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);
229 /** ctor from C-style string. It also accepts complex numbers in GiNaC
230 * notation like "2+5*I". */
231 numeric::numeric(const char *s) : basic(TINFO_numeric)
233 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
234 value = new ::cl_N(0);
235 // parse complex numbers (functional but not completely safe, unfortunately
236 // std::string does not understand regexpese):
237 // ss should represent a simple sum like 2+5*I
239 // make it safe by adding explicit sign
240 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
242 std::string::size_type delim;
244 // chop ss into terms from left to right
246 bool imaginary = false;
247 delim = ss.find_first_of(std::string("+-"),1);
248 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
249 if (delim != std::string::npos &&
250 ss.at(delim-1) == 'E')
251 delim = ss.find_first_of(std::string("+-"),delim+1);
252 term = ss.substr(0,delim);
253 if (delim != std::string::npos)
254 ss = ss.substr(delim);
255 // is the term imaginary?
256 if (term.find("I") != std::string::npos) {
258 term = term.replace(term.find("I"),1,"");
260 if (term.find("*") != std::string::npos)
261 term = term.replace(term.find("*"),1,"");
262 // correct for trivial +/-I without explicit factor on I:
263 if (term.size() == 1)
267 const char *cs = term.c_str();
268 // CLN's short types are not useful within the GiNaC framework, hence
269 // we go straight to the construction of a long float. Simply using
270 // cl_N(s) would require us to use add a CLN exponent mark, otherwise
271 // we would not be save from over-/underflows.
274 *value = *value + ::complex(cl_I(0),::cl_LF(cs));
276 *value = *value + ::cl_LF(cs);
279 *value = *value + ::complex(cl_I(0),::cl_R(cs));
281 *value = *value + ::cl_R(cs);
282 } while(delim != std::string::npos);
284 setflag(status_flags::evaluated|
285 status_flags::hash_calculated);
288 /** Ctor from CLN types. This is for the initiated user or internal use
290 numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
292 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
293 value = new ::cl_N(z);
295 setflag(status_flags::evaluated |
296 status_flags::expanded |
297 status_flags::hash_calculated);
304 /** Construct object from archive_node. */
305 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
307 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
310 // Read number as string
312 if (n.find_string("number", str)) {
314 std::istringstream s(str);
316 std::istrstream s(str.c_str(), str.size() + 1);
318 ::cl_idecoded_float re, im;
322 case 'R': // Integer-decoded real number
323 s >> re.sign >> re.mantissa >> re.exponent;
324 *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
326 case 'C': // Integer-decoded complex number
327 s >> re.sign >> re.mantissa >> re.exponent;
328 s >> im.sign >> im.mantissa >> im.exponent;
329 *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
330 im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
332 default: // Ordinary number
339 setflag(status_flags::evaluated |
340 status_flags::expanded |
341 status_flags::hash_calculated);
344 /** Unarchive the object. */
345 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
347 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
350 /** Archive the object. */
351 void numeric::archive(archive_node &n) const
353 inherited::archive(n);
355 // Write number as string
357 std::ostringstream s;
360 std::ostrstream s(buf, 1024);
362 if (this->is_crational())
365 // Non-rational numbers are written in an integer-decoded format
366 // to preserve the precision
367 if (this->is_real()) {
368 cl_idecoded_float re = integer_decode_float(The(::cl_F)(*value));
370 s << re.sign << " " << re.mantissa << " " << re.exponent;
372 cl_idecoded_float re = integer_decode_float(The(::cl_F)(::realpart(*value)));
373 cl_idecoded_float im = integer_decode_float(The(::cl_F)(::imagpart(*value)));
375 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
376 s << im.sign << " " << im.mantissa << " " << im.exponent;
380 n.add_string("number", s.str());
383 std::string str(buf);
384 n.add_string("number", str);
389 // functions overriding virtual functions from bases classes
394 basic * numeric::duplicate() const
396 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
397 return new numeric(*this);
401 /** Helper function to print a real number in a nicer way than is CLN's
402 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
403 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
404 * long as it only uses cl_LF and no other floating point types.
406 * @see numeric::print() */
407 static void print_real_number(std::ostream & os, const cl_R & num)
409 cl_print_flags ourflags;
410 if (::instanceof(num, ::cl_RA_ring)) {
411 // case 1: integer or rational, nothing special to do:
412 ::print_real(os, ourflags, num);
415 // make CLN believe this number has default_float_format, so it prints
416 // 'E' as exponent marker instead of 'L':
417 ourflags.default_float_format = ::cl_float_format(The(::cl_F)(num));
418 ::print_real(os, ourflags, num);
423 /** This method adds to the output so it blends more consistently together
424 * with the other routines and produces something compatible to ginsh input.
426 * @see print_real_number() */
427 void numeric::print(std::ostream & os, unsigned upper_precedence) const
429 debugmsg("numeric print", LOGLEVEL_PRINT);
430 if (this->is_real()) {
431 // case 1, real: x or -x
432 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
434 print_real_number(os, The(::cl_R)(*value));
437 print_real_number(os, The(::cl_R)(*value));
440 // case 2, imaginary: y*I or -y*I
441 if (::realpart(*value) == 0) {
442 if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
443 if (::imagpart(*value) == -1) {
447 print_real_number(os, The(::cl_R)(::imagpart(*value)));
451 if (::imagpart(*value) == 1) {
454 if (::imagpart (*value) == -1) {
457 print_real_number(os, The(::cl_R)(::imagpart(*value)));
463 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
464 if (precedence <= upper_precedence)
466 print_real_number(os, The(::cl_R)(::realpart(*value)));
467 if (::imagpart(*value) < 0) {
468 if (::imagpart(*value) == -1) {
471 print_real_number(os, The(::cl_R)(::imagpart(*value)));
475 if (::imagpart(*value) == 1) {
479 print_real_number(os, The(::cl_R)(::imagpart(*value)));
483 if (precedence <= upper_precedence)
490 void numeric::printraw(std::ostream & os) const
492 // The method printraw doesn't do much, it simply uses CLN's operator<<()
493 // for output, which is ugly but reliable. e.g: 2+2i
494 debugmsg("numeric printraw", LOGLEVEL_PRINT);
495 os << "numeric(" << *value << ")";
499 void numeric::printtree(std::ostream & os, unsigned indent) const
501 debugmsg("numeric printtree", LOGLEVEL_PRINT);
502 os << std::string(indent,' ') << *value
504 << "hash=" << hashvalue
505 << " (0x" << std::hex << hashvalue << std::dec << ")"
506 << ", flags=" << flags << std::endl;
510 void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
512 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
513 ios::fmtflags oldflags = os.flags();
514 os.setf(ios::scientific);
515 if (this->is_rational() && !this->is_integer()) {
516 if (compare(_num0()) > 0) {
518 if (type == csrc_types::ctype_cl_N)
519 os << "cl_F(\"" << numer().evalf() << "\")";
521 os << numer().to_double();
524 if (type == csrc_types::ctype_cl_N)
525 os << "cl_F(\"" << -numer().evalf() << "\")";
527 os << -numer().to_double();
530 if (type == csrc_types::ctype_cl_N)
531 os << "cl_F(\"" << denom().evalf() << "\")";
533 os << denom().to_double();
536 if (type == csrc_types::ctype_cl_N)
537 os << "cl_F(\"" << evalf() << "\")";
545 bool numeric::info(unsigned inf) const
548 case info_flags::numeric:
549 case info_flags::polynomial:
550 case info_flags::rational_function:
552 case info_flags::real:
554 case info_flags::rational:
555 case info_flags::rational_polynomial:
556 return is_rational();
557 case info_flags::crational:
558 case info_flags::crational_polynomial:
559 return is_crational();
560 case info_flags::integer:
561 case info_flags::integer_polynomial:
563 case info_flags::cinteger:
564 case info_flags::cinteger_polynomial:
565 return is_cinteger();
566 case info_flags::positive:
567 return is_positive();
568 case info_flags::negative:
569 return is_negative();
570 case info_flags::nonnegative:
571 return !is_negative();
572 case info_flags::posint:
573 return is_pos_integer();
574 case info_flags::negint:
575 return is_integer() && is_negative();
576 case info_flags::nonnegint:
577 return is_nonneg_integer();
578 case info_flags::even:
580 case info_flags::odd:
582 case info_flags::prime:
584 case info_flags::algebraic:
590 /** Disassemble real part and imaginary part to scan for the occurrence of a
591 * single number. Also handles the imaginary unit. It ignores the sign on
592 * both this and the argument, which may lead to what might appear as funny
593 * results: (2+I).has(-2) -> true. But this is consistent, since we also
594 * would like to have (-2+I).has(2) -> true and we want to think about the
595 * sign as a multiplicative factor. */
596 bool numeric::has(const ex & other) const
598 if (!is_exactly_of_type(*other.bp, numeric))
600 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
601 if (this->is_equal(o) || this->is_equal(-o))
603 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
604 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
605 this->real().is_equal(-o) || this->imag().is_equal(-o));
607 if (o.is_equal(I)) // e.g scan for I in 42*I
608 return !this->is_real();
609 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
610 return (this->real().has(o*I) || this->imag().has(o*I) ||
611 this->real().has(-o*I) || this->imag().has(-o*I));
617 /** Evaluation of numbers doesn't do anything at all. */
618 ex numeric::eval(int level) const
620 // Warning: if this is ever gonna do something, the ex ctors from all kinds
621 // of numbers should be checking for status_flags::evaluated.
626 /** Cast numeric into a floating-point object. For example exact numeric(1) is
627 * returned as a 1.0000000000000000000000 and so on according to how Digits is
628 * currently set. In case the object already was a floating point number the
629 * precision is trimmed to match the currently set default.
631 * @param level ignored, only needed for overriding basic::evalf.
632 * @return an ex-handle to a numeric. */
633 ex numeric::evalf(int level) const
635 // level can safely be discarded for numeric objects.
636 return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
641 /** Implementation of ex::diff() for a numeric. It always returns 0.
644 ex numeric::derivative(const symbol & s) const
650 int numeric::compare_same_type(const basic & other) const
652 GINAC_ASSERT(is_exactly_of_type(other, numeric));
653 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
655 if (*value == *o.value) {
663 bool numeric::is_equal_same_type(const basic & other) const
665 GINAC_ASSERT(is_exactly_of_type(other,numeric));
666 const numeric *o = static_cast<const numeric *>(&other);
668 return this->is_equal(*o);
672 unsigned numeric::calchash(void) const
674 // Use CLN's hashcode. Warning: It depends only on the number's value, not
675 // its type or precision (i.e. a true equivalence relation on numbers). As
676 // a consequence, 3 and 3.0 share the same hashvalue.
677 return (hashvalue = cl_equal_hashcode(*value) | 0x80000000U);
682 // new virtual functions which can be overridden by derived classes
688 // non-virtual functions in this class
693 /** Numerical addition method. Adds argument to *this and returns result as
694 * a new numeric object. */
695 numeric numeric::add(const numeric & other) const
697 return numeric((*value)+(*other.value));
700 /** Numerical subtraction method. Subtracts argument from *this and returns
701 * result as a new numeric object. */
702 numeric numeric::sub(const numeric & other) const
704 return numeric((*value)-(*other.value));
707 /** Numerical multiplication method. Multiplies *this and argument and returns
708 * result as a new numeric object. */
709 numeric numeric::mul(const numeric & other) const
711 static const numeric * _num1p=&_num1();
714 } else if (&other==_num1p) {
717 return numeric((*value)*(*other.value));
720 /** Numerical division method. Divides *this by argument and returns result as
721 * a new numeric object.
723 * @exception overflow_error (division by zero) */
724 numeric numeric::div(const numeric & other) const
726 if (::zerop(*other.value))
727 throw std::overflow_error("numeric::div(): division by zero");
728 return numeric((*value)/(*other.value));
731 numeric numeric::power(const numeric & other) const
733 static const numeric * _num1p = &_num1();
736 if (::zerop(*value)) {
737 if (::zerop(*other.value))
738 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
739 else if (::zerop(::realpart(*other.value)))
740 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
741 else if (::minusp(::realpart(*other.value)))
742 throw std::overflow_error("numeric::eval(): division by zero");
746 return numeric(::expt(*value,*other.value));
749 /** Inverse of a number. */
750 numeric numeric::inverse(void) const
753 throw std::overflow_error("numeric::inverse(): division by zero");
754 return numeric(::recip(*value)); // -> CLN
757 const numeric & numeric::add_dyn(const numeric & other) const
759 return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
760 setflag(status_flags::dynallocated));
763 const numeric & numeric::sub_dyn(const numeric & other) const
765 return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
766 setflag(status_flags::dynallocated));
769 const numeric & numeric::mul_dyn(const numeric & other) const
771 static const numeric * _num1p=&_num1();
774 } else if (&other==_num1p) {
777 return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
778 setflag(status_flags::dynallocated));
781 const numeric & numeric::div_dyn(const numeric & other) const
783 if (::zerop(*other.value))
784 throw std::overflow_error("division by zero");
785 return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
786 setflag(status_flags::dynallocated));
789 const numeric & numeric::power_dyn(const numeric & other) const
791 static const numeric * _num1p=&_num1();
794 if (::zerop(*value)) {
795 if (::zerop(*other.value))
796 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
797 else if (::zerop(::realpart(*other.value)))
798 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
799 else if (::minusp(::realpart(*other.value)))
800 throw std::overflow_error("numeric::eval(): division by zero");
804 return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
805 setflag(status_flags::dynallocated));
808 const numeric & numeric::operator=(int i)
810 return operator=(numeric(i));
813 const numeric & numeric::operator=(unsigned int i)
815 return operator=(numeric(i));
818 const numeric & numeric::operator=(long i)
820 return operator=(numeric(i));
823 const numeric & numeric::operator=(unsigned long i)
825 return operator=(numeric(i));
828 const numeric & numeric::operator=(double d)
830 return operator=(numeric(d));
833 const numeric & numeric::operator=(const char * s)
835 return operator=(numeric(s));
838 /** Return the complex half-plane (left or right) in which the number lies.
839 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
840 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
842 * @see numeric::compare(const numeric & other) */
843 int numeric::csgn(void) const
847 if (!::zerop(::realpart(*value))) {
848 if (::plusp(::realpart(*value)))
853 if (::plusp(::imagpart(*value)))
860 /** This method establishes a canonical order on all numbers. For complex
861 * numbers this is not possible in a mathematically consistent way but we need
862 * to establish some order and it ought to be fast. So we simply define it
863 * to be compatible with our method csgn.
865 * @return csgn(*this-other)
866 * @see numeric::csgn(void) */
867 int numeric::compare(const numeric & other) const
869 // Comparing two real numbers?
870 if (this->is_real() && other.is_real())
871 // Yes, just compare them
872 return ::cl_compare(The(::cl_R)(*value), The(::cl_R)(*other.value));
874 // No, first compare real parts
875 cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
879 return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
883 bool numeric::is_equal(const numeric & other) const
885 return (*value == *other.value);
888 /** True if object is zero. */
889 bool numeric::is_zero(void) const
891 return ::zerop(*value); // -> CLN
894 /** True if object is not complex and greater than zero. */
895 bool numeric::is_positive(void) const
898 return ::plusp(The(::cl_R)(*value)); // -> CLN
902 /** True if object is not complex and less than zero. */
903 bool numeric::is_negative(void) const
906 return ::minusp(The(::cl_R)(*value)); // -> CLN
910 /** True if object is a non-complex integer. */
911 bool numeric::is_integer(void) const
913 return ::instanceof(*value, ::cl_I_ring); // -> CLN
916 /** True if object is an exact integer greater than zero. */
917 bool numeric::is_pos_integer(void) const
919 return (this->is_integer() && ::plusp(The(::cl_I)(*value))); // -> CLN
922 /** True if object is an exact integer greater or equal zero. */
923 bool numeric::is_nonneg_integer(void) const
925 return (this->is_integer() && !::minusp(The(::cl_I)(*value))); // -> CLN
928 /** True if object is an exact even integer. */
929 bool numeric::is_even(void) const
931 return (this->is_integer() && ::evenp(The(::cl_I)(*value))); // -> CLN
934 /** True if object is an exact odd integer. */
935 bool numeric::is_odd(void) const
937 return (this->is_integer() && ::oddp(The(::cl_I)(*value))); // -> CLN
940 /** Probabilistic primality test.
942 * @return true if object is exact integer and prime. */
943 bool numeric::is_prime(void) const
945 return (this->is_integer() && ::isprobprime(The(::cl_I)(*value))); // -> CLN
948 /** True if object is an exact rational number, may even be complex
949 * (denominator may be unity). */
950 bool numeric::is_rational(void) const
952 return ::instanceof(*value, ::cl_RA_ring); // -> CLN
955 /** True if object is a real integer, rational or float (but not complex). */
956 bool numeric::is_real(void) const
958 return ::instanceof(*value, ::cl_R_ring); // -> CLN
961 bool numeric::operator==(const numeric & other) const
963 return (*value == *other.value); // -> CLN
966 bool numeric::operator!=(const numeric & other) const
968 return (*value != *other.value); // -> CLN
971 /** True if object is element of the domain of integers extended by I, i.e. is
972 * of the form a+b*I, where a and b are integers. */
973 bool numeric::is_cinteger(void) const
975 if (::instanceof(*value, ::cl_I_ring))
977 else if (!this->is_real()) { // complex case, handle n+m*I
978 if (::instanceof(::realpart(*value), ::cl_I_ring) &&
979 ::instanceof(::imagpart(*value), ::cl_I_ring))
985 /** True if object is an exact rational number, may even be complex
986 * (denominator may be unity). */
987 bool numeric::is_crational(void) const
989 if (::instanceof(*value, ::cl_RA_ring))
991 else if (!this->is_real()) { // complex case, handle Q(i):
992 if (::instanceof(::realpart(*value), ::cl_RA_ring) &&
993 ::instanceof(::imagpart(*value), ::cl_RA_ring))
999 /** Numerical comparison: less.
1001 * @exception invalid_argument (complex inequality) */
1002 bool numeric::operator<(const numeric & other) const
1004 if (this->is_real() && other.is_real())
1005 return (The(::cl_R)(*value) < The(::cl_R)(*other.value)); // -> CLN
1006 throw std::invalid_argument("numeric::operator<(): complex inequality");
1007 return false; // make compiler shut up
1010 /** Numerical comparison: less or equal.
1012 * @exception invalid_argument (complex inequality) */
1013 bool numeric::operator<=(const numeric & other) const
1015 if (this->is_real() && other.is_real())
1016 return (The(::cl_R)(*value) <= The(::cl_R)(*other.value)); // -> CLN
1017 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1018 return false; // make compiler shut up
1021 /** Numerical comparison: greater.
1023 * @exception invalid_argument (complex inequality) */
1024 bool numeric::operator>(const numeric & other) const
1026 if (this->is_real() && other.is_real())
1027 return (The(::cl_R)(*value) > The(::cl_R)(*other.value)); // -> CLN
1028 throw std::invalid_argument("numeric::operator>(): complex inequality");
1029 return false; // make compiler shut up
1032 /** Numerical comparison: greater or equal.
1034 * @exception invalid_argument (complex inequality) */
1035 bool numeric::operator>=(const numeric & other) const
1037 if (this->is_real() && other.is_real())
1038 return (The(::cl_R)(*value) >= The(::cl_R)(*other.value)); // -> CLN
1039 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1040 return false; // make compiler shut up
1043 /** Converts numeric types to machine's int. You should check with
1044 * is_integer() if the number is really an integer before calling this method.
1045 * You may also consider checking the range first. */
1046 int numeric::to_int(void) const
1048 GINAC_ASSERT(this->is_integer());
1049 return ::cl_I_to_int(The(::cl_I)(*value)); // -> CLN
1052 /** Converts numeric types to machine's long. You should check with
1053 * is_integer() if the number is really an integer before calling this method.
1054 * You may also consider checking the range first. */
1055 long numeric::to_long(void) const
1057 GINAC_ASSERT(this->is_integer());
1058 return ::cl_I_to_long(The(::cl_I)(*value)); // -> CLN
1061 /** Converts numeric types to machine's double. You should check with is_real()
1062 * if the number is really not complex before calling this method. */
1063 double numeric::to_double(void) const
1065 GINAC_ASSERT(this->is_real());
1066 return ::cl_double_approx(::realpart(*value)); // -> CLN
1069 /** Real part of a number. */
1070 const numeric numeric::real(void) const
1072 return numeric(::realpart(*value)); // -> CLN
1075 /** Imaginary part of a number. */
1076 const numeric numeric::imag(void) const
1078 return numeric(::imagpart(*value)); // -> CLN
1082 // Unfortunately, CLN did not provide an official way to access the numerator
1083 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
1084 // one finds how it works internally in src/rational/cl_RA.h:
1085 struct cl_heap_ratio : cl_heap {
1090 inline cl_heap_ratio* TheRatio (const cl_N& obj)
1091 { return (cl_heap_ratio*)(obj.pointer); }
1092 #endif // ndef SANE_LINKER
1094 /** Numerator. Computes the numerator of rational numbers, rationalized
1095 * numerator of complex if real and imaginary part are both rational numbers
1096 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1098 const numeric numeric::numer(void) const
1100 if (this->is_integer()) {
1101 return numeric(*this);
1104 else if (::instanceof(*value, ::cl_RA_ring)) {
1105 return numeric(::numerator(The(::cl_RA)(*value)));
1107 else if (!this->is_real()) { // complex case, handle Q(i):
1108 cl_R r = ::realpart(*value);
1109 cl_R i = ::imagpart(*value);
1110 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1111 return numeric(*this);
1112 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1113 return numeric(::complex(r*::denominator(The(::cl_RA)(i)), ::numerator(The(::cl_RA)(i))));
1114 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1115 return numeric(::complex(::numerator(The(::cl_RA)(r)), i*::denominator(The(::cl_RA)(r))));
1116 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring)) {
1117 cl_I s = ::lcm(::denominator(The(::cl_RA)(r)), ::denominator(The(::cl_RA)(i)));
1118 return numeric(::complex(::numerator(The(::cl_RA)(r))*(exquo(s,::denominator(The(::cl_RA)(r)))),
1119 ::numerator(The(::cl_RA)(i))*(exquo(s,::denominator(The(::cl_RA)(i))))));
1123 else if (instanceof(*value, ::cl_RA_ring)) {
1124 return numeric(TheRatio(*value)->numerator);
1126 else if (!this->is_real()) { // complex case, handle Q(i):
1127 cl_R r = ::realpart(*value);
1128 cl_R i = ::imagpart(*value);
1129 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1130 return numeric(*this);
1131 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1132 return numeric(::complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
1133 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1134 return numeric(::complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
1135 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring)) {
1136 cl_I s = ::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
1137 return numeric(::complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
1138 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
1141 #endif // def SANE_LINKER
1142 // at least one float encountered
1143 return numeric(*this);
1146 /** Denominator. Computes the denominator of rational numbers, common integer
1147 * denominator of complex if real and imaginary part are both rational numbers
1148 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1149 const numeric numeric::denom(void) const
1151 if (this->is_integer()) {
1155 if (instanceof(*value, ::cl_RA_ring)) {
1156 return numeric(::denominator(The(::cl_RA)(*value)));
1158 if (!this->is_real()) { // complex case, handle Q(i):
1159 cl_R r = ::realpart(*value);
1160 cl_R i = ::imagpart(*value);
1161 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1163 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1164 return numeric(::denominator(The(::cl_RA)(i)));
1165 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1166 return numeric(::denominator(The(::cl_RA)(r)));
1167 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring))
1168 return numeric(::lcm(::denominator(The(::cl_RA)(r)), ::denominator(The(::cl_RA)(i))));
1171 if (instanceof(*value, ::cl_RA_ring)) {
1172 return numeric(TheRatio(*value)->denominator);
1174 if (!this->is_real()) { // complex case, handle Q(i):
1175 cl_R r = ::realpart(*value);
1176 cl_R i = ::imagpart(*value);
1177 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1179 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1180 return numeric(TheRatio(i)->denominator);
1181 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1182 return numeric(TheRatio(r)->denominator);
1183 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring))
1184 return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
1186 #endif // def SANE_LINKER
1187 // at least one float encountered
1191 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1192 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1193 * 2^(n-1) <= x < 2^n.
1195 * @return number of bits (excluding sign) needed to represent that number
1196 * in two's complement if it is an integer, 0 otherwise. */
1197 int numeric::int_length(void) const
1199 if (this->is_integer())
1200 return ::integer_length(The(::cl_I)(*value)); // -> CLN
1207 // static member variables
1212 unsigned numeric::precedence = 30;
1218 const numeric some_numeric;
1219 const type_info & typeid_numeric=typeid(some_numeric);
1220 /** Imaginary unit. This is not a constant but a numeric since we are
1221 * natively handing complex numbers anyways. */
1222 const numeric I = numeric(::complex(cl_I(0),cl_I(1)));
1225 /** Exponential function.
1227 * @return arbitrary precision numerical exp(x). */
1228 const numeric exp(const numeric & x)
1230 return ::exp(*x.value); // -> CLN
1234 /** Natural logarithm.
1236 * @param z complex number
1237 * @return arbitrary precision numerical log(x).
1238 * @exception pole_error("log(): logarithmic pole",0) */
1239 const numeric log(const numeric & z)
1242 throw pole_error("log(): logarithmic pole",0);
1243 return ::log(*z.value); // -> CLN
1247 /** Numeric sine (trigonometric function).
1249 * @return arbitrary precision numerical sin(x). */
1250 const numeric sin(const numeric & x)
1252 return ::sin(*x.value); // -> CLN
1256 /** Numeric cosine (trigonometric function).
1258 * @return arbitrary precision numerical cos(x). */
1259 const numeric cos(const numeric & x)
1261 return ::cos(*x.value); // -> CLN
1265 /** Numeric tangent (trigonometric function).
1267 * @return arbitrary precision numerical tan(x). */
1268 const numeric tan(const numeric & x)
1270 return ::tan(*x.value); // -> CLN
1274 /** Numeric inverse sine (trigonometric function).
1276 * @return arbitrary precision numerical asin(x). */
1277 const numeric asin(const numeric & x)
1279 return ::asin(*x.value); // -> CLN
1283 /** Numeric inverse cosine (trigonometric function).
1285 * @return arbitrary precision numerical acos(x). */
1286 const numeric acos(const numeric & x)
1288 return ::acos(*x.value); // -> CLN
1294 * @param z complex number
1296 * @exception pole_error("atan(): logarithmic pole",0) */
1297 const numeric atan(const numeric & x)
1300 x.real().is_zero() &&
1301 abs(x.imag()).is_equal(_num1()))
1302 throw pole_error("atan(): logarithmic pole",0);
1303 return ::atan(*x.value); // -> CLN
1309 * @param x real number
1310 * @param y real number
1311 * @return atan(y/x) */
1312 const numeric atan(const numeric & y, const numeric & x)
1314 if (x.is_real() && y.is_real())
1315 return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
1317 throw std::invalid_argument("atan(): complex argument");
1321 /** Numeric hyperbolic sine (trigonometric function).
1323 * @return arbitrary precision numerical sinh(x). */
1324 const numeric sinh(const numeric & x)
1326 return ::sinh(*x.value); // -> CLN
1330 /** Numeric hyperbolic cosine (trigonometric function).
1332 * @return arbitrary precision numerical cosh(x). */
1333 const numeric cosh(const numeric & x)
1335 return ::cosh(*x.value); // -> CLN
1339 /** Numeric hyperbolic tangent (trigonometric function).
1341 * @return arbitrary precision numerical tanh(x). */
1342 const numeric tanh(const numeric & x)
1344 return ::tanh(*x.value); // -> CLN
1348 /** Numeric inverse hyperbolic sine (trigonometric function).
1350 * @return arbitrary precision numerical asinh(x). */
1351 const numeric asinh(const numeric & x)
1353 return ::asinh(*x.value); // -> CLN
1357 /** Numeric inverse hyperbolic cosine (trigonometric function).
1359 * @return arbitrary precision numerical acosh(x). */
1360 const numeric acosh(const numeric & x)
1362 return ::acosh(*x.value); // -> CLN
1366 /** Numeric inverse hyperbolic tangent (trigonometric function).
1368 * @return arbitrary precision numerical atanh(x). */
1369 const numeric atanh(const numeric & x)
1371 return ::atanh(*x.value); // -> CLN
1375 /*static ::cl_N Li2_series(const ::cl_N & x,
1376 const ::cl_float_format_t & prec)
1378 // Note: argument must be in the unit circle
1379 // This is very inefficient unless we have fast floating point Bernoulli
1380 // numbers implemented!
1381 ::cl_N c1 = -::log(1-x);
1383 // hard-wire the first two Bernoulli numbers
1384 ::cl_N acc = c1 - ::square(c1)/4;
1386 ::cl_F pisq = ::square(::cl_pi(prec)); // pi^2
1387 ::cl_F piac = ::cl_float(1, prec); // accumulator: pi^(2*i)
1393 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / ::factorial(2*i+1);
1394 // 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));
1397 } while (acc != acc+aug);
1401 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1402 * circle) using a power series. */
1403 static ::cl_N Li2_series(const ::cl_N & x,
1404 const ::cl_float_format_t & prec)
1406 // Note: argument must be in the unit circle
1408 ::cl_N num = ::complex(::cl_float(1, prec), 0);
1413 den = den + i; // 1, 4, 9, 16, ...
1417 } while (acc != acc+aug);
1421 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1422 static ::cl_N Li2_projection(const ::cl_N & x,
1423 const ::cl_float_format_t & prec)
1425 const ::cl_R re = ::realpart(x);
1426 const ::cl_R im = ::imagpart(x);
1427 if (re > ::cl_F(".5"))
1428 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1430 - Li2_series(1-x, prec)
1431 - ::log(x)*::log(1-x));
1432 if ((re <= 0 && ::abs(im) > ::cl_F(".75")) || (re < ::cl_F("-.5")))
1433 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1434 return(-::square(::log(1-x))/2
1435 - Li2_series(x/(x-1), prec));
1436 if (re > 0 && ::abs(im) > ::cl_LF(".75"))
1437 // Li2(x^2)/2 - Li2(-x)
1438 return(Li2_projection(::square(x), prec)/2
1439 - Li2_projection(-x, prec));
1440 return Li2_series(x, prec);
1443 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1444 * the branch cut lies along the positive real axis, starting at 1 and
1445 * continuous with quadrant IV.
1447 * @return arbitrary precision numerical Li2(x). */
1448 const numeric Li2(const numeric & x)
1450 if (::zerop(*x.value))
1453 // what is the desired float format?
1454 // first guess: default format
1455 ::cl_float_format_t prec = ::cl_default_float_format;
1456 // second guess: the argument's format
1457 if (!::instanceof(::realpart(*x.value),cl_RA_ring))
1458 prec = ::cl_float_format(The(::cl_F)(::realpart(*x.value)));
1459 else if (!::instanceof(::imagpart(*x.value),cl_RA_ring))
1460 prec = ::cl_float_format(The(::cl_F)(::imagpart(*x.value)));
1462 if (*x.value==1) // may cause trouble with log(1-x)
1463 return ::cl_zeta(2, prec);
1465 if (::abs(*x.value) > 1)
1466 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1467 return(-::square(::log(-*x.value))/2
1468 - ::cl_zeta(2, prec)
1469 - Li2_projection(::recip(*x.value), prec));
1471 return Li2_projection(*x.value, prec);
1475 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1476 * integer arguments. */
1477 const numeric zeta(const numeric & x)
1479 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1480 // only knows about integer arguments and zeta(3).evalf() automatically
1481 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1482 // being an exact zero for CLN, which can be tested and then we can just
1483 // pass the number casted to an int:
1485 int aux = (int)(::cl_double_approx(::realpart(*x.value)));
1486 if (::zerop(*x.value-aux))
1487 return ::cl_zeta(aux); // -> CLN
1489 std::clog << "zeta(" << x
1490 << "): Does anybody know good way to calculate this numerically?"
1496 /** The Gamma function.
1497 * This is only a stub! */
1498 const numeric lgamma(const numeric & x)
1500 std::clog << "lgamma(" << x
1501 << "): Does anybody know good way to calculate this numerically?"
1505 const numeric tgamma(const numeric & x)
1507 std::clog << "tgamma(" << x
1508 << "): Does anybody know good way to calculate this numerically?"
1514 /** The psi function (aka polygamma function).
1515 * This is only a stub! */
1516 const numeric psi(const numeric & x)
1518 std::clog << "psi(" << x
1519 << "): Does anybody know good way to calculate this numerically?"
1525 /** The psi functions (aka polygamma functions).
1526 * This is only a stub! */
1527 const numeric psi(const numeric & n, const numeric & x)
1529 std::clog << "psi(" << n << "," << x
1530 << "): Does anybody know good way to calculate this numerically?"
1536 /** Factorial combinatorial function.
1538 * @param n integer argument >= 0
1539 * @exception range_error (argument must be integer >= 0) */
1540 const numeric factorial(const numeric & n)
1542 if (!n.is_nonneg_integer())
1543 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1544 return numeric(::factorial(n.to_int())); // -> CLN
1548 /** The double factorial combinatorial function. (Scarcely used, but still
1549 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1551 * @param n integer argument >= -1
1552 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1553 * @exception range_error (argument must be integer >= -1) */
1554 const numeric doublefactorial(const numeric & n)
1556 if (n == numeric(-1)) {
1559 if (!n.is_nonneg_integer()) {
1560 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1562 return numeric(::doublefactorial(n.to_int())); // -> CLN
1566 /** The Binomial coefficients. It computes the binomial coefficients. For
1567 * integer n and k and positive n this is the number of ways of choosing k
1568 * objects from n distinct objects. If n is negative, the formula
1569 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1570 const numeric binomial(const numeric & n, const numeric & k)
1572 if (n.is_integer() && k.is_integer()) {
1573 if (n.is_nonneg_integer()) {
1574 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1575 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1579 return _num_1().power(k)*binomial(k-n-_num1(),k);
1583 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1584 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1588 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1589 * in the expansion of the function x/(e^x-1).
1591 * @return the nth Bernoulli number (a rational number).
1592 * @exception range_error (argument must be integer >= 0) */
1593 const numeric bernoulli(const numeric & nn)
1595 if (!nn.is_integer() || nn.is_negative())
1596 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1600 // The Bernoulli numbers are rational numbers that may be computed using
1603 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1605 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1606 // previous ones, the computation is necessarily very expensive. There are
1607 // several other ways of computing them, a particularly good one being
1611 // for (unsigned i=0; i<n; i++) {
1612 // c = exquo(c*(i-n),(i+2));
1613 // Bern = Bern + c*s/(i+2);
1614 // s = s + expt_pos(cl_I(i+2),n);
1618 // But if somebody works with the n'th Bernoulli number she is likely to
1619 // also need all previous Bernoulli numbers. So we need a complete remember
1620 // table and above divide and conquer algorithm is not suited to build one
1621 // up. The code below is adapted from Pari's function bernvec().
1623 // (There is an interesting relation with the tangent polynomials described
1624 // in `Concrete Mathematics', which leads to a program twice as fast as our
1625 // implementation below, but it requires storing one such polynomial in
1626 // addition to the remember table. This doubles the memory footprint so
1627 // we don't use it.)
1629 // the special cases not covered by the algorithm below
1630 if (nn.is_equal(_num1()))
1635 // store nonvanishing Bernoulli numbers here
1636 static std::vector< ::cl_RA > results;
1637 static int highest_result = 0;
1638 // algorithm not applicable to B(0), so just store it
1639 if (results.size()==0)
1640 results.push_back(::cl_RA(1));
1642 int n = nn.to_long();
1643 for (int i=highest_result; i<n/2; ++i) {
1649 for (int j=i; j>0; --j) {
1650 B = ::cl_I(n*m) * (B+results[j]) / (d1*d2);
1656 B = (1 - ((B+1)/(2*i+3))) / (::cl_I(1)<<(2*i+2));
1657 results.push_back(B);
1660 return results[n/2];
1664 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1665 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1667 * @param n an integer
1668 * @return the nth Fibonacci number F(n) (an integer number)
1669 * @exception range_error (argument must be an integer) */
1670 const numeric fibonacci(const numeric & n)
1672 if (!n.is_integer())
1673 throw std::range_error("numeric::fibonacci(): argument must be integer");
1676 // This is based on an implementation that can be found in CLN's example
1677 // directory. There, it is done recursively, which may be more elegant
1678 // than our non-recursive implementation that has to resort to some bit-
1679 // fiddling. This is, however, a matter of taste.
1680 // The following addition formula holds:
1682 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1684 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1685 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1687 // Replace m by m+1:
1688 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1689 // Now put in m = n, to get
1690 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1691 // F(2n+1) = F(n)^2 + F(n+1)^2
1693 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1696 if (n.is_negative())
1698 return -fibonacci(-n);
1700 return fibonacci(-n);
1704 ::cl_I m = The(::cl_I)(*n.value) >> 1L; // floor(n/2);
1705 for (uintL bit=::integer_length(m); bit>0; --bit) {
1706 // Since a squaring is cheaper than a multiplication, better use
1707 // three squarings instead of one multiplication and two squarings.
1708 ::cl_I u2 = ::square(u);
1709 ::cl_I v2 = ::square(v);
1710 if (::logbitp(bit-1, m)) {
1711 v = ::square(u + v) - u2;
1714 u = v2 - ::square(v - u);
1719 // Here we don't use the squaring formula because one multiplication
1720 // is cheaper than two squarings.
1721 return u * ((v << 1) - u);
1723 return ::square(u) + ::square(v);
1727 /** Absolute value. */
1728 numeric abs(const numeric & x)
1730 return ::abs(*x.value); // -> CLN
1734 /** Modulus (in positive representation).
1735 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1736 * sign of a or is zero. This is different from Maple's modp, where the sign
1737 * of b is ignored. It is in agreement with Mathematica's Mod.
1739 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1740 * integer, 0 otherwise. */
1741 numeric mod(const numeric & a, const numeric & b)
1743 if (a.is_integer() && b.is_integer())
1744 return ::mod(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1746 return _num0(); // Throw?
1750 /** Modulus (in symmetric representation).
1751 * Equivalent to Maple's mods.
1753 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1754 numeric smod(const numeric & a, const numeric & b)
1756 if (a.is_integer() && b.is_integer()) {
1757 cl_I b2 = The(::cl_I)(ceiling1(The(::cl_I)(*b.value) / 2)) - 1;
1758 return ::mod(The(::cl_I)(*a.value) + b2, The(::cl_I)(*b.value)) - b2;
1760 return _num0(); // Throw?
1764 /** Numeric integer remainder.
1765 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1766 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1767 * sign of a or is zero.
1769 * @return remainder of a/b if both are integer, 0 otherwise. */
1770 numeric irem(const numeric & a, const numeric & b)
1772 if (a.is_integer() && b.is_integer())
1773 return ::rem(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1775 return _num0(); // Throw?
1779 /** Numeric integer remainder.
1780 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1781 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1782 * and irem(a,b) has the sign of a or is zero.
1784 * @return remainder of a/b and quotient stored in q if both are integer,
1786 numeric irem(const numeric & a, const numeric & b, numeric & q)
1788 if (a.is_integer() && b.is_integer()) { // -> CLN
1789 cl_I_div_t rem_quo = truncate2(The(::cl_I)(*a.value), The(::cl_I)(*b.value));
1790 q = rem_quo.quotient;
1791 return rem_quo.remainder;
1795 return _num0(); // Throw?
1800 /** Numeric integer quotient.
1801 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1803 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1804 numeric iquo(const numeric & a, const numeric & b)
1806 if (a.is_integer() && b.is_integer())
1807 return truncate1(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1809 return _num0(); // Throw?
1813 /** Numeric integer quotient.
1814 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1815 * r == a - iquo(a,b,r)*b.
1817 * @return truncated quotient of a/b and remainder stored in r if both are
1818 * integer, 0 otherwise. */
1819 numeric iquo(const numeric & a, const numeric & b, numeric & r)
1821 if (a.is_integer() && b.is_integer()) { // -> CLN
1822 cl_I_div_t rem_quo = truncate2(The(::cl_I)(*a.value), The(::cl_I)(*b.value));
1823 r = rem_quo.remainder;
1824 return rem_quo.quotient;
1827 return _num0(); // Throw?
1832 /** Numeric square root.
1833 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1834 * should return integer 2.
1836 * @param z numeric argument
1837 * @return square root of z. Branch cut along negative real axis, the negative
1838 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1839 * where imag(z)>0. */
1840 numeric sqrt(const numeric & z)
1842 return ::sqrt(*z.value); // -> CLN
1846 /** Integer numeric square root. */
1847 numeric isqrt(const numeric & x)
1849 if (x.is_integer()) {
1851 ::isqrt(The(::cl_I)(*x.value), &root); // -> CLN
1854 return _num0(); // Throw?
1858 /** Greatest Common Divisor.
1860 * @return The GCD of two numbers if both are integer, a numerical 1
1861 * if they are not. */
1862 numeric gcd(const numeric & a, const numeric & b)
1864 if (a.is_integer() && b.is_integer())
1865 return ::gcd(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1871 /** Least Common Multiple.
1873 * @return The LCM of two numbers if both are integer, the product of those
1874 * two numbers if they are not. */
1875 numeric lcm(const numeric & a, const numeric & b)
1877 if (a.is_integer() && b.is_integer())
1878 return ::lcm(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1880 return *a.value * *b.value;
1884 /** Floating point evaluation of Archimedes' constant Pi. */
1887 return numeric(::cl_pi(cl_default_float_format)); // -> CLN
1891 /** Floating point evaluation of Euler's constant gamma. */
1894 return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
1898 /** Floating point evaluation of Catalan's constant. */
1899 ex CatalanEvalf(void)
1901 return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
1905 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1906 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1907 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1908 _numeric_digits::_numeric_digits()
1913 cl_default_float_format = ::cl_float_format(17);
1917 _numeric_digits& _numeric_digits::operator=(long prec)
1920 cl_default_float_format = ::cl_float_format(prec);
1925 _numeric_digits::operator long()
1927 return (long)digits;
1931 void _numeric_digits::print(std::ostream & os) const
1933 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1938 std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
1945 // static member variables
1950 bool _numeric_digits::too_late = false;
1953 /** Accuracy in decimal digits. Only object of this type! Can be set using
1954 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1955 _numeric_digits Digits;
1957 #ifndef NO_NAMESPACE_GINAC
1958 } // namespace GiNaC
1959 #endif // ndef NO_NAMESPACE_GINAC