numeric &numeric::operator=(const char * s)
{
- return operator=(numeric(d));
+ return operator=(numeric(s));
}
-const numeric & numeric::operator=(const char * s)
+
+/** Inverse of a number. */
+const numeric numeric::inverse() const
{
- return operator=(numeric(s));
+ if (cln::zerop(value))
+ throw std::overflow_error("numeric::inverse(): division by zero");
+ return numeric(cln::recip(value));
+}
+
+/** Return the step function of a numeric. The imaginary part of it is
+ * ignored because the step function is generally considered real but
+ * a numeric may develop a small imaginary part due to rounding errors.
+ */
+numeric numeric::step() const
+{ cln::cl_R r = cln::realpart(value);
+ if(cln::zerop(r))
+ return numeric(1,2);
+ if(cln::plusp(r))
+ return 1;
+ return 0;
}
/** 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(void) const
-{
- if (this->is_zero())
- return 0;
- if (!::zerop(::realpart(*value))) {
- if (::plusp(::realpart(*value)))
- return 1;
- else
- return -1;
- } else {
- if (::plusp(::imagpart(*value)))
- return 1;
- else
- return -1;
- }
+ * @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(void) */
-int numeric::compare(const numeric & other) const
-{
- // Comparing two real numbers?
- if (this->is_real() && other.is_real())
- // Yes, just compare them
- return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
- else {
- // No, first compare real parts
- cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
- if (real_cmp)
- return real_cmp;
-
- return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
- }
+ * @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
+
+bool numeric::is_equal(const numeric &other) const
{
- return (*value == *other.value);
+ return cln::equal(value, other.value);
}
+
/** True if object is zero. */
-bool numeric::is_zero(void) const
+bool numeric::is_zero() const
{
- return ::zerop(*value); // -> CLN
+ return cln::zerop(value);
}
+
/** True if object is not complex and greater than zero. */
-bool numeric::is_positive(void) const
+bool numeric::is_positive() const
{
- if (this->is_real())
- return ::plusp(The(cl_R)(*value)); // -> CLN
- return false;
+ 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(void) const
+bool numeric::is_negative() const
{
- if (this->is_real())
- return ::minusp(The(cl_R)(*value)); // -> CLN
- return false;
+ 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(void) const
+bool numeric::is_integer() const
{
- return ::instanceof(*value, cl_I_ring); // -> CLN
+ return cln::instanceof(value, cln::cl_I_ring);
}
+
/** True if object is an exact integer greater than zero. */
-bool numeric::is_pos_integer(void) const
+bool numeric::is_pos_integer() const
{
- return (this->is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN
+ 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(void) const
+bool numeric::is_nonneg_integer() const
{
- return (this->is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN
+ 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(void) const
+bool numeric::is_even() const
{
- return (this->is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN
+ 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(void) const
+bool numeric::is_odd() const
{
- return (this->is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN
+ 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(void) const
+bool numeric::is_prime() const
{
- return (this->is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN
+ 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(void) const
+bool numeric::is_rational() const
{
- return ::instanceof(*value, cl_RA_ring); // -> CLN
+ 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(void) const
+bool numeric::is_real() const
{
- return ::instanceof(*value, cl_R_ring); // -> CLN
+ return cln::instanceof(value, cln::cl_R_ring);
}
-bool numeric::operator==(const numeric & other) const
+
+bool numeric::operator==(const numeric &other) const
{
- return (*value == *other.value); // -> CLN
+ return cln::equal(value, other.value);
}
-bool numeric::operator!=(const numeric & other) const
+
+bool numeric::operator!=(const numeric &other) const
{
- return (*value != *other.value); // -> CLN
+ 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(void) const
-{
- if (::instanceof(*value, cl_I_ring))
- return true;
- else if (!this->is_real()) { // complex case, handle n+m*I
- if (::instanceof(::realpart(*value), cl_I_ring) &&
- ::instanceof(::imagpart(*value), cl_I_ring))
- return true;
- }
- return false;
+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(void) const
-{
- if (::instanceof(*value, cl_RA_ring))
- return true;
- else if (!this->is_real()) { // complex case, handle Q(i):
- if (::instanceof(::realpart(*value), cl_RA_ring) &&
- ::instanceof(::imagpart(*value), cl_RA_ring))
- return true;
- }
- return false;
+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
+bool numeric::operator<(const numeric &other) const
{
- if (this->is_real() && other.is_real())
- return (The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
- throw (std::invalid_argument("numeric::operator<(): complex inequality"));
- return false; // make compiler shut up
+ 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
+bool numeric::operator<=(const numeric &other) const
{
- if (this->is_real() && other.is_real())
- return (The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
- throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
- return false; // make compiler shut up
+ 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
+bool numeric::operator>(const numeric &other) const
{
- if (this->is_real() && other.is_real())
- return (The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
- throw (std::invalid_argument("numeric::operator>(): complex inequality"));
- return false; // make compiler shut up
+ 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
+bool numeric::operator>=(const numeric &other) const
{
- if (this->is_real() && other.is_real())
- return (The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
- throw (std::invalid_argument("numeric::operator>=(): complex inequality"));
- return false; // make compiler shut up
+ 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(void) const
+int numeric::to_int() const
{
- GINAC_ASSERT(this->is_integer());
- return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN
+ 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(void) const
+long numeric::to_long() const
{
- GINAC_ASSERT(this->is_integer());
- return ::cl_I_to_long(The(cl_I)(*value)); // -> CLN
+ 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(void) const
+double numeric::to_double() const
{
- GINAC_ASSERT(this->is_real());
- return ::cl_double_approx(::realpart(*value)); // -> CLN
+ 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(void) const
+const numeric numeric::real() const
{
- return numeric(::realpart(*value)); // -> CLN
+ return numeric(cln::realpart(value));
}
+
/** Imaginary part of a number. */
-const numeric numeric::imag(void) const
+const numeric numeric::imag() const
{
- return numeric(::imagpart(*value)); // -> CLN
+ return numeric(cln::imagpart(value));
}
-#ifndef SANE_LINKER
-// Unfortunately, CLN did not provide an official way to access the numerator
-// or denominator of a rational number (cl_RA). Doing some excavations in CLN
-// one finds how it works internally in src/rational/cl_RA.h:
-struct cl_heap_ratio : cl_heap {
- cl_I numerator;
- cl_I denominator;
-};
-
-inline cl_heap_ratio* TheRatio (const cl_N& obj)
-{ return (cl_heap_ratio*)(obj.pointer); }
-#endif // ndef SANE_LINKER
/** 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(void) const
-{
- if (this->is_integer()) {
- return numeric(*this);
- }
-#ifdef SANE_LINKER
- else if (::instanceof(*value, cl_RA_ring)) {
- return numeric(::numerator(The(cl_RA)(*value)));
- }
- else if (!this->is_real()) { // complex case, handle Q(i):
- cl_R r = ::realpart(*value);
- cl_R i = ::imagpart(*value);
- if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
- return numeric(*this);
- if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
- return numeric(::complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i))));
- if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
- return numeric(::complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r))));
- if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring)) {
- cl_I s = ::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)));
- return numeric(::complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))),
- ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i))))));
- }
- }
-#else
- else if (instanceof(*value, cl_RA_ring)) {
- return numeric(TheRatio(*value)->numerator);
- }
- else if (!this->is_real()) { // complex case, handle Q(i):
- cl_R r = ::realpart(*value);
- cl_R i = ::imagpart(*value);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
- return numeric(*this);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
- return numeric(::complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
- return numeric(::complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) {
- cl_I s = ::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
- return numeric(::complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
- TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
- }
- }
-#endif // def SANE_LINKER
- // at least one float encountered
- return numeric(*this);
+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(void) const
-{
- if (this->is_integer()) {
- return _num1();
- }
-#ifdef SANE_LINKER
- if (instanceof(*value, cl_RA_ring)) {
- return numeric(::denominator(The(cl_RA)(*value)));
- }
- if (!this->is_real()) { // complex case, handle Q(i):
- cl_R r = ::realpart(*value);
- cl_R i = ::imagpart(*value);
- if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
- return _num1();
- if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
- return numeric(::denominator(The(cl_RA)(i)));
- if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
- return numeric(::denominator(The(cl_RA)(r)));
- if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring))
- return numeric(::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
- }
-#else
- if (instanceof(*value, cl_RA_ring)) {
- return numeric(TheRatio(*value)->denominator);
- }
- if (!this->is_real()) { // complex case, handle Q(i):
- cl_R r = ::realpart(*value);
- cl_R i = ::imagpart(*value);
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
- return _num1();
- if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
- return numeric(TheRatio(i)->denominator);
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
- return numeric(TheRatio(r)->denominator);
- if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring))
- return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
- }
-#endif // def SANE_LINKER
- // at least one float encountered
- return _num1();
+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(void) const
+int numeric::int_length() const
{
- if (this->is_integer())
- return ::integer_length(The(cl_I)(*value)); // -> CLN
- else
- return 0;
+ if (cln::instanceof(value, cln::cl_I_ring))
+ return cln::integer_length(cln::the<cln::cl_I>(value));
+ else
+ return 0;
}
-
-//////////
-// static member variables
-//////////
-
-// protected
-
-unsigned numeric::precedence = 30;
-
//////////
// global constants
//////////
-const numeric some_numeric;
-const type_info & typeid_numeric=typeid(some_numeric);
/** Imaginary unit. This is not a constant but a numeric since we are
- * natively handing complex numbers anyways. */
-const numeric I = numeric(::complex(cl_I(0),cl_I(1)));
+ * 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)
+const numeric exp(const numeric &x)
{
- return ::exp(*x.value); // -> CLN
+ return cln::exp(x.to_cl_N());
}
/** Natural logarithm.
*
- * @param z complex number
+ * @param x complex number
* @return arbitrary precision numerical log(x).
- * @exception overflow_error (logarithmic singularity) */
-const numeric log(const numeric & z)
+ * @exception pole_error("log(): logarithmic pole",0) */
+const numeric log(const numeric &x)
{
- if (z.is_zero())
- throw (std::overflow_error("log(): logarithmic singularity"));
- return ::log(*z.value); // -> CLN
+ 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)
+const numeric sin(const numeric &x)
{
- return ::sin(*x.value); // -> CLN
+ return cln::sin(x.to_cl_N());
}
/** Numeric cosine (trigonometric function).
*
* @return arbitrary precision numerical cos(x). */
-const numeric cos(const numeric & x)
+const numeric cos(const numeric &x)
{
- return ::cos(*x.value); // -> CLN
+ return cln::cos(x.to_cl_N());
}
/** Numeric tangent (trigonometric function).
*
* @return arbitrary precision numerical tan(x). */
-const numeric tan(const numeric & x)
+const numeric tan(const numeric &x)
{
- return ::tan(*x.value); // -> CLN
+ return cln::tan(x.to_cl_N());
}
-
+
/** Numeric inverse sine (trigonometric function).
*
* @return arbitrary precision numerical asin(x). */
-const numeric asin(const numeric & x)
+const numeric asin(const numeric &x)
{
- return ::asin(*x.value); // -> CLN
+ return cln::asin(x.to_cl_N());
}
/** Numeric inverse cosine (trigonometric function).
*
* @return arbitrary precision numerical acos(x). */
-const numeric acos(const numeric & x)
+const numeric acos(const numeric &x)
{
- return ::acos(*x.value); // -> CLN
+ return cln::acos(x.to_cl_N());
}
-
+
-/** Arcustangent.
+/** Numeric arcustangent.
*
- * @param z complex number
- * @return atan(z)
- * @exception overflow_error (logarithmic singularity) */
-const numeric atan(const numeric & x)
+ * @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()))
- throw (std::overflow_error("atan(): logarithmic singularity"));
- return ::atan(*x.value); // -> CLN
+ 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());
}
-/** Arcustangent.
+/** Numeric arcustangent of two arguments, analytically continued in a suitable way.
*
- * @param x real number
- * @param y real number
- * @return atan(y/x) */
-const numeric atan(const numeric & y, const numeric & x)
-{
- if (x.is_real() && y.is_real())
- return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
- else
- throw (std::invalid_argument("numeric::atan(): complex argument"));
+ * @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)
+const numeric sinh(const numeric &x)
{
- return ::sinh(*x.value); // -> CLN
+ return cln::sinh(x.to_cl_N());
}
/** Numeric hyperbolic cosine (trigonometric function).
*
* @return arbitrary precision numerical cosh(x). */
-const numeric cosh(const numeric & x)
+const numeric cosh(const numeric &x)
{
- return ::cosh(*x.value); // -> CLN
+ return cln::cosh(x.to_cl_N());
}
/** Numeric hyperbolic tangent (trigonometric function).
*
* @return arbitrary precision numerical tanh(x). */
-const numeric tanh(const numeric & x)
+const numeric tanh(const numeric &x)
{
- return ::tanh(*x.value); // -> CLN
+ 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)
+const numeric asinh(const numeric &x)
{
- return ::asinh(*x.value); // -> CLN
+ 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)
+const numeric acosh(const numeric &x)
{
- return ::acosh(*x.value); // -> CLN
+ 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 ::atanh(*x.value); // -> CLN
+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()) {
- int aux = (int)(::cl_double_approx(::realpart(*x.value)));
- if (zerop(*x.value-aux))
- return ::cl_zeta(aux); // -> CLN
- }
- clog << "zeta(" << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
- return numeric(0);
-}
-
+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();
+}
+
+class lanczos_coeffs
+{
+ public:
+ lanczos_coeffs();
+ bool sufficiently_accurate(int digits);
+ int get_order() const { return current_vector->size(); }
+ cln::cl_N calc_lanczos_A(const cln::cl_N &) const;
+ private:
+ // coeffs[0] is used in case Digits <= 20.
+ // coeffs[1] is used in case Digits <= 50.
+ // coeffs[2] is used in case Digits <= 100.
+ // coeffs[3] is used in case Digits <= 200.
+ static std::vector<cln::cl_N> *coeffs;
+ // Pointer to the vector that is currently in use.
+ std::vector<cln::cl_N> *current_vector;
+};
-/** The gamma function.
- * This is only a stub! */
-const numeric gamma(const numeric & x)
-{
- clog << "gamma(" << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
- return numeric(0);
+std::vector<cln::cl_N>* lanczos_coeffs::coeffs = 0;
+
+bool lanczos_coeffs::sufficiently_accurate(int digits)
+{ if (digits<=20) {
+ current_vector = &(coeffs[0]);
+ return true;
+ }
+ if (digits<=50) {
+ current_vector = &(coeffs[1]);
+ return true;
+ }
+ if (digits<=100) {
+ current_vector = &(coeffs[2]);
+ return true;
+ }
+ if (digits<=200) {
+ current_vector = &(coeffs[3]);
+ return true;
+ }
+ return false;
+}
+
+cln::cl_N lanczos_coeffs::calc_lanczos_A(const cln::cl_N &x) const
+{
+ cln::cl_N A = (*current_vector)[0];
+ int size = current_vector->size();
+ for (int i=1; i<size; ++i)
+ A = A + (*current_vector)[i]/(x+cln::cl_I(-1+i));
+ return A;
+}
+
+// The values in this function have been calculated using the program
+// lanczos.cpp in the directory doc/examples. If you want to add more
+// digits, be sure to read the comments in that file.
+lanczos_coeffs::lanczos_coeffs()
+{ if (coeffs)
+ return;
+ /* Use four different arrays for different accuracies. */
+ coeffs = new std::vector<cln::cl_N>[4];
+ std::vector<cln::cl_N> coeffs_12(12);
+ /* twelve coefficients follow. */
+ coeffs_12[0] = "1.000000000000000002194974863102775496587";
+ coeffs_12[1] = "133550.502942477423232096703994753698903";
+ coeffs_12[2] = "-492930.93529936026920053070245469905582";
+ coeffs_12[3] = "741287.473697611642492293025524275986598";
+ coeffs_12[4] = "-585097.37760399665198416642641725036094";
+ coeffs_12[5] = "260425.270330385275465083772352301818652";
+ coeffs_12[6] = "-65413.3533961142651069690504470463782994";
+ coeffs_12[7] = "8801.45963508441793636152568413199291892";
+ coeffs_12[8] = "-564.805024129362118607692062642312799553";
+ coeffs_12[9] = "13.80379833961490898061357227729422691903";
+ coeffs_12[10] = "-0.0807817619724537563116612761921260762075";
+ coeffs_12[11] = "3.47974801622326717770813986587340515986E-5";
+ coeffs[0].swap(coeffs_12);
+ std::vector<cln::cl_N> coeffs_30(30);
+ /* thirty coefficients follow. */
+ coeffs_30[0] = "1.0000000000000000000000000000000000000000000000445658922238202528026977308762";
+ coeffs_30[1] = "1.40445649204966682962030786915579421135474600150789821268713805046080310901683E13";
+ coeffs_30[2] = "-1.4473384178280338809560100504713144673757322488310852336205875273000116908753E14";
+ coeffs_30[3] = "6.9392104219998816400402602197781299548036066538116472480223222192156630720206E14";
+ coeffs_30[4] = "-2.05552680548452350127164925238339710431333013110755662640014074226849466382297E15";
+ coeffs_30[5] = "4.21346047774975891986783355395961145235696863271597017695734168781011785582523E15";
+ coeffs_30[6] = "-6.3439111294220458481092019992445750626799029041090235945435769621790257585491E15";
+ coeffs_30[7] = "7.2684029986336427327225410026373012514882246322145965580608264703248155838791E15";
+ coeffs_30[8] = "-6.4784969409198000751978874152931803231807770528527455966624850088042561231024E15";
+ coeffs_30[9] = "4.5545745239457403086706103662737668418631761744785802123770605916210445083544E15";
+ coeffs_30[10] = "-2.54592491966737919409139938046543941491145224466411852277136834553178078105403E15";
+ coeffs_30[11] = "1.1356718195163150156198936885250451780214219874255251444701005988134747787666E15";
+ coeffs_30[12] = "-4.04275236298036712070700727222520609783336229393218886420197964965371362011123E14";
+ coeffs_30[13] = "1.14472757259832757229433124273590647229089622322597383276758880048004748372644E14";
+ coeffs_30[14] = "-2.56166271828342920179612184110684658183432315551120625854181503468327037516717E13";
+ coeffs_30[15] = "4.4861708254018935131376878973710146069395814469656232761173409397653101421558E12";
+ coeffs_30[16] = "-6.0657495816705687896607821799338217335976369800808791959096705890743701166037E11";
+ coeffs_30[17] = "6.21975328147406581536747878587069711930541459818297675578654403265380823122363E10";
+ coeffs_30[18] = "-4.7255003764027411113501086372508071116675161078057298991208060427341079636661E9";
+ coeffs_30[19] = "2.5814613908651936680441351265410235295992556406609945442133129515256889464315E8";
+ coeffs_30[20] = "-9752115.5047412418881417732027953903591189993329461844657371497174389592441887";
+ coeffs_30[21] = "242056.60372411758318197954509546521913927205056839365620249547101194072057318";
+ coeffs_30[22] = "-3686.17673045938850138289555088011327333352145765167200561022138925168680049115";
+ coeffs_30[23] = "31.3494924501834034405048975310989414795238339283146314931357877820190435258517";
+ coeffs_30[24] = "-0.130254774344853676030752542814176943723937677940441021884132211221409382350105";
+ coeffs_30[25] = "2.16625679868432886771581352257834967866602495378408740265571976698475288337338E-4";
+ coeffs_30[26] = "-1.05077239977528252603869373455592388508233760416601143477182890107978206726294E-7";
+ coeffs_30[27] = "8.5728436055212340846907439451102962820713733082683634385104363203776378266115E-12";
+ coeffs_30[28] = "-3.9175430218003196379961975369936752665267219444417121562332986822123821080906E-17";
+ coeffs_30[29] = "1.06841715008998384033789050831892757796251622802680860264598247667384268519263E-24";
+ coeffs[1].swap(coeffs_30);
+ std::vector<cln::cl_N> coeffs_60(60);
+ /* sixty coefficients follow. */
+ coeffs_60[0] = "1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000007301368866363013444179014835363181183419450549774";
+ coeffs_60[1] = "2.13152397525281235754468356918725048606852617746577461250754322057711822075135461598274984226013367948201688447853106595646692682568953E26";
+ coeffs_60[2] = "-4.548529924829267669336610112411669181387790087825260737133755173032543313325682598833009521765336124891170163525664509845740222794717604E27";
+ coeffs_60[3] = "4.6879437426294973235875133160595324795437824160731608900005486977197800919261614723948577079551305728583507312310069280623018775850412E28";
+ coeffs_60[4] = "-3.10861265267020467624457768823845414206135580030123228715133927538323570190367768297139526311161786169387040978744732051184844409191231E29";
+ coeffs_60[5] = "1.490599577483981276717037178787147902256911799467742317379590487947009001487476793680630580522955117318124168494382267800788736334308E30";
+ coeffs_60[6] = "-5.50755504045738806940255910881807353185463857314393682608295373644157298562106198431098170107741597645409216199785852260920496247655646E30";
+ coeffs_60[7] = "1.631668518639067070100242032960081591016027803392225476881353619523143028349554534276268195490790113905102273979193269720381236708853746E31";
+ coeffs_60[8] = "-3.9823057865511431381368541930378720290638930941334849821428293955264049587073723565727061718251925950255036781219414607001763225298119E31";
+ coeffs_60[9] = "8.16425963140638737297557821827674142140347732117757126331775708561852858085860735359056658172512163756926693444882201094206795155146202E31";
+ coeffs_60[10] = "-1.426548236351667330492229413193359354309705120770113917370333660827270957172393778178051742077714657388432785747112574456061555034588373E32";
+ coeffs_60[11] = "2.14821861694536170414714365485614715949416083667308573285807894910742621740039595483105992136915471547998283891842897000924199509164799E32";
+ coeffs_60[12] = "-2.81233281290021706519566203146379395136352592819625378308636458418501787286411189089807465993150834399778687427813779950602826375635436E32";
+ coeffs_60[13] = "3.222783358826786224404373038021509245352188734386849874296356404770508945395436142634892645963851510893216093037595555902121365717716154E32";
+ coeffs_60[14] = "-3.250409075716999887328836263791911196138647661969351655925350981785153422033954649154242209471752219326556302767677017396179477496948985E32";
+ coeffs_60[15] = "2.897783210826628399578158893643627107049805015801395657097255344786041806868455726759715576609013221857885740543509045196763816109465777E32";
+ coeffs_60[16] = "-2.29136919195969647663887561122314618826917230275433296293059354280077561407373070937197721317435316121212106870152659174216557412788874E32";
+ coeffs_60[17] = "1.611288006928200619663496306945576194382628760891807800193737346171844871295031418730500946186238469256168610033434708290528870722514911E32";
+ coeffs_60[18] = "-1.009632466053186015034182792930705530447465885425278324598880797572411588461783484686932989855033967294215840157892487264656571258327313E32";
+ coeffs_60[19] = "5.64520651042784179741815642438421132518008517154942873706221206276337451930555926854271086501686252334516011905237101877044320182980053E31";
+ coeffs_60[20] = "-2.81912877441595327683492797147781153304080114512116755424671954256427789550109614317215500473322621746416096887803928883800132453510579E31";
+ coeffs_60[21] = "1.257934257434294354026338893625531254891110662111965279263894740714811495074726866375858553579650295684850594211744093582249745250079168E31";
+ coeffs_60[22] = "-5.01544407232599962845688086323662774702854661522104499328570796808858930542190600193190967249971520736397504227594619670310759235566195E30";
+ coeffs_60[23] = "1.786035425040937365122699272239542501767986628253845452136132211710520249195280548478081559036323184490150479070929923213045153333111476E30";
+ coeffs_60[24] = "-5.67605430104368150038863866362066081946938075036837029856903803768657069745962581310398542442108872722631658677177822712376500859930109E29";
+ coeffs_60[25] = "1.607878222558573982505999018371559631909289246981490321219650132406126936263403946310818841465409950661433241956831540547593847161412447E29";
+ coeffs_60[26] = "-4.05332042374309456146169816144083508836132423024788116321074411679252452773181941601763924562378611113519038766273534176937279867894066E28";
+ coeffs_60[27] = "9.07493596543985672039002802030098143847503854224661484396413496012780904911929710460264147600378604646912175235271954302119768907744722E27";
+ coeffs_60[28] = "-1.800074018924350353143489874038038169034914082090587278672411654146678304871125651069902339241049552886098125667720181441150399048551683E27";
+ coeffs_60[29] = "3.154250688078046681602499411296013099183808016176992164829953752437167774310360166977972581670851790753785195101324694758021403186162394E26";
+ coeffs_60[30] = "-4.86629244083379932983782216256143990390210226006560452979433243294026128577640975980482675864760717747936401374948595060083674140963469E25";
+ coeffs_60[31] = "6.58428611248406176613133080039790689602908099995907522692286902207707012485115422092589779128693214784991500936878932461139361901566087E24";
+ coeffs_60[32] = "-7.77846893445970039116628280774361378296946997639645747353868461156972352366479641995295874152354776734003001337605345817120316052066992E23";
+ coeffs_60[33] = "7.98268735994772082084918485121285571015813651374688487489679943603727447378945977989630573952891101472578977333720105112837324185659362E22";
+ coeffs_60[34] = "-7.07562692971089746095546542541499489835693326760069291570193808615779224025348460132750549389189539682228913778397783434269420284483726E21";
+ coeffs_60[35] = "5.381346729881846847476909845563262674288431852755093265786345982700437823098162630059919716651136095720390719236493773958116646152386075E20";
+ coeffs_60[36] = "-3.4856856542678356876484367392130359114150104987588151214926676834365219571876912071608359944324610844909103855562977795837329347647911E19";
+ coeffs_60[37] = "1.90665542883474657677037950113781854248329048412482665873254624417996252139138481002200079466749149325431679310476862249520001277129217E18";
+ coeffs_60[38] = "-8.72254994006151131395107200045641306281165826830744222866994799005490857259177347821280095689079457417603257537321939951004603693393316E16";
+ coeffs_60[39] = "3.30066663941625244322555483012774856710545517350986120571194216206848716066355962922968824538055042855044917677713272771363157100391997E15";
+ coeffs_60[40] = "-1.020092089391030771746960980075254826475625668908623135552682999358854102567810002206013823466362488147261886160954607897574298699485318E14";
+ coeffs_60[41] = "2.537518136375035057088980117582986067754938584307761188810498418760131416720976321039509027979006220650166651208980823946300429957067604E12";
+ coeffs_60[42] = "-4.99523339577986301543863423322168947825482352498610406809585164155176248614834684219539096936869521198401912030883142734471627752449382E10";
+ coeffs_60[43] = "7.62961024898383965152735310352890448678585029645218309944823403624458716639413808284778269959424212699922000610764015063766429510499158E8";
+ coeffs_60[44] = "-8834336.1370238009649936481782352367054397712953420330251745022286767420934395739052638862442455545176778475848478708230456099596423988";
+ coeffs_60[45] = "75445.9196169409678879362111492280315111800786619928588067631801224813888137547544321383450353324917130013984795690223150786036557545929";
+ coeffs_60[46] = "-459.8458738886001056822131294892698769439281099450630714273592488999986769567563218319365007529495798105783705491469742412340762305916056";
+ coeffs_60[47] = "1.922366163948404706136462977961544621491268971185908661903800938507393909575693892375103171073678191394626251633433930639174604982075991";
+ coeffs_60[48] = "-0.00524987734300376305383172698735851896799115189212445098242699916121836353753886238290792298378658233479210271064792489583846726184351881";
+ coeffs_60[49] = "8.81521840386771771843311455937479573971716020932982441671173279504850522350287085310420429874536637110755391716691475171030099411021337E-6";
+ coeffs_60[50] = "-8.42883518072336499031504944519862331274440110738275125460829656821173301216150526266773841539372995424665091651911614576906895281293397E-9";
+ coeffs_60[51] = "4.1559932977982056953309753711587342647729282359841592558743510304569204546713517319749817560490538963802716194154620384631597656968764E-12";
+ coeffs_60[52] = "-9.26494376646923216540342478135986593801117330292329759013854851055518195892306285985326338987592590319793280515888731024676428929933443E-16";
+ coeffs_60[53] = "7.80165274836868312019654872701978288745672229459298320116385383568401529728308916875595120085091565550085090877341856355815270191309086E-20";
+ coeffs_60[54] = "-1.922049272463411538721456378153955404697617250978865956250065913541261535132290272529565880980548519758359440057376306817458561627984943E-24";
+ coeffs_60[55] = "9.46189821976955264154519811789356895736753858729897267240554901027053652869864043679401817030067356960879571432881603836052222728024736E-30";
+ coeffs_60[56] = "-5.06814507370603015985813829025522226614719112357562650414521252967497371724973383019436312018485582224796590023220166954083973156538672E-36";
+ coeffs_60[57] = "1.022249951013180267209479446016461291488484443236553319305574600271584296178678167457933405768832443689762998392188667506451117069946568E-43";
+ coeffs_60[58] = "-1.158776990252157075591666544736990249102708476419363164106801472497162421792350234416969073422311477683246469337273059290064112071625785E-47";
+ coeffs_60[59] = "4.27222387142756413870104074160770434521893587460314314301300261552300727494374933435001642531897059406263033431558827297492879960920275E-49";
+ coeffs[2].swap(coeffs_60);
+ std::vector<cln::cl_N> coeffs_120(120);
+ /* 120 coefficients follow. */
+ coeffs_120[0] = "1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000060166025676976656004344991957470171590616719251813003320122316373430327091055571";
+ coeffs_120[1] = "3.4497260317073952007403696383770947678893302981614719279265682622766639811173298171511730607823612517530376844024218507032522459279662180470113961839690189982241536061314319614353993672315096520499373373015802582693649149063603309572777186560148513524E52";
+ coeffs_120[2] = "-1.4975581565000729527538170857594663742319328925831469933998274880997450758924704742659571258591716460336677591345828722528085692201176737000527729600671680178988361119859420301844184208079614468449296788394212801103162564922199859549237082372776667464E54";
+ coeffs_120[3] = "3.1957762163065481328529158845807843312720427291703934903666695190945338610786360201875291048323381336567812569171891600400186742244091402566230953251621720778096033490814848238417212345597975915378369497445590090951446115848410773972658485451963575288E55";
+ coeffs_120[4] = "-4.4689623509319752841609439083871154399631153121231062689347162975834499076693093642474289117173045421812089871506249999929076992135798925381959196225961791389783472385803138226317976820364502651110639008585046458007356178875618627927171581950486124233E56";
+ coeffs_120[5] = "4.606068718424276543329442566011849623375399823565351941825685310847310447457609082356012685588953435307896055516214072529445026693975872604267789672469025113562486157850515006504573881812473997762948360804814769118883992998548055557441646946685125118E57";
+ coeffs_120[6] = "-3.7314461146854666499272326592212099391213696621869706562566612605818861385928266960370453310708465394226398321257508947092784006446784523328681347046673172481746936234783770854350210504707173921547794426833735429199925024679815789545465854297845328325E58";
+ coeffs_120[7] = "2.474425401670711256989398808079221298913654027234786607507813220440957186918973475366048940039541074278444160674001228864321389049663140487504402096319272526201782217412803784224929141788255724630940381342478088455751340159338461174261577243566175687E59";
+ coeffs_120[8] = "-1.3811875718622847750042362590249762290599823842851465148429257970104907280458901604054390293828410620002370526629527048636126473391278330353375163563724888073254512227198849135923692811222561965740181944727170495185714496890490479692693474125883791901E60";
+ coeffs_120[9] = "6.623089858532754482582703479109160446021743439335073883710993620625687271109284320721410901325182604938578905712329203551531862389936804947105415805829404869727743706364603519433193421234231031076682156125442577335383798263985569601899041876776866622E60";
+ coeffs_120[10] = "-2.7709515004299938864490083840820063124223529009388282231525445615826433364331567602934962481829061542793349831611106716261513624279121506887680318284535361848032886450351898892264386237450622827397559067350672965967202437971930333676917000390477963866E61";
+ coeffs_120[11] = "1.02386112293172223921263435003659366453292875147351461165091656394534393086780717052422266565203902889367201592668259202439166666819852985689989767402099479793087277263747942659943270101657408462079787397068550734516045511611701546009078868077038808757E62";
+ coeffs_120[12] = "-3.3740197731917655541744976218513993073761175468772389726802124778433432226803314067431898210976006853342921093194297198044021414900546886804610561082663076825192459864843102368108908666053756409152492134638014803233805912009476407113691438596300794146E62";
+ coeffs_120[13] = "9.996217786487670355655374796561399578704294298563457268841140703036898520360123177193155340144551120260016445533739357030180277693840431766824113840895797510199955331557143980793267795747200088810293047731873192410526786931879590684673414288653913515E62";
+ coeffs_120[14] = "-2.6804750990199908441443350402311488850281543531194918304012545530803283220192092419107511475988099394746800512008906823331244710178292896561401750166818497729239682879419868799442954945496319510685448344062610897698253544876306888341881254056091234759E63";
+ coeffs_120[15] = "6.5422964482833531603879610057815197301035372862466791995455246163778529556707384145891730234453157337303612060344197138180893720879196243783337539071284141345021864817147590781643393947019750353147151780290464319306645652085743359080495090595200531486E63";
+ coeffs_120[16] = "-1.4604487304348366496825146715570516556564950771546738885215899741781982964860978993963272314092830563320794184042847908967120542212316261920409301852237223308467032419706968676861616456179895880956772385510853673982424825597152850339588189159102980666E64";
+ coeffs_120[17] = "2.9942297466313630831467808691292548682230644559492580161942357031681185068971393754352871129412787966878287389513657398203481589163498279625760316093736277896138061249076616695157422053188087353540756151586375196486093987258640269607104978950906670704E64";
+ coeffs_120[18] = "-5.658399283776588293772725313973093187743120982052603944865098913586526167668102733207163739469977271584007101869254711458133873627143366757941713180350370056955237604551850024423889291598422917971467957836705917204903687959901869098540153925178692732E64";
+ coeffs_120[19] = "9.887368951584101622633538892976576123080629424367489037686110916264512731398396560326756128205833849608930564615629875435100785011872254223744155330328477703592008501954532369042429700051416733748454350165515933757314793533786385104271308839525639768E64";
+ coeffs_120[20] = "-1.6019504550228766725078575508839635707919311420327486864048705642201106895239857903763208049376932672160478820626774934879424498715258948985194011690204294886396827446040036506699933786721588971678753877371518212675519147446728054067639530675249526082E65";
+ coeffs_120[21] = "2.4124568469636899540706437405441629738413207418758399778576327598069435452295650039157974716832514441625728576753250737726840109004878753294786785674578138926529507088264657400701828947949531197915861820274684954206665488761473274445827472596875582911E65";
+ coeffs_120[22] = "-3.3841653726400000079488483558717068873181168418395106876260246491163166726612427450773591871178866824643300679819366574162583413250423974373322308130319007820863363304629451933781204964221002853140392226489420463827400812929748772154909106349410663293E65";
+ coeffs_120[23] = "4.4305670380812288478773282114598811227924298131011853412998479811262358077680067168455361591598296346480072528806092976336961470360354620203822421524751468329936930212919915114854135818230382164555078957880154875221176513434392525189922941290050575762E65";
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+ coeffs_120[86] = "-4.1476244154347831048636005592317388215032295704489937704602030038303705695463546496640638505584602502764898113504560236629804442607426019604639559875021291459916615723777004493344143132459204229291886967479716413925814352313734234340863490128872380307E17";
+ coeffs_120[87] = "4.8067293487250079673131214670887682215073707729621636364424152483295071605326220176372385638491275365750175037404843071051780212494354459897540110089573898336327006157766256896984455454193271731091632286742192439925748114360605084629432813597189767538E15";
+ coeffs_120[88] = "-4.8023544548381246628003457039588616467438691159189277447469028024236284353593054364114519649309416187375157096932150251663679454372678125518452171003992957433311257042292636706448339781439297178835786059318810522834929923770539615271536113963729385909E13";
+ coeffs_120[89] = "4.1055087514683476865343055835875083237542317413651906253058979029083965525058905726360233143503628224856307545474786181299719957472120906835233967660557875100202077212004953379299507351564181758434304881046845705855303854083493519588411179065109026834E11";
+ coeffs_120[90] = "-2.9787503393847675871205038539267895335240592213878943742323972872214441728681744433089698206110260166068266926018988659692353298939109421567999207730700359726920482465669373553804927535369930188390246988033893916611435406224816632683980860607732310186E9";
+ coeffs_120[91] = "1.8178328110729629877907010659834277046059726898311908447099830056830012488194646687474150289147446390570639168063598563291822008033517936194534129929881015025633519502485415000390171249019651579295905194415531994026553693578406432674734610095421683863E7";
+ coeffs_120[92] = "-92391.136314434380495997449781381513978328604842061708454699991154771188446049720445502194923435235472458378926242100033122111143321209059959788378033220861638093951546784186137626553022963832352496255851690092415165826965388502958309163995296640164754";
+ coeffs_120[93] = "386.82763074890451546182061419449593717951707520472938425276820204065379182568600735469831672149785863654956632602671563997131280046154927653332261114114005498875447205079045401364007035880825957300757663780618819785476980699579657587509130753204519233";
+ coeffs_120[94] = "-1.3181204292571874302358432444324779303744749959754136019600954846045028319805074783759764870805734807693739252625657350494147444011046941331047057337345953605042408524072436811726898109072388160378243068564382623631658424851676817690976343859083960324";
+ coeffs_120[95] = "0.003606538673252695455085947121496196507159591230095595764694813152630524319596509155920374890595867709349176662036024214476302717902368680224618116411588086562230407996267622244422187853090635901906175373997993725355114393033631058067900506212434600015";
+ coeffs_120[96] = "-7.805244503909439374422205381130511738566245024242591464192744568789876715121004646510755612128565674260161510215430132815223049297785205382643947556846567064565241387424696940674258789227398935846571768027456535982674711768030751512030174841314425949E-6";
+ coeffs_120[97] = "1.31373705470989377112938364152965446631228819123896570245455699237549295870321627234421140232628798373711221392827979836922621437205363811871692678679625916100572037589291239046725228767017131155814257944742981208252138821140381478767814046301821211856E-8";
+ coeffs_120[98] = "-1.6872873094408224472617181717534409090015431593544429529131126514352910895332010213914243717484771690790552077128803350550170014347729272790464826195676369023970955260051387240496705602732313607409271794413329062030561818907163134089683283286623809325E-11";
+ coeffs_120[99] = "1.6183083251905685095057354853863188515437903228178486856957070037813756492593759658405336450433607296873747595037080703825755020175480385843762609522889527239577435110258291566585028919336090916225831079571865425410181260759913688103716786795647286451E-14";
+ coeffs_120[100] = "-1.13097359411474028225398794102354853670936316496817819635688647804142428962171772690717075128208102537660772310780986623575005236651312181907812813813504742701120603881086064664411899253566047514905519888629604717647221817372977488600336785871295304013E-17";
+ coeffs_120[101] = "5.599216369109121957949255319730053610385733330502739423509794477602247233276045188197007198417289907263120960704056657544648432653622931077692740961599655386871075693202473992087883485704436336279135221721374640982826144708808646466699352755417123702E-21";
+ coeffs_120[102] = "-1.9009180102993021108185348502624676395148544369474718879637745630712451378711342634099259114111847962752555305470572286326367888004493816251811794947276966269738750207359305252041104539066278002044545942171476984766923991983055271262414217352967659228E-24";
+ coeffs_120[103] = "4.261262509940940316499754264112111685174274727656165126333137554124192224955656564229887938745508952447664695831728428607673797269945824475565104978593072684829487175697371245288754204324544164474840153141042852153497051337607734150135978754952561336E-28";
+ coeffs_120[104] = "-6.033854291373449912236926137860325602686312455380825767485673949251953414778800668020214699151728472172651816317924130614791108454134597377848088327850505473503152696524861086193124979489104732214189466703901268332265826882296309653009237279831825243E-32";
+ coeffs_120[105] = "5.1208402745272379096703574714836785944518835939702823617280147111145234914591060871138496110227453241036619229980622243972303295470574470937679143516006222494480144845809123492603651773613707216680534850900104861326332900592715684757980394834998321888E-36";
+ coeffs_120[106] = "-2.4463535717946588550832618025289907099586319384566637643650142186828541109926588999585266911960640972919441499109750654299062004147686492034166034659422424525984094382368955916181276646903453872999065929058429821759475215620044891133652431220664754175E-40";
+ coeffs_120[107] = "6.0973480699773886324239008989591793773608942051497498591908583910660358857815864266160341286217871697703362816166340947142517661604423899536979689047275448159991318658879804351288744125363072102852651926942302209139318098544348348564409845011546432615E-45";
+ coeffs_120[108] = "-7.2234185761285078775026471720270426097727212523472472797635230392183067756271499246654638332288950167477129840028892565652782123508855602380279653475510712205780583313834027906297063690370430285856541927759405826980856379432703473274890527421175151858E-50";
+ coeffs_120[109] = "3.6217112680215791206171182969894344487335819731880124290544082848140757826983885738735436324684863867140575000400288923606439193119990961489053513339202655922248092157737577138929144240507796562250602457839068582279379672722261563501188150876583184441E-55";
+ coeffs_120[110] = "-6.6329300032795486066608594142675837603786558782159646987663521197523704085781830169369726460621246948945196657495305819768951424025780824076252490918306538895670861455244641773606980519824591785816943621538721352987553804824051051144609050417497894495E-61";
+ coeffs_120[111] = "3.6664720904335295532012711597888717227860988776477301054518326674835421172405060906940404374163713097964932859351917152390238690399278248344863365606468942320103392909602843987855082225592776850615943708151738327210634139824601616072015258461809772448E-67";
+ coeffs_120[112] = "-4.7466013179695826928232672846686064011594588664906398407027593213652099998530859940288723349213099851532139911079905393494419637612780994270110734378146177806681489226896952731800026849872070824592339117757940119304241732812925979963178130104280115315E-74";
+ coeffs_120[113] = "1.0163707785221910939390789816391472677729665860532352695801597334766068288835382195560328979864550624486740471947632369344045378626680607890520366137741785540226552923584183986350590955499329375427326072319268396685478606934920507703868118038891818762E-81";
+ coeffs_120[114] = "-3.4814151260242800905467399051937942442621710748397374123807284826536707678408888416026868585492229216524609739211131993326633970334388991812593549702868877534701822990946125111761892723042376117665640296993581745994557803052315791392349639065203872505E-90";
+ coeffs_120[115] = "1.18525924288117432386770939895670573772658621857195305986011196724304231598127227408839423385042572374412446842112646168302015480830234100570192462192015131968307084609177540911503689228342834030959242458698413980031135644018348590823980902427540799814E-91";
+ coeffs_120[116] = "-8.5714961216566153236700116412888006837408819915951896129362859520462766617634320531162919426026429378433105901035364956643086394331335747930198070611009941831387116980941022864465946989065467218665543814574849964435089931072761832853235509961870476035E-93";
+ coeffs_120[117] = "4.5681983751743456413033268196376305093509590040595182930261094908859252761697530924655649930852283295534503341542929581967081012867692190108698698006237799801339418962091877730207560007839789937153876806052229193448161273005984514504886230869730232561E-94";
+ coeffs_120[118] = "-1.5943139155457706045530478744891549581317663177038648406493256399589001327414318955746453934207742828511041930090849236963271943244329753764497401819704943705370596846318480510254313447057477914171472190541408193443142906466279172123681623644325254209E-95";
+ coeffs_120[119] = "2.7319125666863032595604997603472305262880292377469053594326527505796348018540179196191192420176181194669607935656210005192217186286873953583571180312679155204061051208771126804209623533044988888808754656646355388901404252058383561064953226611421609762E-97";
+ coeffs[3].swap(coeffs_120);
+}
+
+
+/** The Gamma function.
+ * Use the Lanczos approximation. If the coefficients used here are not
+ * sufficiently many or sufficiently accurate, more can be calculated
+ * using the program doc/examples/lanczos.cpp. In that case, be sure to
+ * read the comments in that file. */
+const numeric lgamma(const numeric &x)
+{
+ lanczos_coeffs lc;
+ if (lc.sufficiently_accurate(Digits)) {
+ cln::cl_N pi_val = cln::pi(cln::default_float_format);
+ if (x.real() < 0.5)
+ return log(pi_val) - log(sin(pi_val*x.to_cl_N()))
+ - lgamma(numeric(1).sub(x));
+ cln::cl_N A = lc.calc_lanczos_A(x.to_cl_N());
+ cln::cl_N temp = x.to_cl_N() + lc.get_order() - cln::cl_N(1)/2;
+ cln::cl_N result = log(cln::cl_I(2)*pi_val)/2
+ + (x.to_cl_N()-cln::cl_N(1)/2)*log(temp)
+ - temp
+ + log(A);
+ return result;
+ }
+ else
+ throw dunno();
+}
+
+const numeric tgamma(const numeric &x)
+{
+ lanczos_coeffs lc;
+ if (lc.sufficiently_accurate(Digits)) {
+ cln::cl_N pi_val = cln::pi(cln::default_float_format);
+ if (x.real() < 0.5)
+ return pi_val/(sin(pi_val*x))/(tgamma(numeric(1).sub(x)).to_cl_N());
+ cln::cl_N A = lc.calc_lanczos_A(x.to_cl_N());
+ cln::cl_N temp = x.to_cl_N() + lc.get_order() - cln::cl_N(1)/2;
+ cln::cl_N result
+ = sqrt(cln::cl_I(2)*pi_val) * expt(temp, x.to_cl_N()-cln::cl_N(1)/2)
+ * exp(-temp) * A;
+ return result;
+ }
+ else
+ throw dunno();
}
/** The psi function (aka polygamma function).
* This is only a stub! */
-const numeric psi(const numeric & x)
+const numeric psi(const numeric &x)
{
- clog << "psi(" << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
- return numeric(0);
+ throw dunno();
}
/** The psi functions (aka polygamma functions).
* This is only a stub! */
-const numeric psi(const numeric & n, const numeric & x)
+const numeric psi(const numeric &n, const numeric &x)
{
- clog << "psi(" << n << "," << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
- return numeric(0);
+ throw dunno();
}
*
* @param n integer argument >= 0
* @exception range_error (argument must be integer >= 0) */
-const numeric factorial(const numeric & n)
+const numeric factorial(const numeric &n)
{
- if (!n.is_nonneg_integer())
- throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
- return numeric(::factorial(n.to_int())); // -> CLN
+ 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 Gamma(n+1/2) for instance.)
+ * 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)
+const numeric doublefactorial(const numeric &n)
{
- if (n == numeric(-1)) {
- return _num1();
- }
- if (!n.is_nonneg_integer()) {
- throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
- }
- return numeric(::doublefactorial(n.to_int())); // -> CLN
+ 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()));
}
* 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())!=-1)
- return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
- else
- return _num0();
- } else {
- return _num_1().power(k)*binomial(k-n-_num1(),k);
- }
- }
-
- // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit
- throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
+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.");
}
*
* @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"));
- if (nn.is_zero())
- return _num1();
- if (!nn.compare(_num1()))
- return numeric(-1,2);
- if (nn.is_odd())
- return _num0();
- // Until somebody has the Blues and comes up with a much better idea and
- // codes it (preferably in CLN) we make this a remembering function which
- // computes its results using the defining formula
- // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
- // whith B(0) == 1.
- // Be warned, though: the Bernoulli numbers are probably computationally
- // very expensive anyhow and you shouldn't expect miracles to happen.
- static vector<numeric> results;
- static int highest_result = -1;
- int n = nn.sub(_num2()).div(_num2()).to_int();
- if (n <= highest_result)
- return results[n];
- if (results.capacity() < (unsigned)(n+1))
- results.reserve(n+1);
-
- numeric tmp; // used to store the sum
- for (int i=highest_result+1; i<=n; ++i) {
- // the first two elements:
- tmp = numeric(-2*i-1,2);
- // accumulate the remaining elements:
- for (int j=0; j<i; ++j)
- tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
- // divide by -(nn+1) and store result:
- results.push_back(-tmp/numeric(2*i+3));
- }
- highest_result=n;
- return results[n];
+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;
+ // The CLN manual says: "The conversion from `unsigned int' works only
+ // if the argument is < 2^29" (This is for 32 Bit machines. More
+ // generally, cl_value_len is the limiting exponent of 2. We must make
+ // sure that no intermediates are created which exceed this value. The
+ // largest intermediate is (p+3-2*k)*(p/2-k+1) <= (p^2+p)/2.
+ if (p < (1UL<<cl_value_len/2)) {
+ for (unsigned k=1; k<=p/2-1; ++k) {
+ c = cln::exquo(c * ((p+3-2*k) * (p/2-k+1)), (2*k-1)*k);
+ b = b + c*results[k-1];
+ }
+ } else {
+ for (unsigned k=1; k<=p/2-1; ++k) {
+ c = cln::exquo((c * (p+3-2*k)) * (p/2-k+1), cln::cl_I(2*k-1)*k);
+ b = b + c*results[k-1];
+ }
+ }
+ results.push_back(-b/(p+1));
+ }
+ next_r = n+2;
+ return results[n/2-1];
}
* @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"));
- // 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();
- if (n.is_negative())
- if (n.is_even())
- return -fibonacci(-n);
- else
- return fibonacci(-n);
-
- cl_I u(0);
- cl_I v(1);
- cl_I m = The(cl_I)(*n.value) >> 1L; // floor(n/2);
- for (uintL bit=::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.
- cl_I u2 = ::square(u);
- cl_I v2 = ::square(v);
- if (::logbitp(bit-1, m)) {
- v = ::square(u + v) - u2;
- u = u2 + v2;
- } else {
- u = v2 - ::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 ::square(u) + ::square(v);
+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. */
-numeric abs(const numeric & x)
+const numeric abs(const numeric& x)
{
- return ::abs(*x.value); // -> CLN
+ return cln::abs(x.to_cl_N());
}
*
* @return a mod b in the range [0,abs(b)-1] with sign of b if both are
* integer, 0 otherwise. */
-numeric mod(const numeric & a, const numeric & b)
+const numeric mod(const numeric &a, const numeric &b)
{
- if (a.is_integer() && b.is_integer())
- return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- else
- return _num0(); // Throw?
+ 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(m)-1,2), iquo(abs(m),2)]. */
-numeric smod(const numeric & a, const numeric & b)
+ * @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()) {
- cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1;
- return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
- } else
- return _num0(); // Throw?
+ 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;
}
* 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. */
-numeric irem(const numeric & a, const numeric & b)
+ * @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 (a.is_integer() && b.is_integer())
- return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- else
- return _num0(); // Throw?
+ 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.
+ * 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. */
-numeric irem(const numeric & a, const numeric & b, numeric & q)
+ * 0 otherwise.
+ * @exception overflow_error (division by zero) if b is zero. */
+const numeric irem(const numeric &a, const numeric &b, numeric &q)
{
- if (a.is_integer() && b.is_integer()) { // -> CLN
- cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
- q = rem_quo.quotient;
- return rem_quo.remainder;
- }
- else {
- q = _num0();
- return _num0(); // Throw?
- }
+ 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. */
-numeric iquo(const numeric & a, const numeric & b)
+ * @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 (a.is_integer() && b.is_integer())
- return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- else
- return _num0(); // Throw?
+ 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;
}
* r == a - iquo(a,b,r)*b.
*
* @return truncated quotient of a/b and remainder stored in r if both are
- * integer, 0 otherwise. */
-numeric iquo(const numeric & a, const numeric & b, numeric & r)
+ * 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 (a.is_integer() && b.is_integer()) { // -> CLN
- cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
- r = rem_quo.remainder;
- return rem_quo.quotient;
- } else {
- r = _num0();
- return _num0(); // Throw?
- }
-}
-
-
-/** Numeric square root.
- * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
- * should return integer 2.
- *
- * @param z numeric argument
- * @return square root of z. Branch cut along negative real axis, the negative
- * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
- * where imag(z)>0. */
-numeric sqrt(const numeric & z)
-{
- return ::sqrt(*z.value); // -> CLN
-}
-
-
-/** Integer numeric square root. */
-numeric isqrt(const numeric & x)
-{
- if (x.is_integer()) {
- cl_I root;
- ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
- return root;
- } else
- return _num0(); // Throw?
+ 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;
+ }
}
*
* @return The GCD of two numbers if both are integer, a numerical 1
* if they are not. */
-numeric gcd(const numeric & a, const numeric & b)
+const numeric gcd(const numeric &a, const numeric &b)
{
- if (a.is_integer() && b.is_integer())
- return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- else
- return _num1();
+ 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;
}
*
* @return The LCM of two numbers if both are integer, the product of those
* two numbers if they are not. */
-numeric lcm(const numeric & a, const numeric & b)
+const numeric lcm(const numeric &a, const numeric &b)
{
- if (a.is_integer() && b.is_integer())
- return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- else
- return *a.value * *b.value;
+ 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(void)
+ex PiEvalf()
{
- return numeric(::cl_pi(cl_default_float_format)); // -> CLN
+ return numeric(cln::pi(cln::default_float_format));
}
-/** Floating point evaluation of Euler's constant Gamma. */
-ex EulerGammaEvalf(void)
+/** Floating point evaluation of Euler's constant gamma. */
+ex EulerEvalf()
{
- return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
+ return numeric(cln::eulerconst(cln::default_float_format));
}
/** Floating point evaluation of Catalan's constant. */
-ex CatalanEvalf(void)
+ex CatalanEvalf()
{
- return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
+ return numeric(cln::catalanconst(cln::default_float_format));
}
-// It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
-// be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
-// of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
+/** _numeric_digits default ctor, checking for singleton invariance. */
_numeric_digits::_numeric_digits()
- : digits(17)
+ : digits(17)
{
- assert(!too_late);
- too_late = true;
- cl_default_float_format = ::cl_float_format(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);
+
+ // add callbacks for built-in functions
+ // like ... add_callback(Li_lookuptable);
}
+/** Assign a native long to global Digits object. */
_numeric_digits& _numeric_digits::operator=(long prec)
{
- digits=prec;
- cl_default_float_format = ::cl_float_format(prec);
- return *this;
+ long digitsdiff = prec - digits;
+ digits = prec;
+ cln::default_float_format = cln::float_format(prec);
+
+ // call registered callbacks
+ std::vector<digits_changed_callback>::const_iterator it = callbacklist.begin(), end = callbacklist.end();
+ for (; it != end; ++it) {
+ (*it)(digitsdiff);
+ }
+
+ return *this;
}
+/** Convert global Digits object to native type long. */
_numeric_digits::operator long()
{
- return (long)digits;
+ // 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;
}
-void _numeric_digits::print(ostream & os) const
+/** Add a new callback function. */
+void _numeric_digits::add_callback(digits_changed_callback callback)
{
- debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
- os << digits;
+ callbacklist.push_back(callback);
}
-ostream& operator<<(ostream& os, const _numeric_digits & e)
+std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
{
- e.print(os);
- return os;
+ e.print(os);
+ return os;
}
//////////
* assignment from C++ unsigned ints and evaluated like any built-in type. */
_numeric_digits Digits;
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff