3 * This file contains the interface to the underlying bignum package.
4 * Its most important design principle is to completely hide the inner
5 * working of that other package from the user of GiNaC. It must either
6 * provide implementation of arithmetic operators and numerical evaluation
7 * of special functions or implement the interface to the bignum package. */
10 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
12 * This program is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU General Public License as published by
14 * the Free Software Foundation; either version 2 of the License, or
15 * (at your option) any later version.
17 * This program is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU General Public License for more details.
22 * You should have received a copy of the GNU General Public License
23 * along with this program; if not, write to the Free Software
24 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
33 #if defined(HAVE_SSTREAM)
35 #elif defined(HAVE_STRSTREAM)
38 #error Need either sstream or strstream
47 // CLN should pollute the global namespace as little as possible. Hence, we
48 // include most of it here and include only the part needed for properly
49 // declaring cln::cl_number in numeric.h. This can only be safely done in
50 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
51 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
53 #include <cln/output.h>
54 #include <cln/integer_io.h>
55 #include <cln/integer_ring.h>
56 #include <cln/rational_io.h>
57 #include <cln/rational_ring.h>
58 #include <cln/lfloat_class.h>
59 #include <cln/lfloat_io.h>
60 #include <cln/real_io.h>
61 #include <cln/real_ring.h>
62 #include <cln/complex_io.h>
63 #include <cln/complex_ring.h>
64 #include <cln/numtheory.h>
66 #ifndef NO_NAMESPACE_GINAC
68 #endif // ndef NO_NAMESPACE_GINAC
70 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
73 // default constructor, destructor, copy constructor assignment
74 // operator and helpers
79 /** default ctor. Numerically it initializes to an integer zero. */
80 numeric::numeric() : basic(TINFO_numeric)
82 debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
85 setflag(status_flags::evaluated |
86 status_flags::expanded |
87 status_flags::hash_calculated);
92 void numeric::copy(const numeric & other)
98 void numeric::destroy(bool call_parent)
100 if (call_parent) basic::destroy(call_parent);
104 // other constructors
109 numeric::numeric(int i) : basic(TINFO_numeric)
111 debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
112 // Not the whole int-range is available if we don't cast to long
113 // first. This is due to the behaviour of the cl_I-ctor, which
114 // emphasizes efficiency. However, if the integer is small enough,
115 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
116 // using an immediate type:
117 if (cln::cl_immediate_p(i))
118 value = cln::cl_I(i);
120 value = cln::cl_I((long) i);
122 setflag(status_flags::evaluated |
123 status_flags::expanded |
124 status_flags::hash_calculated);
128 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
130 debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
131 // Not the whole uint-range is available if we don't cast to ulong
132 // first. This is due to the behaviour of the cl_I-ctor, which
133 // emphasizes efficiency. However, if the integer is small enough,
134 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
135 // using an immediate type:
136 if (cln::cl_immediate_p(i))
137 value = cln::cl_I(i);
139 value = cln::cl_I((unsigned long) i);
141 setflag(status_flags::evaluated |
142 status_flags::expanded |
143 status_flags::hash_calculated);
147 numeric::numeric(long i) : basic(TINFO_numeric)
149 debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
150 value = cln::cl_I(i);
152 setflag(status_flags::evaluated |
153 status_flags::expanded |
154 status_flags::hash_calculated);
158 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
160 debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
161 value = cln::cl_I(i);
163 setflag(status_flags::evaluated |
164 status_flags::expanded |
165 status_flags::hash_calculated);
168 /** Ctor for rational numerics a/b.
170 * @exception overflow_error (division by zero) */
171 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
173 debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
175 throw std::overflow_error("division by zero");
176 value = cln::cl_I(numer) / cln::cl_I(denom);
178 setflag(status_flags::evaluated |
179 status_flags::expanded |
180 status_flags::hash_calculated);
184 numeric::numeric(double d) : basic(TINFO_numeric)
186 debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
187 // We really want to explicitly use the type cl_LF instead of the
188 // more general cl_F, since that would give us a cl_DF only which
189 // will not be promoted to cl_LF if overflow occurs:
190 value = cln::cl_float(d, cln::default_float_format);
192 setflag(status_flags::evaluated |
193 status_flags::expanded |
194 status_flags::hash_calculated);
197 /** ctor from C-style string. It also accepts complex numbers in GiNaC
198 * notation like "2+5*I". */
199 numeric::numeric(const char *s) : basic(TINFO_numeric)
201 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
202 cln::cl_N ctorval = 0;
203 // parse complex numbers (functional but not completely safe, unfortunately
204 // std::string does not understand regexpese):
205 // ss should represent a simple sum like 2+5*I
207 // make it safe by adding explicit sign
208 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
210 std::string::size_type delim;
212 // chop ss into terms from left to right
214 bool imaginary = false;
215 delim = ss.find_first_of(std::string("+-"),1);
216 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
217 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
218 delim = ss.find_first_of(std::string("+-"),delim+1);
219 term = ss.substr(0,delim);
220 if (delim != std::string::npos)
221 ss = ss.substr(delim);
222 // is the term imaginary?
223 if (term.find("I") != std::string::npos) {
225 term = term.replace(term.find("I"),1,"");
227 if (term.find("*") != std::string::npos)
228 term = term.replace(term.find("*"),1,"");
229 // correct for trivial +/-I without explicit factor on I:
230 if (term.size() == 1)
234 if (term.find(".") != std::string::npos) {
235 // CLN's short type cl_SF is not very useful within the GiNaC
236 // framework where we are mainly interested in the arbitrary
237 // precision type cl_LF. Hence we go straight to the construction
238 // of generic floats. In order to create them we have to convert
239 // our own floating point notation used for output and construction
240 // from char * to CLN's generic notation:
241 // 3.14 --> 3.14e0_<Digits>
242 // 31.4E-1 --> 31.4e-1_<Digits>
244 // No exponent marker? Let's add a trivial one.
245 if (term.find("E") == std::string::npos)
248 term = term.replace(term.find("E"),1,"e");
249 // append _<Digits> to term
250 #if defined(HAVE_SSTREAM)
251 std::ostringstream buf;
252 buf << unsigned(Digits) << std::ends;
253 term += "_" + buf.str();
256 std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
257 term += "_" + string(buf);
259 // construct float using cln::cl_F(const char *) ctor.
261 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
263 ctorval = ctorval + cln::cl_F(term.c_str());
265 // not a floating point number...
267 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
269 ctorval = ctorval + cln::cl_R(term.c_str());
271 } while(delim != std::string::npos);
274 setflag(status_flags::evaluated |
275 status_flags::expanded |
276 status_flags::hash_calculated);
279 /** Ctor from CLN types. This is for the initiated user or internal use
281 numeric::numeric(const cln::cl_N & z) : basic(TINFO_numeric)
283 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
286 setflag(status_flags::evaluated |
287 status_flags::expanded |
288 status_flags::hash_calculated);
295 /** Construct object from archive_node. */
296 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
298 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
299 cln::cl_N ctorval = 0;
301 // Read number as string
303 if (n.find_string("number", str)) {
305 std::istringstream s(str);
307 std::istrstream s(str.c_str(), str.size() + 1);
309 cln::cl_idecoded_float re, im;
313 case 'R': // Integer-decoded real number
314 s >> re.sign >> re.mantissa >> re.exponent;
315 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
317 case 'C': // Integer-decoded complex number
318 s >> re.sign >> re.mantissa >> re.exponent;
319 s >> im.sign >> im.mantissa >> im.exponent;
320 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
321 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
323 default: // Ordinary number
331 setflag(status_flags::evaluated |
332 status_flags::expanded |
333 status_flags::hash_calculated);
336 /** Unarchive the object. */
337 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
339 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
342 /** Archive the object. */
343 void numeric::archive(archive_node &n) const
345 inherited::archive(n);
347 // Write number as string
349 std::ostringstream s;
352 std::ostrstream s(buf, 1024);
354 if (this->is_crational())
355 s << cln::the<cln::cl_N>(value);
357 // Non-rational numbers are written in an integer-decoded format
358 // to preserve the precision
359 if (this->is_real()) {
360 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
362 s << re.sign << " " << re.mantissa << " " << re.exponent;
364 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
365 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
367 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
368 s << im.sign << " " << im.mantissa << " " << im.exponent;
372 n.add_string("number", s.str());
375 std::string str(buf);
376 n.add_string("number", str);
381 // functions overriding virtual functions from bases classes
386 basic * numeric::duplicate() const
388 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
389 return new numeric(*this);
393 /** Helper function to print a real number in a nicer way than is CLN's
394 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
395 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
396 * long as it only uses cl_LF and no other floating point types that we might
397 * want to visibly distinguish from cl_LF.
399 * @see numeric::print() */
400 static void print_real_number(std::ostream & os, const cln::cl_R & num)
402 cln::cl_print_flags ourflags;
403 if (cln::instanceof(num, cln::cl_RA_ring)) {
404 // case 1: integer or rational, nothing special to do:
405 cln::print_real(os, ourflags, num);
408 // make CLN believe this number has default_float_format, so it prints
409 // 'E' as exponent marker instead of 'L':
410 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(num));
411 cln::print_real(os, ourflags, num);
416 /** This method adds to the output so it blends more consistently together
417 * with the other routines and produces something compatible to ginsh input.
419 * @see print_real_number() */
420 void numeric::print(std::ostream & os, unsigned upper_precedence) const
422 debugmsg("numeric print", LOGLEVEL_PRINT);
423 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
424 cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
426 // case 1, real: x or -x
427 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
429 print_real_number(os, r);
432 print_real_number(os, r);
436 // case 2, imaginary: y*I or -y*I
437 if ((precedence<=upper_precedence) && (i < 0)) {
442 print_real_number(os, i);
452 print_real_number(os, i);
458 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
459 if (precedence <= upper_precedence)
461 print_real_number(os, r);
466 print_real_number(os, i);
474 print_real_number(os, i);
478 if (precedence <= upper_precedence)
485 void numeric::printraw(std::ostream & os) const
487 // The method printraw doesn't do much, it simply uses CLN's operator<<()
488 // for output, which is ugly but reliable. e.g: 2+2i
489 debugmsg("numeric printraw", LOGLEVEL_PRINT);
490 os << "numeric(" << cln::the<cln::cl_N>(value) << ")";
494 void numeric::printtree(std::ostream & os, unsigned indent) const
496 debugmsg("numeric printtree", LOGLEVEL_PRINT);
497 os << std::string(indent,' ') << cln::the<cln::cl_N>(value)
499 << "hash=" << hashvalue
500 << " (0x" << std::hex << hashvalue << std::dec << ")"
501 << ", flags=" << flags << std::endl;
505 void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
507 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
508 std::ios::fmtflags oldflags = os.flags();
509 os.setf(std::ios::scientific);
510 if (this->is_rational() && !this->is_integer()) {
511 if (compare(_num0()) > 0) {
513 if (type == csrc_types::ctype_cl_N)
514 os << "cln::cl_F(\"" << numer().evalf() << "\")";
516 os << numer().to_double();
519 if (type == csrc_types::ctype_cl_N)
520 os << "cln::cl_F(\"" << -numer().evalf() << "\")";
522 os << -numer().to_double();
525 if (type == csrc_types::ctype_cl_N)
526 os << "cln::cl_F(\"" << denom().evalf() << "\")";
528 os << denom().to_double();
531 if (type == csrc_types::ctype_cl_N)
532 os << "cln::cl_F(\"" << evalf() << "\")";
540 bool numeric::info(unsigned inf) const
543 case info_flags::numeric:
544 case info_flags::polynomial:
545 case info_flags::rational_function:
547 case info_flags::real:
549 case info_flags::rational:
550 case info_flags::rational_polynomial:
551 return is_rational();
552 case info_flags::crational:
553 case info_flags::crational_polynomial:
554 return is_crational();
555 case info_flags::integer:
556 case info_flags::integer_polynomial:
558 case info_flags::cinteger:
559 case info_flags::cinteger_polynomial:
560 return is_cinteger();
561 case info_flags::positive:
562 return is_positive();
563 case info_flags::negative:
564 return is_negative();
565 case info_flags::nonnegative:
566 return !is_negative();
567 case info_flags::posint:
568 return is_pos_integer();
569 case info_flags::negint:
570 return is_integer() && is_negative();
571 case info_flags::nonnegint:
572 return is_nonneg_integer();
573 case info_flags::even:
575 case info_flags::odd:
577 case info_flags::prime:
579 case info_flags::algebraic:
585 /** Disassemble real part and imaginary part to scan for the occurrence of a
586 * single number. Also handles the imaginary unit. It ignores the sign on
587 * both this and the argument, which may lead to what might appear as funny
588 * results: (2+I).has(-2) -> true. But this is consistent, since we also
589 * would like to have (-2+I).has(2) -> true and we want to think about the
590 * sign as a multiplicative factor. */
591 bool numeric::has(const ex & other) const
593 if (!is_exactly_of_type(*other.bp, numeric))
595 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
596 if (this->is_equal(o) || this->is_equal(-o))
598 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
599 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
600 this->real().is_equal(-o) || this->imag().is_equal(-o));
602 if (o.is_equal(I)) // e.g scan for I in 42*I
603 return !this->is_real();
604 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
605 return (this->real().has(o*I) || this->imag().has(o*I) ||
606 this->real().has(-o*I) || this->imag().has(-o*I));
612 /** Evaluation of numbers doesn't do anything at all. */
613 ex numeric::eval(int level) const
615 // Warning: if this is ever gonna do something, the ex ctors from all kinds
616 // of numbers should be checking for status_flags::evaluated.
621 /** Cast numeric into a floating-point object. For example exact numeric(1) is
622 * returned as a 1.0000000000000000000000 and so on according to how Digits is
623 * currently set. In case the object already was a floating point number the
624 * precision is trimmed to match the currently set default.
626 * @param level ignored, only needed for overriding basic::evalf.
627 * @return an ex-handle to a numeric. */
628 ex numeric::evalf(int level) const
630 // level can safely be discarded for numeric objects.
631 return numeric(cln::cl_float(1.0, cln::default_float_format) *
632 (cln::the<cln::cl_N>(value)));
637 /** Implementation of ex::diff() for a numeric. It always returns 0.
640 ex numeric::derivative(const symbol & s) const
646 int numeric::compare_same_type(const basic & other) const
648 GINAC_ASSERT(is_exactly_of_type(other, numeric));
649 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
651 return this->compare(o);
655 bool numeric::is_equal_same_type(const basic & other) const
657 GINAC_ASSERT(is_exactly_of_type(other,numeric));
658 const numeric *o = static_cast<const numeric *>(&other);
660 return this->is_equal(*o);
664 unsigned numeric::calchash(void) const
666 // Use CLN's hashcode. Warning: It depends only on the number's value, not
667 // its type or precision (i.e. a true equivalence relation on numbers). As
668 // a consequence, 3 and 3.0 share the same hashvalue.
669 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
674 // new virtual functions which can be overridden by derived classes
680 // non-virtual functions in this class
685 /** Numerical addition method. Adds argument to *this and returns result as
686 * a new numeric object. */
687 const numeric numeric::add(const numeric & other) const
689 // Efficiency shortcut: trap the neutral element by pointer.
690 static const numeric * _num0p = &_num0();
693 else if (&other==_num0p)
696 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
700 /** Numerical subtraction method. Subtracts argument from *this and returns
701 * result as a new numeric object. */
702 const numeric numeric::sub(const numeric & other) const
704 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
708 /** Numerical multiplication method. Multiplies *this and argument and returns
709 * result as a new numeric object. */
710 const numeric numeric::mul(const numeric & other) const
712 // Efficiency shortcut: trap the neutral element by pointer.
713 static const numeric * _num1p = &_num1();
716 else if (&other==_num1p)
719 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
723 /** Numerical division method. Divides *this by argument and returns result as
724 * a new numeric object.
726 * @exception overflow_error (division by zero) */
727 const numeric numeric::div(const numeric & other) const
729 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
730 throw std::overflow_error("numeric::div(): division by zero");
731 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
735 const numeric numeric::power(const numeric & other) const
737 // Efficiency shortcut: trap the neutral exponent by pointer.
738 static const numeric * _num1p = &_num1();
742 if (cln::zerop(cln::the<cln::cl_N>(value))) {
743 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
744 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
745 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
746 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
747 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
748 throw std::overflow_error("numeric::eval(): division by zero");
752 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
756 const numeric & numeric::add_dyn(const numeric & other) const
758 // Efficiency shortcut: trap the neutral element by pointer.
759 static const numeric * _num0p = &_num0();
762 else if (&other==_num0p)
765 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
766 setflag(status_flags::dynallocated));
770 const numeric & numeric::sub_dyn(const numeric & other) const
772 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
773 setflag(status_flags::dynallocated));
777 const numeric & numeric::mul_dyn(const numeric & other) const
779 // Efficiency shortcut: trap the neutral element by pointer.
780 static const numeric * _num1p = &_num1();
783 else if (&other==_num1p)
786 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
787 setflag(status_flags::dynallocated));
791 const numeric & numeric::div_dyn(const numeric & other) const
793 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
794 throw std::overflow_error("division by zero");
795 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
796 setflag(status_flags::dynallocated));
800 const numeric & numeric::power_dyn(const numeric & other) const
802 // Efficiency shortcut: trap the neutral exponent by pointer.
803 static const numeric * _num1p=&_num1();
807 if (cln::zerop(cln::the<cln::cl_N>(value))) {
808 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
809 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
810 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
811 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
812 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
813 throw std::overflow_error("numeric::eval(): division by zero");
817 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
818 setflag(status_flags::dynallocated));
822 const numeric & numeric::operator=(int i)
824 return operator=(numeric(i));
828 const numeric & numeric::operator=(unsigned int i)
830 return operator=(numeric(i));
834 const numeric & numeric::operator=(long i)
836 return operator=(numeric(i));
840 const numeric & numeric::operator=(unsigned long i)
842 return operator=(numeric(i));
846 const numeric & numeric::operator=(double d)
848 return operator=(numeric(d));
852 const numeric & numeric::operator=(const char * s)
854 return operator=(numeric(s));
858 /** Inverse of a number. */
859 const numeric numeric::inverse(void) const
861 if (cln::zerop(cln::the<cln::cl_N>(value)))
862 throw std::overflow_error("numeric::inverse(): division by zero");
863 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
867 /** Return the complex half-plane (left or right) in which the number lies.
868 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
869 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
871 * @see numeric::compare(const numeric & other) */
872 int numeric::csgn(void) const
874 if (cln::zerop(cln::the<cln::cl_N>(value)))
876 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
877 if (!cln::zerop(r)) {
883 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
891 /** This method establishes a canonical order on all numbers. For complex
892 * numbers this is not possible in a mathematically consistent way but we need
893 * to establish some order and it ought to be fast. So we simply define it
894 * to be compatible with our method csgn.
896 * @return csgn(*this-other)
897 * @see numeric::csgn(void) */
898 int numeric::compare(const numeric & other) const
900 // Comparing two real numbers?
901 if (cln::instanceof(value, cln::cl_R_ring) &&
902 cln::instanceof(other.value, cln::cl_R_ring))
903 // Yes, so just cln::compare them
904 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
906 // No, first cln::compare real parts...
907 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
910 // ...and then the imaginary parts.
911 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
916 bool numeric::is_equal(const numeric & other) const
918 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
922 /** True if object is zero. */
923 bool numeric::is_zero(void) const
925 return cln::zerop(cln::the<cln::cl_N>(value));
929 /** True if object is not complex and greater than zero. */
930 bool numeric::is_positive(void) const
933 return cln::plusp(cln::the<cln::cl_R>(value));
938 /** True if object is not complex and less than zero. */
939 bool numeric::is_negative(void) const
942 return cln::minusp(cln::the<cln::cl_R>(value));
947 /** True if object is a non-complex integer. */
948 bool numeric::is_integer(void) const
950 return cln::instanceof(value, cln::cl_I_ring);
954 /** True if object is an exact integer greater than zero. */
955 bool numeric::is_pos_integer(void) const
957 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
961 /** True if object is an exact integer greater or equal zero. */
962 bool numeric::is_nonneg_integer(void) const
964 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
968 /** True if object is an exact even integer. */
969 bool numeric::is_even(void) const
971 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
975 /** True if object is an exact odd integer. */
976 bool numeric::is_odd(void) const
978 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
982 /** Probabilistic primality test.
984 * @return true if object is exact integer and prime. */
985 bool numeric::is_prime(void) const
987 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
991 /** True if object is an exact rational number, may even be complex
992 * (denominator may be unity). */
993 bool numeric::is_rational(void) const
995 return cln::instanceof(value, cln::cl_RA_ring);
999 /** True if object is a real integer, rational or float (but not complex). */
1000 bool numeric::is_real(void) const
1002 return cln::instanceof(value, cln::cl_R_ring);
1006 bool numeric::operator==(const numeric & other) const
1008 return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
1012 bool numeric::operator!=(const numeric & other) const
1014 return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
1018 /** True if object is element of the domain of integers extended by I, i.e. is
1019 * of the form a+b*I, where a and b are integers. */
1020 bool numeric::is_cinteger(void) const
1022 if (cln::instanceof(value, cln::cl_I_ring))
1024 else if (!this->is_real()) { // complex case, handle n+m*I
1025 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
1026 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
1033 /** True if object is an exact rational number, may even be complex
1034 * (denominator may be unity). */
1035 bool numeric::is_crational(void) const
1037 if (cln::instanceof(value, cln::cl_RA_ring))
1039 else if (!this->is_real()) { // complex case, handle Q(i):
1040 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
1041 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
1048 /** Numerical comparison: less.
1050 * @exception invalid_argument (complex inequality) */
1051 bool numeric::operator<(const numeric & other) const
1053 if (this->is_real() && other.is_real())
1054 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
1055 throw std::invalid_argument("numeric::operator<(): complex inequality");
1059 /** Numerical comparison: less or equal.
1061 * @exception invalid_argument (complex inequality) */
1062 bool numeric::operator<=(const numeric & other) const
1064 if (this->is_real() && other.is_real())
1065 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
1066 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1070 /** Numerical comparison: greater.
1072 * @exception invalid_argument (complex inequality) */
1073 bool numeric::operator>(const numeric & other) const
1075 if (this->is_real() && other.is_real())
1076 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1077 throw std::invalid_argument("numeric::operator>(): complex inequality");
1081 /** Numerical comparison: greater or equal.
1083 * @exception invalid_argument (complex inequality) */
1084 bool numeric::operator>=(const numeric & other) const
1086 if (this->is_real() && other.is_real())
1087 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1088 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1092 /** Converts numeric types to machine's int. You should check with
1093 * is_integer() if the number is really an integer before calling this method.
1094 * You may also consider checking the range first. */
1095 int numeric::to_int(void) const
1097 GINAC_ASSERT(this->is_integer());
1098 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1102 /** Converts numeric types to machine's long. You should check with
1103 * is_integer() if the number is really an integer before calling this method.
1104 * You may also consider checking the range first. */
1105 long numeric::to_long(void) const
1107 GINAC_ASSERT(this->is_integer());
1108 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1112 /** Converts numeric types to machine's double. You should check with is_real()
1113 * if the number is really not complex before calling this method. */
1114 double numeric::to_double(void) const
1116 GINAC_ASSERT(this->is_real());
1117 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1121 /** Returns a new CLN object of type cl_N, representing the value of *this.
1122 * This method may be used when mixing GiNaC and CLN in one project.
1124 cln::cl_N numeric::to_cl_N(void) const
1126 return cln::cl_N(cln::the<cln::cl_N>(value));
1130 /** Real part of a number. */
1131 const numeric numeric::real(void) const
1133 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1137 /** Imaginary part of a number. */
1138 const numeric numeric::imag(void) const
1140 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1144 /** Numerator. Computes the numerator of rational numbers, rationalized
1145 * numerator of complex if real and imaginary part are both rational numbers
1146 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1148 const numeric numeric::numer(void) const
1150 if (this->is_integer())
1151 return numeric(*this);
1153 else if (cln::instanceof(value, cln::cl_RA_ring))
1154 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1156 else if (!this->is_real()) { // complex case, handle Q(i):
1157 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1158 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1159 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1160 return numeric(*this);
1161 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1162 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1163 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1164 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1165 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1166 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1167 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1168 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1171 // at least one float encountered
1172 return numeric(*this);
1176 /** Denominator. Computes the denominator of rational numbers, common integer
1177 * denominator of complex if real and imaginary part are both rational numbers
1178 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1179 const numeric numeric::denom(void) const
1181 if (this->is_integer())
1184 if (instanceof(value, cln::cl_RA_ring))
1185 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1187 if (!this->is_real()) { // complex case, handle Q(i):
1188 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1189 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1190 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1192 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1193 return numeric(cln::denominator(i));
1194 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1195 return numeric(cln::denominator(r));
1196 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1197 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1199 // at least one float encountered
1204 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1205 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1206 * 2^(n-1) <= x < 2^n.
1208 * @return number of bits (excluding sign) needed to represent that number
1209 * in two's complement if it is an integer, 0 otherwise. */
1210 int numeric::int_length(void) const
1212 if (this->is_integer())
1213 return cln::integer_length(cln::the<cln::cl_I>(value));
1220 // static member variables
1225 unsigned numeric::precedence = 30;
1231 /** Imaginary unit. This is not a constant but a numeric since we are
1232 * natively handing complex numbers anyways. */
1233 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1236 /** Exponential function.
1238 * @return arbitrary precision numerical exp(x). */
1239 const numeric exp(const numeric & x)
1241 return cln::exp(x.to_cl_N());
1245 /** Natural logarithm.
1247 * @param z complex number
1248 * @return arbitrary precision numerical log(x).
1249 * @exception pole_error("log(): logarithmic pole",0) */
1250 const numeric log(const numeric & z)
1253 throw pole_error("log(): logarithmic pole",0);
1254 return cln::log(z.to_cl_N());
1258 /** Numeric sine (trigonometric function).
1260 * @return arbitrary precision numerical sin(x). */
1261 const numeric sin(const numeric & x)
1263 return cln::sin(x.to_cl_N());
1267 /** Numeric cosine (trigonometric function).
1269 * @return arbitrary precision numerical cos(x). */
1270 const numeric cos(const numeric & x)
1272 return cln::cos(x.to_cl_N());
1276 /** Numeric tangent (trigonometric function).
1278 * @return arbitrary precision numerical tan(x). */
1279 const numeric tan(const numeric & x)
1281 return cln::tan(x.to_cl_N());
1285 /** Numeric inverse sine (trigonometric function).
1287 * @return arbitrary precision numerical asin(x). */
1288 const numeric asin(const numeric & x)
1290 return cln::asin(x.to_cl_N());
1294 /** Numeric inverse cosine (trigonometric function).
1296 * @return arbitrary precision numerical acos(x). */
1297 const numeric acos(const numeric & x)
1299 return cln::acos(x.to_cl_N());
1305 * @param z complex number
1307 * @exception pole_error("atan(): logarithmic pole",0) */
1308 const numeric atan(const numeric & x)
1311 x.real().is_zero() &&
1312 abs(x.imag()).is_equal(_num1()))
1313 throw pole_error("atan(): logarithmic pole",0);
1314 return cln::atan(x.to_cl_N());
1320 * @param x real number
1321 * @param y real number
1322 * @return atan(y/x) */
1323 const numeric atan(const numeric & y, const numeric & x)
1325 if (x.is_real() && y.is_real())
1326 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1327 cln::the<cln::cl_R>(y.to_cl_N()));
1329 throw std::invalid_argument("atan(): complex argument");
1333 /** Numeric hyperbolic sine (trigonometric function).
1335 * @return arbitrary precision numerical sinh(x). */
1336 const numeric sinh(const numeric & x)
1338 return cln::sinh(x.to_cl_N());
1342 /** Numeric hyperbolic cosine (trigonometric function).
1344 * @return arbitrary precision numerical cosh(x). */
1345 const numeric cosh(const numeric & x)
1347 return cln::cosh(x.to_cl_N());
1351 /** Numeric hyperbolic tangent (trigonometric function).
1353 * @return arbitrary precision numerical tanh(x). */
1354 const numeric tanh(const numeric & x)
1356 return cln::tanh(x.to_cl_N());
1360 /** Numeric inverse hyperbolic sine (trigonometric function).
1362 * @return arbitrary precision numerical asinh(x). */
1363 const numeric asinh(const numeric & x)
1365 return cln::asinh(x.to_cl_N());
1369 /** Numeric inverse hyperbolic cosine (trigonometric function).
1371 * @return arbitrary precision numerical acosh(x). */
1372 const numeric acosh(const numeric & x)
1374 return cln::acosh(x.to_cl_N());
1378 /** Numeric inverse hyperbolic tangent (trigonometric function).
1380 * @return arbitrary precision numerical atanh(x). */
1381 const numeric atanh(const numeric & x)
1383 return cln::atanh(x.to_cl_N());
1387 /*static cln::cl_N Li2_series(const ::cl_N & x,
1388 const ::float_format_t & prec)
1390 // Note: argument must be in the unit circle
1391 // This is very inefficient unless we have fast floating point Bernoulli
1392 // numbers implemented!
1393 cln::cl_N c1 = -cln::log(1-x);
1395 // hard-wire the first two Bernoulli numbers
1396 cln::cl_N acc = c1 - cln::square(c1)/4;
1398 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1399 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1401 c1 = cln::square(c1);
1405 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1406 // 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));
1409 } while (acc != acc+aug);
1413 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1414 * circle) using a power series. */
1415 static cln::cl_N Li2_series(const cln::cl_N & x,
1416 const cln::float_format_t & prec)
1418 // Note: argument must be in the unit circle
1420 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1425 den = den + i; // 1, 4, 9, 16, ...
1429 } while (acc != acc+aug);
1433 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1434 static cln::cl_N Li2_projection(const cln::cl_N & x,
1435 const cln::float_format_t & prec)
1437 const cln::cl_R re = cln::realpart(x);
1438 const cln::cl_R im = cln::imagpart(x);
1439 if (re > cln::cl_F(".5"))
1440 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1442 - Li2_series(1-x, prec)
1443 - cln::log(x)*cln::log(1-x));
1444 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1445 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1446 return(- cln::square(cln::log(1-x))/2
1447 - Li2_series(x/(x-1), prec));
1448 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1449 // Li2(x^2)/2 - Li2(-x)
1450 return(Li2_projection(cln::square(x), prec)/2
1451 - Li2_projection(-x, prec));
1452 return Li2_series(x, prec);
1455 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1456 * the branch cut lies along the positive real axis, starting at 1 and
1457 * continuous with quadrant IV.
1459 * @return arbitrary precision numerical Li2(x). */
1460 const numeric Li2(const numeric & x)
1465 // what is the desired float format?
1466 // first guess: default format
1467 cln::float_format_t prec = cln::default_float_format;
1468 const cln::cl_N value = x.to_cl_N();
1469 // second guess: the argument's format
1470 if (!x.real().is_rational())
1471 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1472 else if (!x.imag().is_rational())
1473 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1475 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1476 return cln::zeta(2, prec);
1478 if (cln::abs(value) > 1)
1479 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1480 return(- cln::square(cln::log(-value))/2
1481 - cln::zeta(2, prec)
1482 - Li2_projection(cln::recip(value), prec));
1484 return Li2_projection(x.to_cl_N(), prec);
1488 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1489 * integer arguments. */
1490 const numeric zeta(const numeric & x)
1492 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1493 // only knows about integer arguments and zeta(3).evalf() automatically
1494 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1495 // being an exact zero for CLN, which can be tested and then we can just
1496 // pass the number casted to an int:
1498 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1499 if (cln::zerop(x.to_cl_N()-aux))
1500 return cln::zeta(aux);
1502 std::clog << "zeta(" << x
1503 << "): Does anybody know good way to calculate this numerically?"
1509 /** The Gamma function.
1510 * This is only a stub! */
1511 const numeric lgamma(const numeric & x)
1513 std::clog << "lgamma(" << x
1514 << "): Does anybody know good way to calculate this numerically?"
1518 const numeric tgamma(const numeric & x)
1520 std::clog << "tgamma(" << x
1521 << "): Does anybody know good way to calculate this numerically?"
1527 /** The psi function (aka polygamma function).
1528 * This is only a stub! */
1529 const numeric psi(const numeric & x)
1531 std::clog << "psi(" << x
1532 << "): Does anybody know good way to calculate this numerically?"
1538 /** The psi functions (aka polygamma functions).
1539 * This is only a stub! */
1540 const numeric psi(const numeric & n, const numeric & x)
1542 std::clog << "psi(" << n << "," << x
1543 << "): Does anybody know good way to calculate this numerically?"
1549 /** Factorial combinatorial function.
1551 * @param n integer argument >= 0
1552 * @exception range_error (argument must be integer >= 0) */
1553 const numeric factorial(const numeric & n)
1555 if (!n.is_nonneg_integer())
1556 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1557 return numeric(cln::factorial(n.to_int()));
1561 /** The double factorial combinatorial function. (Scarcely used, but still
1562 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1564 * @param n integer argument >= -1
1565 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1566 * @exception range_error (argument must be integer >= -1) */
1567 const numeric doublefactorial(const numeric & n)
1569 if (n == numeric(-1))
1572 if (!n.is_nonneg_integer())
1573 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1575 return numeric(cln::doublefactorial(n.to_int()));
1579 /** The Binomial coefficients. It computes the binomial coefficients. For
1580 * integer n and k and positive n this is the number of ways of choosing k
1581 * objects from n distinct objects. If n is negative, the formula
1582 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1583 const numeric binomial(const numeric & n, const numeric & k)
1585 if (n.is_integer() && k.is_integer()) {
1586 if (n.is_nonneg_integer()) {
1587 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1588 return numeric(cln::binomial(n.to_int(),k.to_int()));
1592 return _num_1().power(k)*binomial(k-n-_num1(),k);
1596 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1597 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1601 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1602 * in the expansion of the function x/(e^x-1).
1604 * @return the nth Bernoulli number (a rational number).
1605 * @exception range_error (argument must be integer >= 0) */
1606 const numeric bernoulli(const numeric & nn)
1608 if (!nn.is_integer() || nn.is_negative())
1609 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1613 // The Bernoulli numbers are rational numbers that may be computed using
1616 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1618 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1619 // previous ones, the computation is necessarily very expensive. There are
1620 // several other ways of computing them, a particularly good one being
1624 // for (unsigned i=0; i<n; i++) {
1625 // c = exquo(c*(i-n),(i+2));
1626 // Bern = Bern + c*s/(i+2);
1627 // s = s + expt_pos(cl_I(i+2),n);
1631 // But if somebody works with the n'th Bernoulli number she is likely to
1632 // also need all previous Bernoulli numbers. So we need a complete remember
1633 // table and above divide and conquer algorithm is not suited to build one
1634 // up. The code below is adapted from Pari's function bernvec().
1636 // (There is an interesting relation with the tangent polynomials described
1637 // in `Concrete Mathematics', which leads to a program twice as fast as our
1638 // implementation below, but it requires storing one such polynomial in
1639 // addition to the remember table. This doubles the memory footprint so
1640 // we don't use it.)
1642 // the special cases not covered by the algorithm below
1643 if (nn.is_equal(_num1()))
1648 // store nonvanishing Bernoulli numbers here
1649 static std::vector< cln::cl_RA > results;
1650 static int highest_result = 0;
1651 // algorithm not applicable to B(0), so just store it
1652 if (results.size()==0)
1653 results.push_back(cln::cl_RA(1));
1655 int n = nn.to_long();
1656 for (int i=highest_result; i<n/2; ++i) {
1662 for (int j=i; j>0; --j) {
1663 B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
1669 B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
1670 results.push_back(B);
1673 return results[n/2];
1677 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1678 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1680 * @param n an integer
1681 * @return the nth Fibonacci number F(n) (an integer number)
1682 * @exception range_error (argument must be an integer) */
1683 const numeric fibonacci(const numeric & n)
1685 if (!n.is_integer())
1686 throw std::range_error("numeric::fibonacci(): argument must be integer");
1689 // The following addition formula holds:
1691 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1693 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1694 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1696 // Replace m by m+1:
1697 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1698 // Now put in m = n, to get
1699 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1700 // F(2n+1) = F(n)^2 + F(n+1)^2
1702 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1705 if (n.is_negative())
1707 return -fibonacci(-n);
1709 return fibonacci(-n);
1713 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1714 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1715 // Since a squaring is cheaper than a multiplication, better use
1716 // three squarings instead of one multiplication and two squarings.
1717 cln::cl_I u2 = cln::square(u);
1718 cln::cl_I v2 = cln::square(v);
1719 if (cln::logbitp(bit-1, m)) {
1720 v = cln::square(u + v) - u2;
1723 u = v2 - cln::square(v - u);
1728 // Here we don't use the squaring formula because one multiplication
1729 // is cheaper than two squarings.
1730 return u * ((v << 1) - u);
1732 return cln::square(u) + cln::square(v);
1736 /** Absolute value. */
1737 const numeric abs(const numeric& x)
1739 return cln::abs(x.to_cl_N());
1743 /** Modulus (in positive representation).
1744 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1745 * sign of a or is zero. This is different from Maple's modp, where the sign
1746 * of b is ignored. It is in agreement with Mathematica's Mod.
1748 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1749 * integer, 0 otherwise. */
1750 const numeric mod(const numeric & a, const numeric & b)
1752 if (a.is_integer() && b.is_integer())
1753 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1754 cln::the<cln::cl_I>(b.to_cl_N()));
1760 /** Modulus (in symmetric representation).
1761 * Equivalent to Maple's mods.
1763 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1764 const numeric smod(const numeric & a, const numeric & b)
1766 if (a.is_integer() && b.is_integer()) {
1767 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1768 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1769 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1775 /** Numeric integer remainder.
1776 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1777 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1778 * sign of a or is zero.
1780 * @return remainder of a/b if both are integer, 0 otherwise. */
1781 const numeric irem(const numeric & a, const numeric & b)
1783 if (a.is_integer() && b.is_integer())
1784 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1785 cln::the<cln::cl_I>(b.to_cl_N()));
1791 /** Numeric integer remainder.
1792 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1793 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1794 * and irem(a,b) has the sign of a or is zero.
1796 * @return remainder of a/b and quotient stored in q if both are integer,
1798 const numeric irem(const numeric & a, const numeric & b, numeric & q)
1800 if (a.is_integer() && b.is_integer()) {
1801 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1802 cln::the<cln::cl_I>(b.to_cl_N()));
1803 q = rem_quo.quotient;
1804 return rem_quo.remainder;
1812 /** Numeric integer quotient.
1813 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1815 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1816 const numeric iquo(const numeric & a, const numeric & b)
1818 if (a.is_integer() && b.is_integer())
1819 return truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1820 cln::the<cln::cl_I>(b.to_cl_N()));
1826 /** Numeric integer quotient.
1827 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1828 * r == a - iquo(a,b,r)*b.
1830 * @return truncated quotient of a/b and remainder stored in r if both are
1831 * integer, 0 otherwise. */
1832 const numeric iquo(const numeric & a, const numeric & b, numeric & r)
1834 if (a.is_integer() && b.is_integer()) {
1835 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1836 cln::the<cln::cl_I>(b.to_cl_N()));
1837 r = rem_quo.remainder;
1838 return rem_quo.quotient;
1846 /** Greatest Common Divisor.
1848 * @return The GCD of two numbers if both are integer, a numerical 1
1849 * if they are not. */
1850 const numeric gcd(const numeric & a, const numeric & b)
1852 if (a.is_integer() && b.is_integer())
1853 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1854 cln::the<cln::cl_I>(b.to_cl_N()));
1860 /** Least Common Multiple.
1862 * @return The LCM of two numbers if both are integer, the product of those
1863 * two numbers if they are not. */
1864 const numeric lcm(const numeric & a, const numeric & b)
1866 if (a.is_integer() && b.is_integer())
1867 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1868 cln::the<cln::cl_I>(b.to_cl_N()));
1874 /** Numeric square root.
1875 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1876 * should return integer 2.
1878 * @param z numeric argument
1879 * @return square root of z. Branch cut along negative real axis, the negative
1880 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1881 * where imag(z)>0. */
1882 const numeric sqrt(const numeric & z)
1884 return cln::sqrt(z.to_cl_N());
1888 /** Integer numeric square root. */
1889 const numeric isqrt(const numeric & x)
1891 if (x.is_integer()) {
1893 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1900 /** Floating point evaluation of Archimedes' constant Pi. */
1903 return numeric(cln::pi(cln::default_float_format));
1907 /** Floating point evaluation of Euler's constant gamma. */
1910 return numeric(cln::eulerconst(cln::default_float_format));
1914 /** Floating point evaluation of Catalan's constant. */
1915 ex CatalanEvalf(void)
1917 return numeric(cln::catalanconst(cln::default_float_format));
1921 _numeric_digits::_numeric_digits()
1924 // It initializes to 17 digits, because in CLN float_format(17) turns out
1925 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1926 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1929 cln::default_float_format = cln::float_format(17);
1933 /** Assign a native long to global Digits object. */
1934 _numeric_digits& _numeric_digits::operator=(long prec)
1937 cln::default_float_format = cln::float_format(prec);
1942 /** Convert global Digits object to native type long. */
1943 _numeric_digits::operator long()
1945 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1946 return (long)digits;
1950 /** Append global Digits object to ostream. */
1951 void _numeric_digits::print(std::ostream & os) const
1953 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1958 std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
1965 // static member variables
1970 bool _numeric_digits::too_late = false;
1973 /** Accuracy in decimal digits. Only object of this type! Can be set using
1974 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1975 _numeric_digits Digits;
1977 #ifndef NO_NAMESPACE_GINAC
1978 } // namespace GiNaC
1979 #endif // ndef NO_NAMESPACE_GINAC