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 not pollute the global namespace, hence we include it here
48 // instead of in some header file where it would propagate to other parts.
49 // Also, we only need a subset of CLN, so we don't include the complete cln.h:
51 #include <cln/cl_output.h>
52 #include <cln/cl_integer_io.h>
53 #include <cln/cl_integer_ring.h>
54 #include <cln/cl_rational_io.h>
55 #include <cln/cl_rational_ring.h>
56 #include <cln/cl_lfloat_class.h>
57 #include <cln/cl_lfloat_io.h>
58 #include <cln/cl_real_io.h>
59 #include <cln/cl_real_ring.h>
60 #include <cln/cl_complex_io.h>
61 #include <cln/cl_complex_ring.h>
62 #include <cln/cl_numtheory.h>
63 #else // def HAVE_CLN_CLN_H
64 #include <cl_output.h>
65 #include <cl_integer_io.h>
66 #include <cl_integer_ring.h>
67 #include <cl_rational_io.h>
68 #include <cl_rational_ring.h>
69 #include <cl_lfloat_class.h>
70 #include <cl_lfloat_io.h>
71 #include <cl_real_io.h>
72 #include <cl_real_ring.h>
73 #include <cl_complex_io.h>
74 #include <cl_complex_ring.h>
75 #include <cl_numtheory.h>
76 #endif // def HAVE_CLN_CLN_H
78 #ifndef NO_NAMESPACE_GINAC
80 #endif // ndef NO_NAMESPACE_GINAC
82 // linker has no problems finding text symbols for numerator or denominator
85 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
88 // default constructor, destructor, copy constructor assignment
89 // operator and helpers
94 /** default ctor. Numerically it initializes to an integer zero. */
95 numeric::numeric() : basic(TINFO_numeric)
97 debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
101 setflag(status_flags::evaluated |
102 status_flags::expanded |
103 status_flags::hash_calculated);
108 debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
112 numeric::numeric(const numeric & other)
114 debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
118 const numeric & numeric::operator=(const numeric & other)
120 debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
121 if (this != &other) {
130 void numeric::copy(const numeric & other)
133 value = new ::cl_N(*other.value);
136 void numeric::destroy(bool call_parent)
139 if (call_parent) basic::destroy(call_parent);
143 // other constructors
148 numeric::numeric(int i) : basic(TINFO_numeric)
150 debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
151 // Not the whole int-range is available if we don't cast to long
152 // first. This is due to the behaviour of the cl_I-ctor, which
153 // emphasizes efficiency:
154 value = new ::cl_I((long) i);
156 setflag(status_flags::evaluated |
157 status_flags::expanded |
158 status_flags::hash_calculated);
162 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
164 debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
165 // Not the whole uint-range is available if we don't cast to ulong
166 // first. This is due to the behaviour of the cl_I-ctor, which
167 // emphasizes efficiency:
168 value = new ::cl_I((unsigned long)i);
170 setflag(status_flags::evaluated |
171 status_flags::expanded |
172 status_flags::hash_calculated);
176 numeric::numeric(long i) : basic(TINFO_numeric)
178 debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
179 value = new ::cl_I(i);
181 setflag(status_flags::evaluated |
182 status_flags::expanded |
183 status_flags::hash_calculated);
187 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
189 debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
190 value = new ::cl_I(i);
192 setflag(status_flags::evaluated |
193 status_flags::expanded |
194 status_flags::hash_calculated);
197 /** Ctor for rational numerics a/b.
199 * @exception overflow_error (division by zero) */
200 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
202 debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
204 throw std::overflow_error("division by zero");
205 value = new ::cl_I(numer);
206 *value = *value / ::cl_I(denom);
208 setflag(status_flags::evaluated |
209 status_flags::expanded |
210 status_flags::hash_calculated);
214 numeric::numeric(double d) : basic(TINFO_numeric)
216 debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
217 // We really want to explicitly use the type cl_LF instead of the
218 // more general cl_F, since that would give us a cl_DF only which
219 // will not be promoted to cl_LF if overflow occurs:
221 *value = cl_float(d, cl_default_float_format);
223 setflag(status_flags::evaluated |
224 status_flags::expanded |
225 status_flags::hash_calculated);
228 /** ctor from C-style string. It also accepts complex numbers in GiNaC
229 * notation like "2+5*I". */
230 numeric::numeric(const char *s) : basic(TINFO_numeric)
232 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
233 value = new ::cl_N(0);
234 // parse complex numbers (functional but not completely safe, unfortunately
235 // std::string does not understand regexpese):
236 // ss should represent a simple sum like 2+5*I
238 // make it safe by adding explicit sign
239 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
241 std::string::size_type delim;
243 // chop ss into terms from left to right
245 bool imaginary = false;
246 delim = ss.find_first_of(std::string("+-"),1);
247 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
248 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
249 delim = ss.find_first_of(std::string("+-"),delim+1);
250 term = ss.substr(0,delim);
251 if (delim != std::string::npos)
252 ss = ss.substr(delim);
253 // is the term imaginary?
254 if (term.find("I") != std::string::npos) {
256 term = term.replace(term.find("I"),1,"");
258 if (term.find("*") != std::string::npos)
259 term = term.replace(term.find("*"),1,"");
260 // correct for trivial +/-I without explicit factor on I:
261 if (term.size() == 1)
265 const char *cs = term.c_str();
266 // CLN's short types are not useful within the GiNaC framework, hence
267 // we go straight to the construction of a long float. Simply using
268 // cl_N(s) would require us to use add a CLN exponent mark, otherwise
269 // we would not be save from over-/underflows.
272 *value = *value + ::complex(cl_I(0),::cl_LF(cs));
274 *value = *value + ::cl_LF(cs);
277 *value = *value + ::complex(cl_I(0),::cl_R(cs));
279 *value = *value + ::cl_R(cs);
280 } while(delim != std::string::npos);
282 setflag(status_flags::evaluated |
283 status_flags::expanded |
284 status_flags::hash_calculated);
287 /** Ctor from CLN types. This is for the initiated user or internal use
289 numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
291 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
292 value = new ::cl_N(z);
294 setflag(status_flags::evaluated |
295 status_flags::expanded |
296 status_flags::hash_calculated);
303 /** Construct object from archive_node. */
304 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
306 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
309 // Read number as string
311 if (n.find_string("number", str)) {
313 std::istringstream s(str);
315 std::istrstream s(str.c_str(), str.size() + 1);
317 ::cl_idecoded_float re, im;
321 case 'R': // Integer-decoded real number
322 s >> re.sign >> re.mantissa >> re.exponent;
323 *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
325 case 'C': // Integer-decoded complex number
326 s >> re.sign >> re.mantissa >> re.exponent;
327 s >> im.sign >> im.mantissa >> im.exponent;
328 *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
329 im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
331 default: // Ordinary number
338 setflag(status_flags::evaluated |
339 status_flags::expanded |
340 status_flags::hash_calculated);
343 /** Unarchive the object. */
344 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
346 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
349 /** Archive the object. */
350 void numeric::archive(archive_node &n) const
352 inherited::archive(n);
354 // Write number as string
356 std::ostringstream s;
359 std::ostrstream s(buf, 1024);
361 if (this->is_crational())
364 // Non-rational numbers are written in an integer-decoded format
365 // to preserve the precision
366 if (this->is_real()) {
367 cl_idecoded_float re = integer_decode_float(The(::cl_F)(*value));
369 s << re.sign << " " << re.mantissa << " " << re.exponent;
371 cl_idecoded_float re = integer_decode_float(The(::cl_F)(::realpart(*value)));
372 cl_idecoded_float im = integer_decode_float(The(::cl_F)(::imagpart(*value)));
374 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
375 s << im.sign << " " << im.mantissa << " " << im.exponent;
379 n.add_string("number", s.str());
382 std::string str(buf);
383 n.add_string("number", str);
388 // functions overriding virtual functions from bases classes
393 basic * numeric::duplicate() const
395 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
396 return new numeric(*this);
400 /** Helper function to print a real number in a nicer way than is CLN's
401 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
402 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
403 * long as it only uses cl_LF and no other floating point types.
405 * @see numeric::print() */
406 static void print_real_number(std::ostream & os, const cl_R & num)
408 cl_print_flags ourflags;
409 if (::instanceof(num, ::cl_RA_ring)) {
410 // case 1: integer or rational, nothing special to do:
411 ::print_real(os, ourflags, num);
414 // make CLN believe this number has default_float_format, so it prints
415 // 'E' as exponent marker instead of 'L':
416 ourflags.default_float_format = ::cl_float_format(The(::cl_F)(num));
417 ::print_real(os, ourflags, num);
422 /** This method adds to the output so it blends more consistently together
423 * with the other routines and produces something compatible to ginsh input.
425 * @see print_real_number() */
426 void numeric::print(std::ostream & os, unsigned upper_precedence) const
428 debugmsg("numeric print", LOGLEVEL_PRINT);
429 if (this->is_real()) {
430 // case 1, real: x or -x
431 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
433 print_real_number(os, The(::cl_R)(*value));
436 print_real_number(os, The(::cl_R)(*value));
439 // case 2, imaginary: y*I or -y*I
440 if (::realpart(*value) == 0) {
441 if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
442 if (::imagpart(*value) == -1) {
446 print_real_number(os, The(::cl_R)(::imagpart(*value)));
450 if (::imagpart(*value) == 1) {
453 if (::imagpart (*value) == -1) {
456 print_real_number(os, The(::cl_R)(::imagpart(*value)));
462 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
463 if (precedence <= upper_precedence)
465 print_real_number(os, The(::cl_R)(::realpart(*value)));
466 if (::imagpart(*value) < 0) {
467 if (::imagpart(*value) == -1) {
470 print_real_number(os, The(::cl_R)(::imagpart(*value)));
474 if (::imagpart(*value) == 1) {
478 print_real_number(os, The(::cl_R)(::imagpart(*value)));
482 if (precedence <= upper_precedence)
489 void numeric::printraw(std::ostream & os) const
491 // The method printraw doesn't do much, it simply uses CLN's operator<<()
492 // for output, which is ugly but reliable. e.g: 2+2i
493 debugmsg("numeric printraw", LOGLEVEL_PRINT);
494 os << "numeric(" << *value << ")";
498 void numeric::printtree(std::ostream & os, unsigned indent) const
500 debugmsg("numeric printtree", LOGLEVEL_PRINT);
501 os << std::string(indent,' ') << *value
503 << "hash=" << hashvalue
504 << " (0x" << std::hex << hashvalue << std::dec << ")"
505 << ", flags=" << flags << std::endl;
509 void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
511 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
512 ios::fmtflags oldflags = os.flags();
513 os.setf(ios::scientific);
514 if (this->is_rational() && !this->is_integer()) {
515 if (compare(_num0()) > 0) {
517 if (type == csrc_types::ctype_cl_N)
518 os << "cl_F(\"" << numer().evalf() << "\")";
520 os << numer().to_double();
523 if (type == csrc_types::ctype_cl_N)
524 os << "cl_F(\"" << -numer().evalf() << "\")";
526 os << -numer().to_double();
529 if (type == csrc_types::ctype_cl_N)
530 os << "cl_F(\"" << denom().evalf() << "\")";
532 os << denom().to_double();
535 if (type == csrc_types::ctype_cl_N)
536 os << "cl_F(\"" << evalf() << "\")";
544 bool numeric::info(unsigned inf) const
547 case info_flags::numeric:
548 case info_flags::polynomial:
549 case info_flags::rational_function:
551 case info_flags::real:
553 case info_flags::rational:
554 case info_flags::rational_polynomial:
555 return is_rational();
556 case info_flags::crational:
557 case info_flags::crational_polynomial:
558 return is_crational();
559 case info_flags::integer:
560 case info_flags::integer_polynomial:
562 case info_flags::cinteger:
563 case info_flags::cinteger_polynomial:
564 return is_cinteger();
565 case info_flags::positive:
566 return is_positive();
567 case info_flags::negative:
568 return is_negative();
569 case info_flags::nonnegative:
570 return !is_negative();
571 case info_flags::posint:
572 return is_pos_integer();
573 case info_flags::negint:
574 return is_integer() && is_negative();
575 case info_flags::nonnegint:
576 return is_nonneg_integer();
577 case info_flags::even:
579 case info_flags::odd:
581 case info_flags::prime:
583 case info_flags::algebraic:
589 /** Disassemble real part and imaginary part to scan for the occurrence of a
590 * single number. Also handles the imaginary unit. It ignores the sign on
591 * both this and the argument, which may lead to what might appear as funny
592 * results: (2+I).has(-2) -> true. But this is consistent, since we also
593 * would like to have (-2+I).has(2) -> true and we want to think about the
594 * sign as a multiplicative factor. */
595 bool numeric::has(const ex & other) const
597 if (!is_exactly_of_type(*other.bp, numeric))
599 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
600 if (this->is_equal(o) || this->is_equal(-o))
602 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
603 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
604 this->real().is_equal(-o) || this->imag().is_equal(-o));
606 if (o.is_equal(I)) // e.g scan for I in 42*I
607 return !this->is_real();
608 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
609 return (this->real().has(o*I) || this->imag().has(o*I) ||
610 this->real().has(-o*I) || this->imag().has(-o*I));
616 /** Evaluation of numbers doesn't do anything at all. */
617 ex numeric::eval(int level) const
619 // Warning: if this is ever gonna do something, the ex ctors from all kinds
620 // of numbers should be checking for status_flags::evaluated.
625 /** Cast numeric into a floating-point object. For example exact numeric(1) is
626 * returned as a 1.0000000000000000000000 and so on according to how Digits is
627 * currently set. In case the object already was a floating point number the
628 * precision is trimmed to match the currently set default.
630 * @param level ignored, only needed for overriding basic::evalf.
631 * @return an ex-handle to a numeric. */
632 ex numeric::evalf(int level) const
634 // level can safely be discarded for numeric objects.
635 return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
640 /** Implementation of ex::diff() for a numeric. It always returns 0.
643 ex numeric::derivative(const symbol & s) const
649 int numeric::compare_same_type(const basic & other) const
651 GINAC_ASSERT(is_exactly_of_type(other, numeric));
652 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
654 if (*value == *o.value) {
662 bool numeric::is_equal_same_type(const basic & other) const
664 GINAC_ASSERT(is_exactly_of_type(other,numeric));
665 const numeric *o = static_cast<const numeric *>(&other);
667 return this->is_equal(*o);
671 unsigned numeric::calchash(void) const
673 // Use CLN's hashcode. Warning: It depends only on the number's value, not
674 // its type or precision (i.e. a true equivalence relation on numbers). As
675 // a consequence, 3 and 3.0 share the same hashvalue.
676 return (hashvalue = cl_equal_hashcode(*value) | 0x80000000U);
681 // new virtual functions which can be overridden by derived classes
687 // non-virtual functions in this class
692 /** Numerical addition method. Adds argument to *this and returns result as
693 * a new numeric object. */
694 numeric numeric::add(const numeric & other) const
696 return numeric((*value)+(*other.value));
699 /** Numerical subtraction method. Subtracts argument from *this and returns
700 * result as a new numeric object. */
701 numeric numeric::sub(const numeric & other) const
703 return numeric((*value)-(*other.value));
706 /** Numerical multiplication method. Multiplies *this and argument and returns
707 * result as a new numeric object. */
708 numeric numeric::mul(const numeric & other) const
710 static const numeric * _num1p=&_num1();
713 } else if (&other==_num1p) {
716 return numeric((*value)*(*other.value));
719 /** Numerical division method. Divides *this by argument and returns result as
720 * a new numeric object.
722 * @exception overflow_error (division by zero) */
723 numeric numeric::div(const numeric & other) const
725 if (::zerop(*other.value))
726 throw std::overflow_error("numeric::div(): division by zero");
727 return numeric((*value)/(*other.value));
730 numeric numeric::power(const numeric & other) const
732 static const numeric * _num1p = &_num1();
735 if (::zerop(*value)) {
736 if (::zerop(*other.value))
737 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
738 else if (::zerop(::realpart(*other.value)))
739 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
740 else if (::minusp(::realpart(*other.value)))
741 throw std::overflow_error("numeric::eval(): division by zero");
745 return numeric(::expt(*value,*other.value));
748 /** Inverse of a number. */
749 numeric numeric::inverse(void) const
752 throw std::overflow_error("numeric::inverse(): division by zero");
753 return numeric(::recip(*value)); // -> CLN
756 const numeric & numeric::add_dyn(const numeric & other) const
758 return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
759 setflag(status_flags::dynallocated));
762 const numeric & numeric::sub_dyn(const numeric & other) const
764 return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
765 setflag(status_flags::dynallocated));
768 const numeric & numeric::mul_dyn(const numeric & other) const
770 static const numeric * _num1p=&_num1();
773 } else if (&other==_num1p) {
776 return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
777 setflag(status_flags::dynallocated));
780 const numeric & numeric::div_dyn(const numeric & other) const
782 if (::zerop(*other.value))
783 throw std::overflow_error("division by zero");
784 return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
785 setflag(status_flags::dynallocated));
788 const numeric & numeric::power_dyn(const numeric & other) const
790 static const numeric * _num1p=&_num1();
793 if (::zerop(*value)) {
794 if (::zerop(*other.value))
795 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
796 else if (::zerop(::realpart(*other.value)))
797 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
798 else if (::minusp(::realpart(*other.value)))
799 throw std::overflow_error("numeric::eval(): division by zero");
803 return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
804 setflag(status_flags::dynallocated));
807 const numeric & numeric::operator=(int i)
809 return operator=(numeric(i));
812 const numeric & numeric::operator=(unsigned int i)
814 return operator=(numeric(i));
817 const numeric & numeric::operator=(long i)
819 return operator=(numeric(i));
822 const numeric & numeric::operator=(unsigned long i)
824 return operator=(numeric(i));
827 const numeric & numeric::operator=(double d)
829 return operator=(numeric(d));
832 const numeric & numeric::operator=(const char * s)
834 return operator=(numeric(s));
837 /** Return the complex half-plane (left or right) in which the number lies.
838 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
839 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
841 * @see numeric::compare(const numeric & other) */
842 int numeric::csgn(void) const
846 if (!::zerop(::realpart(*value))) {
847 if (::plusp(::realpart(*value)))
852 if (::plusp(::imagpart(*value)))
859 /** This method establishes a canonical order on all numbers. For complex
860 * numbers this is not possible in a mathematically consistent way but we need
861 * to establish some order and it ought to be fast. So we simply define it
862 * to be compatible with our method csgn.
864 * @return csgn(*this-other)
865 * @see numeric::csgn(void) */
866 int numeric::compare(const numeric & other) const
868 // Comparing two real numbers?
869 if (this->is_real() && other.is_real())
870 // Yes, just compare them
871 return ::cl_compare(The(::cl_R)(*value), The(::cl_R)(*other.value));
873 // No, first compare real parts
874 cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
878 return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
882 bool numeric::is_equal(const numeric & other) const
884 return (*value == *other.value);
887 /** True if object is zero. */
888 bool numeric::is_zero(void) const
890 return ::zerop(*value); // -> CLN
893 /** True if object is not complex and greater than zero. */
894 bool numeric::is_positive(void) const
897 return ::plusp(The(::cl_R)(*value)); // -> CLN
901 /** True if object is not complex and less than zero. */
902 bool numeric::is_negative(void) const
905 return ::minusp(The(::cl_R)(*value)); // -> CLN
909 /** True if object is a non-complex integer. */
910 bool numeric::is_integer(void) const
912 return ::instanceof(*value, ::cl_I_ring); // -> CLN
915 /** True if object is an exact integer greater than zero. */
916 bool numeric::is_pos_integer(void) const
918 return (this->is_integer() && ::plusp(The(::cl_I)(*value))); // -> CLN
921 /** True if object is an exact integer greater or equal zero. */
922 bool numeric::is_nonneg_integer(void) const
924 return (this->is_integer() && !::minusp(The(::cl_I)(*value))); // -> CLN
927 /** True if object is an exact even integer. */
928 bool numeric::is_even(void) const
930 return (this->is_integer() && ::evenp(The(::cl_I)(*value))); // -> CLN
933 /** True if object is an exact odd integer. */
934 bool numeric::is_odd(void) const
936 return (this->is_integer() && ::oddp(The(::cl_I)(*value))); // -> CLN
939 /** Probabilistic primality test.
941 * @return true if object is exact integer and prime. */
942 bool numeric::is_prime(void) const
944 return (this->is_integer() && ::isprobprime(The(::cl_I)(*value))); // -> CLN
947 /** True if object is an exact rational number, may even be complex
948 * (denominator may be unity). */
949 bool numeric::is_rational(void) const
951 return ::instanceof(*value, ::cl_RA_ring); // -> CLN
954 /** True if object is a real integer, rational or float (but not complex). */
955 bool numeric::is_real(void) const
957 return ::instanceof(*value, ::cl_R_ring); // -> CLN
960 bool numeric::operator==(const numeric & other) const
962 return (*value == *other.value); // -> CLN
965 bool numeric::operator!=(const numeric & other) const
967 return (*value != *other.value); // -> CLN
970 /** True if object is element of the domain of integers extended by I, i.e. is
971 * of the form a+b*I, where a and b are integers. */
972 bool numeric::is_cinteger(void) const
974 if (::instanceof(*value, ::cl_I_ring))
976 else if (!this->is_real()) { // complex case, handle n+m*I
977 if (::instanceof(::realpart(*value), ::cl_I_ring) &&
978 ::instanceof(::imagpart(*value), ::cl_I_ring))
984 /** True if object is an exact rational number, may even be complex
985 * (denominator may be unity). */
986 bool numeric::is_crational(void) const
988 if (::instanceof(*value, ::cl_RA_ring))
990 else if (!this->is_real()) { // complex case, handle Q(i):
991 if (::instanceof(::realpart(*value), ::cl_RA_ring) &&
992 ::instanceof(::imagpart(*value), ::cl_RA_ring))
998 /** Numerical comparison: less.
1000 * @exception invalid_argument (complex inequality) */
1001 bool numeric::operator<(const numeric & other) const
1003 if (this->is_real() && other.is_real())
1004 return (The(::cl_R)(*value) < The(::cl_R)(*other.value)); // -> CLN
1005 throw std::invalid_argument("numeric::operator<(): complex inequality");
1008 /** Numerical comparison: less or equal.
1010 * @exception invalid_argument (complex inequality) */
1011 bool numeric::operator<=(const numeric & other) const
1013 if (this->is_real() && other.is_real())
1014 return (The(::cl_R)(*value) <= The(::cl_R)(*other.value)); // -> CLN
1015 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1016 return false; // make compiler shut up
1019 /** Numerical comparison: greater.
1021 * @exception invalid_argument (complex inequality) */
1022 bool numeric::operator>(const numeric & other) const
1024 if (this->is_real() && other.is_real())
1025 return (The(::cl_R)(*value) > The(::cl_R)(*other.value)); // -> CLN
1026 throw std::invalid_argument("numeric::operator>(): complex inequality");
1029 /** Numerical comparison: greater or equal.
1031 * @exception invalid_argument (complex inequality) */
1032 bool numeric::operator>=(const numeric & other) const
1034 if (this->is_real() && other.is_real())
1035 return (The(::cl_R)(*value) >= The(::cl_R)(*other.value)); // -> CLN
1036 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1039 /** Converts numeric types to machine's int. You should check with
1040 * is_integer() if the number is really an integer before calling this method.
1041 * You may also consider checking the range first. */
1042 int numeric::to_int(void) const
1044 GINAC_ASSERT(this->is_integer());
1045 return ::cl_I_to_int(The(::cl_I)(*value)); // -> CLN
1048 /** Converts numeric types to machine's long. You should check with
1049 * is_integer() if the number is really an integer before calling this method.
1050 * You may also consider checking the range first. */
1051 long numeric::to_long(void) const
1053 GINAC_ASSERT(this->is_integer());
1054 return ::cl_I_to_long(The(::cl_I)(*value)); // -> CLN
1057 /** Converts numeric types to machine's double. You should check with is_real()
1058 * if the number is really not complex before calling this method. */
1059 double numeric::to_double(void) const
1061 GINAC_ASSERT(this->is_real());
1062 return ::cl_double_approx(::realpart(*value)); // -> CLN
1065 /** Real part of a number. */
1066 const numeric numeric::real(void) const
1068 return numeric(::realpart(*value)); // -> CLN
1071 /** Imaginary part of a number. */
1072 const numeric numeric::imag(void) const
1074 return numeric(::imagpart(*value)); // -> CLN
1078 // Unfortunately, CLN did not provide an official way to access the numerator
1079 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
1080 // one finds how it works internally in src/rational/cl_RA.h:
1081 struct cl_heap_ratio : cl_heap {
1086 inline cl_heap_ratio* TheRatio (const cl_N& obj)
1087 { return (cl_heap_ratio*)(obj.pointer); }
1088 #endif // ndef SANE_LINKER
1090 /** Numerator. Computes the numerator of rational numbers, rationalized
1091 * numerator of complex if real and imaginary part are both rational numbers
1092 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1094 const numeric numeric::numer(void) const
1096 if (this->is_integer()) {
1097 return numeric(*this);
1100 else if (::instanceof(*value, ::cl_RA_ring)) {
1101 return numeric(::numerator(The(::cl_RA)(*value)));
1103 else if (!this->is_real()) { // complex case, handle Q(i):
1104 cl_R r = ::realpart(*value);
1105 cl_R i = ::imagpart(*value);
1106 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1107 return numeric(*this);
1108 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1109 return numeric(::complex(r*::denominator(The(::cl_RA)(i)), ::numerator(The(::cl_RA)(i))));
1110 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1111 return numeric(::complex(::numerator(The(::cl_RA)(r)), i*::denominator(The(::cl_RA)(r))));
1112 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring)) {
1113 cl_I s = ::lcm(::denominator(The(::cl_RA)(r)), ::denominator(The(::cl_RA)(i)));
1114 return numeric(::complex(::numerator(The(::cl_RA)(r))*(exquo(s,::denominator(The(::cl_RA)(r)))),
1115 ::numerator(The(::cl_RA)(i))*(exquo(s,::denominator(The(::cl_RA)(i))))));
1119 else if (instanceof(*value, ::cl_RA_ring)) {
1120 return numeric(TheRatio(*value)->numerator);
1122 else if (!this->is_real()) { // complex case, handle Q(i):
1123 cl_R r = ::realpart(*value);
1124 cl_R i = ::imagpart(*value);
1125 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1126 return numeric(*this);
1127 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1128 return numeric(::complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
1129 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1130 return numeric(::complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
1131 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring)) {
1132 cl_I s = ::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
1133 return numeric(::complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
1134 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
1137 #endif // def SANE_LINKER
1138 // at least one float encountered
1139 return numeric(*this);
1142 /** Denominator. Computes the denominator of rational numbers, common integer
1143 * denominator of complex if real and imaginary part are both rational numbers
1144 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1145 const numeric numeric::denom(void) const
1147 if (this->is_integer()) {
1151 if (instanceof(*value, ::cl_RA_ring)) {
1152 return numeric(::denominator(The(::cl_RA)(*value)));
1154 if (!this->is_real()) { // complex case, handle Q(i):
1155 cl_R r = ::realpart(*value);
1156 cl_R i = ::imagpart(*value);
1157 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1159 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1160 return numeric(::denominator(The(::cl_RA)(i)));
1161 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1162 return numeric(::denominator(The(::cl_RA)(r)));
1163 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring))
1164 return numeric(::lcm(::denominator(The(::cl_RA)(r)), ::denominator(The(::cl_RA)(i))));
1167 if (instanceof(*value, ::cl_RA_ring)) {
1168 return numeric(TheRatio(*value)->denominator);
1170 if (!this->is_real()) { // complex case, handle Q(i):
1171 cl_R r = ::realpart(*value);
1172 cl_R i = ::imagpart(*value);
1173 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1175 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1176 return numeric(TheRatio(i)->denominator);
1177 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1178 return numeric(TheRatio(r)->denominator);
1179 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring))
1180 return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
1182 #endif // def SANE_LINKER
1183 // at least one float encountered
1187 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1188 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1189 * 2^(n-1) <= x < 2^n.
1191 * @return number of bits (excluding sign) needed to represent that number
1192 * in two's complement if it is an integer, 0 otherwise. */
1193 int numeric::int_length(void) const
1195 if (this->is_integer())
1196 return ::integer_length(The(::cl_I)(*value)); // -> CLN
1203 // static member variables
1208 unsigned numeric::precedence = 30;
1214 const numeric some_numeric;
1215 const type_info & typeid_numeric=typeid(some_numeric);
1216 /** Imaginary unit. This is not a constant but a numeric since we are
1217 * natively handing complex numbers anyways. */
1218 const numeric I = numeric(::complex(cl_I(0),cl_I(1)));
1221 /** Exponential function.
1223 * @return arbitrary precision numerical exp(x). */
1224 const numeric exp(const numeric & x)
1226 return ::exp(*x.value); // -> CLN
1230 /** Natural logarithm.
1232 * @param z complex number
1233 * @return arbitrary precision numerical log(x).
1234 * @exception pole_error("log(): logarithmic pole",0) */
1235 const numeric log(const numeric & z)
1238 throw pole_error("log(): logarithmic pole",0);
1239 return ::log(*z.value); // -> CLN
1243 /** Numeric sine (trigonometric function).
1245 * @return arbitrary precision numerical sin(x). */
1246 const numeric sin(const numeric & x)
1248 return ::sin(*x.value); // -> CLN
1252 /** Numeric cosine (trigonometric function).
1254 * @return arbitrary precision numerical cos(x). */
1255 const numeric cos(const numeric & x)
1257 return ::cos(*x.value); // -> CLN
1261 /** Numeric tangent (trigonometric function).
1263 * @return arbitrary precision numerical tan(x). */
1264 const numeric tan(const numeric & x)
1266 return ::tan(*x.value); // -> CLN
1270 /** Numeric inverse sine (trigonometric function).
1272 * @return arbitrary precision numerical asin(x). */
1273 const numeric asin(const numeric & x)
1275 return ::asin(*x.value); // -> CLN
1279 /** Numeric inverse cosine (trigonometric function).
1281 * @return arbitrary precision numerical acos(x). */
1282 const numeric acos(const numeric & x)
1284 return ::acos(*x.value); // -> CLN
1290 * @param z complex number
1292 * @exception pole_error("atan(): logarithmic pole",0) */
1293 const numeric atan(const numeric & x)
1296 x.real().is_zero() &&
1297 abs(x.imag()).is_equal(_num1()))
1298 throw pole_error("atan(): logarithmic pole",0);
1299 return ::atan(*x.value); // -> CLN
1305 * @param x real number
1306 * @param y real number
1307 * @return atan(y/x) */
1308 const numeric atan(const numeric & y, const numeric & x)
1310 if (x.is_real() && y.is_real())
1311 return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
1313 throw std::invalid_argument("atan(): complex argument");
1317 /** Numeric hyperbolic sine (trigonometric function).
1319 * @return arbitrary precision numerical sinh(x). */
1320 const numeric sinh(const numeric & x)
1322 return ::sinh(*x.value); // -> CLN
1326 /** Numeric hyperbolic cosine (trigonometric function).
1328 * @return arbitrary precision numerical cosh(x). */
1329 const numeric cosh(const numeric & x)
1331 return ::cosh(*x.value); // -> CLN
1335 /** Numeric hyperbolic tangent (trigonometric function).
1337 * @return arbitrary precision numerical tanh(x). */
1338 const numeric tanh(const numeric & x)
1340 return ::tanh(*x.value); // -> CLN
1344 /** Numeric inverse hyperbolic sine (trigonometric function).
1346 * @return arbitrary precision numerical asinh(x). */
1347 const numeric asinh(const numeric & x)
1349 return ::asinh(*x.value); // -> CLN
1353 /** Numeric inverse hyperbolic cosine (trigonometric function).
1355 * @return arbitrary precision numerical acosh(x). */
1356 const numeric acosh(const numeric & x)
1358 return ::acosh(*x.value); // -> CLN
1362 /** Numeric inverse hyperbolic tangent (trigonometric function).
1364 * @return arbitrary precision numerical atanh(x). */
1365 const numeric atanh(const numeric & x)
1367 return ::atanh(*x.value); // -> CLN
1371 /*static ::cl_N Li2_series(const ::cl_N & x,
1372 const ::cl_float_format_t & prec)
1374 // Note: argument must be in the unit circle
1375 // This is very inefficient unless we have fast floating point Bernoulli
1376 // numbers implemented!
1377 ::cl_N c1 = -::log(1-x);
1379 // hard-wire the first two Bernoulli numbers
1380 ::cl_N acc = c1 - ::square(c1)/4;
1382 ::cl_F pisq = ::square(::cl_pi(prec)); // pi^2
1383 ::cl_F piac = ::cl_float(1, prec); // accumulator: pi^(2*i)
1389 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / ::factorial(2*i+1);
1390 // aug = c2 * ::cl_I(i%2 ? 1 : -1) / ::cl_I(2*i+1) * ::cl_zeta(2*i, prec) / piac / (::cl_I(1)<<(2*i-1));
1393 } while (acc != acc+aug);
1397 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1398 * circle) using a power series. */
1399 static ::cl_N Li2_series(const ::cl_N & x,
1400 const ::cl_float_format_t & prec)
1402 // Note: argument must be in the unit circle
1404 ::cl_N num = ::complex(::cl_float(1, prec), 0);
1409 den = den + i; // 1, 4, 9, 16, ...
1413 } while (acc != acc+aug);
1417 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1418 static ::cl_N Li2_projection(const ::cl_N & x,
1419 const ::cl_float_format_t & prec)
1421 const ::cl_R re = ::realpart(x);
1422 const ::cl_R im = ::imagpart(x);
1423 if (re > ::cl_F(".5"))
1424 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1426 - Li2_series(1-x, prec)
1427 - ::log(x)*::log(1-x));
1428 if ((re <= 0 && ::abs(im) > ::cl_F(".75")) || (re < ::cl_F("-.5")))
1429 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1430 return(- ::square(::log(1-x))/2
1431 - Li2_series(x/(x-1), prec));
1432 if (re > 0 && ::abs(im) > ::cl_LF(".75"))
1433 // Li2(x^2)/2 - Li2(-x)
1434 return(Li2_projection(::square(x), prec)/2
1435 - Li2_projection(-x, prec));
1436 return Li2_series(x, prec);
1439 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1440 * the branch cut lies along the positive real axis, starting at 1 and
1441 * continuous with quadrant IV.
1443 * @return arbitrary precision numerical Li2(x). */
1444 const numeric Li2(const numeric & x)
1446 if (::zerop(*x.value))
1449 // what is the desired float format?
1450 // first guess: default format
1451 ::cl_float_format_t prec = ::cl_default_float_format;
1452 // second guess: the argument's format
1453 if (!::instanceof(::realpart(*x.value),cl_RA_ring))
1454 prec = ::cl_float_format(The(::cl_F)(::realpart(*x.value)));
1455 else if (!::instanceof(::imagpart(*x.value),cl_RA_ring))
1456 prec = ::cl_float_format(The(::cl_F)(::imagpart(*x.value)));
1458 if (*x.value==1) // may cause trouble with log(1-x)
1459 return ::cl_zeta(2, prec);
1461 if (::abs(*x.value) > 1)
1462 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1463 return(- ::square(::log(-*x.value))/2
1464 - ::cl_zeta(2, prec)
1465 - Li2_projection(::recip(*x.value), prec));
1467 return Li2_projection(*x.value, prec);
1471 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1472 * integer arguments. */
1473 const numeric zeta(const numeric & x)
1475 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1476 // only knows about integer arguments and zeta(3).evalf() automatically
1477 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1478 // being an exact zero for CLN, which can be tested and then we can just
1479 // pass the number casted to an int:
1481 int aux = (int)(::cl_double_approx(::realpart(*x.value)));
1482 if (::zerop(*x.value-aux))
1483 return ::cl_zeta(aux); // -> CLN
1485 std::clog << "zeta(" << x
1486 << "): Does anybody know good way to calculate this numerically?"
1492 /** The Gamma function.
1493 * This is only a stub! */
1494 const numeric lgamma(const numeric & x)
1496 std::clog << "lgamma(" << x
1497 << "): Does anybody know good way to calculate this numerically?"
1501 const numeric tgamma(const numeric & x)
1503 std::clog << "tgamma(" << x
1504 << "): Does anybody know good way to calculate this numerically?"
1510 /** The psi function (aka polygamma function).
1511 * This is only a stub! */
1512 const numeric psi(const numeric & x)
1514 std::clog << "psi(" << x
1515 << "): Does anybody know good way to calculate this numerically?"
1521 /** The psi functions (aka polygamma functions).
1522 * This is only a stub! */
1523 const numeric psi(const numeric & n, const numeric & x)
1525 std::clog << "psi(" << n << "," << x
1526 << "): Does anybody know good way to calculate this numerically?"
1532 /** Factorial combinatorial function.
1534 * @param n integer argument >= 0
1535 * @exception range_error (argument must be integer >= 0) */
1536 const numeric factorial(const numeric & n)
1538 if (!n.is_nonneg_integer())
1539 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1540 return numeric(::factorial(n.to_int())); // -> CLN
1544 /** The double factorial combinatorial function. (Scarcely used, but still
1545 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1547 * @param n integer argument >= -1
1548 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1549 * @exception range_error (argument must be integer >= -1) */
1550 const numeric doublefactorial(const numeric & n)
1552 if (n == numeric(-1)) {
1555 if (!n.is_nonneg_integer()) {
1556 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1558 return numeric(::doublefactorial(n.to_int())); // -> CLN
1562 /** The Binomial coefficients. It computes the binomial coefficients. For
1563 * integer n and k and positive n this is the number of ways of choosing k
1564 * objects from n distinct objects. If n is negative, the formula
1565 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1566 const numeric binomial(const numeric & n, const numeric & k)
1568 if (n.is_integer() && k.is_integer()) {
1569 if (n.is_nonneg_integer()) {
1570 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1571 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1575 return _num_1().power(k)*binomial(k-n-_num1(),k);
1579 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1580 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1584 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1585 * in the expansion of the function x/(e^x-1).
1587 * @return the nth Bernoulli number (a rational number).
1588 * @exception range_error (argument must be integer >= 0) */
1589 const numeric bernoulli(const numeric & nn)
1591 if (!nn.is_integer() || nn.is_negative())
1592 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1596 // The Bernoulli numbers are rational numbers that may be computed using
1599 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1601 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1602 // previous ones, the computation is necessarily very expensive. There are
1603 // several other ways of computing them, a particularly good one being
1607 // for (unsigned i=0; i<n; i++) {
1608 // c = exquo(c*(i-n),(i+2));
1609 // Bern = Bern + c*s/(i+2);
1610 // s = s + expt_pos(cl_I(i+2),n);
1614 // But if somebody works with the n'th Bernoulli number she is likely to
1615 // also need all previous Bernoulli numbers. So we need a complete remember
1616 // table and above divide and conquer algorithm is not suited to build one
1617 // up. The code below is adapted from Pari's function bernvec().
1619 // (There is an interesting relation with the tangent polynomials described
1620 // in `Concrete Mathematics', which leads to a program twice as fast as our
1621 // implementation below, but it requires storing one such polynomial in
1622 // addition to the remember table. This doubles the memory footprint so
1623 // we don't use it.)
1625 // the special cases not covered by the algorithm below
1626 if (nn.is_equal(_num1()))
1631 // store nonvanishing Bernoulli numbers here
1632 static std::vector< ::cl_RA > results;
1633 static int highest_result = 0;
1634 // algorithm not applicable to B(0), so just store it
1635 if (results.size()==0)
1636 results.push_back(::cl_RA(1));
1638 int n = nn.to_long();
1639 for (int i=highest_result; i<n/2; ++i) {
1645 for (int j=i; j>0; --j) {
1646 B = ::cl_I(n*m) * (B+results[j]) / (d1*d2);
1652 B = (1 - ((B+1)/(2*i+3))) / (::cl_I(1)<<(2*i+2));
1653 results.push_back(B);
1656 return results[n/2];
1660 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1661 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1663 * @param n an integer
1664 * @return the nth Fibonacci number F(n) (an integer number)
1665 * @exception range_error (argument must be an integer) */
1666 const numeric fibonacci(const numeric & n)
1668 if (!n.is_integer())
1669 throw std::range_error("numeric::fibonacci(): argument must be integer");
1672 // This is based on an implementation that can be found in CLN's example
1673 // directory. There, it is done recursively, which may be more elegant
1674 // than our non-recursive implementation that has to resort to some bit-
1675 // fiddling. This is, however, a matter of taste.
1676 // The following addition formula holds:
1678 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1680 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1681 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1683 // Replace m by m+1:
1684 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1685 // Now put in m = n, to get
1686 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1687 // F(2n+1) = F(n)^2 + F(n+1)^2
1689 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1692 if (n.is_negative())
1694 return -fibonacci(-n);
1696 return fibonacci(-n);
1700 ::cl_I m = The(::cl_I)(*n.value) >> 1L; // floor(n/2);
1701 for (uintL bit=::integer_length(m); bit>0; --bit) {
1702 // Since a squaring is cheaper than a multiplication, better use
1703 // three squarings instead of one multiplication and two squarings.
1704 ::cl_I u2 = ::square(u);
1705 ::cl_I v2 = ::square(v);
1706 if (::logbitp(bit-1, m)) {
1707 v = ::square(u + v) - u2;
1710 u = v2 - ::square(v - u);
1715 // Here we don't use the squaring formula because one multiplication
1716 // is cheaper than two squarings.
1717 return u * ((v << 1) - u);
1719 return ::square(u) + ::square(v);
1723 /** Absolute value. */
1724 numeric abs(const numeric & x)
1726 return ::abs(*x.value); // -> CLN
1730 /** Modulus (in positive representation).
1731 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1732 * sign of a or is zero. This is different from Maple's modp, where the sign
1733 * of b is ignored. It is in agreement with Mathematica's Mod.
1735 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1736 * integer, 0 otherwise. */
1737 numeric mod(const numeric & a, const numeric & b)
1739 if (a.is_integer() && b.is_integer())
1740 return ::mod(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1742 return _num0(); // Throw?
1746 /** Modulus (in symmetric representation).
1747 * Equivalent to Maple's mods.
1749 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1750 numeric smod(const numeric & a, const numeric & b)
1752 if (a.is_integer() && b.is_integer()) {
1753 cl_I b2 = The(::cl_I)(ceiling1(The(::cl_I)(*b.value) >> 1)) - 1;
1754 return ::mod(The(::cl_I)(*a.value) + b2, The(::cl_I)(*b.value)) - b2;
1756 return _num0(); // Throw?
1760 /** Numeric integer remainder.
1761 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1762 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1763 * sign of a or is zero.
1765 * @return remainder of a/b if both are integer, 0 otherwise. */
1766 numeric irem(const numeric & a, const numeric & b)
1768 if (a.is_integer() && b.is_integer())
1769 return ::rem(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1771 return _num0(); // Throw?
1775 /** Numeric integer remainder.
1776 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1777 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1778 * and irem(a,b) has the sign of a or is zero.
1780 * @return remainder of a/b and quotient stored in q if both are integer,
1782 numeric irem(const numeric & a, const numeric & b, numeric & q)
1784 if (a.is_integer() && b.is_integer()) { // -> CLN
1785 cl_I_div_t rem_quo = truncate2(The(::cl_I)(*a.value), The(::cl_I)(*b.value));
1786 q = rem_quo.quotient;
1787 return rem_quo.remainder;
1790 return _num0(); // Throw?
1795 /** Numeric integer quotient.
1796 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1798 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1799 numeric iquo(const numeric & a, const numeric & b)
1801 if (a.is_integer() && b.is_integer())
1802 return truncate1(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1804 return _num0(); // Throw?
1808 /** Numeric integer quotient.
1809 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1810 * r == a - iquo(a,b,r)*b.
1812 * @return truncated quotient of a/b and remainder stored in r if both are
1813 * integer, 0 otherwise. */
1814 numeric iquo(const numeric & a, const numeric & b, numeric & r)
1816 if (a.is_integer() && b.is_integer()) { // -> CLN
1817 cl_I_div_t rem_quo = truncate2(The(::cl_I)(*a.value), The(::cl_I)(*b.value));
1818 r = rem_quo.remainder;
1819 return rem_quo.quotient;
1822 return _num0(); // Throw?
1827 /** Numeric square root.
1828 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1829 * should return integer 2.
1831 * @param z numeric argument
1832 * @return square root of z. Branch cut along negative real axis, the negative
1833 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1834 * where imag(z)>0. */
1835 numeric sqrt(const numeric & z)
1837 return ::sqrt(*z.value); // -> CLN
1841 /** Integer numeric square root. */
1842 numeric isqrt(const numeric & x)
1844 if (x.is_integer()) {
1846 ::isqrt(The(::cl_I)(*x.value), &root); // -> CLN
1849 return _num0(); // Throw?
1853 /** Greatest Common Divisor.
1855 * @return The GCD of two numbers if both are integer, a numerical 1
1856 * if they are not. */
1857 numeric gcd(const numeric & a, const numeric & b)
1859 if (a.is_integer() && b.is_integer())
1860 return ::gcd(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1866 /** Least Common Multiple.
1868 * @return The LCM of two numbers if both are integer, the product of those
1869 * two numbers if they are not. */
1870 numeric lcm(const numeric & a, const numeric & b)
1872 if (a.is_integer() && b.is_integer())
1873 return ::lcm(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1875 return *a.value * *b.value;
1879 /** Floating point evaluation of Archimedes' constant Pi. */
1882 return numeric(::cl_pi(cl_default_float_format)); // -> CLN
1886 /** Floating point evaluation of Euler's constant gamma. */
1889 return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
1893 /** Floating point evaluation of Catalan's constant. */
1894 ex CatalanEvalf(void)
1896 return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
1900 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1901 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1902 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1903 _numeric_digits::_numeric_digits()
1908 cl_default_float_format = ::cl_float_format(17);
1912 _numeric_digits& _numeric_digits::operator=(long prec)
1915 cl_default_float_format = ::cl_float_format(prec);
1920 _numeric_digits::operator long()
1922 return (long)digits;
1926 void _numeric_digits::print(std::ostream & os) const
1928 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1933 std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
1940 // static member variables
1945 bool _numeric_digits::too_late = false;
1948 /** Accuracy in decimal digits. Only object of this type! Can be set using
1949 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1950 _numeric_digits Digits;
1952 #ifndef NO_NAMESPACE_GINAC
1953 } // namespace GiNaC
1954 #endif // ndef NO_NAMESPACE_GINAC