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
42 // CLN should pollute the global namespace as little as possible. Hence, we
43 // include most of it here and include only the part needed for properly
44 // declaring cln::cl_number in numeric.h. This can only be safely done in
45 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
46 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
48 #include <cln/output.h>
49 #include <cln/integer_io.h>
50 #include <cln/integer_ring.h>
51 #include <cln/rational_io.h>
52 #include <cln/rational_ring.h>
53 #include <cln/lfloat_class.h>
54 #include <cln/lfloat_io.h>
55 #include <cln/real_io.h>
56 #include <cln/real_ring.h>
57 #include <cln/complex_io.h>
58 #include <cln/complex_ring.h>
59 #include <cln/numtheory.h>
63 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
66 // default ctor, dtor, copy ctor assignment
67 // operator and helpers
70 /** default ctor. Numerically it initializes to an integer zero. */
71 numeric::numeric() : basic(TINFO_numeric)
73 debugmsg("numeric default ctor", LOGLEVEL_CONSTRUCT);
75 setflag(status_flags::evaluated | status_flags::expanded);
78 void numeric::copy(const numeric &other)
80 inherited::copy(other);
84 DEFAULT_DESTROY(numeric)
92 numeric::numeric(int i) : basic(TINFO_numeric)
94 debugmsg("numeric ctor from int",LOGLEVEL_CONSTRUCT);
95 // Not the whole int-range is available if we don't cast to long
96 // first. This is due to the behaviour of the cl_I-ctor, which
97 // emphasizes efficiency. However, if the integer is small enough,
98 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
99 // using an immediate type:
100 if (cln::cl_immediate_p(i))
101 value = cln::cl_I(i);
103 value = cln::cl_I((long) i);
104 setflag(status_flags::evaluated | status_flags::expanded);
108 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
110 debugmsg("numeric ctor from uint",LOGLEVEL_CONSTRUCT);
111 // Not the whole uint-range is available if we don't cast to ulong
112 // first. This is due to the behaviour of the cl_I-ctor, which
113 // emphasizes efficiency. However, if the integer is small enough,
114 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
115 // using an immediate type:
116 if (cln::cl_immediate_p(i))
117 value = cln::cl_I(i);
119 value = cln::cl_I((unsigned long) i);
120 setflag(status_flags::evaluated | status_flags::expanded);
124 numeric::numeric(long i) : basic(TINFO_numeric)
126 debugmsg("numeric ctor from long",LOGLEVEL_CONSTRUCT);
127 value = cln::cl_I(i);
128 setflag(status_flags::evaluated | status_flags::expanded);
132 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
134 debugmsg("numeric ctor from ulong",LOGLEVEL_CONSTRUCT);
135 value = cln::cl_I(i);
136 setflag(status_flags::evaluated | status_flags::expanded);
139 /** Ctor for rational numerics a/b.
141 * @exception overflow_error (division by zero) */
142 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
144 debugmsg("numeric ctor from long/long",LOGLEVEL_CONSTRUCT);
146 throw std::overflow_error("division by zero");
147 value = cln::cl_I(numer) / cln::cl_I(denom);
148 setflag(status_flags::evaluated | status_flags::expanded);
152 numeric::numeric(double d) : basic(TINFO_numeric)
154 debugmsg("numeric ctor from double",LOGLEVEL_CONSTRUCT);
155 // We really want to explicitly use the type cl_LF instead of the
156 // more general cl_F, since that would give us a cl_DF only which
157 // will not be promoted to cl_LF if overflow occurs:
158 value = cln::cl_float(d, cln::default_float_format);
159 setflag(status_flags::evaluated | status_flags::expanded);
163 /** ctor from C-style string. It also accepts complex numbers in GiNaC
164 * notation like "2+5*I". */
165 numeric::numeric(const char *s) : basic(TINFO_numeric)
167 debugmsg("numeric ctor from string",LOGLEVEL_CONSTRUCT);
168 cln::cl_N ctorval = 0;
169 // parse complex numbers (functional but not completely safe, unfortunately
170 // std::string does not understand regexpese):
171 // ss should represent a simple sum like 2+5*I
173 // make it safe by adding explicit sign
174 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
176 std::string::size_type delim;
178 // chop ss into terms from left to right
180 bool imaginary = false;
181 delim = ss.find_first_of(std::string("+-"),1);
182 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
183 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
184 delim = ss.find_first_of(std::string("+-"),delim+1);
185 term = ss.substr(0,delim);
186 if (delim != std::string::npos)
187 ss = ss.substr(delim);
188 // is the term imaginary?
189 if (term.find("I") != std::string::npos) {
191 term = term.replace(term.find("I"),1,"");
193 if (term.find("*") != std::string::npos)
194 term = term.replace(term.find("*"),1,"");
195 // correct for trivial +/-I without explicit factor on I:
196 if (term.size() == 1)
200 if (term.find(".") != std::string::npos) {
201 // CLN's short type cl_SF is not very useful within the GiNaC
202 // framework where we are mainly interested in the arbitrary
203 // precision type cl_LF. Hence we go straight to the construction
204 // of generic floats. In order to create them we have to convert
205 // our own floating point notation used for output and construction
206 // from char * to CLN's generic notation:
207 // 3.14 --> 3.14e0_<Digits>
208 // 31.4E-1 --> 31.4e-1_<Digits>
210 // No exponent marker? Let's add a trivial one.
211 if (term.find("E") == std::string::npos)
214 term = term.replace(term.find("E"),1,"e");
215 // append _<Digits> to term
216 term += "_" + ToString((unsigned)Digits);
217 // construct float using cln::cl_F(const char *) ctor.
219 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
221 ctorval = ctorval + cln::cl_F(term.c_str());
223 // not a floating point number...
225 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
227 ctorval = ctorval + cln::cl_R(term.c_str());
229 } while(delim != std::string::npos);
231 setflag(status_flags::evaluated | status_flags::expanded);
235 /** Ctor from CLN types. This is for the initiated user or internal use
237 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
239 debugmsg("numeric ctor from cl_N", LOGLEVEL_CONSTRUCT);
241 setflag(status_flags::evaluated | status_flags::expanded);
248 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
250 debugmsg("numeric ctor from archive_node", LOGLEVEL_CONSTRUCT);
251 cln::cl_N ctorval = 0;
253 // Read number as string
255 if (n.find_string("number", str)) {
256 std::istringstream s(str);
257 cln::cl_idecoded_float re, im;
261 case 'R': // Integer-decoded real number
262 s >> re.sign >> re.mantissa >> re.exponent;
263 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
265 case 'C': // Integer-decoded complex number
266 s >> re.sign >> re.mantissa >> re.exponent;
267 s >> im.sign >> im.mantissa >> im.exponent;
268 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
269 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
271 default: // Ordinary number
278 setflag(status_flags::evaluated | status_flags::expanded);
281 void numeric::archive(archive_node &n) const
283 inherited::archive(n);
285 // Write number as string
286 std::ostringstream s;
287 if (this->is_crational())
288 s << cln::the<cln::cl_N>(value);
290 // Non-rational numbers are written in an integer-decoded format
291 // to preserve the precision
292 if (this->is_real()) {
293 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
295 s << re.sign << " " << re.mantissa << " " << re.exponent;
297 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
298 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
300 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
301 s << im.sign << " " << im.mantissa << " " << im.exponent;
304 n.add_string("number", s.str());
307 DEFAULT_UNARCHIVE(numeric)
310 // functions overriding virtual functions from base classes
313 /** Helper function to print a real number in a nicer way than is CLN's
314 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
315 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
316 * long as it only uses cl_LF and no other floating point types that we might
317 * want to visibly distinguish from cl_LF.
319 * @see numeric::print() */
320 static void print_real_number(const print_context & c, const cln::cl_R &x)
322 cln::cl_print_flags ourflags;
323 if (cln::instanceof(x, cln::cl_RA_ring)) {
324 // case 1: integer or rational
325 if (cln::instanceof(x, cln::cl_I_ring) ||
326 !is_a<print_latex>(c)) {
327 cln::print_real(c.s, ourflags, x);
328 } else { // rational output in LaTeX context
330 cln::print_real(c.s, ourflags, cln::numerator(cln::the<cln::cl_RA>(x)));
332 cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
337 // make CLN believe this number has default_float_format, so it prints
338 // 'E' as exponent marker instead of 'L':
339 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
340 cln::print_real(c.s, ourflags, x);
344 /** This method adds to the output so it blends more consistently together
345 * with the other routines and produces something compatible to ginsh input.
347 * @see print_real_number() */
348 void numeric::print(const print_context & c, unsigned level) const
350 debugmsg("numeric print", LOGLEVEL_PRINT);
352 if (is_a<print_tree>(c)) {
354 c.s << std::string(level, ' ') << cln::the<cln::cl_N>(value)
355 << " (" << class_name() << ")"
356 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
359 } else if (is_a<print_csrc>(c)) {
361 std::ios::fmtflags oldflags = c.s.flags();
362 c.s.setf(std::ios::scientific);
363 if (this->is_rational() && !this->is_integer()) {
364 if (compare(_num0) > 0) {
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(\"" << -numer().evalf() << "\")";
375 c.s << -numer().to_double();
378 if (is_a<print_csrc_cl_N>(c))
379 c.s << "cln::cl_F(\"" << denom().evalf() << "\")";
381 c.s << denom().to_double();
384 if (is_a<print_csrc_cl_N>(c))
385 c.s << "cln::cl_F(\"" << evalf() << "\")";
392 const std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
393 const std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
394 const std::string imag_sym = is_a<print_latex>(c) ? "i" : "I";
395 const std::string mul_sym = is_a<print_latex>(c) ? " " : "*";
396 const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
397 const cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
399 // case 1, real: x or -x
400 if ((precedence() <= level) && (!this->is_nonneg_integer())) {
402 print_real_number(c, r);
405 print_real_number(c, r);
409 // case 2, imaginary: y*I or -y*I
410 if ((precedence() <= level) && (i < 0)) {
412 c.s << par_open+imag_sym+par_close;
415 print_real_number(c, i);
416 c.s << mul_sym+imag_sym+par_close;
423 c.s << "-" << imag_sym;
425 print_real_number(c, i);
426 c.s << mul_sym+imag_sym;
431 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
432 if (precedence() <= level)
434 print_real_number(c, r);
439 print_real_number(c, i);
440 c.s << mul_sym+imag_sym;
447 print_real_number(c, i);
448 c.s << mul_sym+imag_sym;
451 if (precedence() <= level)
458 bool numeric::info(unsigned inf) const
461 case info_flags::numeric:
462 case info_flags::polynomial:
463 case info_flags::rational_function:
465 case info_flags::real:
467 case info_flags::rational:
468 case info_flags::rational_polynomial:
469 return is_rational();
470 case info_flags::crational:
471 case info_flags::crational_polynomial:
472 return is_crational();
473 case info_flags::integer:
474 case info_flags::integer_polynomial:
476 case info_flags::cinteger:
477 case info_flags::cinteger_polynomial:
478 return is_cinteger();
479 case info_flags::positive:
480 return is_positive();
481 case info_flags::negative:
482 return is_negative();
483 case info_flags::nonnegative:
484 return !is_negative();
485 case info_flags::posint:
486 return is_pos_integer();
487 case info_flags::negint:
488 return is_integer() && is_negative();
489 case info_flags::nonnegint:
490 return is_nonneg_integer();
491 case info_flags::even:
493 case info_flags::odd:
495 case info_flags::prime:
497 case info_flags::algebraic:
503 /** Disassemble real part and imaginary part to scan for the occurrence of a
504 * single number. Also handles the imaginary unit. It ignores the sign on
505 * both this and the argument, which may lead to what might appear as funny
506 * results: (2+I).has(-2) -> true. But this is consistent, since we also
507 * would like to have (-2+I).has(2) -> true and we want to think about the
508 * sign as a multiplicative factor. */
509 bool numeric::has(const ex &other) const
511 if (!is_ex_exactly_of_type(other, numeric))
513 const numeric &o = ex_to<numeric>(other);
514 if (this->is_equal(o) || this->is_equal(-o))
516 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
517 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
518 this->real().is_equal(-o) || this->imag().is_equal(-o));
520 if (o.is_equal(I)) // e.g scan for I in 42*I
521 return !this->is_real();
522 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
523 return (this->real().has(o*I) || this->imag().has(o*I) ||
524 this->real().has(-o*I) || this->imag().has(-o*I));
530 /** Evaluation of numbers doesn't do anything at all. */
531 ex numeric::eval(int level) const
533 // Warning: if this is ever gonna do something, the ex ctors from all kinds
534 // of numbers should be checking for status_flags::evaluated.
539 /** Cast numeric into a floating-point object. For example exact numeric(1) is
540 * returned as a 1.0000000000000000000000 and so on according to how Digits is
541 * currently set. In case the object already was a floating point number the
542 * precision is trimmed to match the currently set default.
544 * @param level ignored, only needed for overriding basic::evalf.
545 * @return an ex-handle to a numeric. */
546 ex numeric::evalf(int level) const
548 // level can safely be discarded for numeric objects.
549 return numeric(cln::cl_float(1.0, cln::default_float_format) *
550 (cln::the<cln::cl_N>(value)));
555 int numeric::compare_same_type(const basic &other) const
557 GINAC_ASSERT(is_exactly_a<numeric>(other));
558 const numeric &o = static_cast<const numeric &>(other);
560 return this->compare(o);
564 bool numeric::is_equal_same_type(const basic &other) const
566 GINAC_ASSERT(is_exactly_a<numeric>(other));
567 const numeric &o = static_cast<const numeric &>(other);
569 return this->is_equal(o);
573 unsigned numeric::calchash(void) const
575 // Use CLN's hashcode. Warning: It depends only on the number's value, not
576 // its type or precision (i.e. a true equivalence relation on numbers). As
577 // a consequence, 3 and 3.0 share the same hashvalue.
578 setflag(status_flags::hash_calculated);
579 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
584 // new virtual functions which can be overridden by derived classes
590 // non-virtual functions in this class
595 /** Numerical addition method. Adds argument to *this and returns result as
596 * a numeric object. */
597 const numeric numeric::add(const numeric &other) const
599 // Efficiency shortcut: trap the neutral element by pointer.
602 else if (&other==_num0_p)
605 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
609 /** Numerical subtraction method. Subtracts argument from *this and returns
610 * result as a numeric object. */
611 const numeric numeric::sub(const numeric &other) const
613 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
617 /** Numerical multiplication method. Multiplies *this and argument and returns
618 * result as a numeric object. */
619 const numeric numeric::mul(const numeric &other) const
621 // Efficiency shortcut: trap the neutral element by pointer.
624 else if (&other==_num1_p)
627 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
631 /** Numerical division method. Divides *this by argument and returns result as
634 * @exception overflow_error (division by zero) */
635 const numeric numeric::div(const numeric &other) const
637 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
638 throw std::overflow_error("numeric::div(): division by zero");
639 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
643 /** Numerical exponentiation. Raises *this to the power given as argument and
644 * returns result as a numeric object. */
645 const numeric numeric::power(const numeric &other) const
647 // Efficiency shortcut: trap the neutral exponent by pointer.
651 if (cln::zerop(cln::the<cln::cl_N>(value))) {
652 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
653 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
654 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
655 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
656 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
657 throw std::overflow_error("numeric::eval(): division by zero");
661 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
665 const numeric &numeric::add_dyn(const numeric &other) const
667 // Efficiency shortcut: trap the neutral element by pointer.
670 else if (&other==_num0_p)
673 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
674 setflag(status_flags::dynallocated));
678 const numeric &numeric::sub_dyn(const numeric &other) const
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::mul_dyn(const numeric &other) const
687 // Efficiency shortcut: trap the neutral element by pointer.
690 else if (&other==_num1_p)
693 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
694 setflag(status_flags::dynallocated));
698 const numeric &numeric::div_dyn(const numeric &other) const
700 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
701 throw std::overflow_error("division by zero");
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::power_dyn(const numeric &other) const
709 // Efficiency shortcut: trap the neutral exponent by pointer.
713 if (cln::zerop(cln::the<cln::cl_N>(value))) {
714 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
715 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
716 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
717 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
718 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
719 throw std::overflow_error("numeric::eval(): division by zero");
723 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
724 setflag(status_flags::dynallocated));
728 const numeric &numeric::operator=(int i)
730 return operator=(numeric(i));
734 const numeric &numeric::operator=(unsigned int i)
736 return operator=(numeric(i));
740 const numeric &numeric::operator=(long i)
742 return operator=(numeric(i));
746 const numeric &numeric::operator=(unsigned long i)
748 return operator=(numeric(i));
752 const numeric &numeric::operator=(double d)
754 return operator=(numeric(d));
758 const numeric &numeric::operator=(const char * s)
760 return operator=(numeric(s));
764 /** Inverse of a number. */
765 const numeric numeric::inverse(void) const
767 if (cln::zerop(cln::the<cln::cl_N>(value)))
768 throw std::overflow_error("numeric::inverse(): division by zero");
769 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
773 /** Return the complex half-plane (left or right) in which the number lies.
774 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
775 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
777 * @see numeric::compare(const numeric &other) */
778 int numeric::csgn(void) const
780 if (cln::zerop(cln::the<cln::cl_N>(value)))
782 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
783 if (!cln::zerop(r)) {
789 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
797 /** This method establishes a canonical order on all numbers. For complex
798 * numbers this is not possible in a mathematically consistent way but we need
799 * to establish some order and it ought to be fast. So we simply define it
800 * to be compatible with our method csgn.
802 * @return csgn(*this-other)
803 * @see numeric::csgn(void) */
804 int numeric::compare(const numeric &other) const
806 // Comparing two real numbers?
807 if (cln::instanceof(value, cln::cl_R_ring) &&
808 cln::instanceof(other.value, cln::cl_R_ring))
809 // Yes, so just cln::compare them
810 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
812 // No, first cln::compare real parts...
813 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
816 // ...and then the imaginary parts.
817 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
822 bool numeric::is_equal(const numeric &other) const
824 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
828 /** True if object is zero. */
829 bool numeric::is_zero(void) const
831 return cln::zerop(cln::the<cln::cl_N>(value));
835 /** True if object is not complex and greater than zero. */
836 bool numeric::is_positive(void) const
839 return cln::plusp(cln::the<cln::cl_R>(value));
844 /** True if object is not complex and less than zero. */
845 bool numeric::is_negative(void) const
848 return cln::minusp(cln::the<cln::cl_R>(value));
853 /** True if object is a non-complex integer. */
854 bool numeric::is_integer(void) const
856 return cln::instanceof(value, cln::cl_I_ring);
860 /** True if object is an exact integer greater than zero. */
861 bool numeric::is_pos_integer(void) const
863 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
867 /** True if object is an exact integer greater or equal zero. */
868 bool numeric::is_nonneg_integer(void) const
870 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
874 /** True if object is an exact even integer. */
875 bool numeric::is_even(void) const
877 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
881 /** True if object is an exact odd integer. */
882 bool numeric::is_odd(void) const
884 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
888 /** Probabilistic primality test.
890 * @return true if object is exact integer and prime. */
891 bool numeric::is_prime(void) const
893 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
897 /** True if object is an exact rational number, may even be complex
898 * (denominator may be unity). */
899 bool numeric::is_rational(void) const
901 return cln::instanceof(value, cln::cl_RA_ring);
905 /** True if object is a real integer, rational or float (but not complex). */
906 bool numeric::is_real(void) const
908 return cln::instanceof(value, cln::cl_R_ring);
912 bool numeric::operator==(const numeric &other) const
914 return cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
918 bool numeric::operator!=(const numeric &other) const
920 return !cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
924 /** True if object is element of the domain of integers extended by I, i.e. is
925 * of the form a+b*I, where a and b are integers. */
926 bool numeric::is_cinteger(void) const
928 if (cln::instanceof(value, cln::cl_I_ring))
930 else if (!this->is_real()) { // complex case, handle n+m*I
931 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
932 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
939 /** True if object is an exact rational number, may even be complex
940 * (denominator may be unity). */
941 bool numeric::is_crational(void) const
943 if (cln::instanceof(value, cln::cl_RA_ring))
945 else if (!this->is_real()) { // complex case, handle Q(i):
946 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
947 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
954 /** Numerical comparison: less.
956 * @exception invalid_argument (complex inequality) */
957 bool numeric::operator<(const numeric &other) const
959 if (this->is_real() && other.is_real())
960 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
961 throw std::invalid_argument("numeric::operator<(): complex inequality");
965 /** Numerical comparison: less or equal.
967 * @exception invalid_argument (complex inequality) */
968 bool numeric::operator<=(const numeric &other) const
970 if (this->is_real() && other.is_real())
971 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
972 throw std::invalid_argument("numeric::operator<=(): complex inequality");
976 /** Numerical comparison: greater.
978 * @exception invalid_argument (complex inequality) */
979 bool numeric::operator>(const numeric &other) const
981 if (this->is_real() && other.is_real())
982 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
983 throw std::invalid_argument("numeric::operator>(): complex inequality");
987 /** Numerical comparison: greater or equal.
989 * @exception invalid_argument (complex inequality) */
990 bool numeric::operator>=(const numeric &other) const
992 if (this->is_real() && other.is_real())
993 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
994 throw std::invalid_argument("numeric::operator>=(): complex inequality");
998 /** Converts numeric types to machine's int. You should check with
999 * is_integer() if the number is really an integer before calling this method.
1000 * You may also consider checking the range first. */
1001 int numeric::to_int(void) const
1003 GINAC_ASSERT(this->is_integer());
1004 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1008 /** Converts numeric types to machine's long. You should check with
1009 * is_integer() if the number is really an integer before calling this method.
1010 * You may also consider checking the range first. */
1011 long numeric::to_long(void) const
1013 GINAC_ASSERT(this->is_integer());
1014 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1018 /** Converts numeric types to machine's double. You should check with is_real()
1019 * if the number is really not complex before calling this method. */
1020 double numeric::to_double(void) const
1022 GINAC_ASSERT(this->is_real());
1023 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1027 /** Returns a new CLN object of type cl_N, representing the value of *this.
1028 * This method may be used when mixing GiNaC and CLN in one project.
1030 cln::cl_N numeric::to_cl_N(void) const
1032 return cln::cl_N(cln::the<cln::cl_N>(value));
1036 /** Real part of a number. */
1037 const numeric numeric::real(void) const
1039 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1043 /** Imaginary part of a number. */
1044 const numeric numeric::imag(void) const
1046 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1050 /** Numerator. Computes the numerator of rational numbers, rationalized
1051 * numerator of complex if real and imaginary part are both rational numbers
1052 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1054 const numeric numeric::numer(void) const
1056 if (this->is_integer())
1057 return numeric(*this);
1059 else if (cln::instanceof(value, cln::cl_RA_ring))
1060 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1062 else if (!this->is_real()) { // complex case, handle Q(i):
1063 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1064 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1065 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1066 return numeric(*this);
1067 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1068 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1069 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1070 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1071 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1072 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1073 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1074 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1077 // at least one float encountered
1078 return numeric(*this);
1082 /** Denominator. Computes the denominator of rational numbers, common integer
1083 * denominator of complex if real and imaginary part are both rational numbers
1084 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1085 const numeric numeric::denom(void) const
1087 if (this->is_integer())
1090 if (cln::instanceof(value, cln::cl_RA_ring))
1091 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1093 if (!this->is_real()) { // complex case, handle Q(i):
1094 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1095 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1096 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1098 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1099 return numeric(cln::denominator(i));
1100 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1101 return numeric(cln::denominator(r));
1102 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1103 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1105 // at least one float encountered
1110 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1111 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1112 * 2^(n-1) <= x < 2^n.
1114 * @return number of bits (excluding sign) needed to represent that number
1115 * in two's complement if it is an integer, 0 otherwise. */
1116 int numeric::int_length(void) const
1118 if (this->is_integer())
1119 return cln::integer_length(cln::the<cln::cl_I>(value));
1128 /** Imaginary unit. This is not a constant but a numeric since we are
1129 * natively handing complex numbers anyways, so in each expression containing
1130 * an I it is automatically eval'ed away anyhow. */
1131 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1134 /** Exponential function.
1136 * @return arbitrary precision numerical exp(x). */
1137 const numeric exp(const numeric &x)
1139 return cln::exp(x.to_cl_N());
1143 /** Natural logarithm.
1145 * @param z complex number
1146 * @return arbitrary precision numerical log(x).
1147 * @exception pole_error("log(): logarithmic pole",0) */
1148 const numeric log(const numeric &z)
1151 throw pole_error("log(): logarithmic pole",0);
1152 return cln::log(z.to_cl_N());
1156 /** Numeric sine (trigonometric function).
1158 * @return arbitrary precision numerical sin(x). */
1159 const numeric sin(const numeric &x)
1161 return cln::sin(x.to_cl_N());
1165 /** Numeric cosine (trigonometric function).
1167 * @return arbitrary precision numerical cos(x). */
1168 const numeric cos(const numeric &x)
1170 return cln::cos(x.to_cl_N());
1174 /** Numeric tangent (trigonometric function).
1176 * @return arbitrary precision numerical tan(x). */
1177 const numeric tan(const numeric &x)
1179 return cln::tan(x.to_cl_N());
1183 /** Numeric inverse sine (trigonometric function).
1185 * @return arbitrary precision numerical asin(x). */
1186 const numeric asin(const numeric &x)
1188 return cln::asin(x.to_cl_N());
1192 /** Numeric inverse cosine (trigonometric function).
1194 * @return arbitrary precision numerical acos(x). */
1195 const numeric acos(const numeric &x)
1197 return cln::acos(x.to_cl_N());
1203 * @param z complex number
1205 * @exception pole_error("atan(): logarithmic pole",0) */
1206 const numeric atan(const numeric &x)
1209 x.real().is_zero() &&
1210 abs(x.imag()).is_equal(_num1))
1211 throw pole_error("atan(): logarithmic pole",0);
1212 return cln::atan(x.to_cl_N());
1218 * @param x real number
1219 * @param y real number
1220 * @return atan(y/x) */
1221 const numeric atan(const numeric &y, const numeric &x)
1223 if (x.is_real() && y.is_real())
1224 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1225 cln::the<cln::cl_R>(y.to_cl_N()));
1227 throw std::invalid_argument("atan(): complex argument");
1231 /** Numeric hyperbolic sine (trigonometric function).
1233 * @return arbitrary precision numerical sinh(x). */
1234 const numeric sinh(const numeric &x)
1236 return cln::sinh(x.to_cl_N());
1240 /** Numeric hyperbolic cosine (trigonometric function).
1242 * @return arbitrary precision numerical cosh(x). */
1243 const numeric cosh(const numeric &x)
1245 return cln::cosh(x.to_cl_N());
1249 /** Numeric hyperbolic tangent (trigonometric function).
1251 * @return arbitrary precision numerical tanh(x). */
1252 const numeric tanh(const numeric &x)
1254 return cln::tanh(x.to_cl_N());
1258 /** Numeric inverse hyperbolic sine (trigonometric function).
1260 * @return arbitrary precision numerical asinh(x). */
1261 const numeric asinh(const numeric &x)
1263 return cln::asinh(x.to_cl_N());
1267 /** Numeric inverse hyperbolic cosine (trigonometric function).
1269 * @return arbitrary precision numerical acosh(x). */
1270 const numeric acosh(const numeric &x)
1272 return cln::acosh(x.to_cl_N());
1276 /** Numeric inverse hyperbolic tangent (trigonometric function).
1278 * @return arbitrary precision numerical atanh(x). */
1279 const numeric atanh(const numeric &x)
1281 return cln::atanh(x.to_cl_N());
1285 /*static cln::cl_N Li2_series(const ::cl_N &x,
1286 const ::float_format_t &prec)
1288 // Note: argument must be in the unit circle
1289 // This is very inefficient unless we have fast floating point Bernoulli
1290 // numbers implemented!
1291 cln::cl_N c1 = -cln::log(1-x);
1293 // hard-wire the first two Bernoulli numbers
1294 cln::cl_N acc = c1 - cln::square(c1)/4;
1296 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1297 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1299 c1 = cln::square(c1);
1303 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1304 // 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));
1307 } while (acc != acc+aug);
1311 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1312 * circle) using a power series. */
1313 static cln::cl_N Li2_series(const cln::cl_N &x,
1314 const cln::float_format_t &prec)
1316 // Note: argument must be in the unit circle
1318 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1323 den = den + i; // 1, 4, 9, 16, ...
1327 } while (acc != acc+aug);
1331 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1332 static cln::cl_N Li2_projection(const cln::cl_N &x,
1333 const cln::float_format_t &prec)
1335 const cln::cl_R re = cln::realpart(x);
1336 const cln::cl_R im = cln::imagpart(x);
1337 if (re > cln::cl_F(".5"))
1338 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1340 - Li2_series(1-x, prec)
1341 - cln::log(x)*cln::log(1-x));
1342 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1343 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1344 return(- cln::square(cln::log(1-x))/2
1345 - Li2_series(x/(x-1), prec));
1346 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1347 // Li2(x^2)/2 - Li2(-x)
1348 return(Li2_projection(cln::square(x), prec)/2
1349 - Li2_projection(-x, prec));
1350 return Li2_series(x, prec);
1353 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1354 * the branch cut lies along the positive real axis, starting at 1 and
1355 * continuous with quadrant IV.
1357 * @return arbitrary precision numerical Li2(x). */
1358 const numeric Li2(const numeric &x)
1363 // what is the desired float format?
1364 // first guess: default format
1365 cln::float_format_t prec = cln::default_float_format;
1366 const cln::cl_N value = x.to_cl_N();
1367 // second guess: the argument's format
1368 if (!x.real().is_rational())
1369 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1370 else if (!x.imag().is_rational())
1371 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1373 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1374 return cln::zeta(2, prec);
1376 if (cln::abs(value) > 1)
1377 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1378 return(- cln::square(cln::log(-value))/2
1379 - cln::zeta(2, prec)
1380 - Li2_projection(cln::recip(value), prec));
1382 return Li2_projection(x.to_cl_N(), prec);
1386 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1387 * integer arguments. */
1388 const numeric zeta(const numeric &x)
1390 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1391 // only knows about integer arguments and zeta(3).evalf() automatically
1392 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1393 // being an exact zero for CLN, which can be tested and then we can just
1394 // pass the number casted to an int:
1396 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1397 if (cln::zerop(x.to_cl_N()-aux))
1398 return cln::zeta(aux);
1404 /** The Gamma function.
1405 * This is only a stub! */
1406 const numeric lgamma(const numeric &x)
1410 const numeric tgamma(const numeric &x)
1416 /** The psi function (aka polygamma function).
1417 * This is only a stub! */
1418 const numeric psi(const numeric &x)
1424 /** The psi functions (aka polygamma functions).
1425 * This is only a stub! */
1426 const numeric psi(const numeric &n, const numeric &x)
1432 /** Factorial combinatorial function.
1434 * @param n integer argument >= 0
1435 * @exception range_error (argument must be integer >= 0) */
1436 const numeric factorial(const numeric &n)
1438 if (!n.is_nonneg_integer())
1439 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1440 return numeric(cln::factorial(n.to_int()));
1444 /** The double factorial combinatorial function. (Scarcely used, but still
1445 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1447 * @param n integer argument >= -1
1448 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1449 * @exception range_error (argument must be integer >= -1) */
1450 const numeric doublefactorial(const numeric &n)
1452 if (n.is_equal(_num_1))
1455 if (!n.is_nonneg_integer())
1456 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1458 return numeric(cln::doublefactorial(n.to_int()));
1462 /** The Binomial coefficients. It computes the binomial coefficients. For
1463 * integer n and k and positive n this is the number of ways of choosing k
1464 * objects from n distinct objects. If n is negative, the formula
1465 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1466 const numeric binomial(const numeric &n, const numeric &k)
1468 if (n.is_integer() && k.is_integer()) {
1469 if (n.is_nonneg_integer()) {
1470 if (k.compare(n)!=1 && k.compare(_num0)!=-1)
1471 return numeric(cln::binomial(n.to_int(),k.to_int()));
1475 return _num_1.power(k)*binomial(k-n-_num1,k);
1479 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1480 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1484 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1485 * in the expansion of the function x/(e^x-1).
1487 * @return the nth Bernoulli number (a rational number).
1488 * @exception range_error (argument must be integer >= 0) */
1489 const numeric bernoulli(const numeric &nn)
1491 if (!nn.is_integer() || nn.is_negative())
1492 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1496 // The Bernoulli numbers are rational numbers that may be computed using
1499 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1501 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1502 // previous ones, the computation is necessarily very expensive. There are
1503 // several other ways of computing them, a particularly good one being
1507 // for (unsigned i=0; i<n; i++) {
1508 // c = exquo(c*(i-n),(i+2));
1509 // Bern = Bern + c*s/(i+2);
1510 // s = s + expt_pos(cl_I(i+2),n);
1514 // But if somebody works with the n'th Bernoulli number she is likely to
1515 // also need all previous Bernoulli numbers. So we need a complete remember
1516 // table and above divide and conquer algorithm is not suited to build one
1517 // up. The code below is adapted from Pari's function bernvec().
1519 // (There is an interesting relation with the tangent polynomials described
1520 // in `Concrete Mathematics', which leads to a program twice as fast as our
1521 // implementation below, but it requires storing one such polynomial in
1522 // addition to the remember table. This doubles the memory footprint so
1523 // we don't use it.)
1525 // the special cases not covered by the algorithm below
1526 if (nn.is_equal(_num1))
1531 // store nonvanishing Bernoulli numbers here
1532 static std::vector< cln::cl_RA > results;
1533 static int highest_result = 0;
1534 // algorithm not applicable to B(0), so just store it
1535 if (results.empty())
1536 results.push_back(cln::cl_RA(1));
1538 int n = nn.to_long();
1539 for (int i=highest_result; i<n/2; ++i) {
1545 for (int j=i; j>0; --j) {
1546 B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
1552 B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
1553 results.push_back(B);
1556 return results[n/2];
1560 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1561 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1563 * @param n an integer
1564 * @return the nth Fibonacci number F(n) (an integer number)
1565 * @exception range_error (argument must be an integer) */
1566 const numeric fibonacci(const numeric &n)
1568 if (!n.is_integer())
1569 throw std::range_error("numeric::fibonacci(): argument must be integer");
1572 // The following addition formula holds:
1574 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1576 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1577 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1579 // Replace m by m+1:
1580 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1581 // Now put in m = n, to get
1582 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1583 // F(2n+1) = F(n)^2 + F(n+1)^2
1585 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1588 if (n.is_negative())
1590 return -fibonacci(-n);
1592 return fibonacci(-n);
1596 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1597 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1598 // Since a squaring is cheaper than a multiplication, better use
1599 // three squarings instead of one multiplication and two squarings.
1600 cln::cl_I u2 = cln::square(u);
1601 cln::cl_I v2 = cln::square(v);
1602 if (cln::logbitp(bit-1, m)) {
1603 v = cln::square(u + v) - u2;
1606 u = v2 - cln::square(v - u);
1611 // Here we don't use the squaring formula because one multiplication
1612 // is cheaper than two squarings.
1613 return u * ((v << 1) - u);
1615 return cln::square(u) + cln::square(v);
1619 /** Absolute value. */
1620 const numeric abs(const numeric& x)
1622 return cln::abs(x.to_cl_N());
1626 /** Modulus (in positive representation).
1627 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1628 * sign of a or is zero. This is different from Maple's modp, where the sign
1629 * of b is ignored. It is in agreement with Mathematica's Mod.
1631 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1632 * integer, 0 otherwise. */
1633 const numeric mod(const numeric &a, const numeric &b)
1635 if (a.is_integer() && b.is_integer())
1636 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1637 cln::the<cln::cl_I>(b.to_cl_N()));
1643 /** Modulus (in symmetric representation).
1644 * Equivalent to Maple's mods.
1646 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1647 const numeric smod(const numeric &a, const numeric &b)
1649 if (a.is_integer() && b.is_integer()) {
1650 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1651 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1652 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1658 /** Numeric integer remainder.
1659 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1660 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1661 * sign of a or is zero.
1663 * @return remainder of a/b if both are integer, 0 otherwise. */
1664 const numeric irem(const numeric &a, const numeric &b)
1666 if (a.is_integer() && b.is_integer())
1667 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1668 cln::the<cln::cl_I>(b.to_cl_N()));
1674 /** Numeric integer remainder.
1675 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1676 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1677 * and irem(a,b) has the sign of a or is zero.
1679 * @return remainder of a/b and quotient stored in q if both are integer,
1681 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1683 if (a.is_integer() && b.is_integer()) {
1684 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1685 cln::the<cln::cl_I>(b.to_cl_N()));
1686 q = rem_quo.quotient;
1687 return rem_quo.remainder;
1695 /** Numeric integer quotient.
1696 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1698 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1699 const numeric iquo(const numeric &a, const numeric &b)
1701 if (a.is_integer() && b.is_integer())
1702 return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1703 cln::the<cln::cl_I>(b.to_cl_N()));
1709 /** Numeric integer quotient.
1710 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1711 * r == a - iquo(a,b,r)*b.
1713 * @return truncated quotient of a/b and remainder stored in r if both are
1714 * integer, 0 otherwise. */
1715 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1717 if (a.is_integer() && b.is_integer()) {
1718 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1719 cln::the<cln::cl_I>(b.to_cl_N()));
1720 r = rem_quo.remainder;
1721 return rem_quo.quotient;
1729 /** Greatest Common Divisor.
1731 * @return The GCD of two numbers if both are integer, a numerical 1
1732 * if they are not. */
1733 const numeric gcd(const numeric &a, const numeric &b)
1735 if (a.is_integer() && b.is_integer())
1736 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1737 cln::the<cln::cl_I>(b.to_cl_N()));
1743 /** Least Common Multiple.
1745 * @return The LCM of two numbers if both are integer, the product of those
1746 * two numbers if they are not. */
1747 const numeric lcm(const numeric &a, const numeric &b)
1749 if (a.is_integer() && b.is_integer())
1750 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1751 cln::the<cln::cl_I>(b.to_cl_N()));
1757 /** Numeric square root.
1758 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1759 * should return integer 2.
1761 * @param z numeric argument
1762 * @return square root of z. Branch cut along negative real axis, the negative
1763 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1764 * where imag(z)>0. */
1765 const numeric sqrt(const numeric &z)
1767 return cln::sqrt(z.to_cl_N());
1771 /** Integer numeric square root. */
1772 const numeric isqrt(const numeric &x)
1774 if (x.is_integer()) {
1776 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1783 /** Floating point evaluation of Archimedes' constant Pi. */
1786 return numeric(cln::pi(cln::default_float_format));
1790 /** Floating point evaluation of Euler's constant gamma. */
1793 return numeric(cln::eulerconst(cln::default_float_format));
1797 /** Floating point evaluation of Catalan's constant. */
1798 ex CatalanEvalf(void)
1800 return numeric(cln::catalanconst(cln::default_float_format));
1804 /** _numeric_digits default ctor, checking for singleton invariance. */
1805 _numeric_digits::_numeric_digits()
1808 // It initializes to 17 digits, because in CLN float_format(17) turns out
1809 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1810 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1812 throw(std::runtime_error("I told you not to do instantiate me!"));
1814 cln::default_float_format = cln::float_format(17);
1818 /** Assign a native long to global Digits object. */
1819 _numeric_digits& _numeric_digits::operator=(long prec)
1822 cln::default_float_format = cln::float_format(prec);
1827 /** Convert global Digits object to native type long. */
1828 _numeric_digits::operator long()
1830 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1831 return (long)digits;
1835 /** Append global Digits object to ostream. */
1836 void _numeric_digits::print(std::ostream &os) const
1838 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1843 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1850 // static member variables
1855 bool _numeric_digits::too_late = false;
1858 /** Accuracy in decimal digits. Only object of this type! Can be set using
1859 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1860 _numeric_digits Digits;
1862 } // namespace GiNaC