3 * This file contains the interface to the underlying bignum package.
4 * Its most important design principle is to completely hide the inner
5 * working of that other package from the user of GiNaC. It must either
6 * provide implementation of arithmetic operators and numerical evaluation
7 * of special functions or implement the interface to the bignum package. */
10 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
12 * This program is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU General Public License as published by
14 * the Free Software Foundation; either version 2 of the License, or
15 * (at your option) any later version.
17 * This program is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU General Public License for more details.
22 * You should have received a copy of the GNU General Public License
23 * along with this program; if not, write to the Free Software
24 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
33 #if defined(HAVE_SSTREAM)
35 #elif defined(HAVE_STRSTREAM)
38 #error Need either sstream or strstream
47 // CLN should pollute the global namespace as little as possible. Hence, we
48 // include most of it here and include only the part needed for properly
49 // declaring cln::cl_number in numeric.h. This can only be safely done in
50 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
51 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
53 #include <cln/output.h>
54 #include <cln/integer_io.h>
55 #include <cln/integer_ring.h>
56 #include <cln/rational_io.h>
57 #include <cln/rational_ring.h>
58 #include <cln/lfloat_class.h>
59 #include <cln/lfloat_io.h>
60 #include <cln/real_io.h>
61 #include <cln/real_ring.h>
62 #include <cln/complex_io.h>
63 #include <cln/complex_ring.h>
64 #include <cln/numtheory.h>
68 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
71 // default ctor, dtor, copy ctor assignment
72 // operator and helpers
77 /** default ctor. Numerically it initializes to an integer zero. */
78 numeric::numeric() : basic(TINFO_numeric)
80 debugmsg("numeric default ctor", LOGLEVEL_CONSTRUCT);
82 setflag(status_flags::evaluated | status_flags::expanded);
87 /** For use by copy ctor and assignment operator. */
88 void numeric::copy(const numeric &other)
90 inherited::copy(other);
94 void numeric::destroy(bool call_parent)
96 if (call_parent) inherited::destroy(call_parent);
105 numeric::numeric(int i) : basic(TINFO_numeric)
107 debugmsg("numeric ctor from int",LOGLEVEL_CONSTRUCT);
108 // Not the whole int-range is available if we don't cast to long
109 // first. This is due to the behaviour of the cl_I-ctor, which
110 // emphasizes efficiency. However, if the integer is small enough,
111 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
112 // using an immediate type:
113 if (cln::cl_immediate_p(i))
114 value = cln::cl_I(i);
116 value = cln::cl_I((long) i);
117 setflag(status_flags::evaluated | status_flags::expanded);
121 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
123 debugmsg("numeric ctor from uint",LOGLEVEL_CONSTRUCT);
124 // Not the whole uint-range is available if we don't cast to ulong
125 // first. This is due to the behaviour of the cl_I-ctor, which
126 // emphasizes efficiency. However, if the integer is small enough,
127 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
128 // using an immediate type:
129 if (cln::cl_immediate_p(i))
130 value = cln::cl_I(i);
132 value = cln::cl_I((unsigned long) i);
133 setflag(status_flags::evaluated | status_flags::expanded);
137 numeric::numeric(long i) : basic(TINFO_numeric)
139 debugmsg("numeric ctor from long",LOGLEVEL_CONSTRUCT);
140 value = cln::cl_I(i);
141 setflag(status_flags::evaluated | status_flags::expanded);
145 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
147 debugmsg("numeric ctor from ulong",LOGLEVEL_CONSTRUCT);
148 value = cln::cl_I(i);
149 setflag(status_flags::evaluated | status_flags::expanded);
152 /** Ctor for rational numerics a/b.
154 * @exception overflow_error (division by zero) */
155 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
157 debugmsg("numeric ctor from long/long",LOGLEVEL_CONSTRUCT);
159 throw std::overflow_error("division by zero");
160 value = cln::cl_I(numer) / cln::cl_I(denom);
161 setflag(status_flags::evaluated | status_flags::expanded);
165 numeric::numeric(double d) : basic(TINFO_numeric)
167 debugmsg("numeric ctor from double",LOGLEVEL_CONSTRUCT);
168 // We really want to explicitly use the type cl_LF instead of the
169 // more general cl_F, since that would give us a cl_DF only which
170 // will not be promoted to cl_LF if overflow occurs:
171 value = cln::cl_float(d, cln::default_float_format);
172 setflag(status_flags::evaluated | status_flags::expanded);
176 /** ctor from C-style string. It also accepts complex numbers in GiNaC
177 * notation like "2+5*I". */
178 numeric::numeric(const char *s) : basic(TINFO_numeric)
180 debugmsg("numeric ctor from string",LOGLEVEL_CONSTRUCT);
181 cln::cl_N ctorval = 0;
182 // parse complex numbers (functional but not completely safe, unfortunately
183 // std::string does not understand regexpese):
184 // ss should represent a simple sum like 2+5*I
186 // make it safe by adding explicit sign
187 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
189 std::string::size_type delim;
191 // chop ss into terms from left to right
193 bool imaginary = false;
194 delim = ss.find_first_of(std::string("+-"),1);
195 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
196 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
197 delim = ss.find_first_of(std::string("+-"),delim+1);
198 term = ss.substr(0,delim);
199 if (delim != std::string::npos)
200 ss = ss.substr(delim);
201 // is the term imaginary?
202 if (term.find("I") != std::string::npos) {
204 term = term.replace(term.find("I"),1,"");
206 if (term.find("*") != std::string::npos)
207 term = term.replace(term.find("*"),1,"");
208 // correct for trivial +/-I without explicit factor on I:
209 if (term.size() == 1)
213 if (term.find(".") != std::string::npos) {
214 // CLN's short type cl_SF is not very useful within the GiNaC
215 // framework where we are mainly interested in the arbitrary
216 // precision type cl_LF. Hence we go straight to the construction
217 // of generic floats. In order to create them we have to convert
218 // our own floating point notation used for output and construction
219 // from char * to CLN's generic notation:
220 // 3.14 --> 3.14e0_<Digits>
221 // 31.4E-1 --> 31.4e-1_<Digits>
223 // No exponent marker? Let's add a trivial one.
224 if (term.find("E") == std::string::npos)
227 term = term.replace(term.find("E"),1,"e");
228 // append _<Digits> to term
229 #if defined(HAVE_SSTREAM)
230 std::ostringstream buf;
231 buf << unsigned(Digits) << std::ends;
232 term += "_" + buf.str();
235 std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
236 term += "_" + std::string(buf);
238 // construct float using cln::cl_F(const char *) ctor.
240 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
242 ctorval = ctorval + cln::cl_F(term.c_str());
244 // not a floating point number...
246 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
248 ctorval = ctorval + cln::cl_R(term.c_str());
250 } while(delim != std::string::npos);
252 setflag(status_flags::evaluated | status_flags::expanded);
256 /** Ctor from CLN types. This is for the initiated user or internal use
258 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
260 debugmsg("numeric ctor from cl_N", LOGLEVEL_CONSTRUCT);
262 setflag(status_flags::evaluated | status_flags::expanded);
269 /** Construct object from archive_node. */
270 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
272 debugmsg("numeric ctor from archive_node", LOGLEVEL_CONSTRUCT);
273 cln::cl_N ctorval = 0;
275 // Read number as string
277 if (n.find_string("number", str)) {
279 std::istringstream s(str);
281 std::istrstream s(str.c_str(), str.size() + 1);
283 cln::cl_idecoded_float re, im;
287 case 'R': // Integer-decoded real number
288 s >> re.sign >> re.mantissa >> re.exponent;
289 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
291 case 'C': // Integer-decoded complex number
292 s >> re.sign >> re.mantissa >> re.exponent;
293 s >> im.sign >> im.mantissa >> im.exponent;
294 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
295 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
297 default: // Ordinary number
304 setflag(status_flags::evaluated | status_flags::expanded);
307 /** Unarchive the object. */
308 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
310 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
313 /** Archive the object. */
314 void numeric::archive(archive_node &n) const
316 inherited::archive(n);
318 // Write number as string
320 std::ostringstream s;
323 std::ostrstream s(buf, 1024);
325 if (this->is_crational())
326 s << cln::the<cln::cl_N>(value);
328 // Non-rational numbers are written in an integer-decoded format
329 // to preserve the precision
330 if (this->is_real()) {
331 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
333 s << re.sign << " " << re.mantissa << " " << re.exponent;
335 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
336 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
338 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
339 s << im.sign << " " << im.mantissa << " " << im.exponent;
343 n.add_string("number", s.str());
346 std::string str(buf);
347 n.add_string("number", str);
352 // functions overriding virtual functions from bases classes
355 /** Helper function to print a real number in a nicer way than is CLN's
356 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
357 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
358 * long as it only uses cl_LF and no other floating point types that we might
359 * want to visibly distinguish from cl_LF.
361 * @see numeric::print() */
362 static void print_real_number(std::ostream &os, const cln::cl_R &num)
364 cln::cl_print_flags ourflags;
365 if (cln::instanceof(num, cln::cl_RA_ring)) {
366 // case 1: integer or rational, nothing special to do:
367 cln::print_real(os, ourflags, num);
370 // make CLN believe this number has default_float_format, so it prints
371 // 'E' as exponent marker instead of 'L':
372 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(num));
373 cln::print_real(os, ourflags, num);
378 /** This method adds to the output so it blends more consistently together
379 * with the other routines and produces something compatible to ginsh input.
381 * @see print_real_number() */
382 void numeric::print(std::ostream &os, unsigned upper_precedence) const
384 debugmsg("numeric print", LOGLEVEL_PRINT);
385 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
386 cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
388 // case 1, real: x or -x
389 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
391 print_real_number(os, r);
394 print_real_number(os, r);
398 // case 2, imaginary: y*I or -y*I
399 if ((precedence<=upper_precedence) && (i < 0)) {
404 print_real_number(os, i);
414 print_real_number(os, i);
420 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
421 if (precedence <= upper_precedence)
423 print_real_number(os, r);
428 print_real_number(os, i);
436 print_real_number(os, i);
440 if (precedence <= upper_precedence)
447 void numeric::printraw(std::ostream &os) const
449 // The method printraw doesn't do much, it simply uses CLN's operator<<()
450 // for output, which is ugly but reliable. e.g: 2+2i
451 debugmsg("numeric printraw", LOGLEVEL_PRINT);
452 os << class_name() << "(" << cln::the<cln::cl_N>(value) << ")";
456 void numeric::printtree(std::ostream &os, unsigned indent) const
458 debugmsg("numeric printtree", LOGLEVEL_PRINT);
459 os << std::string(indent,' ') << cln::the<cln::cl_N>(value)
461 << "hash=" << hashvalue
462 << " (0x" << std::hex << hashvalue << std::dec << ")"
463 << ", flags=" << flags << std::endl;
467 void numeric::printcsrc(std::ostream &os, unsigned type, unsigned upper_precedence) const
469 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
470 std::ios::fmtflags oldflags = os.flags();
471 os.setf(std::ios::scientific);
472 if (this->is_rational() && !this->is_integer()) {
473 if (compare(_num0()) > 0) {
475 if (type == csrc_types::ctype_cl_N)
476 os << "cln::cl_F(\"" << numer().evalf() << "\")";
478 os << numer().to_double();
481 if (type == csrc_types::ctype_cl_N)
482 os << "cln::cl_F(\"" << -numer().evalf() << "\")";
484 os << -numer().to_double();
487 if (type == csrc_types::ctype_cl_N)
488 os << "cln::cl_F(\"" << denom().evalf() << "\")";
490 os << denom().to_double();
493 if (type == csrc_types::ctype_cl_N)
494 os << "cln::cl_F(\"" << evalf() << "\")";
502 bool numeric::info(unsigned inf) const
505 case info_flags::numeric:
506 case info_flags::polynomial:
507 case info_flags::rational_function:
509 case info_flags::real:
511 case info_flags::rational:
512 case info_flags::rational_polynomial:
513 return is_rational();
514 case info_flags::crational:
515 case info_flags::crational_polynomial:
516 return is_crational();
517 case info_flags::integer:
518 case info_flags::integer_polynomial:
520 case info_flags::cinteger:
521 case info_flags::cinteger_polynomial:
522 return is_cinteger();
523 case info_flags::positive:
524 return is_positive();
525 case info_flags::negative:
526 return is_negative();
527 case info_flags::nonnegative:
528 return !is_negative();
529 case info_flags::posint:
530 return is_pos_integer();
531 case info_flags::negint:
532 return is_integer() && is_negative();
533 case info_flags::nonnegint:
534 return is_nonneg_integer();
535 case info_flags::even:
537 case info_flags::odd:
539 case info_flags::prime:
541 case info_flags::algebraic:
547 /** Disassemble real part and imaginary part to scan for the occurrence of a
548 * single number. Also handles the imaginary unit. It ignores the sign on
549 * both this and the argument, which may lead to what might appear as funny
550 * results: (2+I).has(-2) -> true. But this is consistent, since we also
551 * would like to have (-2+I).has(2) -> true and we want to think about the
552 * sign as a multiplicative factor. */
553 bool numeric::has(const ex &other) const
555 if (!is_exactly_of_type(*other.bp, numeric))
557 const numeric &o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
558 if (this->is_equal(o) || this->is_equal(-o))
560 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
561 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
562 this->real().is_equal(-o) || this->imag().is_equal(-o));
564 if (o.is_equal(I)) // e.g scan for I in 42*I
565 return !this->is_real();
566 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
567 return (this->real().has(o*I) || this->imag().has(o*I) ||
568 this->real().has(-o*I) || this->imag().has(-o*I));
574 /** Evaluation of numbers doesn't do anything at all. */
575 ex numeric::eval(int level) const
577 // Warning: if this is ever gonna do something, the ex ctors from all kinds
578 // of numbers should be checking for status_flags::evaluated.
583 /** Cast numeric into a floating-point object. For example exact numeric(1) is
584 * returned as a 1.0000000000000000000000 and so on according to how Digits is
585 * currently set. In case the object already was a floating point number the
586 * precision is trimmed to match the currently set default.
588 * @param level ignored, only needed for overriding basic::evalf.
589 * @return an ex-handle to a numeric. */
590 ex numeric::evalf(int level) const
592 // level can safely be discarded for numeric objects.
593 return numeric(cln::cl_float(1.0, cln::default_float_format) *
594 (cln::the<cln::cl_N>(value)));
599 int numeric::compare_same_type(const basic &other) const
601 GINAC_ASSERT(is_exactly_of_type(other, numeric));
602 const numeric &o = static_cast<numeric &>(const_cast<basic &>(other));
604 return this->compare(o);
608 bool numeric::is_equal_same_type(const basic &other) const
610 GINAC_ASSERT(is_exactly_of_type(other,numeric));
611 const numeric *o = static_cast<const numeric *>(&other);
613 return this->is_equal(*o);
617 unsigned numeric::calchash(void) const
619 // Use CLN's hashcode. Warning: It depends only on the number's value, not
620 // its type or precision (i.e. a true equivalence relation on numbers). As
621 // a consequence, 3 and 3.0 share the same hashvalue.
622 setflag(status_flags::hash_calculated);
623 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
628 // new virtual functions which can be overridden by derived classes
634 // non-virtual functions in this class
639 /** Numerical addition method. Adds argument to *this and returns result as
640 * a numeric object. */
641 const numeric numeric::add(const numeric &other) const
643 // Efficiency shortcut: trap the neutral element by pointer.
644 static const numeric * _num0p = &_num0();
647 else if (&other==_num0p)
650 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
654 /** Numerical subtraction method. Subtracts argument from *this and returns
655 * result as a numeric object. */
656 const numeric numeric::sub(const numeric &other) const
658 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
662 /** Numerical multiplication method. Multiplies *this and argument and returns
663 * result as a numeric object. */
664 const numeric numeric::mul(const numeric &other) const
666 // Efficiency shortcut: trap the neutral element by pointer.
667 static const numeric * _num1p = &_num1();
670 else if (&other==_num1p)
673 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
677 /** Numerical division method. Divides *this by argument and returns result as
680 * @exception overflow_error (division by zero) */
681 const numeric numeric::div(const numeric &other) const
683 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
684 throw std::overflow_error("numeric::div(): division by zero");
685 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
689 /** Numerical exponentiation. Raises *this to the power given as argument and
690 * returns result as a numeric object. */
691 const numeric numeric::power(const numeric &other) const
693 // Efficiency shortcut: trap the neutral exponent by pointer.
694 static const numeric * _num1p = &_num1();
698 if (cln::zerop(cln::the<cln::cl_N>(value))) {
699 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
700 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
701 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
702 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
703 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
704 throw std::overflow_error("numeric::eval(): division by zero");
708 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
712 const numeric &numeric::add_dyn(const numeric &other) const
714 // Efficiency shortcut: trap the neutral element by pointer.
715 static const numeric * _num0p = &_num0();
718 else if (&other==_num0p)
721 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
722 setflag(status_flags::dynallocated));
726 const numeric &numeric::sub_dyn(const numeric &other) const
728 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
729 setflag(status_flags::dynallocated));
733 const numeric &numeric::mul_dyn(const numeric &other) const
735 // Efficiency shortcut: trap the neutral element by pointer.
736 static const numeric * _num1p = &_num1();
739 else if (&other==_num1p)
742 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
743 setflag(status_flags::dynallocated));
747 const numeric &numeric::div_dyn(const numeric &other) const
749 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
750 throw std::overflow_error("division by zero");
751 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
752 setflag(status_flags::dynallocated));
756 const numeric &numeric::power_dyn(const numeric &other) const
758 // Efficiency shortcut: trap the neutral exponent by pointer.
759 static const numeric * _num1p=&_num1();
763 if (cln::zerop(cln::the<cln::cl_N>(value))) {
764 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
765 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
766 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
767 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
768 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
769 throw std::overflow_error("numeric::eval(): division by zero");
773 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
774 setflag(status_flags::dynallocated));
778 const numeric &numeric::operator=(int i)
780 return operator=(numeric(i));
784 const numeric &numeric::operator=(unsigned int i)
786 return operator=(numeric(i));
790 const numeric &numeric::operator=(long i)
792 return operator=(numeric(i));
796 const numeric &numeric::operator=(unsigned long i)
798 return operator=(numeric(i));
802 const numeric &numeric::operator=(double d)
804 return operator=(numeric(d));
808 const numeric &numeric::operator=(const char * s)
810 return operator=(numeric(s));
814 /** Inverse of a number. */
815 const numeric numeric::inverse(void) const
817 if (cln::zerop(cln::the<cln::cl_N>(value)))
818 throw std::overflow_error("numeric::inverse(): division by zero");
819 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
823 /** Return the complex half-plane (left or right) in which the number lies.
824 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
825 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
827 * @see numeric::compare(const numeric &other) */
828 int numeric::csgn(void) const
830 if (cln::zerop(cln::the<cln::cl_N>(value)))
832 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
833 if (!cln::zerop(r)) {
839 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
847 /** This method establishes a canonical order on all numbers. For complex
848 * numbers this is not possible in a mathematically consistent way but we need
849 * to establish some order and it ought to be fast. So we simply define it
850 * to be compatible with our method csgn.
852 * @return csgn(*this-other)
853 * @see numeric::csgn(void) */
854 int numeric::compare(const numeric &other) const
856 // Comparing two real numbers?
857 if (cln::instanceof(value, cln::cl_R_ring) &&
858 cln::instanceof(other.value, cln::cl_R_ring))
859 // Yes, so just cln::compare them
860 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
862 // No, first cln::compare real parts...
863 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
866 // ...and then the imaginary parts.
867 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
872 bool numeric::is_equal(const numeric &other) const
874 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
878 /** True if object is zero. */
879 bool numeric::is_zero(void) const
881 return cln::zerop(cln::the<cln::cl_N>(value));
885 /** True if object is not complex and greater than zero. */
886 bool numeric::is_positive(void) const
889 return cln::plusp(cln::the<cln::cl_R>(value));
894 /** True if object is not complex and less than zero. */
895 bool numeric::is_negative(void) const
898 return cln::minusp(cln::the<cln::cl_R>(value));
903 /** True if object is a non-complex integer. */
904 bool numeric::is_integer(void) const
906 return cln::instanceof(value, cln::cl_I_ring);
910 /** True if object is an exact integer greater than zero. */
911 bool numeric::is_pos_integer(void) const
913 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
917 /** True if object is an exact integer greater or equal zero. */
918 bool numeric::is_nonneg_integer(void) const
920 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
924 /** True if object is an exact even integer. */
925 bool numeric::is_even(void) const
927 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
931 /** True if object is an exact odd integer. */
932 bool numeric::is_odd(void) const
934 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
938 /** Probabilistic primality test.
940 * @return true if object is exact integer and prime. */
941 bool numeric::is_prime(void) const
943 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
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 cln::instanceof(value, cln::cl_RA_ring);
955 /** True if object is a real integer, rational or float (but not complex). */
956 bool numeric::is_real(void) const
958 return cln::instanceof(value, cln::cl_R_ring);
962 bool numeric::operator==(const numeric &other) const
964 return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
968 bool numeric::operator!=(const numeric &other) const
970 return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
974 /** True if object is element of the domain of integers extended by I, i.e. is
975 * of the form a+b*I, where a and b are integers. */
976 bool numeric::is_cinteger(void) const
978 if (cln::instanceof(value, cln::cl_I_ring))
980 else if (!this->is_real()) { // complex case, handle n+m*I
981 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
982 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
989 /** True if object is an exact rational number, may even be complex
990 * (denominator may be unity). */
991 bool numeric::is_crational(void) const
993 if (cln::instanceof(value, cln::cl_RA_ring))
995 else if (!this->is_real()) { // complex case, handle Q(i):
996 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
997 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
1004 /** Numerical comparison: less.
1006 * @exception invalid_argument (complex inequality) */
1007 bool numeric::operator<(const numeric &other) const
1009 if (this->is_real() && other.is_real())
1010 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
1011 throw std::invalid_argument("numeric::operator<(): complex inequality");
1015 /** Numerical comparison: less or equal.
1017 * @exception invalid_argument (complex inequality) */
1018 bool numeric::operator<=(const numeric &other) const
1020 if (this->is_real() && other.is_real())
1021 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
1022 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1026 /** Numerical comparison: greater.
1028 * @exception invalid_argument (complex inequality) */
1029 bool numeric::operator>(const numeric &other) const
1031 if (this->is_real() && other.is_real())
1032 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1033 throw std::invalid_argument("numeric::operator>(): complex inequality");
1037 /** Numerical comparison: greater or equal.
1039 * @exception invalid_argument (complex inequality) */
1040 bool numeric::operator>=(const numeric &other) const
1042 if (this->is_real() && other.is_real())
1043 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1044 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1048 /** Converts numeric types to machine's int. 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 int numeric::to_int(void) const
1053 GINAC_ASSERT(this->is_integer());
1054 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1058 /** Converts numeric types to machine's long. You should check with
1059 * is_integer() if the number is really an integer before calling this method.
1060 * You may also consider checking the range first. */
1061 long numeric::to_long(void) const
1063 GINAC_ASSERT(this->is_integer());
1064 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1068 /** Converts numeric types to machine's double. You should check with is_real()
1069 * if the number is really not complex before calling this method. */
1070 double numeric::to_double(void) const
1072 GINAC_ASSERT(this->is_real());
1073 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1077 /** Returns a new CLN object of type cl_N, representing the value of *this.
1078 * This method may be used when mixing GiNaC and CLN in one project.
1080 cln::cl_N numeric::to_cl_N(void) const
1082 return cln::cl_N(cln::the<cln::cl_N>(value));
1086 /** Real part of a number. */
1087 const numeric numeric::real(void) const
1089 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1093 /** Imaginary part of a number. */
1094 const numeric numeric::imag(void) const
1096 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1100 /** Numerator. Computes the numerator of rational numbers, rationalized
1101 * numerator of complex if real and imaginary part are both rational numbers
1102 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1104 const numeric numeric::numer(void) const
1106 if (this->is_integer())
1107 return numeric(*this);
1109 else if (cln::instanceof(value, cln::cl_RA_ring))
1110 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1112 else if (!this->is_real()) { // complex case, handle Q(i):
1113 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1114 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1115 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1116 return numeric(*this);
1117 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1118 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1119 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1120 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1121 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1122 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1123 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1124 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1127 // at least one float encountered
1128 return numeric(*this);
1132 /** Denominator. Computes the denominator of rational numbers, common integer
1133 * denominator of complex if real and imaginary part are both rational numbers
1134 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1135 const numeric numeric::denom(void) const
1137 if (this->is_integer())
1140 if (instanceof(value, cln::cl_RA_ring))
1141 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1143 if (!this->is_real()) { // complex case, handle Q(i):
1144 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1145 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1146 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1148 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1149 return numeric(cln::denominator(i));
1150 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1151 return numeric(cln::denominator(r));
1152 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1153 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1155 // at least one float encountered
1160 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1161 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1162 * 2^(n-1) <= x < 2^n.
1164 * @return number of bits (excluding sign) needed to represent that number
1165 * in two's complement if it is an integer, 0 otherwise. */
1166 int numeric::int_length(void) const
1168 if (this->is_integer())
1169 return cln::integer_length(cln::the<cln::cl_I>(value));
1176 // static member variables
1181 unsigned numeric::precedence = 30;
1187 /** Imaginary unit. This is not a constant but a numeric since we are
1188 * natively handing complex numbers anyways, so in each expression containing
1189 * an I it is automatically eval'ed away anyhow. */
1190 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1193 /** Exponential function.
1195 * @return arbitrary precision numerical exp(x). */
1196 const numeric exp(const numeric &x)
1198 return cln::exp(x.to_cl_N());
1202 /** Natural logarithm.
1204 * @param z complex number
1205 * @return arbitrary precision numerical log(x).
1206 * @exception pole_error("log(): logarithmic pole",0) */
1207 const numeric log(const numeric &z)
1210 throw pole_error("log(): logarithmic pole",0);
1211 return cln::log(z.to_cl_N());
1215 /** Numeric sine (trigonometric function).
1217 * @return arbitrary precision numerical sin(x). */
1218 const numeric sin(const numeric &x)
1220 return cln::sin(x.to_cl_N());
1224 /** Numeric cosine (trigonometric function).
1226 * @return arbitrary precision numerical cos(x). */
1227 const numeric cos(const numeric &x)
1229 return cln::cos(x.to_cl_N());
1233 /** Numeric tangent (trigonometric function).
1235 * @return arbitrary precision numerical tan(x). */
1236 const numeric tan(const numeric &x)
1238 return cln::tan(x.to_cl_N());
1242 /** Numeric inverse sine (trigonometric function).
1244 * @return arbitrary precision numerical asin(x). */
1245 const numeric asin(const numeric &x)
1247 return cln::asin(x.to_cl_N());
1251 /** Numeric inverse cosine (trigonometric function).
1253 * @return arbitrary precision numerical acos(x). */
1254 const numeric acos(const numeric &x)
1256 return cln::acos(x.to_cl_N());
1262 * @param z complex number
1264 * @exception pole_error("atan(): logarithmic pole",0) */
1265 const numeric atan(const numeric &x)
1268 x.real().is_zero() &&
1269 abs(x.imag()).is_equal(_num1()))
1270 throw pole_error("atan(): logarithmic pole",0);
1271 return cln::atan(x.to_cl_N());
1277 * @param x real number
1278 * @param y real number
1279 * @return atan(y/x) */
1280 const numeric atan(const numeric &y, const numeric &x)
1282 if (x.is_real() && y.is_real())
1283 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1284 cln::the<cln::cl_R>(y.to_cl_N()));
1286 throw std::invalid_argument("atan(): complex argument");
1290 /** Numeric hyperbolic sine (trigonometric function).
1292 * @return arbitrary precision numerical sinh(x). */
1293 const numeric sinh(const numeric &x)
1295 return cln::sinh(x.to_cl_N());
1299 /** Numeric hyperbolic cosine (trigonometric function).
1301 * @return arbitrary precision numerical cosh(x). */
1302 const numeric cosh(const numeric &x)
1304 return cln::cosh(x.to_cl_N());
1308 /** Numeric hyperbolic tangent (trigonometric function).
1310 * @return arbitrary precision numerical tanh(x). */
1311 const numeric tanh(const numeric &x)
1313 return cln::tanh(x.to_cl_N());
1317 /** Numeric inverse hyperbolic sine (trigonometric function).
1319 * @return arbitrary precision numerical asinh(x). */
1320 const numeric asinh(const numeric &x)
1322 return cln::asinh(x.to_cl_N());
1326 /** Numeric inverse hyperbolic cosine (trigonometric function).
1328 * @return arbitrary precision numerical acosh(x). */
1329 const numeric acosh(const numeric &x)
1331 return cln::acosh(x.to_cl_N());
1335 /** Numeric inverse hyperbolic tangent (trigonometric function).
1337 * @return arbitrary precision numerical atanh(x). */
1338 const numeric atanh(const numeric &x)
1340 return cln::atanh(x.to_cl_N());
1344 /*static cln::cl_N Li2_series(const ::cl_N &x,
1345 const ::float_format_t &prec)
1347 // Note: argument must be in the unit circle
1348 // This is very inefficient unless we have fast floating point Bernoulli
1349 // numbers implemented!
1350 cln::cl_N c1 = -cln::log(1-x);
1352 // hard-wire the first two Bernoulli numbers
1353 cln::cl_N acc = c1 - cln::square(c1)/4;
1355 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1356 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1358 c1 = cln::square(c1);
1362 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1363 // aug = c2 * cln::cl_I(i%2 ? 1 : -1) / cln::cl_I(2*i+1) * cln::cl_zeta(2*i, prec) / piac / (cln::cl_I(1)<<(2*i-1));
1366 } while (acc != acc+aug);
1370 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1371 * circle) using a power series. */
1372 static cln::cl_N Li2_series(const cln::cl_N &x,
1373 const cln::float_format_t &prec)
1375 // Note: argument must be in the unit circle
1377 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1382 den = den + i; // 1, 4, 9, 16, ...
1386 } while (acc != acc+aug);
1390 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1391 static cln::cl_N Li2_projection(const cln::cl_N &x,
1392 const cln::float_format_t &prec)
1394 const cln::cl_R re = cln::realpart(x);
1395 const cln::cl_R im = cln::imagpart(x);
1396 if (re > cln::cl_F(".5"))
1397 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1399 - Li2_series(1-x, prec)
1400 - cln::log(x)*cln::log(1-x));
1401 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1402 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1403 return(- cln::square(cln::log(1-x))/2
1404 - Li2_series(x/(x-1), prec));
1405 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1406 // Li2(x^2)/2 - Li2(-x)
1407 return(Li2_projection(cln::square(x), prec)/2
1408 - Li2_projection(-x, prec));
1409 return Li2_series(x, prec);
1412 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1413 * the branch cut lies along the positive real axis, starting at 1 and
1414 * continuous with quadrant IV.
1416 * @return arbitrary precision numerical Li2(x). */
1417 const numeric Li2(const numeric &x)
1422 // what is the desired float format?
1423 // first guess: default format
1424 cln::float_format_t prec = cln::default_float_format;
1425 const cln::cl_N value = x.to_cl_N();
1426 // second guess: the argument's format
1427 if (!x.real().is_rational())
1428 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1429 else if (!x.imag().is_rational())
1430 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1432 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1433 return cln::zeta(2, prec);
1435 if (cln::abs(value) > 1)
1436 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1437 return(- cln::square(cln::log(-value))/2
1438 - cln::zeta(2, prec)
1439 - Li2_projection(cln::recip(value), prec));
1441 return Li2_projection(x.to_cl_N(), prec);
1445 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1446 * integer arguments. */
1447 const numeric zeta(const numeric &x)
1449 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1450 // only knows about integer arguments and zeta(3).evalf() automatically
1451 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1452 // being an exact zero for CLN, which can be tested and then we can just
1453 // pass the number casted to an int:
1455 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1456 if (cln::zerop(x.to_cl_N()-aux))
1457 return cln::zeta(aux);
1459 std::clog << "zeta(" << x
1460 << "): Does anybody know a good way to calculate this numerically?"
1466 /** The Gamma function.
1467 * This is only a stub! */
1468 const numeric lgamma(const numeric &x)
1470 std::clog << "lgamma(" << x
1471 << "): Does anybody know a good way to calculate this numerically?"
1475 const numeric tgamma(const numeric &x)
1477 std::clog << "tgamma(" << x
1478 << "): Does anybody know a good way to calculate this numerically?"
1484 /** The psi function (aka polygamma function).
1485 * This is only a stub! */
1486 const numeric psi(const numeric &x)
1488 std::clog << "psi(" << x
1489 << "): Does anybody know a good way to calculate this numerically?"
1495 /** The psi functions (aka polygamma functions).
1496 * This is only a stub! */
1497 const numeric psi(const numeric &n, const numeric &x)
1499 std::clog << "psi(" << n << "," << x
1500 << "): Does anybody know a good way to calculate this numerically?"
1506 /** Factorial combinatorial function.
1508 * @param n integer argument >= 0
1509 * @exception range_error (argument must be integer >= 0) */
1510 const numeric factorial(const numeric &n)
1512 if (!n.is_nonneg_integer())
1513 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1514 return numeric(cln::factorial(n.to_int()));
1518 /** The double factorial combinatorial function. (Scarcely used, but still
1519 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1521 * @param n integer argument >= -1
1522 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1523 * @exception range_error (argument must be integer >= -1) */
1524 const numeric doublefactorial(const numeric &n)
1526 if (n.is_equal(_num_1()))
1529 if (!n.is_nonneg_integer())
1530 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1532 return numeric(cln::doublefactorial(n.to_int()));
1536 /** The Binomial coefficients. It computes the binomial coefficients. For
1537 * integer n and k and positive n this is the number of ways of choosing k
1538 * objects from n distinct objects. If n is negative, the formula
1539 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1540 const numeric binomial(const numeric &n, const numeric &k)
1542 if (n.is_integer() && k.is_integer()) {
1543 if (n.is_nonneg_integer()) {
1544 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1545 return numeric(cln::binomial(n.to_int(),k.to_int()));
1549 return _num_1().power(k)*binomial(k-n-_num1(),k);
1553 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1554 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1558 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1559 * in the expansion of the function x/(e^x-1).
1561 * @return the nth Bernoulli number (a rational number).
1562 * @exception range_error (argument must be integer >= 0) */
1563 const numeric bernoulli(const numeric &nn)
1565 if (!nn.is_integer() || nn.is_negative())
1566 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1570 // The Bernoulli numbers are rational numbers that may be computed using
1573 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1575 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1576 // previous ones, the computation is necessarily very expensive. There are
1577 // several other ways of computing them, a particularly good one being
1581 // for (unsigned i=0; i<n; i++) {
1582 // c = exquo(c*(i-n),(i+2));
1583 // Bern = Bern + c*s/(i+2);
1584 // s = s + expt_pos(cl_I(i+2),n);
1588 // But if somebody works with the n'th Bernoulli number she is likely to
1589 // also need all previous Bernoulli numbers. So we need a complete remember
1590 // table and above divide and conquer algorithm is not suited to build one
1591 // up. The code below is adapted from Pari's function bernvec().
1593 // (There is an interesting relation with the tangent polynomials described
1594 // in `Concrete Mathematics', which leads to a program twice as fast as our
1595 // implementation below, but it requires storing one such polynomial in
1596 // addition to the remember table. This doubles the memory footprint so
1597 // we don't use it.)
1599 // the special cases not covered by the algorithm below
1600 if (nn.is_equal(_num1()))
1605 // store nonvanishing Bernoulli numbers here
1606 static std::vector< cln::cl_RA > results;
1607 static int highest_result = 0;
1608 // algorithm not applicable to B(0), so just store it
1609 if (results.size()==0)
1610 results.push_back(cln::cl_RA(1));
1612 int n = nn.to_long();
1613 for (int i=highest_result; i<n/2; ++i) {
1619 for (int j=i; j>0; --j) {
1620 B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
1626 B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
1627 results.push_back(B);
1630 return results[n/2];
1634 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1635 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1637 * @param n an integer
1638 * @return the nth Fibonacci number F(n) (an integer number)
1639 * @exception range_error (argument must be an integer) */
1640 const numeric fibonacci(const numeric &n)
1642 if (!n.is_integer())
1643 throw std::range_error("numeric::fibonacci(): argument must be integer");
1646 // The following addition formula holds:
1648 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1650 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1651 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1653 // Replace m by m+1:
1654 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1655 // Now put in m = n, to get
1656 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1657 // F(2n+1) = F(n)^2 + F(n+1)^2
1659 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1662 if (n.is_negative())
1664 return -fibonacci(-n);
1666 return fibonacci(-n);
1670 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1671 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1672 // Since a squaring is cheaper than a multiplication, better use
1673 // three squarings instead of one multiplication and two squarings.
1674 cln::cl_I u2 = cln::square(u);
1675 cln::cl_I v2 = cln::square(v);
1676 if (cln::logbitp(bit-1, m)) {
1677 v = cln::square(u + v) - u2;
1680 u = v2 - cln::square(v - u);
1685 // Here we don't use the squaring formula because one multiplication
1686 // is cheaper than two squarings.
1687 return u * ((v << 1) - u);
1689 return cln::square(u) + cln::square(v);
1693 /** Absolute value. */
1694 const numeric abs(const numeric& x)
1696 return cln::abs(x.to_cl_N());
1700 /** Modulus (in positive representation).
1701 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1702 * sign of a or is zero. This is different from Maple's modp, where the sign
1703 * of b is ignored. It is in agreement with Mathematica's Mod.
1705 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1706 * integer, 0 otherwise. */
1707 const numeric mod(const numeric &a, const numeric &b)
1709 if (a.is_integer() && b.is_integer())
1710 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1711 cln::the<cln::cl_I>(b.to_cl_N()));
1717 /** Modulus (in symmetric representation).
1718 * Equivalent to Maple's mods.
1720 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1721 const numeric smod(const numeric &a, const numeric &b)
1723 if (a.is_integer() && b.is_integer()) {
1724 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1725 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1726 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1732 /** Numeric integer remainder.
1733 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1734 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1735 * sign of a or is zero.
1737 * @return remainder of a/b if both are integer, 0 otherwise. */
1738 const numeric irem(const numeric &a, const numeric &b)
1740 if (a.is_integer() && b.is_integer())
1741 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1742 cln::the<cln::cl_I>(b.to_cl_N()));
1748 /** Numeric integer remainder.
1749 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1750 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1751 * and irem(a,b) has the sign of a or is zero.
1753 * @return remainder of a/b and quotient stored in q if both are integer,
1755 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1757 if (a.is_integer() && b.is_integer()) {
1758 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1759 cln::the<cln::cl_I>(b.to_cl_N()));
1760 q = rem_quo.quotient;
1761 return rem_quo.remainder;
1769 /** Numeric integer quotient.
1770 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1772 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1773 const numeric iquo(const numeric &a, const numeric &b)
1775 if (a.is_integer() && b.is_integer())
1776 return truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1777 cln::the<cln::cl_I>(b.to_cl_N()));
1783 /** Numeric integer quotient.
1784 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1785 * r == a - iquo(a,b,r)*b.
1787 * @return truncated quotient of a/b and remainder stored in r if both are
1788 * integer, 0 otherwise. */
1789 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1791 if (a.is_integer() && b.is_integer()) {
1792 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1793 cln::the<cln::cl_I>(b.to_cl_N()));
1794 r = rem_quo.remainder;
1795 return rem_quo.quotient;
1803 /** Greatest Common Divisor.
1805 * @return The GCD of two numbers if both are integer, a numerical 1
1806 * if they are not. */
1807 const numeric gcd(const numeric &a, const numeric &b)
1809 if (a.is_integer() && b.is_integer())
1810 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1811 cln::the<cln::cl_I>(b.to_cl_N()));
1817 /** Least Common Multiple.
1819 * @return The LCM of two numbers if both are integer, the product of those
1820 * two numbers if they are not. */
1821 const numeric lcm(const numeric &a, const numeric &b)
1823 if (a.is_integer() && b.is_integer())
1824 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1825 cln::the<cln::cl_I>(b.to_cl_N()));
1831 /** Numeric square root.
1832 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1833 * should return integer 2.
1835 * @param z numeric argument
1836 * @return square root of z. Branch cut along negative real axis, the negative
1837 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1838 * where imag(z)>0. */
1839 const numeric sqrt(const numeric &z)
1841 return cln::sqrt(z.to_cl_N());
1845 /** Integer numeric square root. */
1846 const numeric isqrt(const numeric &x)
1848 if (x.is_integer()) {
1850 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1857 /** Floating point evaluation of Archimedes' constant Pi. */
1860 return numeric(cln::pi(cln::default_float_format));
1864 /** Floating point evaluation of Euler's constant gamma. */
1867 return numeric(cln::eulerconst(cln::default_float_format));
1871 /** Floating point evaluation of Catalan's constant. */
1872 ex CatalanEvalf(void)
1874 return numeric(cln::catalanconst(cln::default_float_format));
1878 /** _numeric_digits default ctor, checking for singleton invariance. */
1879 _numeric_digits::_numeric_digits()
1882 // It initializes to 17 digits, because in CLN float_format(17) turns out
1883 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1884 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1886 throw(std::runtime_error("I told you not to do instantiate me!"));
1888 cln::default_float_format = cln::float_format(17);
1892 /** Assign a native long to global Digits object. */
1893 _numeric_digits& _numeric_digits::operator=(long prec)
1896 cln::default_float_format = cln::float_format(prec);
1901 /** Convert global Digits object to native type long. */
1902 _numeric_digits::operator long()
1904 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1905 return (long)digits;
1909 /** Append global Digits object to ostream. */
1910 void _numeric_digits::print(std::ostream &os) const
1912 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1917 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1924 // static member variables
1929 bool _numeric_digits::too_late = false;
1932 /** Accuracy in decimal digits. Only object of this type! Can be set using
1933 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1934 _numeric_digits Digits;
1936 } // namespace GiNaC