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
75 /** default ctor. Numerically it initializes to an integer zero. */
76 numeric::numeric() : basic(TINFO_numeric)
78 debugmsg("numeric default ctor", LOGLEVEL_CONSTRUCT);
80 setflag(status_flags::evaluated | status_flags::expanded);
83 void numeric::copy(const numeric &other)
85 inherited::copy(other);
89 DEFAULT_DESTROY(numeric)
97 numeric::numeric(int i) : basic(TINFO_numeric)
99 debugmsg("numeric ctor from int",LOGLEVEL_CONSTRUCT);
100 // Not the whole int-range is available if we don't cast to long
101 // first. This is due to the behaviour of the cl_I-ctor, which
102 // emphasizes efficiency. However, if the integer is small enough,
103 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
104 // using an immediate type:
105 if (cln::cl_immediate_p(i))
106 value = cln::cl_I(i);
108 value = cln::cl_I((long) i);
109 setflag(status_flags::evaluated | status_flags::expanded);
113 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
115 debugmsg("numeric ctor from uint",LOGLEVEL_CONSTRUCT);
116 // Not the whole uint-range is available if we don't cast to ulong
117 // first. This is due to the behaviour of the cl_I-ctor, which
118 // emphasizes efficiency. However, if the integer is small enough,
119 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
120 // using an immediate type:
121 if (cln::cl_immediate_p(i))
122 value = cln::cl_I(i);
124 value = cln::cl_I((unsigned long) i);
125 setflag(status_flags::evaluated | status_flags::expanded);
129 numeric::numeric(long i) : basic(TINFO_numeric)
131 debugmsg("numeric ctor from long",LOGLEVEL_CONSTRUCT);
132 value = cln::cl_I(i);
133 setflag(status_flags::evaluated | status_flags::expanded);
137 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
139 debugmsg("numeric ctor from ulong",LOGLEVEL_CONSTRUCT);
140 value = cln::cl_I(i);
141 setflag(status_flags::evaluated | status_flags::expanded);
144 /** Ctor for rational numerics a/b.
146 * @exception overflow_error (division by zero) */
147 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
149 debugmsg("numeric ctor from long/long",LOGLEVEL_CONSTRUCT);
151 throw std::overflow_error("division by zero");
152 value = cln::cl_I(numer) / cln::cl_I(denom);
153 setflag(status_flags::evaluated | status_flags::expanded);
157 numeric::numeric(double d) : basic(TINFO_numeric)
159 debugmsg("numeric ctor from double",LOGLEVEL_CONSTRUCT);
160 // We really want to explicitly use the type cl_LF instead of the
161 // more general cl_F, since that would give us a cl_DF only which
162 // will not be promoted to cl_LF if overflow occurs:
163 value = cln::cl_float(d, cln::default_float_format);
164 setflag(status_flags::evaluated | status_flags::expanded);
168 /** ctor from C-style string. It also accepts complex numbers in GiNaC
169 * notation like "2+5*I". */
170 numeric::numeric(const char *s) : basic(TINFO_numeric)
172 debugmsg("numeric ctor from string",LOGLEVEL_CONSTRUCT);
173 cln::cl_N ctorval = 0;
174 // parse complex numbers (functional but not completely safe, unfortunately
175 // std::string does not understand regexpese):
176 // ss should represent a simple sum like 2+5*I
178 // make it safe by adding explicit sign
179 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
181 std::string::size_type delim;
183 // chop ss into terms from left to right
185 bool imaginary = false;
186 delim = ss.find_first_of(std::string("+-"),1);
187 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
188 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
189 delim = ss.find_first_of(std::string("+-"),delim+1);
190 term = ss.substr(0,delim);
191 if (delim != std::string::npos)
192 ss = ss.substr(delim);
193 // is the term imaginary?
194 if (term.find("I") != std::string::npos) {
196 term = term.replace(term.find("I"),1,"");
198 if (term.find("*") != std::string::npos)
199 term = term.replace(term.find("*"),1,"");
200 // correct for trivial +/-I without explicit factor on I:
201 if (term.size() == 1)
205 if (term.find(".") != std::string::npos) {
206 // CLN's short type cl_SF is not very useful within the GiNaC
207 // framework where we are mainly interested in the arbitrary
208 // precision type cl_LF. Hence we go straight to the construction
209 // of generic floats. In order to create them we have to convert
210 // our own floating point notation used for output and construction
211 // from char * to CLN's generic notation:
212 // 3.14 --> 3.14e0_<Digits>
213 // 31.4E-1 --> 31.4e-1_<Digits>
215 // No exponent marker? Let's add a trivial one.
216 if (term.find("E") == std::string::npos)
219 term = term.replace(term.find("E"),1,"e");
220 // append _<Digits> to term
221 #if defined(HAVE_SSTREAM)
222 std::ostringstream buf;
223 buf << unsigned(Digits) << std::ends;
224 term += "_" + buf.str();
227 std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
228 term += "_" + std::string(buf);
230 // construct float using cln::cl_F(const char *) ctor.
232 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
234 ctorval = ctorval + cln::cl_F(term.c_str());
236 // not a floating point number...
238 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
240 ctorval = ctorval + cln::cl_R(term.c_str());
242 } while(delim != std::string::npos);
244 setflag(status_flags::evaluated | status_flags::expanded);
248 /** Ctor from CLN types. This is for the initiated user or internal use
250 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
252 debugmsg("numeric ctor from cl_N", LOGLEVEL_CONSTRUCT);
254 setflag(status_flags::evaluated | status_flags::expanded);
261 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
263 debugmsg("numeric ctor from archive_node", LOGLEVEL_CONSTRUCT);
264 cln::cl_N ctorval = 0;
266 // Read number as string
268 if (n.find_string("number", str)) {
270 std::istringstream s(str);
272 std::istrstream s(str.c_str(), str.size() + 1);
274 cln::cl_idecoded_float re, im;
278 case 'R': // Integer-decoded real number
279 s >> re.sign >> re.mantissa >> re.exponent;
280 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
282 case 'C': // Integer-decoded complex number
283 s >> re.sign >> re.mantissa >> re.exponent;
284 s >> im.sign >> im.mantissa >> im.exponent;
285 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
286 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
288 default: // Ordinary number
295 setflag(status_flags::evaluated | status_flags::expanded);
298 void numeric::archive(archive_node &n) const
300 inherited::archive(n);
302 // Write number as string
304 std::ostringstream s;
307 std::ostrstream s(buf, 1024);
309 if (this->is_crational())
310 s << cln::the<cln::cl_N>(value);
312 // Non-rational numbers are written in an integer-decoded format
313 // to preserve the precision
314 if (this->is_real()) {
315 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
317 s << re.sign << " " << re.mantissa << " " << re.exponent;
319 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
320 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
322 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
323 s << im.sign << " " << im.mantissa << " " << im.exponent;
327 n.add_string("number", s.str());
330 std::string str(buf);
331 n.add_string("number", str);
335 DEFAULT_UNARCHIVE(numeric)
338 // functions overriding virtual functions from bases classes
341 /** Helper function to print a real number in a nicer way than is CLN's
342 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
343 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
344 * long as it only uses cl_LF and no other floating point types that we might
345 * want to visibly distinguish from cl_LF.
347 * @see numeric::print() */
348 static void print_real_number(std::ostream &os, const cln::cl_R &num)
350 cln::cl_print_flags ourflags;
351 if (cln::instanceof(num, cln::cl_RA_ring)) {
352 // case 1: integer or rational, nothing special to do:
353 cln::print_real(os, ourflags, num);
356 // make CLN believe this number has default_float_format, so it prints
357 // 'E' as exponent marker instead of 'L':
358 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(num));
359 cln::print_real(os, ourflags, num);
364 /** This method adds to the output so it blends more consistently together
365 * with the other routines and produces something compatible to ginsh input.
367 * @see print_real_number() */
368 void numeric::print(std::ostream &os, unsigned upper_precedence) const
370 debugmsg("numeric print", LOGLEVEL_PRINT);
371 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
372 cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
374 // case 1, real: x or -x
375 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
377 print_real_number(os, r);
380 print_real_number(os, r);
384 // case 2, imaginary: y*I or -y*I
385 if ((precedence<=upper_precedence) && (i < 0)) {
390 print_real_number(os, i);
400 print_real_number(os, i);
406 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
407 if (precedence <= upper_precedence)
409 print_real_number(os, r);
414 print_real_number(os, i);
422 print_real_number(os, i);
426 if (precedence <= upper_precedence)
433 void numeric::printraw(std::ostream &os) const
435 // The method printraw doesn't do much, it simply uses CLN's operator<<()
436 // for output, which is ugly but reliable. e.g: 2+2i
437 debugmsg("numeric printraw", LOGLEVEL_PRINT);
438 os << class_name() << "(" << cln::the<cln::cl_N>(value) << ")";
442 void numeric::printtree(std::ostream &os, unsigned indent) const
444 debugmsg("numeric printtree", LOGLEVEL_PRINT);
445 os << std::string(indent,' ') << cln::the<cln::cl_N>(value)
447 << "hash=" << hashvalue
448 << " (0x" << std::hex << hashvalue << std::dec << ")"
449 << ", flags=" << flags << std::endl;
453 void numeric::printcsrc(std::ostream &os, unsigned type, unsigned upper_precedence) const
455 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
456 std::ios::fmtflags oldflags = os.flags();
457 os.setf(std::ios::scientific);
458 if (this->is_rational() && !this->is_integer()) {
459 if (compare(_num0()) > 0) {
461 if (type == csrc_types::ctype_cl_N)
462 os << "cln::cl_F(\"" << numer().evalf() << "\")";
464 os << numer().to_double();
467 if (type == csrc_types::ctype_cl_N)
468 os << "cln::cl_F(\"" << -numer().evalf() << "\")";
470 os << -numer().to_double();
473 if (type == csrc_types::ctype_cl_N)
474 os << "cln::cl_F(\"" << denom().evalf() << "\")";
476 os << denom().to_double();
479 if (type == csrc_types::ctype_cl_N)
480 os << "cln::cl_F(\"" << evalf() << "\")";
488 bool numeric::info(unsigned inf) const
491 case info_flags::numeric:
492 case info_flags::polynomial:
493 case info_flags::rational_function:
495 case info_flags::real:
497 case info_flags::rational:
498 case info_flags::rational_polynomial:
499 return is_rational();
500 case info_flags::crational:
501 case info_flags::crational_polynomial:
502 return is_crational();
503 case info_flags::integer:
504 case info_flags::integer_polynomial:
506 case info_flags::cinteger:
507 case info_flags::cinteger_polynomial:
508 return is_cinteger();
509 case info_flags::positive:
510 return is_positive();
511 case info_flags::negative:
512 return is_negative();
513 case info_flags::nonnegative:
514 return !is_negative();
515 case info_flags::posint:
516 return is_pos_integer();
517 case info_flags::negint:
518 return is_integer() && is_negative();
519 case info_flags::nonnegint:
520 return is_nonneg_integer();
521 case info_flags::even:
523 case info_flags::odd:
525 case info_flags::prime:
527 case info_flags::algebraic:
533 /** Disassemble real part and imaginary part to scan for the occurrence of a
534 * single number. Also handles the imaginary unit. It ignores the sign on
535 * both this and the argument, which may lead to what might appear as funny
536 * results: (2+I).has(-2) -> true. But this is consistent, since we also
537 * would like to have (-2+I).has(2) -> true and we want to think about the
538 * sign as a multiplicative factor. */
539 bool numeric::has(const ex &other) const
541 if (!is_exactly_of_type(*other.bp, numeric))
543 const numeric &o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
544 if (this->is_equal(o) || this->is_equal(-o))
546 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
547 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
548 this->real().is_equal(-o) || this->imag().is_equal(-o));
550 if (o.is_equal(I)) // e.g scan for I in 42*I
551 return !this->is_real();
552 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
553 return (this->real().has(o*I) || this->imag().has(o*I) ||
554 this->real().has(-o*I) || this->imag().has(-o*I));
560 /** Evaluation of numbers doesn't do anything at all. */
561 ex numeric::eval(int level) const
563 // Warning: if this is ever gonna do something, the ex ctors from all kinds
564 // of numbers should be checking for status_flags::evaluated.
569 /** Cast numeric into a floating-point object. For example exact numeric(1) is
570 * returned as a 1.0000000000000000000000 and so on according to how Digits is
571 * currently set. In case the object already was a floating point number the
572 * precision is trimmed to match the currently set default.
574 * @param level ignored, only needed for overriding basic::evalf.
575 * @return an ex-handle to a numeric. */
576 ex numeric::evalf(int level) const
578 // level can safely be discarded for numeric objects.
579 return numeric(cln::cl_float(1.0, cln::default_float_format) *
580 (cln::the<cln::cl_N>(value)));
585 int numeric::compare_same_type(const basic &other) const
587 GINAC_ASSERT(is_exactly_of_type(other, numeric));
588 const numeric &o = static_cast<numeric &>(const_cast<basic &>(other));
590 return this->compare(o);
594 bool numeric::is_equal_same_type(const basic &other) const
596 GINAC_ASSERT(is_exactly_of_type(other,numeric));
597 const numeric *o = static_cast<const numeric *>(&other);
599 return this->is_equal(*o);
603 unsigned numeric::calchash(void) const
605 // Use CLN's hashcode. Warning: It depends only on the number's value, not
606 // its type or precision (i.e. a true equivalence relation on numbers). As
607 // a consequence, 3 and 3.0 share the same hashvalue.
608 setflag(status_flags::hash_calculated);
609 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
614 // new virtual functions which can be overridden by derived classes
620 // non-virtual functions in this class
625 /** Numerical addition method. Adds argument to *this and returns result as
626 * a numeric object. */
627 const numeric numeric::add(const numeric &other) const
629 // Efficiency shortcut: trap the neutral element by pointer.
630 static const numeric * _num0p = &_num0();
633 else if (&other==_num0p)
636 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
640 /** Numerical subtraction method. Subtracts argument from *this and returns
641 * result as a numeric object. */
642 const numeric numeric::sub(const numeric &other) const
644 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
648 /** Numerical multiplication method. Multiplies *this and argument and returns
649 * result as a numeric object. */
650 const numeric numeric::mul(const numeric &other) const
652 // Efficiency shortcut: trap the neutral element by pointer.
653 static const numeric * _num1p = &_num1();
656 else if (&other==_num1p)
659 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
663 /** Numerical division method. Divides *this by argument and returns result as
666 * @exception overflow_error (division by zero) */
667 const numeric numeric::div(const numeric &other) const
669 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
670 throw std::overflow_error("numeric::div(): division by zero");
671 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
675 /** Numerical exponentiation. Raises *this to the power given as argument and
676 * returns result as a numeric object. */
677 const numeric numeric::power(const numeric &other) const
679 // Efficiency shortcut: trap the neutral exponent by pointer.
680 static const numeric * _num1p = &_num1();
684 if (cln::zerop(cln::the<cln::cl_N>(value))) {
685 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
686 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
687 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
688 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
689 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
690 throw std::overflow_error("numeric::eval(): division by zero");
694 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
698 const numeric &numeric::add_dyn(const numeric &other) const
700 // Efficiency shortcut: trap the neutral element by pointer.
701 static const numeric * _num0p = &_num0();
704 else if (&other==_num0p)
707 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
708 setflag(status_flags::dynallocated));
712 const numeric &numeric::sub_dyn(const numeric &other) const
714 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
715 setflag(status_flags::dynallocated));
719 const numeric &numeric::mul_dyn(const numeric &other) const
721 // Efficiency shortcut: trap the neutral element by pointer.
722 static const numeric * _num1p = &_num1();
725 else if (&other==_num1p)
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::div_dyn(const numeric &other) const
735 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
736 throw std::overflow_error("division by zero");
737 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
738 setflag(status_flags::dynallocated));
742 const numeric &numeric::power_dyn(const numeric &other) const
744 // Efficiency shortcut: trap the neutral exponent by pointer.
745 static const numeric * _num1p=&_num1();
749 if (cln::zerop(cln::the<cln::cl_N>(value))) {
750 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
751 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
752 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
753 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
754 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
755 throw std::overflow_error("numeric::eval(): division by zero");
759 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
760 setflag(status_flags::dynallocated));
764 const numeric &numeric::operator=(int i)
766 return operator=(numeric(i));
770 const numeric &numeric::operator=(unsigned int i)
772 return operator=(numeric(i));
776 const numeric &numeric::operator=(long i)
778 return operator=(numeric(i));
782 const numeric &numeric::operator=(unsigned long i)
784 return operator=(numeric(i));
788 const numeric &numeric::operator=(double d)
790 return operator=(numeric(d));
794 const numeric &numeric::operator=(const char * s)
796 return operator=(numeric(s));
800 /** Inverse of a number. */
801 const numeric numeric::inverse(void) const
803 if (cln::zerop(cln::the<cln::cl_N>(value)))
804 throw std::overflow_error("numeric::inverse(): division by zero");
805 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
809 /** Return the complex half-plane (left or right) in which the number lies.
810 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
811 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
813 * @see numeric::compare(const numeric &other) */
814 int numeric::csgn(void) const
816 if (cln::zerop(cln::the<cln::cl_N>(value)))
818 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
819 if (!cln::zerop(r)) {
825 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
833 /** This method establishes a canonical order on all numbers. For complex
834 * numbers this is not possible in a mathematically consistent way but we need
835 * to establish some order and it ought to be fast. So we simply define it
836 * to be compatible with our method csgn.
838 * @return csgn(*this-other)
839 * @see numeric::csgn(void) */
840 int numeric::compare(const numeric &other) const
842 // Comparing two real numbers?
843 if (cln::instanceof(value, cln::cl_R_ring) &&
844 cln::instanceof(other.value, cln::cl_R_ring))
845 // Yes, so just cln::compare them
846 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
848 // No, first cln::compare real parts...
849 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
852 // ...and then the imaginary parts.
853 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
858 bool numeric::is_equal(const numeric &other) const
860 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
864 /** True if object is zero. */
865 bool numeric::is_zero(void) const
867 return cln::zerop(cln::the<cln::cl_N>(value));
871 /** True if object is not complex and greater than zero. */
872 bool numeric::is_positive(void) const
875 return cln::plusp(cln::the<cln::cl_R>(value));
880 /** True if object is not complex and less than zero. */
881 bool numeric::is_negative(void) const
884 return cln::minusp(cln::the<cln::cl_R>(value));
889 /** True if object is a non-complex integer. */
890 bool numeric::is_integer(void) const
892 return cln::instanceof(value, cln::cl_I_ring);
896 /** True if object is an exact integer greater than zero. */
897 bool numeric::is_pos_integer(void) const
899 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
903 /** True if object is an exact integer greater or equal zero. */
904 bool numeric::is_nonneg_integer(void) const
906 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
910 /** True if object is an exact even integer. */
911 bool numeric::is_even(void) const
913 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
917 /** True if object is an exact odd integer. */
918 bool numeric::is_odd(void) const
920 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
924 /** Probabilistic primality test.
926 * @return true if object is exact integer and prime. */
927 bool numeric::is_prime(void) const
929 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
933 /** True if object is an exact rational number, may even be complex
934 * (denominator may be unity). */
935 bool numeric::is_rational(void) const
937 return cln::instanceof(value, cln::cl_RA_ring);
941 /** True if object is a real integer, rational or float (but not complex). */
942 bool numeric::is_real(void) const
944 return cln::instanceof(value, cln::cl_R_ring);
948 bool numeric::operator==(const numeric &other) const
950 return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
954 bool numeric::operator!=(const numeric &other) const
956 return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
960 /** True if object is element of the domain of integers extended by I, i.e. is
961 * of the form a+b*I, where a and b are integers. */
962 bool numeric::is_cinteger(void) const
964 if (cln::instanceof(value, cln::cl_I_ring))
966 else if (!this->is_real()) { // complex case, handle n+m*I
967 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
968 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
975 /** True if object is an exact rational number, may even be complex
976 * (denominator may be unity). */
977 bool numeric::is_crational(void) const
979 if (cln::instanceof(value, cln::cl_RA_ring))
981 else if (!this->is_real()) { // complex case, handle Q(i):
982 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
983 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
990 /** Numerical comparison: less.
992 * @exception invalid_argument (complex inequality) */
993 bool numeric::operator<(const numeric &other) const
995 if (this->is_real() && other.is_real())
996 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
997 throw std::invalid_argument("numeric::operator<(): complex inequality");
1001 /** Numerical comparison: less or equal.
1003 * @exception invalid_argument (complex inequality) */
1004 bool numeric::operator<=(const numeric &other) const
1006 if (this->is_real() && other.is_real())
1007 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
1008 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1012 /** Numerical comparison: greater.
1014 * @exception invalid_argument (complex inequality) */
1015 bool numeric::operator>(const numeric &other) const
1017 if (this->is_real() && other.is_real())
1018 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1019 throw std::invalid_argument("numeric::operator>(): complex inequality");
1023 /** Numerical comparison: greater or equal.
1025 * @exception invalid_argument (complex inequality) */
1026 bool numeric::operator>=(const numeric &other) const
1028 if (this->is_real() && other.is_real())
1029 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1030 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1034 /** Converts numeric types to machine's int. You should check with
1035 * is_integer() if the number is really an integer before calling this method.
1036 * You may also consider checking the range first. */
1037 int numeric::to_int(void) const
1039 GINAC_ASSERT(this->is_integer());
1040 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1044 /** Converts numeric types to machine's long. You should check with
1045 * is_integer() if the number is really an integer before calling this method.
1046 * You may also consider checking the range first. */
1047 long numeric::to_long(void) const
1049 GINAC_ASSERT(this->is_integer());
1050 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1054 /** Converts numeric types to machine's double. You should check with is_real()
1055 * if the number is really not complex before calling this method. */
1056 double numeric::to_double(void) const
1058 GINAC_ASSERT(this->is_real());
1059 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1063 /** Returns a new CLN object of type cl_N, representing the value of *this.
1064 * This method may be used when mixing GiNaC and CLN in one project.
1066 cln::cl_N numeric::to_cl_N(void) const
1068 return cln::cl_N(cln::the<cln::cl_N>(value));
1072 /** Real part of a number. */
1073 const numeric numeric::real(void) const
1075 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1079 /** Imaginary part of a number. */
1080 const numeric numeric::imag(void) const
1082 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1086 /** Numerator. Computes the numerator of rational numbers, rationalized
1087 * numerator of complex if real and imaginary part are both rational numbers
1088 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1090 const numeric numeric::numer(void) const
1092 if (this->is_integer())
1093 return numeric(*this);
1095 else if (cln::instanceof(value, cln::cl_RA_ring))
1096 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1098 else if (!this->is_real()) { // complex case, handle Q(i):
1099 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1100 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1101 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1102 return numeric(*this);
1103 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1104 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1105 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1106 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1107 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1108 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1109 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1110 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1113 // at least one float encountered
1114 return numeric(*this);
1118 /** Denominator. Computes the denominator of rational numbers, common integer
1119 * denominator of complex if real and imaginary part are both rational numbers
1120 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1121 const numeric numeric::denom(void) const
1123 if (this->is_integer())
1126 if (instanceof(value, cln::cl_RA_ring))
1127 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1129 if (!this->is_real()) { // complex case, handle Q(i):
1130 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1131 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1132 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1134 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1135 return numeric(cln::denominator(i));
1136 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1137 return numeric(cln::denominator(r));
1138 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1139 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1141 // at least one float encountered
1146 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1147 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1148 * 2^(n-1) <= x < 2^n.
1150 * @return number of bits (excluding sign) needed to represent that number
1151 * in two's complement if it is an integer, 0 otherwise. */
1152 int numeric::int_length(void) const
1154 if (this->is_integer())
1155 return cln::integer_length(cln::the<cln::cl_I>(value));
1162 // static member variables
1167 unsigned numeric::precedence = 30;
1173 /** Imaginary unit. This is not a constant but a numeric since we are
1174 * natively handing complex numbers anyways, so in each expression containing
1175 * an I it is automatically eval'ed away anyhow. */
1176 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1179 /** Exponential function.
1181 * @return arbitrary precision numerical exp(x). */
1182 const numeric exp(const numeric &x)
1184 return cln::exp(x.to_cl_N());
1188 /** Natural logarithm.
1190 * @param z complex number
1191 * @return arbitrary precision numerical log(x).
1192 * @exception pole_error("log(): logarithmic pole",0) */
1193 const numeric log(const numeric &z)
1196 throw pole_error("log(): logarithmic pole",0);
1197 return cln::log(z.to_cl_N());
1201 /** Numeric sine (trigonometric function).
1203 * @return arbitrary precision numerical sin(x). */
1204 const numeric sin(const numeric &x)
1206 return cln::sin(x.to_cl_N());
1210 /** Numeric cosine (trigonometric function).
1212 * @return arbitrary precision numerical cos(x). */
1213 const numeric cos(const numeric &x)
1215 return cln::cos(x.to_cl_N());
1219 /** Numeric tangent (trigonometric function).
1221 * @return arbitrary precision numerical tan(x). */
1222 const numeric tan(const numeric &x)
1224 return cln::tan(x.to_cl_N());
1228 /** Numeric inverse sine (trigonometric function).
1230 * @return arbitrary precision numerical asin(x). */
1231 const numeric asin(const numeric &x)
1233 return cln::asin(x.to_cl_N());
1237 /** Numeric inverse cosine (trigonometric function).
1239 * @return arbitrary precision numerical acos(x). */
1240 const numeric acos(const numeric &x)
1242 return cln::acos(x.to_cl_N());
1248 * @param z complex number
1250 * @exception pole_error("atan(): logarithmic pole",0) */
1251 const numeric atan(const numeric &x)
1254 x.real().is_zero() &&
1255 abs(x.imag()).is_equal(_num1()))
1256 throw pole_error("atan(): logarithmic pole",0);
1257 return cln::atan(x.to_cl_N());
1263 * @param x real number
1264 * @param y real number
1265 * @return atan(y/x) */
1266 const numeric atan(const numeric &y, const numeric &x)
1268 if (x.is_real() && y.is_real())
1269 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1270 cln::the<cln::cl_R>(y.to_cl_N()));
1272 throw std::invalid_argument("atan(): complex argument");
1276 /** Numeric hyperbolic sine (trigonometric function).
1278 * @return arbitrary precision numerical sinh(x). */
1279 const numeric sinh(const numeric &x)
1281 return cln::sinh(x.to_cl_N());
1285 /** Numeric hyperbolic cosine (trigonometric function).
1287 * @return arbitrary precision numerical cosh(x). */
1288 const numeric cosh(const numeric &x)
1290 return cln::cosh(x.to_cl_N());
1294 /** Numeric hyperbolic tangent (trigonometric function).
1296 * @return arbitrary precision numerical tanh(x). */
1297 const numeric tanh(const numeric &x)
1299 return cln::tanh(x.to_cl_N());
1303 /** Numeric inverse hyperbolic sine (trigonometric function).
1305 * @return arbitrary precision numerical asinh(x). */
1306 const numeric asinh(const numeric &x)
1308 return cln::asinh(x.to_cl_N());
1312 /** Numeric inverse hyperbolic cosine (trigonometric function).
1314 * @return arbitrary precision numerical acosh(x). */
1315 const numeric acosh(const numeric &x)
1317 return cln::acosh(x.to_cl_N());
1321 /** Numeric inverse hyperbolic tangent (trigonometric function).
1323 * @return arbitrary precision numerical atanh(x). */
1324 const numeric atanh(const numeric &x)
1326 return cln::atanh(x.to_cl_N());
1330 /*static cln::cl_N Li2_series(const ::cl_N &x,
1331 const ::float_format_t &prec)
1333 // Note: argument must be in the unit circle
1334 // This is very inefficient unless we have fast floating point Bernoulli
1335 // numbers implemented!
1336 cln::cl_N c1 = -cln::log(1-x);
1338 // hard-wire the first two Bernoulli numbers
1339 cln::cl_N acc = c1 - cln::square(c1)/4;
1341 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1342 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1344 c1 = cln::square(c1);
1348 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1349 // 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));
1352 } while (acc != acc+aug);
1356 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1357 * circle) using a power series. */
1358 static cln::cl_N Li2_series(const cln::cl_N &x,
1359 const cln::float_format_t &prec)
1361 // Note: argument must be in the unit circle
1363 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1368 den = den + i; // 1, 4, 9, 16, ...
1372 } while (acc != acc+aug);
1376 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1377 static cln::cl_N Li2_projection(const cln::cl_N &x,
1378 const cln::float_format_t &prec)
1380 const cln::cl_R re = cln::realpart(x);
1381 const cln::cl_R im = cln::imagpart(x);
1382 if (re > cln::cl_F(".5"))
1383 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1385 - Li2_series(1-x, prec)
1386 - cln::log(x)*cln::log(1-x));
1387 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1388 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1389 return(- cln::square(cln::log(1-x))/2
1390 - Li2_series(x/(x-1), prec));
1391 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1392 // Li2(x^2)/2 - Li2(-x)
1393 return(Li2_projection(cln::square(x), prec)/2
1394 - Li2_projection(-x, prec));
1395 return Li2_series(x, prec);
1398 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1399 * the branch cut lies along the positive real axis, starting at 1 and
1400 * continuous with quadrant IV.
1402 * @return arbitrary precision numerical Li2(x). */
1403 const numeric Li2(const numeric &x)
1408 // what is the desired float format?
1409 // first guess: default format
1410 cln::float_format_t prec = cln::default_float_format;
1411 const cln::cl_N value = x.to_cl_N();
1412 // second guess: the argument's format
1413 if (!x.real().is_rational())
1414 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1415 else if (!x.imag().is_rational())
1416 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1418 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1419 return cln::zeta(2, prec);
1421 if (cln::abs(value) > 1)
1422 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1423 return(- cln::square(cln::log(-value))/2
1424 - cln::zeta(2, prec)
1425 - Li2_projection(cln::recip(value), prec));
1427 return Li2_projection(x.to_cl_N(), prec);
1431 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1432 * integer arguments. */
1433 const numeric zeta(const numeric &x)
1435 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1436 // only knows about integer arguments and zeta(3).evalf() automatically
1437 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1438 // being an exact zero for CLN, which can be tested and then we can just
1439 // pass the number casted to an int:
1441 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1442 if (cln::zerop(x.to_cl_N()-aux))
1443 return cln::zeta(aux);
1445 std::clog << "zeta(" << x
1446 << "): Does anybody know a good way to calculate this numerically?"
1452 /** The Gamma function.
1453 * This is only a stub! */
1454 const numeric lgamma(const numeric &x)
1456 std::clog << "lgamma(" << x
1457 << "): Does anybody know a good way to calculate this numerically?"
1461 const numeric tgamma(const numeric &x)
1463 std::clog << "tgamma(" << x
1464 << "): Does anybody know a good way to calculate this numerically?"
1470 /** The psi function (aka polygamma function).
1471 * This is only a stub! */
1472 const numeric psi(const numeric &x)
1474 std::clog << "psi(" << x
1475 << "): Does anybody know a good way to calculate this numerically?"
1481 /** The psi functions (aka polygamma functions).
1482 * This is only a stub! */
1483 const numeric psi(const numeric &n, const numeric &x)
1485 std::clog << "psi(" << n << "," << x
1486 << "): Does anybody know a good way to calculate this numerically?"
1492 /** Factorial combinatorial function.
1494 * @param n integer argument >= 0
1495 * @exception range_error (argument must be integer >= 0) */
1496 const numeric factorial(const numeric &n)
1498 if (!n.is_nonneg_integer())
1499 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1500 return numeric(cln::factorial(n.to_int()));
1504 /** The double factorial combinatorial function. (Scarcely used, but still
1505 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1507 * @param n integer argument >= -1
1508 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1509 * @exception range_error (argument must be integer >= -1) */
1510 const numeric doublefactorial(const numeric &n)
1512 if (n.is_equal(_num_1()))
1515 if (!n.is_nonneg_integer())
1516 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1518 return numeric(cln::doublefactorial(n.to_int()));
1522 /** The Binomial coefficients. It computes the binomial coefficients. For
1523 * integer n and k and positive n this is the number of ways of choosing k
1524 * objects from n distinct objects. If n is negative, the formula
1525 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1526 const numeric binomial(const numeric &n, const numeric &k)
1528 if (n.is_integer() && k.is_integer()) {
1529 if (n.is_nonneg_integer()) {
1530 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1531 return numeric(cln::binomial(n.to_int(),k.to_int()));
1535 return _num_1().power(k)*binomial(k-n-_num1(),k);
1539 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1540 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1544 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1545 * in the expansion of the function x/(e^x-1).
1547 * @return the nth Bernoulli number (a rational number).
1548 * @exception range_error (argument must be integer >= 0) */
1549 const numeric bernoulli(const numeric &nn)
1551 if (!nn.is_integer() || nn.is_negative())
1552 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1556 // The Bernoulli numbers are rational numbers that may be computed using
1559 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1561 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1562 // previous ones, the computation is necessarily very expensive. There are
1563 // several other ways of computing them, a particularly good one being
1567 // for (unsigned i=0; i<n; i++) {
1568 // c = exquo(c*(i-n),(i+2));
1569 // Bern = Bern + c*s/(i+2);
1570 // s = s + expt_pos(cl_I(i+2),n);
1574 // But if somebody works with the n'th Bernoulli number she is likely to
1575 // also need all previous Bernoulli numbers. So we need a complete remember
1576 // table and above divide and conquer algorithm is not suited to build one
1577 // up. The code below is adapted from Pari's function bernvec().
1579 // (There is an interesting relation with the tangent polynomials described
1580 // in `Concrete Mathematics', which leads to a program twice as fast as our
1581 // implementation below, but it requires storing one such polynomial in
1582 // addition to the remember table. This doubles the memory footprint so
1583 // we don't use it.)
1585 // the special cases not covered by the algorithm below
1586 if (nn.is_equal(_num1()))
1591 // store nonvanishing Bernoulli numbers here
1592 static std::vector< cln::cl_RA > results;
1593 static int highest_result = 0;
1594 // algorithm not applicable to B(0), so just store it
1595 if (results.size()==0)
1596 results.push_back(cln::cl_RA(1));
1598 int n = nn.to_long();
1599 for (int i=highest_result; i<n/2; ++i) {
1605 for (int j=i; j>0; --j) {
1606 B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
1612 B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
1613 results.push_back(B);
1616 return results[n/2];
1620 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1621 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1623 * @param n an integer
1624 * @return the nth Fibonacci number F(n) (an integer number)
1625 * @exception range_error (argument must be an integer) */
1626 const numeric fibonacci(const numeric &n)
1628 if (!n.is_integer())
1629 throw std::range_error("numeric::fibonacci(): argument must be integer");
1632 // The following addition formula holds:
1634 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1636 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1637 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1639 // Replace m by m+1:
1640 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1641 // Now put in m = n, to get
1642 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1643 // F(2n+1) = F(n)^2 + F(n+1)^2
1645 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1648 if (n.is_negative())
1650 return -fibonacci(-n);
1652 return fibonacci(-n);
1656 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1657 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1658 // Since a squaring is cheaper than a multiplication, better use
1659 // three squarings instead of one multiplication and two squarings.
1660 cln::cl_I u2 = cln::square(u);
1661 cln::cl_I v2 = cln::square(v);
1662 if (cln::logbitp(bit-1, m)) {
1663 v = cln::square(u + v) - u2;
1666 u = v2 - cln::square(v - u);
1671 // Here we don't use the squaring formula because one multiplication
1672 // is cheaper than two squarings.
1673 return u * ((v << 1) - u);
1675 return cln::square(u) + cln::square(v);
1679 /** Absolute value. */
1680 const numeric abs(const numeric& x)
1682 return cln::abs(x.to_cl_N());
1686 /** Modulus (in positive representation).
1687 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1688 * sign of a or is zero. This is different from Maple's modp, where the sign
1689 * of b is ignored. It is in agreement with Mathematica's Mod.
1691 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1692 * integer, 0 otherwise. */
1693 const numeric mod(const numeric &a, const numeric &b)
1695 if (a.is_integer() && b.is_integer())
1696 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1697 cln::the<cln::cl_I>(b.to_cl_N()));
1703 /** Modulus (in symmetric representation).
1704 * Equivalent to Maple's mods.
1706 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1707 const numeric smod(const numeric &a, const numeric &b)
1709 if (a.is_integer() && b.is_integer()) {
1710 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1711 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1712 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1718 /** Numeric integer remainder.
1719 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1720 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1721 * sign of a or is zero.
1723 * @return remainder of a/b if both are integer, 0 otherwise. */
1724 const numeric irem(const numeric &a, const numeric &b)
1726 if (a.is_integer() && b.is_integer())
1727 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1728 cln::the<cln::cl_I>(b.to_cl_N()));
1734 /** Numeric integer remainder.
1735 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1736 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1737 * and irem(a,b) has the sign of a or is zero.
1739 * @return remainder of a/b and quotient stored in q if both are integer,
1741 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1743 if (a.is_integer() && b.is_integer()) {
1744 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1745 cln::the<cln::cl_I>(b.to_cl_N()));
1746 q = rem_quo.quotient;
1747 return rem_quo.remainder;
1755 /** Numeric integer quotient.
1756 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1758 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1759 const numeric iquo(const numeric &a, const numeric &b)
1761 if (a.is_integer() && b.is_integer())
1762 return truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1763 cln::the<cln::cl_I>(b.to_cl_N()));
1769 /** Numeric integer quotient.
1770 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1771 * r == a - iquo(a,b,r)*b.
1773 * @return truncated quotient of a/b and remainder stored in r if both are
1774 * integer, 0 otherwise. */
1775 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1777 if (a.is_integer() && b.is_integer()) {
1778 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1779 cln::the<cln::cl_I>(b.to_cl_N()));
1780 r = rem_quo.remainder;
1781 return rem_quo.quotient;
1789 /** Greatest Common Divisor.
1791 * @return The GCD of two numbers if both are integer, a numerical 1
1792 * if they are not. */
1793 const numeric gcd(const numeric &a, const numeric &b)
1795 if (a.is_integer() && b.is_integer())
1796 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1797 cln::the<cln::cl_I>(b.to_cl_N()));
1803 /** Least Common Multiple.
1805 * @return The LCM of two numbers if both are integer, the product of those
1806 * two numbers if they are not. */
1807 const numeric lcm(const numeric &a, const numeric &b)
1809 if (a.is_integer() && b.is_integer())
1810 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1811 cln::the<cln::cl_I>(b.to_cl_N()));
1817 /** Numeric square root.
1818 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1819 * should return integer 2.
1821 * @param z numeric argument
1822 * @return square root of z. Branch cut along negative real axis, the negative
1823 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1824 * where imag(z)>0. */
1825 const numeric sqrt(const numeric &z)
1827 return cln::sqrt(z.to_cl_N());
1831 /** Integer numeric square root. */
1832 const numeric isqrt(const numeric &x)
1834 if (x.is_integer()) {
1836 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1843 /** Floating point evaluation of Archimedes' constant Pi. */
1846 return numeric(cln::pi(cln::default_float_format));
1850 /** Floating point evaluation of Euler's constant gamma. */
1853 return numeric(cln::eulerconst(cln::default_float_format));
1857 /** Floating point evaluation of Catalan's constant. */
1858 ex CatalanEvalf(void)
1860 return numeric(cln::catalanconst(cln::default_float_format));
1864 /** _numeric_digits default ctor, checking for singleton invariance. */
1865 _numeric_digits::_numeric_digits()
1868 // It initializes to 17 digits, because in CLN float_format(17) turns out
1869 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1870 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1872 throw(std::runtime_error("I told you not to do instantiate me!"));
1874 cln::default_float_format = cln::float_format(17);
1878 /** Assign a native long to global Digits object. */
1879 _numeric_digits& _numeric_digits::operator=(long prec)
1882 cln::default_float_format = cln::float_format(prec);
1887 /** Convert global Digits object to native type long. */
1888 _numeric_digits::operator long()
1890 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1891 return (long)digits;
1895 /** Append global Digits object to ostream. */
1896 void _numeric_digits::print(std::ostream &os) const
1898 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1903 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1910 // static member variables
1915 bool _numeric_digits::too_late = false;
1918 /** Accuracy in decimal digits. Only object of this type! Can be set using
1919 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1920 _numeric_digits Digits;
1922 } // namespace GiNaC