* of special functions or implement the interface to the bignum package. */
/*
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2000 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
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
+#include "config.h"
+
#include <vector>
#include <stdexcept>
+#include <string>
-#include "numeric.h"
-#include "ex.h"
-#include "config.h"
-#include "debugmsg.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:
-#ifdef HAVE_CLN_CLN_H
-#include <CLN/cln.h>
+#if defined(HAVE_SSTREAM)
+#include <sstream>
+#elif defined(HAVE_STRSTREAM)
+#include <strstream>
#else
-#include <cln.h>
+#error Need either sstream or strstream
#endif
+#include "numeric.h"
+#include "ex.h"
+#include "archive.h"
+#include "debugmsg.h"
+#include "utils.h"
+
+// CLN should pollute the global namespace as little as possible. Hence, we
+// include most of it here and include only the part needed for properly
+// declaring cln::cl_number in numeric.h. This can only be safely done in
+// namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
+// subset of CLN, so we don't include the complete <cln/cln.h> but only the
+// essential stuff:
+#include <cln/output.h>
+#include <cln/integer_io.h>
+#include <cln/integer_ring.h>
+#include <cln/rational_io.h>
+#include <cln/rational_ring.h>
+#include <cln/lfloat_class.h>
+#include <cln/lfloat_io.h>
+#include <cln/real_io.h>
+#include <cln/real_ring.h>
+#include <cln/complex_io.h>
+#include <cln/complex_ring.h>
+#include <cln/numtheory.h>
+
+#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
+#endif // ndef NO_NAMESPACE_GINAC
-// linker has no problems finding text symbols for numerator or denominator
-//#define SANE_LINKER
+GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
//////////
// default constructor, destructor, copy constructor assignment
/** default ctor. Numerically it initializes to an integer zero. */
numeric::numeric() : basic(TINFO_numeric)
{
- debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
- value = new cl_N;
- *value=cl_I(0);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
+ value = cln::cl_I(0);
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
}
numeric::~numeric()
{
- debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
- destroy(0);
+ debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
+ destroy(false);
}
-numeric::numeric(numeric const & other)
+numeric::numeric(const numeric & other)
{
- debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
- copy(other);
+ debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
+ copy(other);
}
-numeric const & numeric::operator=(numeric const & other)
+const numeric & numeric::operator=(const numeric & other)
{
- debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
- if (this != &other) {
- destroy(1);
- copy(other);
- }
- return *this;
+ debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
+ if (this != &other) {
+ destroy(true);
+ copy(other);
+ }
+ return *this;
}
// protected
-void numeric::copy(numeric const & other)
+void numeric::copy(const numeric & other)
{
- basic::copy(other);
- value = new cl_N(*other.value);
+ basic::copy(other);
+ value = other.value;
}
void numeric::destroy(bool call_parent)
{
- delete value;
- if (call_parent) basic::destroy(call_parent);
+ if (call_parent) basic::destroy(call_parent);
}
//////////
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);
+ 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. However, if the integer is small enough,
+ // i.e. satisfies cl_immediate_p(), we save space and dereferences by
+ // using an immediate type:
+ if (cln::cl_immediate_p(i))
+ value = cln::cl_I(i);
+ else
+ value = cln::cl_I((long) i);
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
}
+
numeric::numeric(unsigned int i) : basic(TINFO_numeric)
{
- 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);
+ 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. However, if the integer is small enough,
+ // i.e. satisfies cl_immediate_p(), we save space and dereferences by
+ // using an immediate type:
+ if (cln::cl_immediate_p(i))
+ value = cln::cl_I(i);
+ else
+ value = cln::cl_I((unsigned long) i);
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
}
+
numeric::numeric(long i) : basic(TINFO_numeric)
{
- debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
- value = new cl_I(i);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
+ value = cln::cl_I(i);
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
}
+
numeric::numeric(unsigned long i) : basic(TINFO_numeric)
{
- debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
- value = new cl_I(i);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
+ value = cln::cl_I(i);
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
}
/** Ctor 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);
+ debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
+ if (!denom)
+ throw std::overflow_error("division by zero");
+ value = cln::cl_I(numer) / cln::cl_I(denom);
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ 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(char const *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);
+ 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 = cln::cl_float(d, cln::default_float_format);
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
+}
+
+/** ctor from C-style string. It also accepts complex numbers in GiNaC
+ * notation like "2+5*I". */
+numeric::numeric(const char *s) : basic(TINFO_numeric)
+{
+ debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
+ 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);
+ // make it safe by adding explicit sign
+ if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
+ ss = '+' + ss;
+ std::string::size_type delim;
+ 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 = term.replace(term.find("I"),1,"");
+ // erase '*':
+ if (term.find("*") != std::string::npos)
+ term = term.replace(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) {
+ // 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
+#if defined(HAVE_SSTREAM)
+ std::ostringstream buf;
+ buf << unsigned(Digits) << std::ends;
+ term += "_" + buf.str();
+#else
+ char buf[14];
+ std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
+ term += "_" + string(buf);
+#endif
+ // 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 {
+ // 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;
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
}
/** Ctor from CLN types. This is for the initiated user or internal use
* only. */
-numeric::numeric(cl_N const & z) : basic(TINFO_numeric)
+numeric::numeric(const cln::cl_N & z) : basic(TINFO_numeric)
{
- debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
- value = new cl_N(z);
- calchash();
- setflag(status_flags::evaluated|
- status_flags::hash_calculated);
+ debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
+ value = z;
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ status_flags::hash_calculated);
+}
+
+//////////
+// 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);
+ cln::cl_N ctorval = 0;
+
+ // Read number as string
+ std::string str;
+ if (n.find_string("number", str)) {
+#ifdef HAVE_SSTREAM
+ std::istringstream s(str);
+#else
+ std::istrstream s(str.c_str(), str.size() + 1);
+#endif
+ cln::cl_idecoded_float re, im;
+ char c;
+ s.get(c);
+ switch (c) {
+ case 'R': // Integer-decoded real number
+ s >> re.sign >> re.mantissa >> re.exponent;
+ ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
+ break;
+ case 'C': // Integer-decoded complex number
+ s >> re.sign >> re.mantissa >> re.exponent;
+ s >> im.sign >> im.mantissa >> im.exponent;
+ ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
+ im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
+ break;
+ default: // Ordinary number
+ s.putback(c);
+ s >> ctorval;
+ break;
+ }
+ }
+ value = ctorval;
+ calchash();
+ setflag(status_flags::evaluated |
+ status_flags::expanded |
+ 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);
+}
+
+/** Archive the object. */
+void numeric::archive(archive_node &n) const
+{
+ inherited::archive(n);
+
+ // Write number as string
+#ifdef HAVE_SSTREAM
+ std::ostringstream s;
+#else
+ char buf[1024];
+ std::ostrstream s(buf, 1024);
+#endif
+ if (this->is_crational())
+ s << cln::the<cln::cl_N>(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;
+ }
+ }
+#ifdef HAVE_SSTREAM
+ n.add_string("number", s.str());
+#else
+ s << ends;
+ std::string str(buf);
+ n.add_string("number", str);
+#endif
}
//////////
basic * numeric::duplicate() const
{
- debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
- return new numeric(*this);
-}
-
-// The method printraw doesn't do much, it simply uses CLN's operator<<() for
-// output, which is ugly but reliable. Examples:
-// 2+2i
-void numeric::printraw(ostream & os) const
-{
- debugmsg("numeric printraw", LOGLEVEL_PRINT);
- os << "numeric(" << *value << ")";
-}
-
-// The method print adds to the output so it blends more consistently together
-// with the other routines and produces something compatible to Maple input.
-void numeric::print(ostream & os, unsigned upper_precedence) const
-{
- debugmsg("numeric print", LOGLEVEL_PRINT);
- if (is_real()) {
- // case 1, real: x or -x
- if ((precedence<=upper_precedence) && (!is_pos_integer())) {
- os << "(" << *value << ")";
- } else {
- os << *value;
- }
- } 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 << ")";
- }
- }
+ debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
+ return new numeric(*this);
+}
+
+
+/** 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(std::ostream & os, const cln::cl_R & num)
+{
+ cln::cl_print_flags ourflags;
+ if (cln::instanceof(num, cln::cl_RA_ring)) {
+ // case 1: integer or rational, nothing special to do:
+ cln::print_real(os, ourflags, num);
+ } 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>(num));
+ cln::print_real(os, ourflags, num);
+ }
+ return;
+}
+
+/** This method adds to the output so it blends more consistently together
+ * with the other routines and produces something compatible to ginsh input.
+ *
+ * @see print_real_number() */
+void numeric::print(std::ostream & os, unsigned upper_precedence) const
+{
+ debugmsg("numeric print", LOGLEVEL_PRINT);
+ cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
+ cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
+ if (cln::zerop(i)) {
+ // case 1, real: x or -x
+ if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
+ os << "(";
+ print_real_number(os, r);
+ os << ")";
+ } else {
+ print_real_number(os, r);
+ }
+ } else {
+ if (cln::zerop(r)) {
+ // case 2, imaginary: y*I or -y*I
+ if ((precedence<=upper_precedence) && (i < 0)) {
+ if (i == -1) {
+ os << "(-I)";
+ } else {
+ os << "(";
+ print_real_number(os, i);
+ os << "*I)";
+ }
+ } else {
+ if (i == 1) {
+ os << "I";
+ } else {
+ if (i == -1) {
+ os << "-I";
+ } else {
+ print_real_number(os, i);
+ os << "*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 << "(";
+ print_real_number(os, r);
+ if (i < 0) {
+ if (i == -1) {
+ os << "-I";
+ } else {
+ print_real_number(os, i);
+ os << "*I";
+ }
+ } else {
+ if (i == 1) {
+ os << "+I";
+ } else {
+ os << "+";
+ print_real_number(os, i);
+ os << "*I";
+ }
+ }
+ if (precedence <= upper_precedence)
+ os << ")";
+ }
+ }
+}
+
+
+void numeric::printraw(std::ostream & os) 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(" << cln::the<cln::cl_N>(value) << ")";
+}
+
+
+void numeric::printtree(std::ostream & os, unsigned indent) const
+{
+ debugmsg("numeric printtree", LOGLEVEL_PRINT);
+ os << std::string(indent,' ') << cln::the<cln::cl_N>(value)
+ << " (numeric): "
+ << "hash=" << hashvalue
+ << " (0x" << std::hex << hashvalue << std::dec << ")"
+ << ", flags=" << flags << std::endl;
+}
+
+
+void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
+{
+ debugmsg("numeric print csrc", LOGLEVEL_PRINT);
+ std::ios::fmtflags oldflags = os.flags();
+ os.setf(std::ios::scientific);
+ if (this->is_rational() && !this->is_integer()) {
+ if (compare(_num0()) > 0) {
+ os << "(";
+ if (type == csrc_types::ctype_cl_N)
+ os << "cln::cl_F(\"" << numer().evalf() << "\")";
+ else
+ os << numer().to_double();
+ } else {
+ os << "-(";
+ if (type == csrc_types::ctype_cl_N)
+ os << "cln::cl_F(\"" << -numer().evalf() << "\")";
+ else
+ os << -numer().to_double();
+ }
+ os << "/";
+ if (type == csrc_types::ctype_cl_N)
+ os << "cln::cl_F(\"" << denom().evalf() << "\")";
+ else
+ os << denom().to_double();
+ os << ")";
+ } else {
+ if (type == csrc_types::ctype_cl_N)
+ os << "cln::cl_F(\"" << evalf() << "\")";
+ else
+ os << to_double();
+ }
+ os.flags(oldflags);
}
+
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::integer:
- case info_flags::integer_polynomial:
- return is_integer();
- case info_flags::positive:
- return is_positive();
- case info_flags::negative:
- return is_negative();
- case info_flags::nonnegative:
- return compare(numZERO())>=0;
- case info_flags::posint:
- return is_pos_integer();
- case info_flags::negint:
- return is_integer() && (compare(numZERO())<0);
- 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:
+ 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;
+}
+
+/** Disassemble real part and imaginary part to scan for the occurrence of a
+ * single number. Also handles the imaginary unit. It ignores the sign on
+ * both this and the argument, which may lead to what might appear as funny
+ * results: (2+I).has(-2) -> true. But this is consistent, since we also
+ * would like to have (-2+I).has(2) -> true and we want to think about the
+ * sign as a multiplicative factor. */
+bool numeric::has(const ex & other) const
+{
+ if (!is_exactly_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;
+}
+
+
+/** Evaluation of numbers doesn't do anything at all. */
+ex numeric::eval(int level) const
+{
+ // Warning: if this is ever gonna do something, the ex ctors from all kinds
+ // of numbers should be checking for status_flags::evaluated.
+ return this->hold();
}
+
/** Cast numeric into a floating-point object. For example exact numeric(1) is
* returned as a 1.0000000000000000000000 and so on according to how Digits is
- * currently set.
+ * 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.
- * @return an ex-handle to a numeric. */
+ * @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) *
+ (cln::the<cln::cl_N>(value)));
}
// protected
-int numeric::compare_same_type(basic const & other) const
+/** Implementation of ex::diff() for a numeric. It always returns 0.
+ *
+ * @see ex::diff */
+ex numeric::derivative(const symbol & s) const
{
- GINAC_ASSERT(is_exactly_of_type(other, numeric));
- numeric const & o = static_cast<numeric &>(const_cast<basic &>(other));
+ return _ex0();
+}
- if (*value == *o.value) {
- return 0;
- }
- return compare(o);
+int numeric::compare_same_type(const basic & other) const
+{
+ GINAC_ASSERT(is_exactly_of_type(other, numeric));
+ const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
+
+ return this->compare(o);
}
-bool numeric::is_equal_same_type(basic const & other) const
+
+bool numeric::is_equal_same_type(const basic & other) const
{
- GINAC_ASSERT(is_exactly_of_type(other,numeric));
- numeric const *o = static_cast<numeric const *>(&other);
-
- return is_equal(*o);
+ GINAC_ASSERT(is_exactly_of_type(other,numeric));
+ const numeric *o = static_cast<const numeric *>(&other);
+
+ return this->is_equal(*o);
}
-/*
+
unsigned numeric::calchash(void) 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);
+ // Use CLN's hashcode. Warning: It 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.
+ return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
}
-*/
//////////
/** Numerical addition method. Adds argument to *this and returns result as
* a new numeric object. */
-numeric numeric::add(numeric const & other) const
+const numeric numeric::add(const numeric & other) const
{
- return numeric((*value)+(*other.value));
+ // Efficiency shortcut: trap the neutral element by pointer.
+ static const numeric * _num0p = &_num0();
+ if (this==_num0p)
+ return other;
+ else if (&other==_num0p)
+ return *this;
+
+ return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
}
+
/** Numerical subtraction method. Subtracts argument from *this and returns
* result as a new numeric object. */
-numeric numeric::sub(numeric const & other) const
+const numeric numeric::sub(const numeric & other) const
{
- return numeric((*value)-(*other.value));
+ return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
}
+
/** Numerical multiplication method. Multiplies *this and argument and returns
* result as a new numeric object. */
-numeric numeric::mul(numeric const & other) const
+const numeric numeric::mul(const numeric & other) const
{
- static const numeric * numONEp=&numONE();
- if (this==numONEp) {
- return other;
- } else if (&other==numONEp) {
- return *this;
- }
- return numeric((*value)*(*other.value));
+ // Efficiency shortcut: trap the neutral element by pointer.
+ static const numeric * _num1p = &_num1();
+ if (this==_num1p)
+ return other;
+ else if (&other==_num1p)
+ return *this;
+
+ return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
}
+
/** Numerical division method. Divides *this by argument and returns result as
* a new numeric object.
*
* @exception overflow_error (division by zero) */
-numeric numeric::div(numeric const & other) const
+const numeric numeric::div(const numeric & other) const
{
- if (zerop(*other.value))
- throw (std::overflow_error("division by zero"));
- return numeric((*value)/(*other.value));
+ if (cln::zerop(cln::the<cln::cl_N>(other.value)))
+ throw std::overflow_error("numeric::div(): division by zero");
+ return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
}
-numeric numeric::power(numeric const & other) const
+
+const numeric numeric::power(const numeric & other) const
{
- static const numeric * numONEp=&numONE();
- if (&other==numONEp) {
- return *this;
- }
- if (zerop(*value) && other.is_real() && minusp(realpart(*other.value)))
- throw (std::overflow_error("division by zero"));
- return numeric(expt(*value,*other.value));
+ // Efficiency shortcut: trap the neutral exponent by pointer.
+ static const numeric * _num1p = &_num1();
+ if (&other==_num1p)
+ return *this;
+
+ if (cln::zerop(cln::the<cln::cl_N>(value))) {
+ if (cln::zerop(cln::the<cln::cl_N>(other.value)))
+ throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
+ else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
+ throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
+ else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
+ throw std::overflow_error("numeric::eval(): division by zero");
+ else
+ return _num0();
+ }
+ return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
}
-/** Inverse of a number. */
-numeric numeric::inverse(void) const
+
+const numeric & numeric::add_dyn(const numeric & other) const
{
- return numeric(recip(*value)); // -> CLN
+ // Efficiency shortcut: trap the neutral element by pointer.
+ static const numeric * _num0p = &_num0();
+ if (this==_num0p)
+ return other;
+ else if (&other==_num0p)
+ return *this;
+
+ return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
+ setflag(status_flags::dynallocated));
}
-numeric const & numeric::add_dyn(numeric const & other) const
+
+const numeric & numeric::sub_dyn(const numeric & other) const
{
- return static_cast<numeric const &>((new numeric((*value)+(*other.value)))->
- setflag(status_flags::dynallocated));
+ return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
+ setflag(status_flags::dynallocated));
}
-numeric const & numeric::sub_dyn(numeric const & other) const
+
+const numeric & numeric::mul_dyn(const numeric & other) const
{
- return static_cast<numeric const &>((new numeric((*value)-(*other.value)))->
- setflag(status_flags::dynallocated));
+ // Efficiency shortcut: trap the neutral element by pointer.
+ static const numeric * _num1p = &_num1();
+ if (this==_num1p)
+ return other;
+ else if (&other==_num1p)
+ return *this;
+
+ return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
+ setflag(status_flags::dynallocated));
}
-numeric const & numeric::mul_dyn(numeric const & other) const
+
+const numeric & numeric::div_dyn(const numeric & other) const
{
- static const numeric * numONEp=&numONE();
- if (this==numONEp) {
- return other;
- } else if (&other==numONEp) {
- return *this;
- }
- return static_cast<numeric const &>((new numeric((*value)*(*other.value)))->
- setflag(status_flags::dynallocated));
+ if (cln::zerop(cln::the<cln::cl_N>(other.value)))
+ throw std::overflow_error("division by zero");
+ return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
+ setflag(status_flags::dynallocated));
}
-numeric const & numeric::div_dyn(numeric const & other) const
+
+const numeric & numeric::power_dyn(const numeric & other) const
{
- if (zerop(*other.value))
- throw (std::overflow_error("division by zero"));
- return static_cast<numeric const &>((new numeric((*value)/(*other.value)))->
- setflag(status_flags::dynallocated));
+ // Efficiency shortcut: trap the neutral exponent by pointer.
+ static const numeric * _num1p=&_num1();
+ if (&other==_num1p)
+ return *this;
+
+ if (cln::zerop(cln::the<cln::cl_N>(value))) {
+ if (cln::zerop(cln::the<cln::cl_N>(other.value)))
+ throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
+ else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
+ throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
+ else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
+ throw std::overflow_error("numeric::eval(): division by zero");
+ else
+ return _num0();
+ }
+ return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
+ setflag(status_flags::dynallocated));
}
-numeric const & numeric::power_dyn(numeric const & other) const
+
+const numeric & numeric::operator=(int i)
{
- static const numeric * numONEp=&numONE();
- if (&other==numONEp) {
- return *this;
- }
- // The ifs are only a workaround for a bug in CLN. It gets stuck otherwise:
- if ( !other.is_integer() &&
- other.is_rational() &&
- (*this).is_nonneg_integer() ) {
- if ( !zerop(*value) ) {
- return static_cast<numeric const &>((new numeric(exp(*other.value * log(*value))))->
- setflag(status_flags::dynallocated));
- } else {
- if ( !zerop(*other.value) ) { // 0^(n/m)
- return static_cast<numeric const &>((new numeric(0))->
- setflag(status_flags::dynallocated));
- } else { // raise FPE (0^0 requested)
- return static_cast<numeric const &>((new numeric(1/(*other.value)))->
- setflag(status_flags::dynallocated));
- }
- }
- } else { // default -> CLN
- return static_cast<numeric const &>((new numeric(expt(*value,*other.value)))->
- setflag(status_flags::dynallocated));
- }
+ return operator=(numeric(i));
}
-numeric const & numeric::operator=(int i)
+
+const numeric & numeric::operator=(unsigned int i)
{
- return operator=(numeric(i));
+ return operator=(numeric(i));
}
-numeric const & numeric::operator=(unsigned int i)
+
+const numeric & numeric::operator=(long i)
{
- return operator=(numeric(i));
+ return operator=(numeric(i));
}
-numeric const & numeric::operator=(long i)
+
+const numeric & numeric::operator=(unsigned long i)
{
- return operator=(numeric(i));
+ return operator=(numeric(i));
}
-numeric const & numeric::operator=(unsigned long i)
+
+const numeric & numeric::operator=(double d)
{
- return operator=(numeric(i));
+ return operator=(numeric(d));
}
-numeric const & numeric::operator=(double d)
+
+const numeric & numeric::operator=(const char * s)
{
- return operator=(numeric(d));
+ return operator=(numeric(s));
}
-numeric const & numeric::operator=(char const * s)
+
+/** Inverse of a number. */
+const numeric numeric::inverse(void) const
{
- return operator=(numeric(s));
+ if (cln::zerop(cln::the<cln::cl_N>(value)))
+ throw std::overflow_error("numeric::inverse(): division by zero");
+ return numeric(cln::recip(cln::the<cln::cl_N>(value)));
}
+
/** Return the complex half-plane (left or right) in which the number lies.
* csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
* csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
*
- * @see numeric::compare(numeric const & other) */
+ * @see numeric::compare(const numeric & other) */
int numeric::csgn(void) const
{
- if (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;
- }
+ if (cln::zerop(cln::the<cln::cl_N>(value)))
+ return 0;
+ cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
+ if (!cln::zerop(r)) {
+ if (cln::plusp(r))
+ return 1;
+ else
+ return -1;
+ } else {
+ if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(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
*
* @return csgn(*this-other)
* @see numeric::csgn(void) */
-int numeric::compare(numeric const & other) const
+int numeric::compare(const numeric & other) const
{
- // Comparing two real numbers?
- if (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));
- }
+ // 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(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
+ if (real_cmp)
+ return real_cmp;
+ // ...and then the imaginary parts.
+ return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
+ }
}
-bool numeric::is_equal(numeric const & other) const
+
+bool numeric::is_equal(const numeric & other) const
{
- return (*value == *other.value);
+ return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
}
+
/** True if object is zero. */
bool numeric::is_zero(void) const
{
- return zerop(*value); // -> CLN
+ return cln::zerop(cln::the<cln::cl_N>(value));
}
+
/** True if object is not complex and greater than zero. */
bool numeric::is_positive(void) const
{
- if (is_real()) {
- return plusp(The(cl_R)(*value)); // -> CLN
- }
- return false;
+ if (this->is_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
{
- if (is_real()) {
- return minusp(The(cl_R)(*value)); // -> CLN
- }
- return false;
+ if (this->is_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
{
- return (bool)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
{
- return (is_integer() &&
- plusp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && 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
{
- return (is_integer() &&
- !minusp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
}
+
/** True if object is an exact even integer. */
bool numeric::is_even(void) const
{
- return (is_integer() &&
- evenp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
}
+
/** True if object is an exact odd integer. */
bool numeric::is_odd(void) const
{
- return (is_integer() &&
- oddp(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && 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
{
- return (is_integer() &&
- isprobprime(The(cl_I)(*value))); // -> CLN
+ return (this->is_integer() && 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
{
- if (instanceof(*value, cl_RA_ring)) {
- return true;
- } else if (!is_real()) { // complex case, handle Q(i):
- if ( instanceof(realpart(*value), cl_RA_ring) &&
- instanceof(imagpart(*value), cl_RA_ring) )
- return true;
- }
- return false;
+ 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
{
- return (bool)instanceof(*value, cl_R_ring); // -> CLN
+ return cln::instanceof(value, cln::cl_R_ring);
+}
+
+
+bool numeric::operator==(const numeric & other) const
+{
+ return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
}
-bool numeric::operator==(numeric const & other) const
+
+bool numeric::operator!=(const numeric & other) const
{
- return (*value == *other.value); // -> CLN
+ return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
}
-bool numeric::operator!=(numeric const & other) const
+
+/** 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
{
- return (*value != *other.value); // -> CLN
+ 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(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
+ cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(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 (cln::instanceof(value, cln::cl_RA_ring))
+ return true;
+ else if (!this->is_real()) { // complex case, handle Q(i):
+ if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
+ cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
+ return true;
+ }
+ return false;
+}
+
+
/** Numerical comparison: less.
*
* @exception invalid_argument (complex inequality) */
-bool numeric::operator<(numeric const & other) const
+bool numeric::operator<(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(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<=(numeric const & other) const
+bool numeric::operator<=(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(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>(numeric const & other) const
+bool numeric::operator>(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(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>=(numeric const & other) const
+bool numeric::operator>=(const numeric & other) const
{
- if ( is_real() && other.is_real() ) {
- return (bool)(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. */
+
+/** 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
{
- GINAC_ASSERT(is_integer());
- return cl_I_to_int(The(cl_I)(*value));
+ 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
+{
+ 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
{
- GINAC_ASSERT(is_real());
- return cl_double_approx(realpart(*value));
+ GINAC_ASSERT(this->is_real());
+ return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
}
+
/** Real part of a number. */
-numeric numeric::real(void) const
+const numeric numeric::real(void) const
{
- return numeric(realpart(*value)); // -> CLN
+ return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
}
+
/** Imaginary part of a number. */
-numeric numeric::imag(void) const
+const numeric numeric::imag(void) const
{
- return numeric(imagpart(*value)); // -> CLN
+ return numeric(cln::imagpart(cln::the<cln::cl_N>(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 itself in all other cases. */
-numeric numeric::numer(void) const
-{
- if (is_integer()) {
- return numeric(*this);
- }
-#ifdef SANE_LINKER
- else if (instanceof(*value, cl_RA_ring)) {
- return numeric(numerator(The(cl_RA)(*value)));
- }
- else if (!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 (!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);
+ * (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);
+
+ 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(cln::the<cln::cl_N>(value)));
+ const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(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. */
-numeric numeric::denom(void) const
-{
- if (is_integer()) {
- return numONE();
- }
-#ifdef SANE_LINKER
- if (instanceof(*value, cl_RA_ring)) {
- return numeric(denominator(The(cl_RA)(*value)));
- }
- if (!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 numONE();
- 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 (!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 numONE();
- 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 numONE();
+const numeric numeric::denom(void) const
+{
+ if (this->is_integer())
+ return _num1();
+
+ if (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(cln::the<cln::cl_N>(value)));
+ const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
+ if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
+ return _num1();
+ 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();
}
+
/** 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.
* in two's complement if it is an integer, 0 otherwise. */
int numeric::int_length(void) const
{
- if (is_integer()) {
- return integer_length(The(cl_I)(*value)); // -> CLN
- } else {
- return 0;
- }
+ if (this->is_integer())
+ return cln::integer_length(cln::the<cln::cl_I>(value));
+ else
+ return 0;
+}
+
+
+/** Returns a new CLN object of type cl_N, representing the value of *this.
+ * This method is useful for casting when mixing GiNaC and CLN in one project.
+ */
+numeric::operator cln::cl_N() const
+{
+ return cln::cl_N(cln::the<cln::cl_N>(value));
}
//////////
const numeric some_numeric;
-type_info const & typeid_numeric=typeid(some_numeric);
+const std::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)));
-
-//////////
-// global functions
-//////////
+const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
-numeric const & numZERO(void)
-{
- const static ex eZERO = ex((new numeric(0))->setflag(status_flags::dynallocated));
- const static numeric * nZERO = static_cast<const numeric *>(eZERO.bp);
- return *nZERO;
-}
-
-numeric const & numONE(void)
-{
- const static ex eONE = ex((new numeric(1))->setflag(status_flags::dynallocated));
- const static numeric * nONE = static_cast<const numeric *>(eONE.bp);
- return *nONE;
-}
-
-numeric const & numTWO(void)
-{
- const static ex eTWO = ex((new numeric(2))->setflag(status_flags::dynallocated));
- const static numeric * nTWO = static_cast<const numeric *>(eTWO.bp);
- return *nTWO;
-}
-
-numeric const & numTHREE(void)
-{
- const static ex eTHREE = ex((new numeric(3))->setflag(status_flags::dynallocated));
- const static numeric * nTHREE = static_cast<const numeric *>(eTHREE.bp);
- return *nTHREE;
-}
-
-numeric const & numMINUSONE(void)
-{
- const static ex eMINUSONE = ex((new numeric(-1))->setflag(status_flags::dynallocated));
- const static numeric * nMINUSONE = static_cast<const numeric *>(eMINUSONE.bp);
- return *nMINUSONE;
-}
-
-numeric const & numHALF(void)
-{
- const static ex eHALF = ex((new numeric(1, 2))->setflag(status_flags::dynallocated));
- const static numeric * nHALF = static_cast<const numeric *>(eHALF.bp);
- return *nHALF;
-}
/** Exponential function.
*
* @return arbitrary precision numerical exp(x). */
-numeric exp(numeric const & x)
+const numeric exp(const numeric & x)
{
- return ::exp(*x.value); // -> CLN
+ return cln::exp(cln::cl_N(x));
}
+
/** Natural logarithm.
*
* @param z complex number
* @return arbitrary precision numerical log(x).
- * @exception overflow_error (logarithmic singularity) */
-numeric log(numeric const & z)
+ * @exception pole_error("log(): logarithmic pole",0) */
+const numeric log(const numeric & z)
{
- if (z.is_zero())
- throw (std::overflow_error("log(): logarithmic singularity"));
- return ::log(*z.value); // -> CLN
+ if (z.is_zero())
+ throw pole_error("log(): logarithmic pole",0);
+ return cln::log(cln::cl_N(z));
}
+
/** Numeric sine (trigonometric function).
*
* @return arbitrary precision numerical sin(x). */
-numeric sin(numeric const & x)
+const numeric sin(const numeric & x)
{
- return ::sin(*x.value); // -> CLN
+ return cln::sin(cln::cl_N(x));
}
+
/** Numeric cosine (trigonometric function).
*
* @return arbitrary precision numerical cos(x). */
-numeric cos(numeric const & x)
+const numeric cos(const numeric & x)
{
- return ::cos(*x.value); // -> CLN
+ return cln::cos(cln::cl_N(x));
}
-
+
+
/** Numeric tangent (trigonometric function).
*
* @return arbitrary precision numerical tan(x). */
-numeric tan(numeric const & x)
+const numeric tan(const numeric & x)
{
- return ::tan(*x.value); // -> CLN
+ return cln::tan(cln::cl_N(x));
}
-
+
+
/** Numeric inverse sine (trigonometric function).
*
* @return arbitrary precision numerical asin(x). */
-numeric asin(numeric const & x)
+const numeric asin(const numeric & x)
{
- return ::asin(*x.value); // -> CLN
+ return cln::asin(cln::cl_N(x));
}
-
+
+
/** Numeric inverse cosine (trigonometric function).
*
* @return arbitrary precision numerical acos(x). */
-numeric acos(numeric const & x)
+const numeric acos(const numeric & x)
{
- return ::acos(*x.value); // -> CLN
+ return cln::acos(cln::cl_N(x));
}
-
-/** Arcustangents.
+
+
+/** Arcustangent.
*
* @param z complex number
* @return atan(z)
- * @exception overflow_error (logarithmic singularity) */
-numeric atan(numeric const & x)
+ * @exception pole_error("atan(): logarithmic pole",0) */
+const numeric atan(const numeric & x)
{
- if (!x.is_real() &&
- x.real().is_zero() &&
- !abs(x.imag()).is_equal(numONE()))
- 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()))
+ throw pole_error("atan(): logarithmic pole",0);
+ return cln::atan(cln::cl_N(x));
}
-/** Arcustangents.
+
+/** Arcustangent.
*
* @param x real number
* @param y real number
* @return atan(y/x) */
-numeric atan(numeric const & y, numeric const & 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"));
+ if (x.is_real() && y.is_real())
+ return cln::atan(cln::the<cln::cl_R>(cln::cl_N(x)),
+ cln::the<cln::cl_R>(cln::cl_N(y)));
+ else
+ throw std::invalid_argument("atan(): complex argument");
}
+
/** Numeric hyperbolic sine (trigonometric function).
*
* @return arbitrary precision numerical sinh(x). */
-numeric sinh(numeric const & x)
+const numeric sinh(const numeric & x)
{
- return ::sinh(*x.value); // -> CLN
+ return cln::sinh(cln::cl_N(x));
}
+
/** Numeric hyperbolic cosine (trigonometric function).
*
* @return arbitrary precision numerical cosh(x). */
-numeric cosh(numeric const & x)
+const numeric cosh(const numeric & x)
{
- return ::cosh(*x.value); // -> CLN
+ return cln::cosh(cln::cl_N(x));
}
-
+
+
/** Numeric hyperbolic tangent (trigonometric function).
*
* @return arbitrary precision numerical tanh(x). */
-numeric tanh(numeric const & x)
+const numeric tanh(const numeric & x)
{
- return ::tanh(*x.value); // -> CLN
+ return cln::tanh(cln::cl_N(x));
}
-
+
+
/** Numeric inverse hyperbolic sine (trigonometric function).
*
* @return arbitrary precision numerical asinh(x). */
-numeric asinh(numeric const & x)
+const numeric asinh(const numeric & x)
{
- return ::asinh(*x.value); // -> CLN
+ return cln::asinh(cln::cl_N(x));
}
+
/** Numeric inverse hyperbolic cosine (trigonometric function).
*
* @return arbitrary precision numerical acosh(x). */
-numeric acosh(numeric const & x)
+const numeric acosh(const numeric & x)
{
- return ::acosh(*x.value); // -> CLN
+ return cln::acosh(cln::cl_N(x));
}
+
/** Numeric inverse hyperbolic tangent (trigonometric function).
*
* @return arbitrary precision numerical atanh(x). */
-numeric atanh(numeric const & x)
-{
- return ::atanh(*x.value); // -> CLN
+const numeric atanh(const numeric & x)
+{
+ return cln::atanh(cln::cl_N(x));
+}
+
+
+/*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();
+
+ // what is the desired float format?
+ // first guess: default format
+ cln::float_format_t prec = cln::default_float_format;
+ const cln::cl_N value = cln::cl_N(x);
+ // 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 (cln::the<cln::cl_N>(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(cln::cl_N(x), prec);
}
+
/** Numeric evaluation of Riemann's Zeta function. Currently works only for
* integer arguments. */
-numeric zeta(numeric const & x)
+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>(cln::cl_N(x))));
+ if (cln::zerop(cln::cl_N(x)-aux))
+ return cln::zeta(aux);
+ }
+ std::clog << "zeta(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
+ return numeric(0);
+}
+
+
+/** The Gamma function.
+ * This is only a stub! */
+const numeric lgamma(const numeric & x)
+{
+ std::clog << "lgamma(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
+ return numeric(0);
+}
+const numeric tgamma(const numeric & x)
{
- if (x.is_integer())
- return ::cl_zeta(x.to_int()); // -> CLN
- else
- clog << "zeta(): Does anybody know good way to calculate this numerically?" << endl;
- return numeric(0);
+ std::clog << "tgamma(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
+ return numeric(0);
}
-/** The gamma function.
+
+/** The psi function (aka polygamma function).
* This is only a stub! */
-numeric gamma(numeric const & x)
+const numeric psi(const numeric & x)
{
- clog << "gamma(): Does anybody know good way to calculate this numerically?" << endl;
- return numeric(0);
+ std::clog << "psi(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
+ return numeric(0);
}
-/** The psi function (aka polygamma function).
+
+/** The psi functions (aka polygamma functions).
* This is only a stub! */
-numeric psi(numeric const & n, numeric const & x)
+const numeric psi(const numeric & n, const numeric & x)
{
- clog << "psi(): Does anybody know good way to calculate this numerically?" << endl;
- return numeric(0);
+ std::clog << "psi(" << n << "," << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
+ return numeric(0);
}
+
/** Factorial combinatorial function.
*
+ * @param n integer argument >= 0
* @exception range_error (argument must be integer >= 0) */
-numeric factorial(numeric const & nn)
+const numeric factorial(const numeric & n)
{
- if ( !nn.is_nonneg_integer() ) {
- throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
- }
-
- return numeric(::factorial(nn.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)!!
+ * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
* @exception range_error (argument must be integer >= -1) */
-numeric doublefactorial(numeric const & nn)
-{
- // META-NOTE: The whole shit here will become obsolete and may be moved
- // out once CLN learns about double factorial, which should be as soon as
- // 1.0.3 rolls out!
-
- // We store the results separately for even and odd arguments. This has
- // the advantage that we don't have to compute any even result at all if
- // the function is always called with odd arguments and vice versa. There
- // is no tradeoff involved in this, it is guaranteed to save time as well
- // as memory. (If this is not enough justification consider the Gamma
- // function of half integer arguments: it only needs odd doublefactorials.)
- static vector<numeric> evenresults;
- static int highest_evenresult = -1;
- static vector<numeric> oddresults;
- static int highest_oddresult = -1;
-
- if (nn == numeric(-1)) {
- return numONE();
- }
- if (!nn.is_nonneg_integer()) {
- throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
- }
- if (nn.is_even()) {
- int n = nn.div(numTWO()).to_int();
- if (n <= highest_evenresult) {
- return evenresults[n];
- }
- if (evenresults.capacity() < (unsigned)(n+1)) {
- evenresults.reserve(n+1);
- }
- if (highest_evenresult < 0) {
- evenresults.push_back(numONE());
- highest_evenresult=0;
- }
- for (int i=highest_evenresult+1; i<=n; i++) {
- evenresults.push_back(numeric(evenresults[i-1].mul(numeric(i*2))));
- }
- highest_evenresult=n;
- return evenresults[n];
- } else {
- int n = nn.sub(numONE()).div(numTWO()).to_int();
- if (n <= highest_oddresult) {
- return oddresults[n];
- }
- if (oddresults.capacity() < (unsigned)n) {
- oddresults.reserve(n+1);
- }
- if (highest_oddresult < 0) {
- oddresults.push_back(numONE());
- highest_oddresult=0;
- }
- for (int i=highest_oddresult+1; i<=n; i++) {
- oddresults.push_back(numeric(oddresults[i-1].mul(numeric(i*2+1))));
- }
- highest_oddresult=n;
- return oddresults[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(cln::doublefactorial(n.to_int()));
}
+
/** The Binomial coefficients. It computes the binomial coefficients. For
* integer n and k and positive n this is the number of ways of choosing k
* objects from n distinct objects. If n is negative, the formula
* binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
-numeric binomial(numeric const & n, numeric const & k)
-{
- if (n.is_integer() && k.is_integer()) {
- if (n.is_nonneg_integer()) {
- if (k.compare(n)!=1 && k.compare(numZERO())!=-1)
- return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
- else
- return numZERO();
- } else {
- return numMINUSONE().power(k)*binomial(k-n-numONE(),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())!=-1)
+ return numeric(cln::binomial(n.to_int(),k.to_int()));
+ 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.");
}
+
/** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
* in the expansion of the function x/(e^x-1).
*
* @return the nth Bernoulli number (a rational number).
* @exception range_error (argument must be integer >= 0) */
-numeric bernoulli(numeric const & nn)
-{
- if (!nn.is_integer() || nn.is_negative())
- throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
- if (nn.is_zero())
- return numONE();
- if (!nn.compare(numONE()))
- return numeric(-1,2);
- if (nn.is_odd())
- return numZERO();
- // 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 formula
- // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
- // whith B(0) == 1.
- static vector<numeric> results;
- static int highest_result = -1;
- int n = nn.sub(numTWO()).div(numTWO()).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 code below is adapted from Pari's function bernvec().
+ //
+ // (There is an interesting relation with the tangent polynomials described
+ // in `Concrete Mathematics', which leads to a program twice as fast 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.)
+
+ // the special cases not covered by the algorithm below
+ if (nn.is_equal(_num1()))
+ return _num_1_2();
+ if (nn.is_odd())
+ return _num0();
+
+ // store nonvanishing Bernoulli numbers here
+ static std::vector< cln::cl_RA > results;
+ static int highest_result = 0;
+ // algorithm not applicable to B(0), so just store it
+ if (results.size()==0)
+ results.push_back(cln::cl_RA(1));
+
+ int n = nn.to_long();
+ for (int i=highest_result; i<n/2; ++i) {
+ cln::cl_RA B = 0;
+ long n = 8;
+ long m = 5;
+ long d1 = i;
+ long d2 = 2*i-1;
+ for (int j=i; j>0; --j) {
+ B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
+ n += 4;
+ m += 2;
+ d1 -= 1;
+ d2 -= 2;
+ }
+ B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
+ results.push_back(B);
+ ++highest_result;
+ }
+ return results[n/2];
+}
+
+
+/** Fibonacci number. The nth Fibonacci number F(n) is defined by the
+ * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
+ *
+ * @param n an integer
+ * @return the nth Fibonacci number F(n) (an integer number)
+ * @exception range_error (argument must be an integer) */
+const numeric fibonacci(const numeric & n)
+{
+ if (!n.is_integer())
+ throw std::range_error("numeric::fibonacci(): argument must be integer");
+ // Method:
+ //
+ // The following addition formula holds:
+ //
+ // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
+ //
+ // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
+ // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
+ // agree.)
+ // Replace m by m+1:
+ // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
+ // Now put in m = n, to get
+ // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
+ // F(2n+1) = F(n)^2 + F(n+1)^2
+ // hence
+ // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
+ if (n.is_zero())
+ return _num0();
+ 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>(cln::cl_N(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(numeric const & x)
+const numeric abs(const numeric& x)
{
- return ::abs(*x.value); // -> CLN
+ return cln::abs(cln::cl_N(x));
}
+
/** Modulus (in positive representation).
* In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
* sign of a or is zero. This is different from Maple's modp, where the sign
*
* @return a mod b in the range [0,abs(b)-1] with sign of b if both are
* integer, 0 otherwise. */
-numeric mod(numeric const & a, numeric const & 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 numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer())
+ return cln::mod(cln::the<cln::cl_I>(cln::cl_N(a)),
+ cln::the<cln::cl_I>(cln::cl_N(b)));
+ else
+ return _num0();
}
+
/** 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(numeric const & a, numeric const & b)
+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 numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer()) {
+ const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(cln::cl_N(b)) >> 1) - 1;
+ return cln::mod(cln::the<cln::cl_I>(cln::cl_N(a)) + b2,
+ cln::the<cln::cl_I>(cln::cl_N(b))) - b2;
+ } else
+ return _num0();
}
+
/** Numeric integer remainder.
* Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
* In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
* sign of a or is zero.
*
* @return remainder of a/b if both are integer, 0 otherwise. */
-numeric irem(numeric const & a, numeric const & b)
+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 numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer())
+ return cln::rem(cln::the<cln::cl_I>(cln::cl_N(a)),
+ cln::the<cln::cl_I>(cln::cl_N(b)));
+ else
+ return _num0();
}
+
/** 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,
*
* @return remainder of a/b and quotient stored in q if both are integer,
* 0 otherwise. */
-numeric irem(numeric const & a, numeric const & b, numeric & q)
+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 = numZERO();
- return numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer()) {
+ const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(cln::cl_N(a)),
+ cln::the<cln::cl_I>(cln::cl_N(b)));
+ q = rem_quo.quotient;
+ return rem_quo.remainder;
+ } else {
+ q = _num0();
+ return _num0();
+ }
}
+
/** 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(numeric const & a, numeric const & b)
+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 numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer())
+ return truncate1(cln::the<cln::cl_I>(cln::cl_N(a)),
+ cln::the<cln::cl_I>(cln::cl_N(b)));
+ else
+ return _num0();
}
+
/** Numeric integer quotient.
* Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
* r == a - iquo(a,b,r)*b.
*
* @return truncated quotient of a/b and remainder stored in r if both are
* integer, 0 otherwise. */
-numeric iquo(numeric const & a, numeric const & b, numeric & r)
+const numeric iquo(const numeric & a, const numeric & b, numeric & r)
+{
+ if (a.is_integer() && b.is_integer()) {
+ const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(cln::cl_N(a)),
+ cln::the<cln::cl_I>(cln::cl_N(b)));
+ r = rem_quo.remainder;
+ return rem_quo.quotient;
+ } else {
+ r = _num0();
+ return _num0();
+ }
+}
+
+
+/** Greatest Common Divisor.
+ *
+ * @return The GCD of two numbers if both are integer, a numerical 1
+ * if they are not. */
+const numeric gcd(const numeric & a, const numeric & b)
+{
+ if (a.is_integer() && b.is_integer())
+ return cln::gcd(cln::the<cln::cl_I>(cln::cl_N(a)),
+ cln::the<cln::cl_I>(cln::cl_N(b)));
+ else
+ return _num1();
+}
+
+
+/** Least Common Multiple.
+ *
+ * @return The LCM of two numbers if both are integer, the product of those
+ * two numbers if they are not. */
+const numeric lcm(const numeric & a, const numeric & b)
{
- if (a.is_integer() && b.is_integer()) { // -> 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 = numZERO();
- return numZERO(); // Throw?
- }
+ if (a.is_integer() && b.is_integer())
+ return cln::lcm(cln::the<cln::cl_I>(cln::cl_N(a)),
+ cln::the<cln::cl_I>(cln::cl_N(b)));
+ else
+ return a.mul(b);
}
+
/** Numeric square root.
* If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
* should return integer 2.
* @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(numeric const & z)
+const numeric sqrt(const numeric & z)
{
- return ::sqrt(*z.value); // -> CLN
+ return cln::sqrt(cln::cl_N(z));
}
+
/** Integer numeric square root. */
-numeric isqrt(numeric const & x)
+const numeric isqrt(const numeric & x)
{
if (x.is_integer()) {
- cl_I root;
- ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
+ cln::cl_I root;
+ cln::isqrt(cln::the<cln::cl_I>(cln::cl_N(x)), &root);
return root;
} else
- return numZERO(); // Throw?
+ return _num0();
}
-/** Greatest Common Divisor.
- *
- * @return The GCD of two numbers if both are integer, a numerical 1
- * if they are not. */
-numeric gcd(numeric const & a, numeric const & b)
-{
- if (a.is_integer() && b.is_integer())
- return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
- else
- return numONE();
-}
-
-/** Least Common Multiple.
- *
- * @return The LCM of two numbers if both are integer, the product of those
- * two numbers if they are not. */
-numeric lcm(numeric const & a, numeric const & 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;
-}
+/** Floating point evaluation of Archimedes' constant Pi. */
ex PiEvalf(void)
{
- return numeric(cl_pi(cl_default_float_format)); // -> CLN
+ return numeric(cln::pi(cln::default_float_format));
}
-ex EulerGammaEvalf(void)
+
+/** Floating point evaluation of Euler's constant gamma. */
+ex EulerEvalf(void)
{
- 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)
{
- 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::_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.
+ assert(!too_late);
+ too_late = true;
+ cln::default_float_format = cln::float_format(17);
}
+
+/** Assign a native long to global Digits object. */
_numeric_digits& _numeric_digits::operator=(long prec)
{
- digits=prec;
- cl_default_float_format = cl_float_format(prec);
- return *this;
+ digits = prec;
+ cln::default_float_format = cln::float_format(prec);
+ 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;
}
-void _numeric_digits::print(ostream & os) const
+
+/** Append global Digits object to ostream. */
+void _numeric_digits::print(std::ostream & os) const
{
- debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
- os << digits;
+ debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
+ os << digits;
}
-ostream& operator<<(ostream& os, _numeric_digits const & e)
+
+std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
{
- e.print(os);
- return os;
+ e.print(os);
+ return os;
}
//////////
bool _numeric_digits::too_late = false;
+
/** Accuracy in decimal digits. Only object of this type! Can be set using
* assignment from C++ unsigned ints and evaluated like any built-in type. */
_numeric_digits Digits;
+#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
+#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff