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
48 // CLN should pollute the global namespace as little as possible. Hence, we
49 // include most of it here and include only the part needed for properly
50 // declaring cln::cl_number in numeric.h. This can only be safely done in
51 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
52 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
54 #include <cln/output.h>
55 #include <cln/integer_io.h>
56 #include <cln/integer_ring.h>
57 #include <cln/rational_io.h>
58 #include <cln/rational_ring.h>
59 #include <cln/lfloat_class.h>
60 #include <cln/lfloat_io.h>
61 #include <cln/real_io.h>
62 #include <cln/real_ring.h>
63 #include <cln/complex_io.h>
64 #include <cln/complex_ring.h>
65 #include <cln/numtheory.h>
69 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
72 // default ctor, dtor, copy ctor assignment
73 // operator and helpers
76 /** default ctor. Numerically it initializes to an integer zero. */
77 numeric::numeric() : basic(TINFO_numeric)
79 debugmsg("numeric default ctor", LOGLEVEL_CONSTRUCT);
81 setflag(status_flags::evaluated | status_flags::expanded);
84 void numeric::copy(const numeric &other)
86 inherited::copy(other);
90 DEFAULT_DESTROY(numeric)
98 numeric::numeric(int i) : basic(TINFO_numeric)
100 debugmsg("numeric ctor from int",LOGLEVEL_CONSTRUCT);
101 // Not the whole int-range is available if we don't cast to long
102 // first. This is due to the behaviour of the cl_I-ctor, which
103 // emphasizes efficiency. However, if the integer is small enough,
104 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
105 // using an immediate type:
106 if (cln::cl_immediate_p(i))
107 value = cln::cl_I(i);
109 value = cln::cl_I((long) i);
110 setflag(status_flags::evaluated | status_flags::expanded);
114 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
116 debugmsg("numeric ctor from uint",LOGLEVEL_CONSTRUCT);
117 // Not the whole uint-range is available if we don't cast to ulong
118 // first. This is due to the behaviour of the cl_I-ctor, which
119 // emphasizes efficiency. However, if the integer is small enough,
120 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
121 // using an immediate type:
122 if (cln::cl_immediate_p(i))
123 value = cln::cl_I(i);
125 value = cln::cl_I((unsigned long) i);
126 setflag(status_flags::evaluated | status_flags::expanded);
130 numeric::numeric(long i) : basic(TINFO_numeric)
132 debugmsg("numeric ctor from long",LOGLEVEL_CONSTRUCT);
133 value = cln::cl_I(i);
134 setflag(status_flags::evaluated | status_flags::expanded);
138 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
140 debugmsg("numeric ctor from ulong",LOGLEVEL_CONSTRUCT);
141 value = cln::cl_I(i);
142 setflag(status_flags::evaluated | status_flags::expanded);
145 /** Ctor for rational numerics a/b.
147 * @exception overflow_error (division by zero) */
148 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
150 debugmsg("numeric ctor from long/long",LOGLEVEL_CONSTRUCT);
152 throw std::overflow_error("division by zero");
153 value = cln::cl_I(numer) / cln::cl_I(denom);
154 setflag(status_flags::evaluated | status_flags::expanded);
158 numeric::numeric(double d) : basic(TINFO_numeric)
160 debugmsg("numeric ctor from double",LOGLEVEL_CONSTRUCT);
161 // We really want to explicitly use the type cl_LF instead of the
162 // more general cl_F, since that would give us a cl_DF only which
163 // will not be promoted to cl_LF if overflow occurs:
164 value = cln::cl_float(d, cln::default_float_format);
165 setflag(status_flags::evaluated | status_flags::expanded);
169 /** ctor from C-style string. It also accepts complex numbers in GiNaC
170 * notation like "2+5*I". */
171 numeric::numeric(const char *s) : basic(TINFO_numeric)
173 debugmsg("numeric ctor from string",LOGLEVEL_CONSTRUCT);
174 cln::cl_N ctorval = 0;
175 // parse complex numbers (functional but not completely safe, unfortunately
176 // std::string does not understand regexpese):
177 // ss should represent a simple sum like 2+5*I
179 // make it safe by adding explicit sign
180 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
182 std::string::size_type delim;
184 // chop ss into terms from left to right
186 bool imaginary = false;
187 delim = ss.find_first_of(std::string("+-"),1);
188 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
189 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
190 delim = ss.find_first_of(std::string("+-"),delim+1);
191 term = ss.substr(0,delim);
192 if (delim != std::string::npos)
193 ss = ss.substr(delim);
194 // is the term imaginary?
195 if (term.find("I") != std::string::npos) {
197 term = term.replace(term.find("I"),1,"");
199 if (term.find("*") != std::string::npos)
200 term = term.replace(term.find("*"),1,"");
201 // correct for trivial +/-I without explicit factor on I:
202 if (term.size() == 1)
206 if (term.find(".") != std::string::npos) {
207 // CLN's short type cl_SF is not very useful within the GiNaC
208 // framework where we are mainly interested in the arbitrary
209 // precision type cl_LF. Hence we go straight to the construction
210 // of generic floats. In order to create them we have to convert
211 // our own floating point notation used for output and construction
212 // from char * to CLN's generic notation:
213 // 3.14 --> 3.14e0_<Digits>
214 // 31.4E-1 --> 31.4e-1_<Digits>
216 // No exponent marker? Let's add a trivial one.
217 if (term.find("E") == std::string::npos)
220 term = term.replace(term.find("E"),1,"e");
221 // append _<Digits> to term
222 #if defined(HAVE_SSTREAM)
223 std::ostringstream buf;
224 buf << unsigned(Digits) << std::ends;
225 term += "_" + buf.str();
228 std::ostrstream(buf,sizeof(buf)) << unsigned(Digits) << std::ends;
229 term += "_" + std::string(buf);
231 // construct float using cln::cl_F(const char *) ctor.
233 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
235 ctorval = ctorval + cln::cl_F(term.c_str());
237 // not a floating point number...
239 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
241 ctorval = ctorval + cln::cl_R(term.c_str());
243 } while(delim != std::string::npos);
245 setflag(status_flags::evaluated | status_flags::expanded);
249 /** Ctor from CLN types. This is for the initiated user or internal use
251 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
253 debugmsg("numeric ctor from cl_N", LOGLEVEL_CONSTRUCT);
255 setflag(status_flags::evaluated | status_flags::expanded);
262 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
264 debugmsg("numeric ctor from archive_node", LOGLEVEL_CONSTRUCT);
265 cln::cl_N ctorval = 0;
267 // Read number as string
269 if (n.find_string("number", str)) {
271 std::istringstream s(str);
273 std::istrstream s(str.c_str(), str.size() + 1);
275 cln::cl_idecoded_float re, im;
279 case 'R': // Integer-decoded real number
280 s >> re.sign >> re.mantissa >> re.exponent;
281 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
283 case 'C': // Integer-decoded complex number
284 s >> re.sign >> re.mantissa >> re.exponent;
285 s >> im.sign >> im.mantissa >> im.exponent;
286 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
287 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
289 default: // Ordinary number
296 setflag(status_flags::evaluated | status_flags::expanded);
299 void numeric::archive(archive_node &n) const
301 inherited::archive(n);
303 // Write number as string
305 std::ostringstream s;
308 std::ostrstream s(buf, 1024);
310 if (this->is_crational())
311 s << cln::the<cln::cl_N>(value);
313 // Non-rational numbers are written in an integer-decoded format
314 // to preserve the precision
315 if (this->is_real()) {
316 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
318 s << re.sign << " " << re.mantissa << " " << re.exponent;
320 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
321 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
323 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
324 s << im.sign << " " << im.mantissa << " " << im.exponent;
328 n.add_string("number", s.str());
331 std::string str(buf);
332 n.add_string("number", str);
336 DEFAULT_UNARCHIVE(numeric)
339 // functions overriding virtual functions from bases classes
342 /** Helper function to print a real number in a nicer way than is CLN's
343 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
344 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
345 * long as it only uses cl_LF and no other floating point types that we might
346 * want to visibly distinguish from cl_LF.
348 * @see numeric::print() */
349 static void print_real_number(std::ostream &os, const cln::cl_R &num)
351 cln::cl_print_flags ourflags;
352 if (cln::instanceof(num, cln::cl_RA_ring)) {
353 // case 1: integer or rational, nothing special to do:
354 cln::print_real(os, ourflags, num);
357 // make CLN believe this number has default_float_format, so it prints
358 // 'E' as exponent marker instead of 'L':
359 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(num));
360 cln::print_real(os, ourflags, num);
365 /** This method adds to the output so it blends more consistently together
366 * with the other routines and produces something compatible to ginsh input.
368 * @see print_real_number() */
369 void numeric::print(const print_context & c, unsigned level) const
371 debugmsg("numeric print", LOGLEVEL_PRINT);
373 if (is_of_type(c, print_tree)) {
375 c.s << std::string(level, ' ') << cln::the<cln::cl_N>(value)
376 << " (" << class_name() << ")"
377 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
380 } else if (is_of_type(c, print_csrc)) {
382 std::ios::fmtflags oldflags = c.s.flags();
383 c.s.setf(std::ios::scientific);
384 if (this->is_rational() && !this->is_integer()) {
385 if (compare(_num0()) > 0) {
387 if (is_of_type(c, print_csrc_cl_N))
388 c.s << "cln::cl_F(\"" << numer().evalf() << "\")";
390 c.s << numer().to_double();
393 if (is_of_type(c, print_csrc_cl_N))
394 c.s << "cln::cl_F(\"" << -numer().evalf() << "\")";
396 c.s << -numer().to_double();
399 if (is_of_type(c, print_csrc_cl_N))
400 c.s << "cln::cl_F(\"" << denom().evalf() << "\")";
402 c.s << denom().to_double();
405 if (is_of_type(c, print_csrc_cl_N))
406 c.s << "cln::cl_F(\"" << evalf() << "\")";
414 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
415 cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
417 // case 1, real: x or -x
418 if ((precedence <= level) && (!this->is_nonneg_integer())) {
420 print_real_number(c.s, r);
423 print_real_number(c.s, r);
427 // case 2, imaginary: y*I or -y*I
428 if ((precedence <= level) && (i < 0)) {
433 print_real_number(c.s, i);
443 print_real_number(c.s, i);
449 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
450 if (precedence <= level)
452 print_real_number(c.s, r);
457 print_real_number(c.s, i);
465 print_real_number(c.s, i);
469 if (precedence <= level)
476 bool numeric::info(unsigned inf) const
479 case info_flags::numeric:
480 case info_flags::polynomial:
481 case info_flags::rational_function:
483 case info_flags::real:
485 case info_flags::rational:
486 case info_flags::rational_polynomial:
487 return is_rational();
488 case info_flags::crational:
489 case info_flags::crational_polynomial:
490 return is_crational();
491 case info_flags::integer:
492 case info_flags::integer_polynomial:
494 case info_flags::cinteger:
495 case info_flags::cinteger_polynomial:
496 return is_cinteger();
497 case info_flags::positive:
498 return is_positive();
499 case info_flags::negative:
500 return is_negative();
501 case info_flags::nonnegative:
502 return !is_negative();
503 case info_flags::posint:
504 return is_pos_integer();
505 case info_flags::negint:
506 return is_integer() && is_negative();
507 case info_flags::nonnegint:
508 return is_nonneg_integer();
509 case info_flags::even:
511 case info_flags::odd:
513 case info_flags::prime:
515 case info_flags::algebraic:
521 /** Disassemble real part and imaginary part to scan for the occurrence of a
522 * single number. Also handles the imaginary unit. It ignores the sign on
523 * both this and the argument, which may lead to what might appear as funny
524 * results: (2+I).has(-2) -> true. But this is consistent, since we also
525 * would like to have (-2+I).has(2) -> true and we want to think about the
526 * sign as a multiplicative factor. */
527 bool numeric::has(const ex &other) const
529 if (!is_exactly_of_type(*other.bp, numeric))
531 const numeric &o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
532 if (this->is_equal(o) || this->is_equal(-o))
534 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
535 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
536 this->real().is_equal(-o) || this->imag().is_equal(-o));
538 if (o.is_equal(I)) // e.g scan for I in 42*I
539 return !this->is_real();
540 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
541 return (this->real().has(o*I) || this->imag().has(o*I) ||
542 this->real().has(-o*I) || this->imag().has(-o*I));
548 /** Evaluation of numbers doesn't do anything at all. */
549 ex numeric::eval(int level) const
551 // Warning: if this is ever gonna do something, the ex ctors from all kinds
552 // of numbers should be checking for status_flags::evaluated.
557 /** Cast numeric into a floating-point object. For example exact numeric(1) is
558 * returned as a 1.0000000000000000000000 and so on according to how Digits is
559 * currently set. In case the object already was a floating point number the
560 * precision is trimmed to match the currently set default.
562 * @param level ignored, only needed for overriding basic::evalf.
563 * @return an ex-handle to a numeric. */
564 ex numeric::evalf(int level) const
566 // level can safely be discarded for numeric objects.
567 return numeric(cln::cl_float(1.0, cln::default_float_format) *
568 (cln::the<cln::cl_N>(value)));
573 int numeric::compare_same_type(const basic &other) const
575 GINAC_ASSERT(is_exactly_of_type(other, numeric));
576 const numeric &o = static_cast<numeric &>(const_cast<basic &>(other));
578 return this->compare(o);
582 bool numeric::is_equal_same_type(const basic &other) const
584 GINAC_ASSERT(is_exactly_of_type(other,numeric));
585 const numeric *o = static_cast<const numeric *>(&other);
587 return this->is_equal(*o);
591 unsigned numeric::calchash(void) const
593 // Use CLN's hashcode. Warning: It depends only on the number's value, not
594 // its type or precision (i.e. a true equivalence relation on numbers). As
595 // a consequence, 3 and 3.0 share the same hashvalue.
596 setflag(status_flags::hash_calculated);
597 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
602 // new virtual functions which can be overridden by derived classes
608 // non-virtual functions in this class
613 /** Numerical addition method. Adds argument to *this and returns result as
614 * a numeric object. */
615 const numeric numeric::add(const numeric &other) const
617 // Efficiency shortcut: trap the neutral element by pointer.
618 static const numeric * _num0p = &_num0();
621 else if (&other==_num0p)
624 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
628 /** Numerical subtraction method. Subtracts argument from *this and returns
629 * result as a numeric object. */
630 const numeric numeric::sub(const numeric &other) const
632 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
636 /** Numerical multiplication method. Multiplies *this and argument and returns
637 * result as a numeric object. */
638 const numeric numeric::mul(const numeric &other) const
640 // Efficiency shortcut: trap the neutral element by pointer.
641 static const numeric * _num1p = &_num1();
644 else if (&other==_num1p)
647 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
651 /** Numerical division method. Divides *this by argument and returns result as
654 * @exception overflow_error (division by zero) */
655 const numeric numeric::div(const numeric &other) const
657 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
658 throw std::overflow_error("numeric::div(): division by zero");
659 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
663 /** Numerical exponentiation. Raises *this to the power given as argument and
664 * returns result as a numeric object. */
665 const numeric numeric::power(const numeric &other) const
667 // Efficiency shortcut: trap the neutral exponent by pointer.
668 static const numeric * _num1p = &_num1();
672 if (cln::zerop(cln::the<cln::cl_N>(value))) {
673 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
674 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
675 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
676 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
677 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
678 throw std::overflow_error("numeric::eval(): division by zero");
682 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
686 const numeric &numeric::add_dyn(const numeric &other) const
688 // Efficiency shortcut: trap the neutral element by pointer.
689 static const numeric * _num0p = &_num0();
692 else if (&other==_num0p)
695 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
696 setflag(status_flags::dynallocated));
700 const numeric &numeric::sub_dyn(const numeric &other) const
702 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
703 setflag(status_flags::dynallocated));
707 const numeric &numeric::mul_dyn(const numeric &other) const
709 // Efficiency shortcut: trap the neutral element by pointer.
710 static const numeric * _num1p = &_num1();
713 else if (&other==_num1p)
716 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
717 setflag(status_flags::dynallocated));
721 const numeric &numeric::div_dyn(const numeric &other) const
723 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
724 throw std::overflow_error("division by zero");
725 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
726 setflag(status_flags::dynallocated));
730 const numeric &numeric::power_dyn(const numeric &other) const
732 // Efficiency shortcut: trap the neutral exponent by pointer.
733 static const numeric * _num1p=&_num1();
737 if (cln::zerop(cln::the<cln::cl_N>(value))) {
738 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
739 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
740 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
741 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
742 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
743 throw std::overflow_error("numeric::eval(): division by zero");
747 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
748 setflag(status_flags::dynallocated));
752 const numeric &numeric::operator=(int i)
754 return operator=(numeric(i));
758 const numeric &numeric::operator=(unsigned int i)
760 return operator=(numeric(i));
764 const numeric &numeric::operator=(long i)
766 return operator=(numeric(i));
770 const numeric &numeric::operator=(unsigned long i)
772 return operator=(numeric(i));
776 const numeric &numeric::operator=(double d)
778 return operator=(numeric(d));
782 const numeric &numeric::operator=(const char * s)
784 return operator=(numeric(s));
788 /** Inverse of a number. */
789 const numeric numeric::inverse(void) const
791 if (cln::zerop(cln::the<cln::cl_N>(value)))
792 throw std::overflow_error("numeric::inverse(): division by zero");
793 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
797 /** Return the complex half-plane (left or right) in which the number lies.
798 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
799 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
801 * @see numeric::compare(const numeric &other) */
802 int numeric::csgn(void) const
804 if (cln::zerop(cln::the<cln::cl_N>(value)))
806 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
807 if (!cln::zerop(r)) {
813 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
821 /** This method establishes a canonical order on all numbers. For complex
822 * numbers this is not possible in a mathematically consistent way but we need
823 * to establish some order and it ought to be fast. So we simply define it
824 * to be compatible with our method csgn.
826 * @return csgn(*this-other)
827 * @see numeric::csgn(void) */
828 int numeric::compare(const numeric &other) const
830 // Comparing two real numbers?
831 if (cln::instanceof(value, cln::cl_R_ring) &&
832 cln::instanceof(other.value, cln::cl_R_ring))
833 // Yes, so just cln::compare them
834 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
836 // No, first cln::compare real parts...
837 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
840 // ...and then the imaginary parts.
841 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
846 bool numeric::is_equal(const numeric &other) const
848 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
852 /** True if object is zero. */
853 bool numeric::is_zero(void) const
855 return cln::zerop(cln::the<cln::cl_N>(value));
859 /** True if object is not complex and greater than zero. */
860 bool numeric::is_positive(void) const
863 return cln::plusp(cln::the<cln::cl_R>(value));
868 /** True if object is not complex and less than zero. */
869 bool numeric::is_negative(void) const
872 return cln::minusp(cln::the<cln::cl_R>(value));
877 /** True if object is a non-complex integer. */
878 bool numeric::is_integer(void) const
880 return cln::instanceof(value, cln::cl_I_ring);
884 /** True if object is an exact integer greater than zero. */
885 bool numeric::is_pos_integer(void) const
887 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
891 /** True if object is an exact integer greater or equal zero. */
892 bool numeric::is_nonneg_integer(void) const
894 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
898 /** True if object is an exact even integer. */
899 bool numeric::is_even(void) const
901 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
905 /** True if object is an exact odd integer. */
906 bool numeric::is_odd(void) const
908 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
912 /** Probabilistic primality test.
914 * @return true if object is exact integer and prime. */
915 bool numeric::is_prime(void) const
917 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
921 /** True if object is an exact rational number, may even be complex
922 * (denominator may be unity). */
923 bool numeric::is_rational(void) const
925 return cln::instanceof(value, cln::cl_RA_ring);
929 /** True if object is a real integer, rational or float (but not complex). */
930 bool numeric::is_real(void) const
932 return cln::instanceof(value, cln::cl_R_ring);
936 bool numeric::operator==(const numeric &other) const
938 return equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
942 bool numeric::operator!=(const numeric &other) const
944 return !equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
948 /** True if object is element of the domain of integers extended by I, i.e. is
949 * of the form a+b*I, where a and b are integers. */
950 bool numeric::is_cinteger(void) const
952 if (cln::instanceof(value, cln::cl_I_ring))
954 else if (!this->is_real()) { // complex case, handle n+m*I
955 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
956 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
963 /** True if object is an exact rational number, may even be complex
964 * (denominator may be unity). */
965 bool numeric::is_crational(void) const
967 if (cln::instanceof(value, cln::cl_RA_ring))
969 else if (!this->is_real()) { // complex case, handle Q(i):
970 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
971 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
978 /** Numerical comparison: less.
980 * @exception invalid_argument (complex inequality) */
981 bool numeric::operator<(const numeric &other) const
983 if (this->is_real() && other.is_real())
984 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
985 throw std::invalid_argument("numeric::operator<(): complex inequality");
989 /** Numerical comparison: less or equal.
991 * @exception invalid_argument (complex inequality) */
992 bool numeric::operator<=(const numeric &other) const
994 if (this->is_real() && other.is_real())
995 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
996 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1000 /** Numerical comparison: greater.
1002 * @exception invalid_argument (complex inequality) */
1003 bool numeric::operator>(const numeric &other) const
1005 if (this->is_real() && other.is_real())
1006 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1007 throw std::invalid_argument("numeric::operator>(): complex inequality");
1011 /** Numerical comparison: greater or equal.
1013 * @exception invalid_argument (complex inequality) */
1014 bool numeric::operator>=(const numeric &other) const
1016 if (this->is_real() && other.is_real())
1017 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1018 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1022 /** Converts numeric types to machine's int. You should check with
1023 * is_integer() if the number is really an integer before calling this method.
1024 * You may also consider checking the range first. */
1025 int numeric::to_int(void) const
1027 GINAC_ASSERT(this->is_integer());
1028 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1032 /** Converts numeric types to machine's long. You should check with
1033 * is_integer() if the number is really an integer before calling this method.
1034 * You may also consider checking the range first. */
1035 long numeric::to_long(void) const
1037 GINAC_ASSERT(this->is_integer());
1038 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1042 /** Converts numeric types to machine's double. You should check with is_real()
1043 * if the number is really not complex before calling this method. */
1044 double numeric::to_double(void) const
1046 GINAC_ASSERT(this->is_real());
1047 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1051 /** Returns a new CLN object of type cl_N, representing the value of *this.
1052 * This method may be used when mixing GiNaC and CLN in one project.
1054 cln::cl_N numeric::to_cl_N(void) const
1056 return cln::cl_N(cln::the<cln::cl_N>(value));
1060 /** Real part of a number. */
1061 const numeric numeric::real(void) const
1063 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1067 /** Imaginary part of a number. */
1068 const numeric numeric::imag(void) const
1070 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1074 /** Numerator. Computes the numerator of rational numbers, rationalized
1075 * numerator of complex if real and imaginary part are both rational numbers
1076 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1078 const numeric numeric::numer(void) const
1080 if (this->is_integer())
1081 return numeric(*this);
1083 else if (cln::instanceof(value, cln::cl_RA_ring))
1084 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1086 else if (!this->is_real()) { // complex case, handle Q(i):
1087 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1088 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1089 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1090 return numeric(*this);
1091 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1092 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1093 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1094 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1095 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1096 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1097 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1098 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1101 // at least one float encountered
1102 return numeric(*this);
1106 /** Denominator. Computes the denominator of rational numbers, common integer
1107 * denominator of complex if real and imaginary part are both rational numbers
1108 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1109 const numeric numeric::denom(void) const
1111 if (this->is_integer())
1114 if (instanceof(value, cln::cl_RA_ring))
1115 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1117 if (!this->is_real()) { // complex case, handle Q(i):
1118 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1119 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1120 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1122 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1123 return numeric(cln::denominator(i));
1124 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1125 return numeric(cln::denominator(r));
1126 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1127 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1129 // at least one float encountered
1134 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1135 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1136 * 2^(n-1) <= x < 2^n.
1138 * @return number of bits (excluding sign) needed to represent that number
1139 * in two's complement if it is an integer, 0 otherwise. */
1140 int numeric::int_length(void) const
1142 if (this->is_integer())
1143 return cln::integer_length(cln::the<cln::cl_I>(value));
1150 // static member variables
1155 unsigned numeric::precedence = 30;
1161 /** Imaginary unit. This is not a constant but a numeric since we are
1162 * natively handing complex numbers anyways, so in each expression containing
1163 * an I it is automatically eval'ed away anyhow. */
1164 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1167 /** Exponential function.
1169 * @return arbitrary precision numerical exp(x). */
1170 const numeric exp(const numeric &x)
1172 return cln::exp(x.to_cl_N());
1176 /** Natural logarithm.
1178 * @param z complex number
1179 * @return arbitrary precision numerical log(x).
1180 * @exception pole_error("log(): logarithmic pole",0) */
1181 const numeric log(const numeric &z)
1184 throw pole_error("log(): logarithmic pole",0);
1185 return cln::log(z.to_cl_N());
1189 /** Numeric sine (trigonometric function).
1191 * @return arbitrary precision numerical sin(x). */
1192 const numeric sin(const numeric &x)
1194 return cln::sin(x.to_cl_N());
1198 /** Numeric cosine (trigonometric function).
1200 * @return arbitrary precision numerical cos(x). */
1201 const numeric cos(const numeric &x)
1203 return cln::cos(x.to_cl_N());
1207 /** Numeric tangent (trigonometric function).
1209 * @return arbitrary precision numerical tan(x). */
1210 const numeric tan(const numeric &x)
1212 return cln::tan(x.to_cl_N());
1216 /** Numeric inverse sine (trigonometric function).
1218 * @return arbitrary precision numerical asin(x). */
1219 const numeric asin(const numeric &x)
1221 return cln::asin(x.to_cl_N());
1225 /** Numeric inverse cosine (trigonometric function).
1227 * @return arbitrary precision numerical acos(x). */
1228 const numeric acos(const numeric &x)
1230 return cln::acos(x.to_cl_N());
1236 * @param z complex number
1238 * @exception pole_error("atan(): logarithmic pole",0) */
1239 const numeric atan(const numeric &x)
1242 x.real().is_zero() &&
1243 abs(x.imag()).is_equal(_num1()))
1244 throw pole_error("atan(): logarithmic pole",0);
1245 return cln::atan(x.to_cl_N());
1251 * @param x real number
1252 * @param y real number
1253 * @return atan(y/x) */
1254 const numeric atan(const numeric &y, const numeric &x)
1256 if (x.is_real() && y.is_real())
1257 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1258 cln::the<cln::cl_R>(y.to_cl_N()));
1260 throw std::invalid_argument("atan(): complex argument");
1264 /** Numeric hyperbolic sine (trigonometric function).
1266 * @return arbitrary precision numerical sinh(x). */
1267 const numeric sinh(const numeric &x)
1269 return cln::sinh(x.to_cl_N());
1273 /** Numeric hyperbolic cosine (trigonometric function).
1275 * @return arbitrary precision numerical cosh(x). */
1276 const numeric cosh(const numeric &x)
1278 return cln::cosh(x.to_cl_N());
1282 /** Numeric hyperbolic tangent (trigonometric function).
1284 * @return arbitrary precision numerical tanh(x). */
1285 const numeric tanh(const numeric &x)
1287 return cln::tanh(x.to_cl_N());
1291 /** Numeric inverse hyperbolic sine (trigonometric function).
1293 * @return arbitrary precision numerical asinh(x). */
1294 const numeric asinh(const numeric &x)
1296 return cln::asinh(x.to_cl_N());
1300 /** Numeric inverse hyperbolic cosine (trigonometric function).
1302 * @return arbitrary precision numerical acosh(x). */
1303 const numeric acosh(const numeric &x)
1305 return cln::acosh(x.to_cl_N());
1309 /** Numeric inverse hyperbolic tangent (trigonometric function).
1311 * @return arbitrary precision numerical atanh(x). */
1312 const numeric atanh(const numeric &x)
1314 return cln::atanh(x.to_cl_N());
1318 /*static cln::cl_N Li2_series(const ::cl_N &x,
1319 const ::float_format_t &prec)
1321 // Note: argument must be in the unit circle
1322 // This is very inefficient unless we have fast floating point Bernoulli
1323 // numbers implemented!
1324 cln::cl_N c1 = -cln::log(1-x);
1326 // hard-wire the first two Bernoulli numbers
1327 cln::cl_N acc = c1 - cln::square(c1)/4;
1329 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1330 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1332 c1 = cln::square(c1);
1336 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1337 // 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));
1340 } while (acc != acc+aug);
1344 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1345 * circle) using a power series. */
1346 static cln::cl_N Li2_series(const cln::cl_N &x,
1347 const cln::float_format_t &prec)
1349 // Note: argument must be in the unit circle
1351 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1356 den = den + i; // 1, 4, 9, 16, ...
1360 } while (acc != acc+aug);
1364 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1365 static cln::cl_N Li2_projection(const cln::cl_N &x,
1366 const cln::float_format_t &prec)
1368 const cln::cl_R re = cln::realpart(x);
1369 const cln::cl_R im = cln::imagpart(x);
1370 if (re > cln::cl_F(".5"))
1371 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1373 - Li2_series(1-x, prec)
1374 - cln::log(x)*cln::log(1-x));
1375 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1376 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1377 return(- cln::square(cln::log(1-x))/2
1378 - Li2_series(x/(x-1), prec));
1379 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1380 // Li2(x^2)/2 - Li2(-x)
1381 return(Li2_projection(cln::square(x), prec)/2
1382 - Li2_projection(-x, prec));
1383 return Li2_series(x, prec);
1386 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1387 * the branch cut lies along the positive real axis, starting at 1 and
1388 * continuous with quadrant IV.
1390 * @return arbitrary precision numerical Li2(x). */
1391 const numeric Li2(const numeric &x)
1396 // what is the desired float format?
1397 // first guess: default format
1398 cln::float_format_t prec = cln::default_float_format;
1399 const cln::cl_N value = x.to_cl_N();
1400 // second guess: the argument's format
1401 if (!x.real().is_rational())
1402 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1403 else if (!x.imag().is_rational())
1404 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1406 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1407 return cln::zeta(2, prec);
1409 if (cln::abs(value) > 1)
1410 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1411 return(- cln::square(cln::log(-value))/2
1412 - cln::zeta(2, prec)
1413 - Li2_projection(cln::recip(value), prec));
1415 return Li2_projection(x.to_cl_N(), prec);
1419 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1420 * integer arguments. */
1421 const numeric zeta(const numeric &x)
1423 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1424 // only knows about integer arguments and zeta(3).evalf() automatically
1425 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1426 // being an exact zero for CLN, which can be tested and then we can just
1427 // pass the number casted to an int:
1429 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1430 if (cln::zerop(x.to_cl_N()-aux))
1431 return cln::zeta(aux);
1433 std::clog << "zeta(" << x
1434 << "): Does anybody know a good way to calculate this numerically?"
1440 /** The Gamma function.
1441 * This is only a stub! */
1442 const numeric lgamma(const numeric &x)
1444 std::clog << "lgamma(" << x
1445 << "): Does anybody know a good way to calculate this numerically?"
1449 const numeric tgamma(const numeric &x)
1451 std::clog << "tgamma(" << x
1452 << "): Does anybody know a good way to calculate this numerically?"
1458 /** The psi function (aka polygamma function).
1459 * This is only a stub! */
1460 const numeric psi(const numeric &x)
1462 std::clog << "psi(" << x
1463 << "): Does anybody know a good way to calculate this numerically?"
1469 /** The psi functions (aka polygamma functions).
1470 * This is only a stub! */
1471 const numeric psi(const numeric &n, const numeric &x)
1473 std::clog << "psi(" << n << "," << x
1474 << "): Does anybody know a good way to calculate this numerically?"
1480 /** Factorial combinatorial function.
1482 * @param n integer argument >= 0
1483 * @exception range_error (argument must be integer >= 0) */
1484 const numeric factorial(const numeric &n)
1486 if (!n.is_nonneg_integer())
1487 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1488 return numeric(cln::factorial(n.to_int()));
1492 /** The double factorial combinatorial function. (Scarcely used, but still
1493 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1495 * @param n integer argument >= -1
1496 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1497 * @exception range_error (argument must be integer >= -1) */
1498 const numeric doublefactorial(const numeric &n)
1500 if (n.is_equal(_num_1()))
1503 if (!n.is_nonneg_integer())
1504 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1506 return numeric(cln::doublefactorial(n.to_int()));
1510 /** The Binomial coefficients. It computes the binomial coefficients. For
1511 * integer n and k and positive n this is the number of ways of choosing k
1512 * objects from n distinct objects. If n is negative, the formula
1513 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1514 const numeric binomial(const numeric &n, const numeric &k)
1516 if (n.is_integer() && k.is_integer()) {
1517 if (n.is_nonneg_integer()) {
1518 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1519 return numeric(cln::binomial(n.to_int(),k.to_int()));
1523 return _num_1().power(k)*binomial(k-n-_num1(),k);
1527 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1528 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1532 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1533 * in the expansion of the function x/(e^x-1).
1535 * @return the nth Bernoulli number (a rational number).
1536 * @exception range_error (argument must be integer >= 0) */
1537 const numeric bernoulli(const numeric &nn)
1539 if (!nn.is_integer() || nn.is_negative())
1540 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1544 // The Bernoulli numbers are rational numbers that may be computed using
1547 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1549 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1550 // previous ones, the computation is necessarily very expensive. There are
1551 // several other ways of computing them, a particularly good one being
1555 // for (unsigned i=0; i<n; i++) {
1556 // c = exquo(c*(i-n),(i+2));
1557 // Bern = Bern + c*s/(i+2);
1558 // s = s + expt_pos(cl_I(i+2),n);
1562 // But if somebody works with the n'th Bernoulli number she is likely to
1563 // also need all previous Bernoulli numbers. So we need a complete remember
1564 // table and above divide and conquer algorithm is not suited to build one
1565 // up. The code below is adapted from Pari's function bernvec().
1567 // (There is an interesting relation with the tangent polynomials described
1568 // in `Concrete Mathematics', which leads to a program twice as fast as our
1569 // implementation below, but it requires storing one such polynomial in
1570 // addition to the remember table. This doubles the memory footprint so
1571 // we don't use it.)
1573 // the special cases not covered by the algorithm below
1574 if (nn.is_equal(_num1()))
1579 // store nonvanishing Bernoulli numbers here
1580 static std::vector< cln::cl_RA > results;
1581 static int highest_result = 0;
1582 // algorithm not applicable to B(0), so just store it
1583 if (results.size()==0)
1584 results.push_back(cln::cl_RA(1));
1586 int n = nn.to_long();
1587 for (int i=highest_result; i<n/2; ++i) {
1593 for (int j=i; j>0; --j) {
1594 B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
1600 B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
1601 results.push_back(B);
1604 return results[n/2];
1608 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1609 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1611 * @param n an integer
1612 * @return the nth Fibonacci number F(n) (an integer number)
1613 * @exception range_error (argument must be an integer) */
1614 const numeric fibonacci(const numeric &n)
1616 if (!n.is_integer())
1617 throw std::range_error("numeric::fibonacci(): argument must be integer");
1620 // The following addition formula holds:
1622 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1624 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1625 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1627 // Replace m by m+1:
1628 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1629 // Now put in m = n, to get
1630 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1631 // F(2n+1) = F(n)^2 + F(n+1)^2
1633 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1636 if (n.is_negative())
1638 return -fibonacci(-n);
1640 return fibonacci(-n);
1644 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1645 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1646 // Since a squaring is cheaper than a multiplication, better use
1647 // three squarings instead of one multiplication and two squarings.
1648 cln::cl_I u2 = cln::square(u);
1649 cln::cl_I v2 = cln::square(v);
1650 if (cln::logbitp(bit-1, m)) {
1651 v = cln::square(u + v) - u2;
1654 u = v2 - cln::square(v - u);
1659 // Here we don't use the squaring formula because one multiplication
1660 // is cheaper than two squarings.
1661 return u * ((v << 1) - u);
1663 return cln::square(u) + cln::square(v);
1667 /** Absolute value. */
1668 const numeric abs(const numeric& x)
1670 return cln::abs(x.to_cl_N());
1674 /** Modulus (in positive representation).
1675 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1676 * sign of a or is zero. This is different from Maple's modp, where the sign
1677 * of b is ignored. It is in agreement with Mathematica's Mod.
1679 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1680 * integer, 0 otherwise. */
1681 const numeric mod(const numeric &a, const numeric &b)
1683 if (a.is_integer() && b.is_integer())
1684 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1685 cln::the<cln::cl_I>(b.to_cl_N()));
1691 /** Modulus (in symmetric representation).
1692 * Equivalent to Maple's mods.
1694 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1695 const numeric smod(const numeric &a, const numeric &b)
1697 if (a.is_integer() && b.is_integer()) {
1698 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1699 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1700 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1706 /** Numeric integer remainder.
1707 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1708 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1709 * sign of a or is zero.
1711 * @return remainder of a/b if both are integer, 0 otherwise. */
1712 const numeric irem(const numeric &a, const numeric &b)
1714 if (a.is_integer() && b.is_integer())
1715 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1716 cln::the<cln::cl_I>(b.to_cl_N()));
1722 /** Numeric integer remainder.
1723 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1724 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1725 * and irem(a,b) has the sign of a or is zero.
1727 * @return remainder of a/b and quotient stored in q if both are integer,
1729 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1731 if (a.is_integer() && b.is_integer()) {
1732 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1733 cln::the<cln::cl_I>(b.to_cl_N()));
1734 q = rem_quo.quotient;
1735 return rem_quo.remainder;
1743 /** Numeric integer quotient.
1744 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1746 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1747 const numeric iquo(const numeric &a, const numeric &b)
1749 if (a.is_integer() && b.is_integer())
1750 return truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1751 cln::the<cln::cl_I>(b.to_cl_N()));
1757 /** Numeric integer quotient.
1758 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1759 * r == a - iquo(a,b,r)*b.
1761 * @return truncated quotient of a/b and remainder stored in r if both are
1762 * integer, 0 otherwise. */
1763 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1765 if (a.is_integer() && b.is_integer()) {
1766 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1767 cln::the<cln::cl_I>(b.to_cl_N()));
1768 r = rem_quo.remainder;
1769 return rem_quo.quotient;
1777 /** Greatest Common Divisor.
1779 * @return The GCD of two numbers if both are integer, a numerical 1
1780 * if they are not. */
1781 const numeric gcd(const numeric &a, const numeric &b)
1783 if (a.is_integer() && b.is_integer())
1784 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1785 cln::the<cln::cl_I>(b.to_cl_N()));
1791 /** Least Common Multiple.
1793 * @return The LCM of two numbers if both are integer, the product of those
1794 * two numbers if they are not. */
1795 const numeric lcm(const numeric &a, const numeric &b)
1797 if (a.is_integer() && b.is_integer())
1798 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1799 cln::the<cln::cl_I>(b.to_cl_N()));
1805 /** Numeric square root.
1806 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1807 * should return integer 2.
1809 * @param z numeric argument
1810 * @return square root of z. Branch cut along negative real axis, the negative
1811 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1812 * where imag(z)>0. */
1813 const numeric sqrt(const numeric &z)
1815 return cln::sqrt(z.to_cl_N());
1819 /** Integer numeric square root. */
1820 const numeric isqrt(const numeric &x)
1822 if (x.is_integer()) {
1824 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1831 /** Floating point evaluation of Archimedes' constant Pi. */
1834 return numeric(cln::pi(cln::default_float_format));
1838 /** Floating point evaluation of Euler's constant gamma. */
1841 return numeric(cln::eulerconst(cln::default_float_format));
1845 /** Floating point evaluation of Catalan's constant. */
1846 ex CatalanEvalf(void)
1848 return numeric(cln::catalanconst(cln::default_float_format));
1852 /** _numeric_digits default ctor, checking for singleton invariance. */
1853 _numeric_digits::_numeric_digits()
1856 // It initializes to 17 digits, because in CLN float_format(17) turns out
1857 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1858 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1860 throw(std::runtime_error("I told you not to do instantiate me!"));
1862 cln::default_float_format = cln::float_format(17);
1866 /** Assign a native long to global Digits object. */
1867 _numeric_digits& _numeric_digits::operator=(long prec)
1870 cln::default_float_format = cln::float_format(prec);
1875 /** Convert global Digits object to native type long. */
1876 _numeric_digits::operator long()
1878 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1879 return (long)digits;
1883 /** Append global Digits object to ostream. */
1884 void _numeric_digits::print(std::ostream &os) const
1886 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1891 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1898 // static member variables
1903 bool _numeric_digits::too_late = false;
1906 /** Accuracy in decimal digits. Only object of this type! Can be set using
1907 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1908 _numeric_digits Digits;
1910 } // namespace GiNaC