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) 1999-2002 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
41 // CLN should pollute the global namespace as little as possible. Hence, we
42 // include most of it here and include only the part needed for properly
43 // declaring cln::cl_number in numeric.h. This can only be safely done in
44 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
45 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
47 #include <cln/output.h>
48 #include <cln/integer_io.h>
49 #include <cln/integer_ring.h>
50 #include <cln/rational_io.h>
51 #include <cln/rational_ring.h>
52 #include <cln/lfloat_class.h>
53 #include <cln/lfloat_io.h>
54 #include <cln/real_io.h>
55 #include <cln/real_ring.h>
56 #include <cln/complex_io.h>
57 #include <cln/complex_ring.h>
58 #include <cln/numtheory.h>
62 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
65 // default ctor, dtor, copy ctor, assignment operator and helpers
68 /** default ctor. Numerically it initializes to an integer zero. */
69 numeric::numeric() : basic(TINFO_numeric)
72 setflag(status_flags::evaluated | status_flags::expanded);
75 void numeric::copy(const numeric &other)
77 inherited::copy(other);
81 DEFAULT_DESTROY(numeric)
89 numeric::numeric(int i) : basic(TINFO_numeric)
91 // Not the whole int-range is available if we don't cast to long
92 // first. This is due to the behaviour of the cl_I-ctor, which
93 // emphasizes efficiency. However, if the integer is small enough
94 // we save space and dereferences by using an immediate type.
95 // (C.f. <cln/object.h>)
96 if (i < (1U<<cl_value_len-1))
99 value = cln::cl_I((long) i);
100 setflag(status_flags::evaluated | status_flags::expanded);
104 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
106 // Not the whole uint-range is available if we don't cast to ulong
107 // first. This is due to the behaviour of the cl_I-ctor, which
108 // emphasizes efficiency. However, if the integer is small enough
109 // we save space and dereferences by using an immediate type.
110 // (C.f. <cln/object.h>)
111 if (i < (1U<<cl_value_len-1))
112 value = cln::cl_I(i);
114 value = cln::cl_I((unsigned long) i);
115 setflag(status_flags::evaluated | status_flags::expanded);
119 numeric::numeric(long i) : basic(TINFO_numeric)
121 value = cln::cl_I(i);
122 setflag(status_flags::evaluated | status_flags::expanded);
126 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
128 value = cln::cl_I(i);
129 setflag(status_flags::evaluated | status_flags::expanded);
132 /** Ctor for rational numerics a/b.
134 * @exception overflow_error (division by zero) */
135 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
138 throw std::overflow_error("division by zero");
139 value = cln::cl_I(numer) / cln::cl_I(denom);
140 setflag(status_flags::evaluated | status_flags::expanded);
144 numeric::numeric(double d) : basic(TINFO_numeric)
146 // We really want to explicitly use the type cl_LF instead of the
147 // more general cl_F, since that would give us a cl_DF only which
148 // will not be promoted to cl_LF if overflow occurs:
149 value = cln::cl_float(d, cln::default_float_format);
150 setflag(status_flags::evaluated | status_flags::expanded);
154 /** ctor from C-style string. It also accepts complex numbers in GiNaC
155 * notation like "2+5*I". */
156 numeric::numeric(const char *s) : basic(TINFO_numeric)
158 cln::cl_N ctorval = 0;
159 // parse complex numbers (functional but not completely safe, unfortunately
160 // std::string does not understand regexpese):
161 // ss should represent a simple sum like 2+5*I
163 std::string::size_type delim;
165 // make this implementation safe by adding explicit sign
166 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
169 // We use 'E' as exponent marker in the output, but some people insist on
170 // writing 'e' at input, so let's substitute them right at the beginning:
171 while ((delim = ss.find("e"))!=std::string::npos)
172 ss.replace(delim,1,"E");
176 // chop ss into terms from left to right
178 bool imaginary = false;
179 delim = ss.find_first_of(std::string("+-"),1);
180 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
181 if (delim!=std::string::npos && ss.at(delim-1)=='E')
182 delim = ss.find_first_of(std::string("+-"),delim+1);
183 term = ss.substr(0,delim);
184 if (delim!=std::string::npos)
185 ss = ss.substr(delim);
186 // is the term imaginary?
187 if (term.find("I")!=std::string::npos) {
189 term.erase(term.find("I"),1);
191 if (term.find("*")!=std::string::npos)
192 term.erase(term.find("*"),1);
193 // correct for trivial +/-I without explicit factor on I:
198 if (term.find('.')!=std::string::npos || term.find('E')!=std::string::npos) {
199 // CLN's short type cl_SF is not very useful within the GiNaC
200 // framework where we are mainly interested in the arbitrary
201 // precision type cl_LF. Hence we go straight to the construction
202 // of generic floats. In order to create them we have to convert
203 // our own floating point notation used for output and construction
204 // from char * to CLN's generic notation:
205 // 3.14 --> 3.14e0_<Digits>
206 // 31.4E-1 --> 31.4e-1_<Digits>
208 // No exponent marker? Let's add a trivial one.
209 if (term.find("E")==std::string::npos)
212 term = term.replace(term.find("E"),1,"e");
213 // append _<Digits> to term
214 term += "_" + ToString((unsigned)Digits);
215 // construct float using cln::cl_F(const char *) ctor.
217 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
219 ctorval = ctorval + cln::cl_F(term.c_str());
221 // this is not a floating point number...
223 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
225 ctorval = ctorval + cln::cl_R(term.c_str());
227 } while (delim != std::string::npos);
229 setflag(status_flags::evaluated | status_flags::expanded);
233 /** Ctor from CLN types. This is for the initiated user or internal use
235 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
238 setflag(status_flags::evaluated | status_flags::expanded);
245 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
247 cln::cl_N ctorval = 0;
249 // Read number as string
251 if (n.find_string("number", str)) {
252 std::istringstream s(str);
253 cln::cl_idecoded_float re, im;
257 case 'R': // Integer-decoded real number
258 s >> re.sign >> re.mantissa >> re.exponent;
259 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
261 case 'C': // Integer-decoded complex number
262 s >> re.sign >> re.mantissa >> re.exponent;
263 s >> im.sign >> im.mantissa >> im.exponent;
264 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
265 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
267 default: // Ordinary number
274 setflag(status_flags::evaluated | status_flags::expanded);
277 void numeric::archive(archive_node &n) const
279 inherited::archive(n);
281 // Write number as string
282 std::ostringstream s;
283 if (this->is_crational())
284 s << cln::the<cln::cl_N>(value);
286 // Non-rational numbers are written in an integer-decoded format
287 // to preserve the precision
288 if (this->is_real()) {
289 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
291 s << re.sign << " " << re.mantissa << " " << re.exponent;
293 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
294 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
296 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
297 s << im.sign << " " << im.mantissa << " " << im.exponent;
300 n.add_string("number", s.str());
303 DEFAULT_UNARCHIVE(numeric)
306 // functions overriding virtual functions from base classes
309 /** Helper function to print a real number in a nicer way than is CLN's
310 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
311 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
312 * long as it only uses cl_LF and no other floating point types that we might
313 * want to visibly distinguish from cl_LF.
315 * @see numeric::print() */
316 static void print_real_number(const print_context & c, const cln::cl_R &x)
318 cln::cl_print_flags ourflags;
319 if (cln::instanceof(x, cln::cl_RA_ring)) {
320 // case 1: integer or rational
321 if (cln::instanceof(x, cln::cl_I_ring) ||
322 !is_a<print_latex>(c)) {
323 cln::print_real(c.s, ourflags, x);
324 } else { // rational output in LaTeX context
326 cln::print_real(c.s, ourflags, cln::numerator(cln::the<cln::cl_RA>(x)));
328 cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
333 // make CLN believe this number has default_float_format, so it prints
334 // 'E' as exponent marker instead of 'L':
335 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
336 cln::print_real(c.s, ourflags, x);
340 /** This method adds to the output so it blends more consistently together
341 * with the other routines and produces something compatible to ginsh input.
343 * @see print_real_number() */
344 void numeric::print(const print_context & c, unsigned level) const
346 if (is_a<print_tree>(c)) {
348 c.s << std::string(level, ' ') << cln::the<cln::cl_N>(value)
349 << " (" << class_name() << ")"
350 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
353 } else if (is_a<print_csrc>(c)) {
355 std::ios::fmtflags oldflags = c.s.flags();
356 c.s.setf(std::ios::scientific);
357 if (this->is_rational() && !this->is_integer()) {
358 if (compare(_num0) > 0) {
360 if (is_a<print_csrc_cl_N>(c))
361 c.s << "cln::cl_F(\"" << numer().evalf() << "\")";
363 c.s << numer().to_double();
366 if (is_a<print_csrc_cl_N>(c))
367 c.s << "cln::cl_F(\"" << -numer().evalf() << "\")";
369 c.s << -numer().to_double();
372 if (is_a<print_csrc_cl_N>(c))
373 c.s << "cln::cl_F(\"" << denom().evalf() << "\")";
375 c.s << denom().to_double();
378 if (is_a<print_csrc_cl_N>(c))
379 c.s << "cln::cl_F(\"" << evalf() << "\")";
386 const std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
387 const std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
388 const std::string imag_sym = is_a<print_latex>(c) ? "i" : "I";
389 const std::string mul_sym = is_a<print_latex>(c) ? " " : "*";
390 const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
391 const cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
392 if (is_a<print_python_repr>(c))
393 c.s << class_name() << "('";
395 // case 1, real: x or -x
396 if ((precedence() <= level) && (!this->is_nonneg_integer())) {
398 print_real_number(c, r);
401 print_real_number(c, r);
405 // case 2, imaginary: y*I or -y*I
409 if (precedence()<=level)
412 c.s << "-" << imag_sym;
414 print_real_number(c, i);
415 c.s << mul_sym+imag_sym;
417 if (precedence()<=level)
421 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
422 if (precedence() <= level)
424 print_real_number(c, r);
429 print_real_number(c, i);
430 c.s << mul_sym+imag_sym;
437 print_real_number(c, i);
438 c.s << mul_sym+imag_sym;
441 if (precedence() <= level)
445 if (is_a<print_python_repr>(c))
450 bool numeric::info(unsigned inf) const
453 case info_flags::numeric:
454 case info_flags::polynomial:
455 case info_flags::rational_function:
457 case info_flags::real:
459 case info_flags::rational:
460 case info_flags::rational_polynomial:
461 return is_rational();
462 case info_flags::crational:
463 case info_flags::crational_polynomial:
464 return is_crational();
465 case info_flags::integer:
466 case info_flags::integer_polynomial:
468 case info_flags::cinteger:
469 case info_flags::cinteger_polynomial:
470 return is_cinteger();
471 case info_flags::positive:
472 return is_positive();
473 case info_flags::negative:
474 return is_negative();
475 case info_flags::nonnegative:
476 return !is_negative();
477 case info_flags::posint:
478 return is_pos_integer();
479 case info_flags::negint:
480 return is_integer() && is_negative();
481 case info_flags::nonnegint:
482 return is_nonneg_integer();
483 case info_flags::even:
485 case info_flags::odd:
487 case info_flags::prime:
489 case info_flags::algebraic:
495 int numeric::degree(const ex & s) const
500 int numeric::ldegree(const ex & s) const
505 ex numeric::coeff(const ex & s, int n) const
507 return n==0 ? *this : _ex0;
510 /** Disassemble real part and imaginary part to scan for the occurrence of a
511 * single number. Also handles the imaginary unit. It ignores the sign on
512 * both this and the argument, which may lead to what might appear as funny
513 * results: (2+I).has(-2) -> true. But this is consistent, since we also
514 * would like to have (-2+I).has(2) -> true and we want to think about the
515 * sign as a multiplicative factor. */
516 bool numeric::has(const ex &other) const
518 if (!is_ex_exactly_of_type(other, numeric))
520 const numeric &o = ex_to<numeric>(other);
521 if (this->is_equal(o) || this->is_equal(-o))
523 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
524 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
525 this->real().is_equal(-o) || this->imag().is_equal(-o));
527 if (o.is_equal(I)) // e.g scan for I in 42*I
528 return !this->is_real();
529 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
530 return (this->real().has(o*I) || this->imag().has(o*I) ||
531 this->real().has(-o*I) || this->imag().has(-o*I));
537 /** Evaluation of numbers doesn't do anything at all. */
538 ex numeric::eval(int level) const
540 // Warning: if this is ever gonna do something, the ex ctors from all kinds
541 // of numbers should be checking for status_flags::evaluated.
546 /** Cast numeric into a floating-point object. For example exact numeric(1) is
547 * returned as a 1.0000000000000000000000 and so on according to how Digits is
548 * currently set. In case the object already was a floating point number the
549 * precision is trimmed to match the currently set default.
551 * @param level ignored, only needed for overriding basic::evalf.
552 * @return an ex-handle to a numeric. */
553 ex numeric::evalf(int level) const
555 // level can safely be discarded for numeric objects.
556 return numeric(cln::cl_float(1.0, cln::default_float_format) *
557 (cln::the<cln::cl_N>(value)));
562 int numeric::compare_same_type(const basic &other) const
564 GINAC_ASSERT(is_exactly_a<numeric>(other));
565 const numeric &o = static_cast<const numeric &>(other);
567 return this->compare(o);
571 bool numeric::is_equal_same_type(const basic &other) const
573 GINAC_ASSERT(is_exactly_a<numeric>(other));
574 const numeric &o = static_cast<const numeric &>(other);
576 return this->is_equal(o);
580 unsigned numeric::calchash(void) const
582 // Use CLN's hashcode. Warning: It depends only on the number's value, not
583 // its type or precision (i.e. a true equivalence relation on numbers). As
584 // a consequence, 3 and 3.0 share the same hashvalue.
585 setflag(status_flags::hash_calculated);
586 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
591 // new virtual functions which can be overridden by derived classes
597 // non-virtual functions in this class
602 /** Numerical addition method. Adds argument to *this and returns result as
603 * a numeric object. */
604 const numeric numeric::add(const numeric &other) const
606 // Efficiency shortcut: trap the neutral element by pointer.
609 else if (&other==_num0_p)
612 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
616 /** Numerical subtraction method. Subtracts argument from *this and returns
617 * result as a numeric object. */
618 const numeric numeric::sub(const numeric &other) const
620 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
624 /** Numerical multiplication method. Multiplies *this and argument and returns
625 * result as a numeric object. */
626 const numeric numeric::mul(const numeric &other) const
628 // Efficiency shortcut: trap the neutral element by pointer.
631 else if (&other==_num1_p)
634 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
638 /** Numerical division method. Divides *this by argument and returns result as
641 * @exception overflow_error (division by zero) */
642 const numeric numeric::div(const numeric &other) const
644 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
645 throw std::overflow_error("numeric::div(): division by zero");
646 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
650 /** Numerical exponentiation. Raises *this to the power given as argument and
651 * returns result as a numeric object. */
652 const numeric numeric::power(const numeric &other) const
654 // Efficiency shortcut: trap the neutral exponent by pointer.
658 if (cln::zerop(cln::the<cln::cl_N>(value))) {
659 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
660 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
661 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
662 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
663 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
664 throw std::overflow_error("numeric::eval(): division by zero");
668 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
672 const numeric &numeric::add_dyn(const numeric &other) const
674 // Efficiency shortcut: trap the neutral element by pointer.
677 else if (&other==_num0_p)
680 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
681 setflag(status_flags::dynallocated));
685 const numeric &numeric::sub_dyn(const numeric &other) const
687 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
688 setflag(status_flags::dynallocated));
692 const numeric &numeric::mul_dyn(const numeric &other) const
694 // Efficiency shortcut: trap the neutral element by pointer.
697 else if (&other==_num1_p)
700 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
701 setflag(status_flags::dynallocated));
705 const numeric &numeric::div_dyn(const numeric &other) const
707 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
708 throw std::overflow_error("division by zero");
709 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
710 setflag(status_flags::dynallocated));
714 const numeric &numeric::power_dyn(const numeric &other) const
716 // Efficiency shortcut: trap the neutral exponent by pointer.
720 if (cln::zerop(cln::the<cln::cl_N>(value))) {
721 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
722 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
723 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
724 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
725 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
726 throw std::overflow_error("numeric::eval(): division by zero");
730 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
731 setflag(status_flags::dynallocated));
735 const numeric &numeric::operator=(int i)
737 return operator=(numeric(i));
741 const numeric &numeric::operator=(unsigned int i)
743 return operator=(numeric(i));
747 const numeric &numeric::operator=(long i)
749 return operator=(numeric(i));
753 const numeric &numeric::operator=(unsigned long i)
755 return operator=(numeric(i));
759 const numeric &numeric::operator=(double d)
761 return operator=(numeric(d));
765 const numeric &numeric::operator=(const char * s)
767 return operator=(numeric(s));
771 /** Inverse of a number. */
772 const numeric numeric::inverse(void) const
774 if (cln::zerop(cln::the<cln::cl_N>(value)))
775 throw std::overflow_error("numeric::inverse(): division by zero");
776 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
780 /** Return the complex half-plane (left or right) in which the number lies.
781 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
782 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
784 * @see numeric::compare(const numeric &other) */
785 int numeric::csgn(void) const
787 if (cln::zerop(cln::the<cln::cl_N>(value)))
789 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
790 if (!cln::zerop(r)) {
796 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
804 /** This method establishes a canonical order on all numbers. For complex
805 * numbers this is not possible in a mathematically consistent way but we need
806 * to establish some order and it ought to be fast. So we simply define it
807 * to be compatible with our method csgn.
809 * @return csgn(*this-other)
810 * @see numeric::csgn(void) */
811 int numeric::compare(const numeric &other) const
813 // Comparing two real numbers?
814 if (cln::instanceof(value, cln::cl_R_ring) &&
815 cln::instanceof(other.value, cln::cl_R_ring))
816 // Yes, so just cln::compare them
817 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
819 // No, first cln::compare real parts...
820 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
823 // ...and then the imaginary parts.
824 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
829 bool numeric::is_equal(const numeric &other) const
831 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
835 /** True if object is zero. */
836 bool numeric::is_zero(void) const
838 return cln::zerop(cln::the<cln::cl_N>(value));
842 /** True if object is not complex and greater than zero. */
843 bool numeric::is_positive(void) const
846 return cln::plusp(cln::the<cln::cl_R>(value));
851 /** True if object is not complex and less than zero. */
852 bool numeric::is_negative(void) const
855 return cln::minusp(cln::the<cln::cl_R>(value));
860 /** True if object is a non-complex integer. */
861 bool numeric::is_integer(void) const
863 return cln::instanceof(value, cln::cl_I_ring);
867 /** True if object is an exact integer greater than zero. */
868 bool numeric::is_pos_integer(void) const
870 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
874 /** True if object is an exact integer greater or equal zero. */
875 bool numeric::is_nonneg_integer(void) const
877 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
881 /** True if object is an exact even integer. */
882 bool numeric::is_even(void) const
884 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
888 /** True if object is an exact odd integer. */
889 bool numeric::is_odd(void) const
891 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
895 /** Probabilistic primality test.
897 * @return true if object is exact integer and prime. */
898 bool numeric::is_prime(void) const
900 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
904 /** True if object is an exact rational number, may even be complex
905 * (denominator may be unity). */
906 bool numeric::is_rational(void) const
908 return cln::instanceof(value, cln::cl_RA_ring);
912 /** True if object is a real integer, rational or float (but not complex). */
913 bool numeric::is_real(void) const
915 return cln::instanceof(value, cln::cl_R_ring);
919 bool numeric::operator==(const numeric &other) const
921 return cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
925 bool numeric::operator!=(const numeric &other) const
927 return !cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
931 /** True if object is element of the domain of integers extended by I, i.e. is
932 * of the form a+b*I, where a and b are integers. */
933 bool numeric::is_cinteger(void) const
935 if (cln::instanceof(value, cln::cl_I_ring))
937 else if (!this->is_real()) { // complex case, handle n+m*I
938 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
939 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
946 /** True if object is an exact rational number, may even be complex
947 * (denominator may be unity). */
948 bool numeric::is_crational(void) const
950 if (cln::instanceof(value, cln::cl_RA_ring))
952 else if (!this->is_real()) { // complex case, handle Q(i):
953 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
954 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
961 /** Numerical comparison: less.
963 * @exception invalid_argument (complex inequality) */
964 bool numeric::operator<(const numeric &other) const
966 if (this->is_real() && other.is_real())
967 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
968 throw std::invalid_argument("numeric::operator<(): complex inequality");
972 /** Numerical comparison: less or equal.
974 * @exception invalid_argument (complex inequality) */
975 bool numeric::operator<=(const numeric &other) const
977 if (this->is_real() && other.is_real())
978 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
979 throw std::invalid_argument("numeric::operator<=(): complex inequality");
983 /** Numerical comparison: greater.
985 * @exception invalid_argument (complex inequality) */
986 bool numeric::operator>(const numeric &other) const
988 if (this->is_real() && other.is_real())
989 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
990 throw std::invalid_argument("numeric::operator>(): complex inequality");
994 /** Numerical comparison: greater or equal.
996 * @exception invalid_argument (complex inequality) */
997 bool numeric::operator>=(const numeric &other) const
999 if (this->is_real() && other.is_real())
1000 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1001 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1005 /** Converts numeric types to machine's int. You should check with
1006 * is_integer() if the number is really an integer before calling this method.
1007 * You may also consider checking the range first. */
1008 int numeric::to_int(void) const
1010 GINAC_ASSERT(this->is_integer());
1011 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1015 /** Converts numeric types to machine's long. You should check with
1016 * is_integer() if the number is really an integer before calling this method.
1017 * You may also consider checking the range first. */
1018 long numeric::to_long(void) const
1020 GINAC_ASSERT(this->is_integer());
1021 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1025 /** Converts numeric types to machine's double. You should check with is_real()
1026 * if the number is really not complex before calling this method. */
1027 double numeric::to_double(void) const
1029 GINAC_ASSERT(this->is_real());
1030 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1034 /** Returns a new CLN object of type cl_N, representing the value of *this.
1035 * This method may be used when mixing GiNaC and CLN in one project.
1037 cln::cl_N numeric::to_cl_N(void) const
1039 return cln::cl_N(cln::the<cln::cl_N>(value));
1043 /** Real part of a number. */
1044 const numeric numeric::real(void) const
1046 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1050 /** Imaginary part of a number. */
1051 const numeric numeric::imag(void) const
1053 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1057 /** Numerator. Computes the numerator of rational numbers, rationalized
1058 * numerator of complex if real and imaginary part are both rational numbers
1059 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1061 const numeric numeric::numer(void) const
1063 if (this->is_integer())
1064 return numeric(*this);
1066 else if (cln::instanceof(value, cln::cl_RA_ring))
1067 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1069 else if (!this->is_real()) { // complex case, handle Q(i):
1070 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1071 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1072 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1073 return numeric(*this);
1074 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1075 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1076 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1077 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1078 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1079 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1080 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1081 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1084 // at least one float encountered
1085 return numeric(*this);
1089 /** Denominator. Computes the denominator of rational numbers, common integer
1090 * denominator of complex if real and imaginary part are both rational numbers
1091 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1092 const numeric numeric::denom(void) const
1094 if (this->is_integer())
1097 if (cln::instanceof(value, cln::cl_RA_ring))
1098 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1100 if (!this->is_real()) { // complex case, handle Q(i):
1101 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1102 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1103 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1105 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1106 return numeric(cln::denominator(i));
1107 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1108 return numeric(cln::denominator(r));
1109 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1110 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1112 // at least one float encountered
1117 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1118 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1119 * 2^(n-1) <= x < 2^n.
1121 * @return number of bits (excluding sign) needed to represent that number
1122 * in two's complement if it is an integer, 0 otherwise. */
1123 int numeric::int_length(void) const
1125 if (this->is_integer())
1126 return cln::integer_length(cln::the<cln::cl_I>(value));
1135 /** Imaginary unit. This is not a constant but a numeric since we are
1136 * natively handing complex numbers anyways, so in each expression containing
1137 * an I it is automatically eval'ed away anyhow. */
1138 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1141 /** Exponential function.
1143 * @return arbitrary precision numerical exp(x). */
1144 const numeric exp(const numeric &x)
1146 return cln::exp(x.to_cl_N());
1150 /** Natural logarithm.
1152 * @param z complex number
1153 * @return arbitrary precision numerical log(x).
1154 * @exception pole_error("log(): logarithmic pole",0) */
1155 const numeric log(const numeric &z)
1158 throw pole_error("log(): logarithmic pole",0);
1159 return cln::log(z.to_cl_N());
1163 /** Numeric sine (trigonometric function).
1165 * @return arbitrary precision numerical sin(x). */
1166 const numeric sin(const numeric &x)
1168 return cln::sin(x.to_cl_N());
1172 /** Numeric cosine (trigonometric function).
1174 * @return arbitrary precision numerical cos(x). */
1175 const numeric cos(const numeric &x)
1177 return cln::cos(x.to_cl_N());
1181 /** Numeric tangent (trigonometric function).
1183 * @return arbitrary precision numerical tan(x). */
1184 const numeric tan(const numeric &x)
1186 return cln::tan(x.to_cl_N());
1190 /** Numeric inverse sine (trigonometric function).
1192 * @return arbitrary precision numerical asin(x). */
1193 const numeric asin(const numeric &x)
1195 return cln::asin(x.to_cl_N());
1199 /** Numeric inverse cosine (trigonometric function).
1201 * @return arbitrary precision numerical acos(x). */
1202 const numeric acos(const numeric &x)
1204 return cln::acos(x.to_cl_N());
1210 * @param z complex number
1212 * @exception pole_error("atan(): logarithmic pole",0) */
1213 const numeric atan(const numeric &x)
1216 x.real().is_zero() &&
1217 abs(x.imag()).is_equal(_num1))
1218 throw pole_error("atan(): logarithmic pole",0);
1219 return cln::atan(x.to_cl_N());
1225 * @param x real number
1226 * @param y real number
1227 * @return atan(y/x) */
1228 const numeric atan(const numeric &y, const numeric &x)
1230 if (x.is_real() && y.is_real())
1231 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1232 cln::the<cln::cl_R>(y.to_cl_N()));
1234 throw std::invalid_argument("atan(): complex argument");
1238 /** Numeric hyperbolic sine (trigonometric function).
1240 * @return arbitrary precision numerical sinh(x). */
1241 const numeric sinh(const numeric &x)
1243 return cln::sinh(x.to_cl_N());
1247 /** Numeric hyperbolic cosine (trigonometric function).
1249 * @return arbitrary precision numerical cosh(x). */
1250 const numeric cosh(const numeric &x)
1252 return cln::cosh(x.to_cl_N());
1256 /** Numeric hyperbolic tangent (trigonometric function).
1258 * @return arbitrary precision numerical tanh(x). */
1259 const numeric tanh(const numeric &x)
1261 return cln::tanh(x.to_cl_N());
1265 /** Numeric inverse hyperbolic sine (trigonometric function).
1267 * @return arbitrary precision numerical asinh(x). */
1268 const numeric asinh(const numeric &x)
1270 return cln::asinh(x.to_cl_N());
1274 /** Numeric inverse hyperbolic cosine (trigonometric function).
1276 * @return arbitrary precision numerical acosh(x). */
1277 const numeric acosh(const numeric &x)
1279 return cln::acosh(x.to_cl_N());
1283 /** Numeric inverse hyperbolic tangent (trigonometric function).
1285 * @return arbitrary precision numerical atanh(x). */
1286 const numeric atanh(const numeric &x)
1288 return cln::atanh(x.to_cl_N());
1292 /*static cln::cl_N Li2_series(const ::cl_N &x,
1293 const ::float_format_t &prec)
1295 // Note: argument must be in the unit circle
1296 // This is very inefficient unless we have fast floating point Bernoulli
1297 // numbers implemented!
1298 cln::cl_N c1 = -cln::log(1-x);
1300 // hard-wire the first two Bernoulli numbers
1301 cln::cl_N acc = c1 - cln::square(c1)/4;
1303 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1304 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1306 c1 = cln::square(c1);
1310 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1311 // 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));
1314 } while (acc != acc+aug);
1318 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1319 * circle) using a power series. */
1320 static cln::cl_N Li2_series(const cln::cl_N &x,
1321 const cln::float_format_t &prec)
1323 // Note: argument must be in the unit circle
1325 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1330 den = den + i; // 1, 4, 9, 16, ...
1334 } while (acc != acc+aug);
1338 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1339 static cln::cl_N Li2_projection(const cln::cl_N &x,
1340 const cln::float_format_t &prec)
1342 const cln::cl_R re = cln::realpart(x);
1343 const cln::cl_R im = cln::imagpart(x);
1344 if (re > cln::cl_F(".5"))
1345 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1347 - Li2_series(1-x, prec)
1348 - cln::log(x)*cln::log(1-x));
1349 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1350 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1351 return(- cln::square(cln::log(1-x))/2
1352 - Li2_series(x/(x-1), prec));
1353 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1354 // Li2(x^2)/2 - Li2(-x)
1355 return(Li2_projection(cln::square(x), prec)/2
1356 - Li2_projection(-x, prec));
1357 return Li2_series(x, prec);
1360 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1361 * the branch cut lies along the positive real axis, starting at 1 and
1362 * continuous with quadrant IV.
1364 * @return arbitrary precision numerical Li2(x). */
1365 const numeric Li2(const numeric &x)
1370 // what is the desired float format?
1371 // first guess: default format
1372 cln::float_format_t prec = cln::default_float_format;
1373 const cln::cl_N value = x.to_cl_N();
1374 // second guess: the argument's format
1375 if (!x.real().is_rational())
1376 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1377 else if (!x.imag().is_rational())
1378 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1380 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1381 return cln::zeta(2, prec);
1383 if (cln::abs(value) > 1)
1384 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1385 return(- cln::square(cln::log(-value))/2
1386 - cln::zeta(2, prec)
1387 - Li2_projection(cln::recip(value), prec));
1389 return Li2_projection(x.to_cl_N(), prec);
1393 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1394 * integer arguments. */
1395 const numeric zeta(const numeric &x)
1397 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1398 // only knows about integer arguments and zeta(3).evalf() automatically
1399 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1400 // being an exact zero for CLN, which can be tested and then we can just
1401 // pass the number casted to an int:
1403 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1404 if (cln::zerop(x.to_cl_N()-aux))
1405 return cln::zeta(aux);
1411 /** The Gamma function.
1412 * This is only a stub! */
1413 const numeric lgamma(const numeric &x)
1417 const numeric tgamma(const numeric &x)
1423 /** The psi function (aka polygamma function).
1424 * This is only a stub! */
1425 const numeric psi(const numeric &x)
1431 /** The psi functions (aka polygamma functions).
1432 * This is only a stub! */
1433 const numeric psi(const numeric &n, const numeric &x)
1439 /** Factorial combinatorial function.
1441 * @param n integer argument >= 0
1442 * @exception range_error (argument must be integer >= 0) */
1443 const numeric factorial(const numeric &n)
1445 if (!n.is_nonneg_integer())
1446 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1447 return numeric(cln::factorial(n.to_int()));
1451 /** The double factorial combinatorial function. (Scarcely used, but still
1452 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1454 * @param n integer argument >= -1
1455 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1456 * @exception range_error (argument must be integer >= -1) */
1457 const numeric doublefactorial(const numeric &n)
1459 if (n.is_equal(_num_1))
1462 if (!n.is_nonneg_integer())
1463 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1465 return numeric(cln::doublefactorial(n.to_int()));
1469 /** The Binomial coefficients. It computes the binomial coefficients. For
1470 * integer n and k and positive n this is the number of ways of choosing k
1471 * objects from n distinct objects. If n is negative, the formula
1472 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1473 const numeric binomial(const numeric &n, const numeric &k)
1475 if (n.is_integer() && k.is_integer()) {
1476 if (n.is_nonneg_integer()) {
1477 if (k.compare(n)!=1 && k.compare(_num0)!=-1)
1478 return numeric(cln::binomial(n.to_int(),k.to_int()));
1482 return _num_1.power(k)*binomial(k-n-_num1,k);
1486 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1487 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1491 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1492 * in the expansion of the function x/(e^x-1).
1494 * @return the nth Bernoulli number (a rational number).
1495 * @exception range_error (argument must be integer >= 0) */
1496 const numeric bernoulli(const numeric &nn)
1498 if (!nn.is_integer() || nn.is_negative())
1499 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1503 // The Bernoulli numbers are rational numbers that may be computed using
1506 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1508 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1509 // previous ones, the computation is necessarily very expensive. There are
1510 // several other ways of computing them, a particularly good one being
1514 // for (unsigned i=0; i<n; i++) {
1515 // c = exquo(c*(i-n),(i+2));
1516 // Bern = Bern + c*s/(i+2);
1517 // s = s + expt_pos(cl_I(i+2),n);
1521 // But if somebody works with the n'th Bernoulli number she is likely to
1522 // also need all previous Bernoulli numbers. So we need a complete remember
1523 // table and above divide and conquer algorithm is not suited to build one
1524 // up. The formula below accomplishes this. It is a modification of the
1525 // defining formula above but the computation of the binomial coefficients
1526 // is carried along in an inline fashion. It also honors the fact that
1527 // B_n is zero when n is odd and greater than 1.
1529 // (There is an interesting relation with the tangent polynomials described
1530 // in `Concrete Mathematics', which leads to a program a little faster as
1531 // our implementation below, but it requires storing one such polynomial in
1532 // addition to the remember table. This doubles the memory footprint so
1533 // we don't use it.)
1535 const unsigned n = nn.to_int();
1537 // the special cases not covered by the algorithm below
1539 return (n==1) ? _num_1_2 : _num0;
1543 // store nonvanishing Bernoulli numbers here
1544 static std::vector< cln::cl_RA > results;
1545 static unsigned next_r = 0;
1547 // algorithm not applicable to B(2), so just store it
1549 results.push_back(cln::recip(cln::cl_RA(6)));
1553 return results[n/2-1];
1555 results.reserve(n/2);
1556 for (unsigned p=next_r; p<=n; p+=2) {
1557 cln::cl_I c = 1; // seed for binonmial coefficients
1558 cln::cl_RA b = cln::cl_RA(1-p)/2;
1559 const unsigned p3 = p+3;
1560 const unsigned pm = p-2;
1562 // test if intermediate unsigned int can be represented by immediate
1563 // objects by CLN (i.e. < 2^29 for 32 Bit machines, see <cln/object.h>)
1564 if (p < (1UL<<cl_value_len/2)) {
1565 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1566 c = cln::exquo(c * ((p3-i) * p_2), (i-1)*k);
1567 b = b + c*results[k-1];
1570 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1571 c = cln::exquo((c * (p3-i)) * p_2, cln::cl_I(i-1)*k);
1572 b = b + c*results[k-1];
1575 results.push_back(-b/(p+1));
1578 return results[n/2-1];
1582 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1583 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1585 * @param n an integer
1586 * @return the nth Fibonacci number F(n) (an integer number)
1587 * @exception range_error (argument must be an integer) */
1588 const numeric fibonacci(const numeric &n)
1590 if (!n.is_integer())
1591 throw std::range_error("numeric::fibonacci(): argument must be integer");
1594 // The following addition formula holds:
1596 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1598 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1599 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1601 // Replace m by m+1:
1602 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1603 // Now put in m = n, to get
1604 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1605 // F(2n+1) = F(n)^2 + F(n+1)^2
1607 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1610 if (n.is_negative())
1612 return -fibonacci(-n);
1614 return fibonacci(-n);
1618 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1619 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1620 // Since a squaring is cheaper than a multiplication, better use
1621 // three squarings instead of one multiplication and two squarings.
1622 cln::cl_I u2 = cln::square(u);
1623 cln::cl_I v2 = cln::square(v);
1624 if (cln::logbitp(bit-1, m)) {
1625 v = cln::square(u + v) - u2;
1628 u = v2 - cln::square(v - u);
1633 // Here we don't use the squaring formula because one multiplication
1634 // is cheaper than two squarings.
1635 return u * ((v << 1) - u);
1637 return cln::square(u) + cln::square(v);
1641 /** Absolute value. */
1642 const numeric abs(const numeric& x)
1644 return cln::abs(x.to_cl_N());
1648 /** Modulus (in positive representation).
1649 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1650 * sign of a or is zero. This is different from Maple's modp, where the sign
1651 * of b is ignored. It is in agreement with Mathematica's Mod.
1653 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1654 * integer, 0 otherwise. */
1655 const numeric mod(const numeric &a, const numeric &b)
1657 if (a.is_integer() && b.is_integer())
1658 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1659 cln::the<cln::cl_I>(b.to_cl_N()));
1665 /** Modulus (in symmetric representation).
1666 * Equivalent to Maple's mods.
1668 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1669 const numeric smod(const numeric &a, const numeric &b)
1671 if (a.is_integer() && b.is_integer()) {
1672 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1673 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1674 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1680 /** Numeric integer remainder.
1681 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1682 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1683 * sign of a or is zero.
1685 * @return remainder of a/b if both are integer, 0 otherwise. */
1686 const numeric irem(const numeric &a, const numeric &b)
1688 if (a.is_integer() && b.is_integer())
1689 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1690 cln::the<cln::cl_I>(b.to_cl_N()));
1696 /** Numeric integer remainder.
1697 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1698 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1699 * and irem(a,b) has the sign of a or is zero.
1701 * @return remainder of a/b and quotient stored in q if both are integer,
1703 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1705 if (a.is_integer() && b.is_integer()) {
1706 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1707 cln::the<cln::cl_I>(b.to_cl_N()));
1708 q = rem_quo.quotient;
1709 return rem_quo.remainder;
1717 /** Numeric integer quotient.
1718 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1720 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1721 const numeric iquo(const numeric &a, const numeric &b)
1723 if (a.is_integer() && b.is_integer())
1724 return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1725 cln::the<cln::cl_I>(b.to_cl_N()));
1731 /** Numeric integer quotient.
1732 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1733 * r == a - iquo(a,b,r)*b.
1735 * @return truncated quotient of a/b and remainder stored in r if both are
1736 * integer, 0 otherwise. */
1737 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1739 if (a.is_integer() && b.is_integer()) {
1740 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1741 cln::the<cln::cl_I>(b.to_cl_N()));
1742 r = rem_quo.remainder;
1743 return rem_quo.quotient;
1751 /** Greatest Common Divisor.
1753 * @return The GCD of two numbers if both are integer, a numerical 1
1754 * if they are not. */
1755 const numeric gcd(const numeric &a, const numeric &b)
1757 if (a.is_integer() && b.is_integer())
1758 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1759 cln::the<cln::cl_I>(b.to_cl_N()));
1765 /** Least Common Multiple.
1767 * @return The LCM of two numbers if both are integer, the product of those
1768 * two numbers if they are not. */
1769 const numeric lcm(const numeric &a, const numeric &b)
1771 if (a.is_integer() && b.is_integer())
1772 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1773 cln::the<cln::cl_I>(b.to_cl_N()));
1779 /** Numeric square root.
1780 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1781 * should return integer 2.
1783 * @param z numeric argument
1784 * @return square root of z. Branch cut along negative real axis, the negative
1785 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1786 * where imag(z)>0. */
1787 const numeric sqrt(const numeric &z)
1789 return cln::sqrt(z.to_cl_N());
1793 /** Integer numeric square root. */
1794 const numeric isqrt(const numeric &x)
1796 if (x.is_integer()) {
1798 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1805 /** Floating point evaluation of Archimedes' constant Pi. */
1808 return numeric(cln::pi(cln::default_float_format));
1812 /** Floating point evaluation of Euler's constant gamma. */
1815 return numeric(cln::eulerconst(cln::default_float_format));
1819 /** Floating point evaluation of Catalan's constant. */
1820 ex CatalanEvalf(void)
1822 return numeric(cln::catalanconst(cln::default_float_format));
1826 /** _numeric_digits default ctor, checking for singleton invariance. */
1827 _numeric_digits::_numeric_digits()
1830 // It initializes to 17 digits, because in CLN float_format(17) turns out
1831 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1832 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1834 throw(std::runtime_error("I told you not to do instantiate me!"));
1836 cln::default_float_format = cln::float_format(17);
1840 /** Assign a native long to global Digits object. */
1841 _numeric_digits& _numeric_digits::operator=(long prec)
1844 cln::default_float_format = cln::float_format(prec);
1849 /** Convert global Digits object to native type long. */
1850 _numeric_digits::operator long()
1852 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1853 return (long)digits;
1857 /** Append global Digits object to ostream. */
1858 void _numeric_digits::print(std::ostream &os) const
1864 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1871 // static member variables
1876 bool _numeric_digits::too_late = false;
1879 /** Accuracy in decimal digits. Only object of this type! Can be set using
1880 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1881 _numeric_digits Digits;
1883 } // namespace GiNaC