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) 1999-2000 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 debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
96 numeric::numeric(const numeric & other)
98 debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
102 const numeric & numeric::operator=(const numeric & other)
104 debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
105 if (this != &other) {
114 void numeric::copy(const numeric & other)
120 void numeric::destroy(bool call_parent)
122 if (call_parent) basic::destroy(call_parent);
126 // other constructors
131 numeric::numeric(int i) : basic(TINFO_numeric)
133 debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
134 // Not the whole int-range is available if we don't cast to long
135 // first. This is due to the behaviour of the cl_I-ctor, which
136 // emphasizes efficiency. However, if the integer is small enough,
137 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
138 // using an immediate type:
139 if (cln::cl_immediate_p(i))
140 value = cln::cl_I(i);
142 value = cln::cl_I((long) i);
144 setflag(status_flags::evaluated |
145 status_flags::expanded |
146 status_flags::hash_calculated);
150 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
152 debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
153 // Not the whole uint-range is available if we don't cast to ulong
154 // first. This is due to the behaviour of the cl_I-ctor, which
155 // emphasizes efficiency. However, if the integer is small enough,
156 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
157 // using an immediate type:
158 if (cln::cl_immediate_p(i))
159 value = cln::cl_I(i);
161 value = cln::cl_I((unsigned long) i);
163 setflag(status_flags::evaluated |
164 status_flags::expanded |
165 status_flags::hash_calculated);
169 numeric::numeric(long i) : basic(TINFO_numeric)
171 debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
172 value = cln::cl_I(i);
174 setflag(status_flags::evaluated |
175 status_flags::expanded |
176 status_flags::hash_calculated);
180 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
182 debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
183 value = cln::cl_I(i);
185 setflag(status_flags::evaluated |
186 status_flags::expanded |
187 status_flags::hash_calculated);
190 /** Ctor for rational numerics a/b.
192 * @exception overflow_error (division by zero) */
193 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
195 debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
197 throw std::overflow_error("division by zero");
198 value = cln::cl_I(numer) / cln::cl_I(denom);
200 setflag(status_flags::evaluated |
201 status_flags::expanded |
202 status_flags::hash_calculated);
206 numeric::numeric(double d) : basic(TINFO_numeric)
208 debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
209 // We really want to explicitly use the type cl_LF instead of the
210 // more general cl_F, since that would give us a cl_DF only which
211 // will not be promoted to cl_LF if overflow occurs:
212 value = cln::cl_float(d, cln::default_float_format);
214 setflag(status_flags::evaluated |
215 status_flags::expanded |
216 status_flags::hash_calculated);
219 /** ctor from C-style string. It also accepts complex numbers in GiNaC
220 * notation like "2+5*I". */
221 numeric::numeric(const char *s) : basic(TINFO_numeric)
223 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
224 cln::cl_N ctorval = 0;
225 // parse complex numbers (functional but not completely safe, unfortunately
226 // std::string does not understand regexpese):
227 // ss should represent a simple sum like 2+5*I
229 // make it safe by adding explicit sign
230 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
232 std::string::size_type delim;
234 // chop ss into terms from left to right
236 bool imaginary = false;
237 delim = ss.find_first_of(std::string("+-"),1);
238 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
239 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
240 delim = ss.find_first_of(std::string("+-"),delim+1);
241 term = ss.substr(0,delim);
242 if (delim != std::string::npos)
243 ss = ss.substr(delim);
244 // is the term imaginary?
245 if (term.find("I") != std::string::npos) {
247 term = term.replace(term.find("I"),1,"");
249 if (term.find("*") != std::string::npos)
250 term = term.replace(term.find("*"),1,"");
251 // correct for trivial +/-I without explicit factor on I:
252 if (term.size() == 1)
256 if (term.find(".") != std::string::npos) {
257 // CLN's short type cl_SF is not very useful within the GiNaC
258 // framework where we are mainly interested in the arbitrary
259 // precision type cl_LF. Hence we go straight to the construction
260 // of generic floats. In order to create them we have to convert
261 // our own floating point notation used for output and construction
262 // from char * to CLN's generic notation:
263 // 3.14 --> 3.14e0_<Digits>
264 // 31.4E-1 --> 31.4e-1_<Digits>
266 // No exponent marker? Let's add a trivial one.
267 if (term.find("E") == std::string::npos)
270 term = term.replace(term.find("E"),1,"e");
271 // append _<Digits> to term
272 #if defined(HAVE_SSTREAM)
273 std::ostringstream buf;
274 buf << unsigned(Digits) << std::ends;
275 term += "_" + buf.str();
278 std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
279 term += "_" + string(buf);
281 // construct float using cln::cl_F(const char *) ctor.
283 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
285 ctorval = ctorval + cln::cl_F(term.c_str());
287 // not a floating point number...
289 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
291 ctorval = ctorval + cln::cl_R(term.c_str());
293 } while(delim != std::string::npos);
296 setflag(status_flags::evaluated |
297 status_flags::expanded |
298 status_flags::hash_calculated);
301 /** Ctor from CLN types. This is for the initiated user or internal use
303 numeric::numeric(const cln::cl_N & z) : basic(TINFO_numeric)
305 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
308 setflag(status_flags::evaluated |
309 status_flags::expanded |
310 status_flags::hash_calculated);
317 /** Construct object from archive_node. */
318 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
320 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
321 cln::cl_N ctorval = 0;
323 // Read number as string
325 if (n.find_string("number", str)) {
327 std::istringstream s(str);
329 std::istrstream s(str.c_str(), str.size() + 1);
331 cln::cl_idecoded_float re, im;
335 case 'R': // Integer-decoded real number
336 s >> re.sign >> re.mantissa >> re.exponent;
337 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
339 case 'C': // Integer-decoded complex number
340 s >> re.sign >> re.mantissa >> re.exponent;
341 s >> im.sign >> im.mantissa >> im.exponent;
342 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
343 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
345 default: // Ordinary number
353 setflag(status_flags::evaluated |
354 status_flags::expanded |
355 status_flags::hash_calculated);
358 /** Unarchive the object. */
359 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
361 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
364 /** Archive the object. */
365 void numeric::archive(archive_node &n) const
367 inherited::archive(n);
369 // Write number as string
371 std::ostringstream s;
374 std::ostrstream s(buf, 1024);
376 if (this->is_crational())
377 s << cln::the<cln::cl_N>(value);
379 // Non-rational numbers are written in an integer-decoded format
380 // to preserve the precision
381 if (this->is_real()) {
382 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
384 s << re.sign << " " << re.mantissa << " " << re.exponent;
386 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
387 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
389 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
390 s << im.sign << " " << im.mantissa << " " << im.exponent;
394 n.add_string("number", s.str());
397 std::string str(buf);
398 n.add_string("number", str);
403 // functions overriding virtual functions from bases classes
408 basic * numeric::duplicate() const
410 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
411 return new numeric(*this);
415 /** Helper function to print a real number in a nicer way than is CLN's
416 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
417 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
418 * long as it only uses cl_LF and no other floating point types that we might
419 * want to visibly distinguish from cl_LF.
421 * @see numeric::print() */
422 static void print_real_number(std::ostream & os, const cln::cl_R & num)
424 cln::cl_print_flags ourflags;
425 if (cln::instanceof(num, cln::cl_RA_ring)) {
426 // case 1: integer or rational, nothing special to do:
427 cln::print_real(os, ourflags, num);
430 // make CLN believe this number has default_float_format, so it prints
431 // 'E' as exponent marker instead of 'L':
432 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(num));
433 cln::print_real(os, ourflags, num);
438 /** This method adds to the output so it blends more consistently together
439 * with the other routines and produces something compatible to ginsh input.
441 * @see print_real_number() */
442 void numeric::print(std::ostream & os, unsigned upper_precedence) const
444 debugmsg("numeric print", LOGLEVEL_PRINT);
445 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
446 cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
448 // case 1, real: x or -x
449 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
451 print_real_number(os, r);
454 print_real_number(os, r);
458 // case 2, imaginary: y*I or -y*I
459 if ((precedence<=upper_precedence) && (i < 0)) {
464 print_real_number(os, i);
474 print_real_number(os, i);
480 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
481 if (precedence <= upper_precedence)
483 print_real_number(os, r);
488 print_real_number(os, i);
496 print_real_number(os, i);
500 if (precedence <= upper_precedence)
507 void numeric::printraw(std::ostream & os) const
509 // The method printraw doesn't do much, it simply uses CLN's operator<<()
510 // for output, which is ugly but reliable. e.g: 2+2i
511 debugmsg("numeric printraw", LOGLEVEL_PRINT);
512 os << "numeric(" << cln::the<cln::cl_N>(value) << ")";
516 void numeric::printtree(std::ostream & os, unsigned indent) const
518 debugmsg("numeric printtree", LOGLEVEL_PRINT);
519 os << std::string(indent,' ') << cln::the<cln::cl_N>(value)
521 << "hash=" << hashvalue
522 << " (0x" << std::hex << hashvalue << std::dec << ")"
523 << ", flags=" << flags << std::endl;
527 void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
529 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
530 std::ios::fmtflags oldflags = os.flags();
531 os.setf(std::ios::scientific);
532 if (this->is_rational() && !this->is_integer()) {
533 if (compare(_num0()) > 0) {
535 if (type == csrc_types::ctype_cl_N)
536 os << "cln::cl_F(\"" << numer().evalf() << "\")";
538 os << numer().to_double();
541 if (type == csrc_types::ctype_cl_N)
542 os << "cln::cl_F(\"" << -numer().evalf() << "\")";
544 os << -numer().to_double();
547 if (type == csrc_types::ctype_cl_N)
548 os << "cln::cl_F(\"" << denom().evalf() << "\")";
550 os << denom().to_double();
553 if (type == csrc_types::ctype_cl_N)
554 os << "cln::cl_F(\"" << evalf() << "\")";
562 bool numeric::info(unsigned inf) const
565 case info_flags::numeric:
566 case info_flags::polynomial:
567 case info_flags::rational_function:
569 case info_flags::real:
571 case info_flags::rational:
572 case info_flags::rational_polynomial:
573 return is_rational();
574 case info_flags::crational:
575 case info_flags::crational_polynomial:
576 return is_crational();
577 case info_flags::integer:
578 case info_flags::integer_polynomial:
580 case info_flags::cinteger:
581 case info_flags::cinteger_polynomial:
582 return is_cinteger();
583 case info_flags::positive:
584 return is_positive();
585 case info_flags::negative:
586 return is_negative();
587 case info_flags::nonnegative:
588 return !is_negative();
589 case info_flags::posint:
590 return is_pos_integer();
591 case info_flags::negint:
592 return is_integer() && is_negative();
593 case info_flags::nonnegint:
594 return is_nonneg_integer();
595 case info_flags::even:
597 case info_flags::odd:
599 case info_flags::prime:
601 case info_flags::algebraic:
607 /** Disassemble real part and imaginary part to scan for the occurrence of a
608 * single number. Also handles the imaginary unit. It ignores the sign on
609 * both this and the argument, which may lead to what might appear as funny
610 * results: (2+I).has(-2) -> true. But this is consistent, since we also
611 * would like to have (-2+I).has(2) -> true and we want to think about the
612 * sign as a multiplicative factor. */
613 bool numeric::has(const ex & other) const
615 if (!is_exactly_of_type(*other.bp, numeric))
617 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
618 if (this->is_equal(o) || this->is_equal(-o))
620 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
621 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
622 this->real().is_equal(-o) || this->imag().is_equal(-o));
624 if (o.is_equal(I)) // e.g scan for I in 42*I
625 return !this->is_real();
626 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
627 return (this->real().has(o*I) || this->imag().has(o*I) ||
628 this->real().has(-o*I) || this->imag().has(-o*I));
634 /** Evaluation of numbers doesn't do anything at all. */
635 ex numeric::eval(int level) const
637 // Warning: if this is ever gonna do something, the ex ctors from all kinds
638 // of numbers should be checking for status_flags::evaluated.
643 /** Cast numeric into a floating-point object. For example exact numeric(1) is
644 * returned as a 1.0000000000000000000000 and so on according to how Digits is
645 * currently set. In case the object already was a floating point number the
646 * precision is trimmed to match the currently set default.
648 * @param level ignored, only needed for overriding basic::evalf.
649 * @return an ex-handle to a numeric. */
650 ex numeric::evalf(int level) const
652 // level can safely be discarded for numeric objects.
653 return numeric(cln::cl_float(1.0, cln::default_float_format) *
654 (cln::the<cln::cl_N>(value)));
659 /** Implementation of ex::diff() for a numeric. It always returns 0.
662 ex numeric::derivative(const symbol & s) const
668 int numeric::compare_same_type(const basic & other) const
670 GINAC_ASSERT(is_exactly_of_type(other, numeric));
671 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
673 return this->compare(o);
677 bool numeric::is_equal_same_type(const basic & other) const
679 GINAC_ASSERT(is_exactly_of_type(other,numeric));
680 const numeric *o = static_cast<const numeric *>(&other);
682 return this->is_equal(*o);
686 unsigned numeric::calchash(void) const
688 // Use CLN's hashcode. Warning: It depends only on the number's value, not
689 // its type or precision (i.e. a true equivalence relation on numbers). As
690 // a consequence, 3 and 3.0 share the same hashvalue.
691 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
696 // new virtual functions which can be overridden by derived classes
702 // non-virtual functions in this class
707 /** Numerical addition method. Adds argument to *this and returns result as
708 * a new numeric object. */
709 const numeric numeric::add(const numeric & other) const
711 // Efficiency shortcut: trap the neutral element by pointer.
712 static const numeric * _num0p = &_num0();
715 else if (&other==_num0p)
718 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
722 /** Numerical subtraction method. Subtracts argument from *this and returns
723 * result as a new numeric object. */
724 const numeric numeric::sub(const numeric & other) const
726 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
730 /** Numerical multiplication method. Multiplies *this and argument and returns
731 * result as a new numeric object. */
732 const numeric numeric::mul(const numeric & other) const
734 // Efficiency shortcut: trap the neutral element by pointer.
735 static const numeric * _num1p = &_num1();
738 else if (&other==_num1p)
741 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
745 /** Numerical division method. Divides *this by argument and returns result as
746 * a new numeric object.
748 * @exception overflow_error (division by zero) */
749 const numeric numeric::div(const numeric & other) const
751 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
752 throw std::overflow_error("numeric::div(): division by zero");
753 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
757 const numeric numeric::power(const numeric & other) const
759 // Efficiency shortcut: trap the neutral exponent by pointer.
760 static const numeric * _num1p = &_num1();
764 if (cln::zerop(cln::the<cln::cl_N>(value))) {
765 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
766 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
767 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
768 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
769 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
770 throw std::overflow_error("numeric::eval(): division by zero");
774 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
778 const numeric & numeric::add_dyn(const numeric & other) const
780 // Efficiency shortcut: trap the neutral element by pointer.
781 static const numeric * _num0p = &_num0();
784 else if (&other==_num0p)
787 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
788 setflag(status_flags::dynallocated));
792 const numeric & numeric::sub_dyn(const numeric & other) const
794 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
795 setflag(status_flags::dynallocated));
799 const numeric & numeric::mul_dyn(const numeric & other) const
801 // Efficiency shortcut: trap the neutral element by pointer.
802 static const numeric * _num1p = &_num1();
805 else if (&other==_num1p)
808 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
809 setflag(status_flags::dynallocated));
813 const numeric & numeric::div_dyn(const numeric & other) const
815 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
816 throw std::overflow_error("division by zero");
817 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
818 setflag(status_flags::dynallocated));
822 const numeric & numeric::power_dyn(const numeric & other) const
824 // Efficiency shortcut: trap the neutral exponent by pointer.
825 static const numeric * _num1p=&_num1();
829 if (cln::zerop(cln::the<cln::cl_N>(value))) {
830 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
831 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
832 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
833 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
834 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
835 throw std::overflow_error("numeric::eval(): division by zero");
839 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
840 setflag(status_flags::dynallocated));
844 const numeric & numeric::operator=(int i)
846 return operator=(numeric(i));
850 const numeric & numeric::operator=(unsigned int i)
852 return operator=(numeric(i));
856 const numeric & numeric::operator=(long i)
858 return operator=(numeric(i));
862 const numeric & numeric::operator=(unsigned long i)
864 return operator=(numeric(i));
868 const numeric & numeric::operator=(double d)
870 return operator=(numeric(d));
874 const numeric & numeric::operator=(const char * s)
876 return operator=(numeric(s));
880 /** Inverse of a number. */
881 const numeric numeric::inverse(void) const
883 if (cln::zerop(cln::the<cln::cl_N>(value)))
884 throw std::overflow_error("numeric::inverse(): division by zero");
885 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
889 /** Return the complex half-plane (left or right) in which the number lies.
890 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
891 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
893 * @see numeric::compare(const numeric & other) */
894 int numeric::csgn(void) const
896 if (cln::zerop(cln::the<cln::cl_N>(value)))
898 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
899 if (!cln::zerop(r)) {
905 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
913 /** This method establishes a canonical order on all numbers. For complex
914 * numbers this is not possible in a mathematically consistent way but we need
915 * to establish some order and it ought to be fast. So we simply define it
916 * to be compatible with our method csgn.
918 * @return csgn(*this-other)
919 * @see numeric::csgn(void) */
920 int numeric::compare(const numeric & other) const
922 // Comparing two real numbers?
923 if (cln::instanceof(value, cln::cl_R_ring) &&
924 cln::instanceof(other.value, cln::cl_R_ring))
925 // Yes, so just cln::compare them
926 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
928 // No, first cln::compare real parts...
929 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
932 // ...and then the imaginary parts.
933 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
938 bool numeric::is_equal(const numeric & other) const
940 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
944 /** True if object is zero. */
945 bool numeric::is_zero(void) const
947 return cln::zerop(cln::the<cln::cl_N>(value));
951 /** True if object is not complex and greater than zero. */
952 bool numeric::is_positive(void) const
955 return cln::plusp(cln::the<cln::cl_R>(value));
960 /** True if object is not complex and less than zero. */
961 bool numeric::is_negative(void) const
964 return cln::minusp(cln::the<cln::cl_R>(value));
969 /** True if object is a non-complex integer. */
970 bool numeric::is_integer(void) const
972 return cln::instanceof(value, cln::cl_I_ring);
976 /** True if object is an exact integer greater than zero. */
977 bool numeric::is_pos_integer(void) const
979 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
983 /** True if object is an exact integer greater or equal zero. */
984 bool numeric::is_nonneg_integer(void) const
986 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
990 /** True if object is an exact even integer. */
991 bool numeric::is_even(void) const
993 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
997 /** True if object is an exact odd integer. */
998 bool numeric::is_odd(void) const
1000 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
1004 /** Probabilistic primality test.
1006 * @return true if object is exact integer and prime. */
1007 bool numeric::is_prime(void) const
1009 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
1013 /** True if object is an exact rational number, may even be complex
1014 * (denominator may be unity). */
1015 bool numeric::is_rational(void) const
1017 return cln::instanceof(value, cln::cl_RA_ring);
1021 /** True if object is a real integer, rational or float (but not complex). */
1022 bool numeric::is_real(void) const
1024 return cln::instanceof(value, cln::cl_R_ring);
1028 bool numeric::operator==(const numeric & other) const
1030 return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
1034 bool numeric::operator!=(const numeric & other) const
1036 return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
1040 /** True if object is element of the domain of integers extended by I, i.e. is
1041 * of the form a+b*I, where a and b are integers. */
1042 bool numeric::is_cinteger(void) const
1044 if (cln::instanceof(value, cln::cl_I_ring))
1046 else if (!this->is_real()) { // complex case, handle n+m*I
1047 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
1048 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
1055 /** True if object is an exact rational number, may even be complex
1056 * (denominator may be unity). */
1057 bool numeric::is_crational(void) const
1059 if (cln::instanceof(value, cln::cl_RA_ring))
1061 else if (!this->is_real()) { // complex case, handle Q(i):
1062 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
1063 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
1070 /** Numerical comparison: less.
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: less 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 /** Numerical comparison: greater.
1094 * @exception invalid_argument (complex inequality) */
1095 bool numeric::operator>(const numeric & other) const
1097 if (this->is_real() && other.is_real())
1098 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1099 throw std::invalid_argument("numeric::operator>(): complex inequality");
1103 /** Numerical comparison: greater or equal.
1105 * @exception invalid_argument (complex inequality) */
1106 bool numeric::operator>=(const numeric & other) const
1108 if (this->is_real() && other.is_real())
1109 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1110 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1114 /** Converts numeric types to machine's int. You should check with
1115 * is_integer() if the number is really an integer before calling this method.
1116 * You may also consider checking the range first. */
1117 int numeric::to_int(void) const
1119 GINAC_ASSERT(this->is_integer());
1120 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1124 /** Converts numeric types to machine's long. You should check with
1125 * is_integer() if the number is really an integer before calling this method.
1126 * You may also consider checking the range first. */
1127 long numeric::to_long(void) const
1129 GINAC_ASSERT(this->is_integer());
1130 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1134 /** Converts numeric types to machine's double. You should check with is_real()
1135 * if the number is really not complex before calling this method. */
1136 double numeric::to_double(void) const
1138 GINAC_ASSERT(this->is_real());
1139 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1143 /** Real part of a number. */
1144 const numeric numeric::real(void) const
1146 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1150 /** Imaginary part of a number. */
1151 const numeric numeric::imag(void) const
1153 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1157 /** Numerator. Computes the numerator of rational numbers, rationalized
1158 * numerator of complex if real and imaginary part are both rational numbers
1159 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1161 const numeric numeric::numer(void) const
1163 if (this->is_integer())
1164 return numeric(*this);
1166 else if (cln::instanceof(value, cln::cl_RA_ring))
1167 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1169 else if (!this->is_real()) { // complex case, handle Q(i):
1170 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1171 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1172 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1173 return numeric(*this);
1174 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1175 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1176 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1177 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1178 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1179 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1180 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1181 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1184 // at least one float encountered
1185 return numeric(*this);
1189 /** Denominator. Computes the denominator of rational numbers, common integer
1190 * denominator of complex if real and imaginary part are both rational numbers
1191 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1192 const numeric numeric::denom(void) const
1194 if (this->is_integer())
1197 if (instanceof(value, cln::cl_RA_ring))
1198 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1200 if (!this->is_real()) { // complex case, handle Q(i):
1201 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1202 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1203 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1205 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1206 return numeric(cln::denominator(i));
1207 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1208 return numeric(cln::denominator(r));
1209 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1210 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1212 // at least one float encountered
1217 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1218 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1219 * 2^(n-1) <= x < 2^n.
1221 * @return number of bits (excluding sign) needed to represent that number
1222 * in two's complement if it is an integer, 0 otherwise. */
1223 int numeric::int_length(void) const
1225 if (this->is_integer())
1226 return cln::integer_length(cln::the<cln::cl_I>(value));
1232 /** Returns a new CLN object of type cl_N, representing the value of *this.
1233 * This method is useful for casting when mixing GiNaC and CLN in one project.
1235 numeric::operator cln::cl_N() const
1237 return cln::cl_N(cln::the<cln::cl_N>(value));
1242 // static member variables
1247 unsigned numeric::precedence = 30;
1253 const numeric some_numeric;
1254 const std::type_info & typeid_numeric = typeid(some_numeric);
1255 /** Imaginary unit. This is not a constant but a numeric since we are
1256 * natively handing complex numbers anyways. */
1257 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1260 /** Exponential function.
1262 * @return arbitrary precision numerical exp(x). */
1263 const numeric exp(const numeric & x)
1265 return cln::exp(cln::cl_N(x));
1269 /** Natural logarithm.
1271 * @param z complex number
1272 * @return arbitrary precision numerical log(x).
1273 * @exception pole_error("log(): logarithmic pole",0) */
1274 const numeric log(const numeric & z)
1277 throw pole_error("log(): logarithmic pole",0);
1278 return cln::log(cln::cl_N(z));
1282 /** Numeric sine (trigonometric function).
1284 * @return arbitrary precision numerical sin(x). */
1285 const numeric sin(const numeric & x)
1287 return cln::sin(cln::cl_N(x));
1291 /** Numeric cosine (trigonometric function).
1293 * @return arbitrary precision numerical cos(x). */
1294 const numeric cos(const numeric & x)
1296 return cln::cos(cln::cl_N(x));
1300 /** Numeric tangent (trigonometric function).
1302 * @return arbitrary precision numerical tan(x). */
1303 const numeric tan(const numeric & x)
1305 return cln::tan(cln::cl_N(x));
1309 /** Numeric inverse sine (trigonometric function).
1311 * @return arbitrary precision numerical asin(x). */
1312 const numeric asin(const numeric & x)
1314 return cln::asin(cln::cl_N(x));
1318 /** Numeric inverse cosine (trigonometric function).
1320 * @return arbitrary precision numerical acos(x). */
1321 const numeric acos(const numeric & x)
1323 return cln::acos(cln::cl_N(x));
1329 * @param z complex number
1331 * @exception pole_error("atan(): logarithmic pole",0) */
1332 const numeric atan(const numeric & x)
1335 x.real().is_zero() &&
1336 abs(x.imag()).is_equal(_num1()))
1337 throw pole_error("atan(): logarithmic pole",0);
1338 return cln::atan(cln::cl_N(x));
1344 * @param x real number
1345 * @param y real number
1346 * @return atan(y/x) */
1347 const numeric atan(const numeric & y, const numeric & x)
1349 if (x.is_real() && y.is_real())
1350 return cln::atan(cln::the<cln::cl_R>(cln::cl_N(x)),
1351 cln::the<cln::cl_R>(cln::cl_N(y)));
1353 throw std::invalid_argument("atan(): complex argument");
1357 /** Numeric hyperbolic sine (trigonometric function).
1359 * @return arbitrary precision numerical sinh(x). */
1360 const numeric sinh(const numeric & x)
1362 return cln::sinh(cln::cl_N(x));
1366 /** Numeric hyperbolic cosine (trigonometric function).
1368 * @return arbitrary precision numerical cosh(x). */
1369 const numeric cosh(const numeric & x)
1371 return cln::cosh(cln::cl_N(x));
1375 /** Numeric hyperbolic tangent (trigonometric function).
1377 * @return arbitrary precision numerical tanh(x). */
1378 const numeric tanh(const numeric & x)
1380 return cln::tanh(cln::cl_N(x));
1384 /** Numeric inverse hyperbolic sine (trigonometric function).
1386 * @return arbitrary precision numerical asinh(x). */
1387 const numeric asinh(const numeric & x)
1389 return cln::asinh(cln::cl_N(x));
1393 /** Numeric inverse hyperbolic cosine (trigonometric function).
1395 * @return arbitrary precision numerical acosh(x). */
1396 const numeric acosh(const numeric & x)
1398 return cln::acosh(cln::cl_N(x));
1402 /** Numeric inverse hyperbolic tangent (trigonometric function).
1404 * @return arbitrary precision numerical atanh(x). */
1405 const numeric atanh(const numeric & x)
1407 return cln::atanh(cln::cl_N(x));
1411 /*static cln::cl_N Li2_series(const ::cl_N & x,
1412 const ::float_format_t & prec)
1414 // Note: argument must be in the unit circle
1415 // This is very inefficient unless we have fast floating point Bernoulli
1416 // numbers implemented!
1417 cln::cl_N c1 = -cln::log(1-x);
1419 // hard-wire the first two Bernoulli numbers
1420 cln::cl_N acc = c1 - cln::square(c1)/4;
1422 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1423 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1425 c1 = cln::square(c1);
1429 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1430 // 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));
1433 } while (acc != acc+aug);
1437 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1438 * circle) using a power series. */
1439 static cln::cl_N Li2_series(const cln::cl_N & x,
1440 const cln::float_format_t & prec)
1442 // Note: argument must be in the unit circle
1444 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1449 den = den + i; // 1, 4, 9, 16, ...
1453 } while (acc != acc+aug);
1457 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1458 static cln::cl_N Li2_projection(const cln::cl_N & x,
1459 const cln::float_format_t & prec)
1461 const cln::cl_R re = cln::realpart(x);
1462 const cln::cl_R im = cln::imagpart(x);
1463 if (re > cln::cl_F(".5"))
1464 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1466 - Li2_series(1-x, prec)
1467 - cln::log(x)*cln::log(1-x));
1468 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1469 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1470 return(- cln::square(cln::log(1-x))/2
1471 - Li2_series(x/(x-1), prec));
1472 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1473 // Li2(x^2)/2 - Li2(-x)
1474 return(Li2_projection(cln::square(x), prec)/2
1475 - Li2_projection(-x, prec));
1476 return Li2_series(x, prec);
1479 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1480 * the branch cut lies along the positive real axis, starting at 1 and
1481 * continuous with quadrant IV.
1483 * @return arbitrary precision numerical Li2(x). */
1484 const numeric Li2(const numeric & x)
1489 // what is the desired float format?
1490 // first guess: default format
1491 cln::float_format_t prec = cln::default_float_format;
1492 const cln::cl_N value = cln::cl_N(x);
1493 // second guess: the argument's format
1494 if (!x.real().is_rational())
1495 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1496 else if (!x.imag().is_rational())
1497 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1499 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1500 return cln::zeta(2, prec);
1502 if (cln::abs(value) > 1)
1503 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1504 return(- cln::square(cln::log(-value))/2
1505 - cln::zeta(2, prec)
1506 - Li2_projection(cln::recip(value), prec));
1508 return Li2_projection(cln::cl_N(x), prec);
1512 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1513 * integer arguments. */
1514 const numeric zeta(const numeric & x)
1516 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1517 // only knows about integer arguments and zeta(3).evalf() automatically
1518 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1519 // being an exact zero for CLN, which can be tested and then we can just
1520 // pass the number casted to an int:
1522 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(cln::cl_N(x))));
1523 if (cln::zerop(cln::cl_N(x)-aux))
1524 return cln::zeta(aux);
1526 std::clog << "zeta(" << x
1527 << "): Does anybody know good way to calculate this numerically?"
1533 /** The Gamma function.
1534 * This is only a stub! */
1535 const numeric lgamma(const numeric & x)
1537 std::clog << "lgamma(" << x
1538 << "): Does anybody know good way to calculate this numerically?"
1542 const numeric tgamma(const numeric & x)
1544 std::clog << "tgamma(" << x
1545 << "): Does anybody know good way to calculate this numerically?"
1551 /** The psi function (aka polygamma function).
1552 * This is only a stub! */
1553 const numeric psi(const numeric & x)
1555 std::clog << "psi(" << x
1556 << "): Does anybody know good way to calculate this numerically?"
1562 /** The psi functions (aka polygamma functions).
1563 * This is only a stub! */
1564 const numeric psi(const numeric & n, const numeric & x)
1566 std::clog << "psi(" << n << "," << x
1567 << "): Does anybody know good way to calculate this numerically?"
1573 /** Factorial combinatorial function.
1575 * @param n integer argument >= 0
1576 * @exception range_error (argument must be integer >= 0) */
1577 const numeric factorial(const numeric & n)
1579 if (!n.is_nonneg_integer())
1580 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1581 return numeric(cln::factorial(n.to_int()));
1585 /** The double factorial combinatorial function. (Scarcely used, but still
1586 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1588 * @param n integer argument >= -1
1589 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1590 * @exception range_error (argument must be integer >= -1) */
1591 const numeric doublefactorial(const numeric & n)
1593 if (n == numeric(-1))
1596 if (!n.is_nonneg_integer())
1597 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1599 return numeric(cln::doublefactorial(n.to_int()));
1603 /** The Binomial coefficients. It computes the binomial coefficients. For
1604 * integer n and k and positive n this is the number of ways of choosing k
1605 * objects from n distinct objects. If n is negative, the formula
1606 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1607 const numeric binomial(const numeric & n, const numeric & k)
1609 if (n.is_integer() && k.is_integer()) {
1610 if (n.is_nonneg_integer()) {
1611 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1612 return numeric(cln::binomial(n.to_int(),k.to_int()));
1616 return _num_1().power(k)*binomial(k-n-_num1(),k);
1620 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1621 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1625 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1626 * in the expansion of the function x/(e^x-1).
1628 * @return the nth Bernoulli number (a rational number).
1629 * @exception range_error (argument must be integer >= 0) */
1630 const numeric bernoulli(const numeric & nn)
1632 if (!nn.is_integer() || nn.is_negative())
1633 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1637 // The Bernoulli numbers are rational numbers that may be computed using
1640 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1642 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1643 // previous ones, the computation is necessarily very expensive. There are
1644 // several other ways of computing them, a particularly good one being
1648 // for (unsigned i=0; i<n; i++) {
1649 // c = exquo(c*(i-n),(i+2));
1650 // Bern = Bern + c*s/(i+2);
1651 // s = s + expt_pos(cl_I(i+2),n);
1655 // But if somebody works with the n'th Bernoulli number she is likely to
1656 // also need all previous Bernoulli numbers. So we need a complete remember
1657 // table and above divide and conquer algorithm is not suited to build one
1658 // up. The code below is adapted from Pari's function bernvec().
1660 // (There is an interesting relation with the tangent polynomials described
1661 // in `Concrete Mathematics', which leads to a program twice as fast as our
1662 // implementation below, but it requires storing one such polynomial in
1663 // addition to the remember table. This doubles the memory footprint so
1664 // we don't use it.)
1666 // the special cases not covered by the algorithm below
1667 if (nn.is_equal(_num1()))
1672 // store nonvanishing Bernoulli numbers here
1673 static std::vector< cln::cl_RA > results;
1674 static int highest_result = 0;
1675 // algorithm not applicable to B(0), so just store it
1676 if (results.size()==0)
1677 results.push_back(cln::cl_RA(1));
1679 int n = nn.to_long();
1680 for (int i=highest_result; i<n/2; ++i) {
1686 for (int j=i; j>0; --j) {
1687 B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
1693 B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
1694 results.push_back(B);
1697 return results[n/2];
1701 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1702 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1704 * @param n an integer
1705 * @return the nth Fibonacci number F(n) (an integer number)
1706 * @exception range_error (argument must be an integer) */
1707 const numeric fibonacci(const numeric & n)
1709 if (!n.is_integer())
1710 throw std::range_error("numeric::fibonacci(): argument must be integer");
1713 // The following addition formula holds:
1715 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1717 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1718 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1720 // Replace m by m+1:
1721 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1722 // Now put in m = n, to get
1723 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1724 // F(2n+1) = F(n)^2 + F(n+1)^2
1726 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1729 if (n.is_negative())
1731 return -fibonacci(-n);
1733 return fibonacci(-n);
1737 cln::cl_I m = cln::the<cln::cl_I>(cln::cl_N(n)) >> 1L; // floor(n/2);
1738 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1739 // Since a squaring is cheaper than a multiplication, better use
1740 // three squarings instead of one multiplication and two squarings.
1741 cln::cl_I u2 = cln::square(u);
1742 cln::cl_I v2 = cln::square(v);
1743 if (cln::logbitp(bit-1, m)) {
1744 v = cln::square(u + v) - u2;
1747 u = v2 - cln::square(v - u);
1752 // Here we don't use the squaring formula because one multiplication
1753 // is cheaper than two squarings.
1754 return u * ((v << 1) - u);
1756 return cln::square(u) + cln::square(v);
1760 /** Absolute value. */
1761 const numeric abs(const numeric& x)
1763 return cln::abs(cln::cl_N(x));
1767 /** Modulus (in positive representation).
1768 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1769 * sign of a or is zero. This is different from Maple's modp, where the sign
1770 * of b is ignored. It is in agreement with Mathematica's Mod.
1772 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1773 * integer, 0 otherwise. */
1774 const numeric mod(const numeric & a, const numeric & b)
1776 if (a.is_integer() && b.is_integer())
1777 return cln::mod(cln::the<cln::cl_I>(cln::cl_N(a)),
1778 cln::the<cln::cl_I>(cln::cl_N(b)));
1784 /** Modulus (in symmetric representation).
1785 * Equivalent to Maple's mods.
1787 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1788 const numeric smod(const numeric & a, const numeric & b)
1790 if (a.is_integer() && b.is_integer()) {
1791 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(cln::cl_N(b)) >> 1) - 1;
1792 return cln::mod(cln::the<cln::cl_I>(cln::cl_N(a)) + b2,
1793 cln::the<cln::cl_I>(cln::cl_N(b))) - b2;
1799 /** Numeric integer remainder.
1800 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1801 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1802 * sign of a or is zero.
1804 * @return remainder of a/b if both are integer, 0 otherwise. */
1805 const numeric irem(const numeric & a, const numeric & b)
1807 if (a.is_integer() && b.is_integer())
1808 return cln::rem(cln::the<cln::cl_I>(cln::cl_N(a)),
1809 cln::the<cln::cl_I>(cln::cl_N(b)));
1815 /** Numeric integer remainder.
1816 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1817 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1818 * and irem(a,b) has the sign of a or is zero.
1820 * @return remainder of a/b and quotient stored in q if both are integer,
1822 const numeric irem(const numeric & a, const numeric & b, numeric & q)
1824 if (a.is_integer() && b.is_integer()) {
1825 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(cln::cl_N(a)),
1826 cln::the<cln::cl_I>(cln::cl_N(b)));
1827 q = rem_quo.quotient;
1828 return rem_quo.remainder;
1836 /** Numeric integer quotient.
1837 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1839 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1840 const numeric iquo(const numeric & a, const numeric & b)
1842 if (a.is_integer() && b.is_integer())
1843 return truncate1(cln::the<cln::cl_I>(cln::cl_N(a)),
1844 cln::the<cln::cl_I>(cln::cl_N(b)));
1850 /** Numeric integer quotient.
1851 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1852 * r == a - iquo(a,b,r)*b.
1854 * @return truncated quotient of a/b and remainder stored in r if both are
1855 * integer, 0 otherwise. */
1856 const numeric iquo(const numeric & a, const numeric & b, numeric & r)
1858 if (a.is_integer() && b.is_integer()) {
1859 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(cln::cl_N(a)),
1860 cln::the<cln::cl_I>(cln::cl_N(b)));
1861 r = rem_quo.remainder;
1862 return rem_quo.quotient;
1870 /** Greatest Common Divisor.
1872 * @return The GCD of two numbers if both are integer, a numerical 1
1873 * if they are not. */
1874 const numeric gcd(const numeric & a, const numeric & b)
1876 if (a.is_integer() && b.is_integer())
1877 return cln::gcd(cln::the<cln::cl_I>(cln::cl_N(a)),
1878 cln::the<cln::cl_I>(cln::cl_N(b)));
1884 /** Least Common Multiple.
1886 * @return The LCM of two numbers if both are integer, the product of those
1887 * two numbers if they are not. */
1888 const numeric lcm(const numeric & a, const numeric & b)
1890 if (a.is_integer() && b.is_integer())
1891 return cln::lcm(cln::the<cln::cl_I>(cln::cl_N(a)),
1892 cln::the<cln::cl_I>(cln::cl_N(b)));
1898 /** Numeric square root.
1899 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1900 * should return integer 2.
1902 * @param z numeric argument
1903 * @return square root of z. Branch cut along negative real axis, the negative
1904 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1905 * where imag(z)>0. */
1906 const numeric sqrt(const numeric & z)
1908 return cln::sqrt(cln::cl_N(z));
1912 /** Integer numeric square root. */
1913 const numeric isqrt(const numeric & x)
1915 if (x.is_integer()) {
1917 cln::isqrt(cln::the<cln::cl_I>(cln::cl_N(x)), &root);
1924 /** Floating point evaluation of Archimedes' constant Pi. */
1927 return numeric(cln::pi(cln::default_float_format));
1931 /** Floating point evaluation of Euler's constant gamma. */
1934 return numeric(cln::eulerconst(cln::default_float_format));
1938 /** Floating point evaluation of Catalan's constant. */
1939 ex CatalanEvalf(void)
1941 return numeric(cln::catalanconst(cln::default_float_format));
1945 _numeric_digits::_numeric_digits()
1948 // It initializes to 17 digits, because in CLN float_format(17) turns out
1949 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1950 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1953 cln::default_float_format = cln::float_format(17);
1957 /** Assign a native long to global Digits object. */
1958 _numeric_digits& _numeric_digits::operator=(long prec)
1961 cln::default_float_format = cln::float_format(prec);
1966 /** Convert global Digits object to native type long. */
1967 _numeric_digits::operator long()
1969 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1970 return (long)digits;
1974 /** Append global Digits object to ostream. */
1975 void _numeric_digits::print(std::ostream & os) const
1977 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1982 std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
1989 // static member variables
1994 bool _numeric_digits::too_late = false;
1997 /** Accuracy in decimal digits. Only object of this type! Can be set using
1998 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1999 _numeric_digits Digits;
2001 #ifndef NO_NAMESPACE_GINAC
2002 } // namespace GiNaC
2003 #endif // ndef NO_NAMESPACE_GINAC