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) 1999-2003 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
328 cln::print_real(c.s, ourflags, cln::abs(cln::numerator(cln::the<cln::cl_RA>(x))));
330 cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
335 // make CLN believe this number has default_float_format, so it prints
336 // 'E' as exponent marker instead of 'L':
337 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
338 cln::print_real(c.s, ourflags, x);
342 /** This method adds to the output so it blends more consistently together
343 * with the other routines and produces something compatible to ginsh input.
345 * @see print_real_number() */
346 void numeric::print(const print_context & c, unsigned level) const
348 if (is_a<print_tree>(c)) {
350 c.s << std::string(level, ' ') << cln::the<cln::cl_N>(value)
351 << " (" << class_name() << ")"
352 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
355 } else if (is_a<print_csrc>(c)) {
357 std::ios::fmtflags oldflags = c.s.flags();
358 c.s.setf(std::ios::scientific);
359 int oldprec = c.s.precision();
360 if (is_a<print_csrc_double>(c))
364 if (is_a<print_csrc_cl_N>(c) && this->is_integer()) {
365 c.s << "cln::cl_I(\"";
366 const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
367 print_real_number(c,r);
369 } else if (this->is_rational() && !this->is_integer()) {
370 if (compare(_num0) > 0) {
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(\"" << -numer().evalf() << "\")";
381 c.s << -numer().to_double();
384 if (is_a<print_csrc_cl_N>(c))
385 c.s << "cln::cl_F(\"" << denom().evalf() << "\")";
387 c.s << denom().to_double();
390 if (is_a<print_csrc_cl_N>(c))
391 c.s << "cln::cl_F(\"" << evalf() << "_" << Digits << "\")";
396 c.s.precision(oldprec);
399 const std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
400 const std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
401 const std::string imag_sym = is_a<print_latex>(c) ? "i" : "I";
402 const std::string mul_sym = is_a<print_latex>(c) ? " " : "*";
403 const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
404 const cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
405 if (is_a<print_python_repr>(c))
406 c.s << class_name() << "('";
408 // case 1, real: x or -x
409 if ((precedence() <= level) && (!this->is_nonneg_integer())) {
411 print_real_number(c, r);
414 print_real_number(c, r);
418 // case 2, imaginary: y*I or -y*I
422 if (precedence()<=level)
425 c.s << "-" << imag_sym;
427 print_real_number(c, i);
428 c.s << mul_sym+imag_sym;
430 if (precedence()<=level)
434 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
435 if (precedence() <= level)
437 print_real_number(c, r);
442 print_real_number(c, i);
443 c.s << mul_sym+imag_sym;
450 print_real_number(c, i);
451 c.s << mul_sym+imag_sym;
454 if (precedence() <= level)
458 if (is_a<print_python_repr>(c))
463 bool numeric::info(unsigned inf) const
466 case info_flags::numeric:
467 case info_flags::polynomial:
468 case info_flags::rational_function:
470 case info_flags::real:
472 case info_flags::rational:
473 case info_flags::rational_polynomial:
474 return is_rational();
475 case info_flags::crational:
476 case info_flags::crational_polynomial:
477 return is_crational();
478 case info_flags::integer:
479 case info_flags::integer_polynomial:
481 case info_flags::cinteger:
482 case info_flags::cinteger_polynomial:
483 return is_cinteger();
484 case info_flags::positive:
485 return is_positive();
486 case info_flags::negative:
487 return is_negative();
488 case info_flags::nonnegative:
489 return !is_negative();
490 case info_flags::posint:
491 return is_pos_integer();
492 case info_flags::negint:
493 return is_integer() && is_negative();
494 case info_flags::nonnegint:
495 return is_nonneg_integer();
496 case info_flags::even:
498 case info_flags::odd:
500 case info_flags::prime:
502 case info_flags::algebraic:
508 int numeric::degree(const ex & s) const
513 int numeric::ldegree(const ex & s) const
518 ex numeric::coeff(const ex & s, int n) const
520 return n==0 ? *this : _ex0;
523 /** Disassemble real part and imaginary part to scan for the occurrence of a
524 * single number. Also handles the imaginary unit. It ignores the sign on
525 * both this and the argument, which may lead to what might appear as funny
526 * results: (2+I).has(-2) -> true. But this is consistent, since we also
527 * would like to have (-2+I).has(2) -> true and we want to think about the
528 * sign as a multiplicative factor. */
529 bool numeric::has(const ex &other) const
531 if (!is_ex_exactly_of_type(other, numeric))
533 const numeric &o = ex_to<numeric>(other);
534 if (this->is_equal(o) || this->is_equal(-o))
536 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
537 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
538 this->real().is_equal(-o) || this->imag().is_equal(-o));
540 if (o.is_equal(I)) // e.g scan for I in 42*I
541 return !this->is_real();
542 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
543 return (this->real().has(o*I) || this->imag().has(o*I) ||
544 this->real().has(-o*I) || this->imag().has(-o*I));
550 /** Evaluation of numbers doesn't do anything at all. */
551 ex numeric::eval(int level) const
553 // Warning: if this is ever gonna do something, the ex ctors from all kinds
554 // of numbers should be checking for status_flags::evaluated.
559 /** Cast numeric into a floating-point object. For example exact numeric(1) is
560 * returned as a 1.0000000000000000000000 and so on according to how Digits is
561 * currently set. In case the object already was a floating point number the
562 * precision is trimmed to match the currently set default.
564 * @param level ignored, only needed for overriding basic::evalf.
565 * @return an ex-handle to a numeric. */
566 ex numeric::evalf(int level) const
568 // level can safely be discarded for numeric objects.
569 return numeric(cln::cl_float(1.0, cln::default_float_format) *
570 (cln::the<cln::cl_N>(value)));
575 int numeric::compare_same_type(const basic &other) const
577 GINAC_ASSERT(is_exactly_a<numeric>(other));
578 const numeric &o = static_cast<const numeric &>(other);
580 return this->compare(o);
584 bool numeric::is_equal_same_type(const basic &other) const
586 GINAC_ASSERT(is_exactly_a<numeric>(other));
587 const numeric &o = static_cast<const numeric &>(other);
589 return this->is_equal(o);
593 unsigned numeric::calchash(void) const
595 // Use CLN's hashcode. Warning: It depends only on the number's value, not
596 // its type or precision (i.e. a true equivalence relation on numbers). As
597 // a consequence, 3 and 3.0 share the same hashvalue.
598 setflag(status_flags::hash_calculated);
599 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
604 // new virtual functions which can be overridden by derived classes
610 // non-virtual functions in this class
615 /** Numerical addition method. Adds argument to *this and returns result as
616 * a numeric object. */
617 const numeric numeric::add(const numeric &other) const
619 // Efficiency shortcut: trap the neutral element by pointer.
622 else if (&other==_num0_p)
625 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
629 /** Numerical subtraction method. Subtracts argument from *this and returns
630 * result as a numeric object. */
631 const numeric numeric::sub(const numeric &other) const
633 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
637 /** Numerical multiplication method. Multiplies *this and argument and returns
638 * result as a numeric object. */
639 const numeric numeric::mul(const numeric &other) const
641 // Efficiency shortcut: trap the neutral element by pointer.
644 else if (&other==_num1_p)
647 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
651 /** Numerical division method. Divides *this by argument and returns result as
654 * @exception overflow_error (division by zero) */
655 const numeric numeric::div(const numeric &other) const
657 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
658 throw std::overflow_error("numeric::div(): division by zero");
659 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
663 /** Numerical exponentiation. Raises *this to the power given as argument and
664 * returns result as a numeric object. */
665 const numeric numeric::power(const numeric &other) const
667 // Efficiency shortcut: trap the neutral exponent by pointer.
671 if (cln::zerop(cln::the<cln::cl_N>(value))) {
672 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
673 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
674 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
675 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
676 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
677 throw std::overflow_error("numeric::eval(): division by zero");
681 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
685 const numeric &numeric::add_dyn(const numeric &other) const
687 // Efficiency shortcut: trap the neutral element by pointer.
690 else if (&other==_num0_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::sub_dyn(const numeric &other) const
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::mul_dyn(const numeric &other) const
707 // Efficiency shortcut: trap the neutral element by pointer.
710 else if (&other==_num1_p)
713 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
714 setflag(status_flags::dynallocated));
718 const numeric &numeric::div_dyn(const numeric &other) const
720 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
721 throw std::overflow_error("division by zero");
722 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
723 setflag(status_flags::dynallocated));
727 const numeric &numeric::power_dyn(const numeric &other) const
729 // Efficiency shortcut: trap the neutral exponent by pointer.
733 if (cln::zerop(cln::the<cln::cl_N>(value))) {
734 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
735 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
736 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
737 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
738 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
739 throw std::overflow_error("numeric::eval(): division by zero");
743 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
744 setflag(status_flags::dynallocated));
748 const numeric &numeric::operator=(int i)
750 return operator=(numeric(i));
754 const numeric &numeric::operator=(unsigned int i)
756 return operator=(numeric(i));
760 const numeric &numeric::operator=(long i)
762 return operator=(numeric(i));
766 const numeric &numeric::operator=(unsigned long i)
768 return operator=(numeric(i));
772 const numeric &numeric::operator=(double d)
774 return operator=(numeric(d));
778 const numeric &numeric::operator=(const char * s)
780 return operator=(numeric(s));
784 /** Inverse of a number. */
785 const numeric numeric::inverse(void) const
787 if (cln::zerop(cln::the<cln::cl_N>(value)))
788 throw std::overflow_error("numeric::inverse(): division by zero");
789 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
793 /** Return the complex half-plane (left or right) in which the number lies.
794 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
795 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
797 * @see numeric::compare(const numeric &other) */
798 int numeric::csgn(void) const
800 if (cln::zerop(cln::the<cln::cl_N>(value)))
802 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
803 if (!cln::zerop(r)) {
809 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
817 /** This method establishes a canonical order on all numbers. For complex
818 * numbers this is not possible in a mathematically consistent way but we need
819 * to establish some order and it ought to be fast. So we simply define it
820 * to be compatible with our method csgn.
822 * @return csgn(*this-other)
823 * @see numeric::csgn(void) */
824 int numeric::compare(const numeric &other) const
826 // Comparing two real numbers?
827 if (cln::instanceof(value, cln::cl_R_ring) &&
828 cln::instanceof(other.value, cln::cl_R_ring))
829 // Yes, so just cln::compare them
830 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
832 // No, first cln::compare real parts...
833 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
836 // ...and then the imaginary parts.
837 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
842 bool numeric::is_equal(const numeric &other) const
844 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
848 /** True if object is zero. */
849 bool numeric::is_zero(void) const
851 return cln::zerop(cln::the<cln::cl_N>(value));
855 /** True if object is not complex and greater than zero. */
856 bool numeric::is_positive(void) const
859 return cln::plusp(cln::the<cln::cl_R>(value));
864 /** True if object is not complex and less than zero. */
865 bool numeric::is_negative(void) const
868 return cln::minusp(cln::the<cln::cl_R>(value));
873 /** True if object is a non-complex integer. */
874 bool numeric::is_integer(void) const
876 return cln::instanceof(value, cln::cl_I_ring);
880 /** True if object is an exact integer greater than zero. */
881 bool numeric::is_pos_integer(void) const
883 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
887 /** True if object is an exact integer greater or equal zero. */
888 bool numeric::is_nonneg_integer(void) const
890 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
894 /** True if object is an exact even integer. */
895 bool numeric::is_even(void) const
897 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
901 /** True if object is an exact odd integer. */
902 bool numeric::is_odd(void) const
904 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
908 /** Probabilistic primality test.
910 * @return true if object is exact integer and prime. */
911 bool numeric::is_prime(void) const
913 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
917 /** True if object is an exact rational number, may even be complex
918 * (denominator may be unity). */
919 bool numeric::is_rational(void) const
921 return cln::instanceof(value, cln::cl_RA_ring);
925 /** True if object is a real integer, rational or float (but not complex). */
926 bool numeric::is_real(void) const
928 return cln::instanceof(value, cln::cl_R_ring);
932 bool numeric::operator==(const numeric &other) const
934 return cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
938 bool numeric::operator!=(const numeric &other) const
940 return !cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
944 /** True if object is element of the domain of integers extended by I, i.e. is
945 * of the form a+b*I, where a and b are integers. */
946 bool numeric::is_cinteger(void) const
948 if (cln::instanceof(value, cln::cl_I_ring))
950 else if (!this->is_real()) { // complex case, handle n+m*I
951 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
952 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
959 /** True if object is an exact rational number, may even be complex
960 * (denominator may be unity). */
961 bool numeric::is_crational(void) const
963 if (cln::instanceof(value, cln::cl_RA_ring))
965 else if (!this->is_real()) { // complex case, handle Q(i):
966 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
967 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
974 /** Numerical comparison: less.
976 * @exception invalid_argument (complex inequality) */
977 bool numeric::operator<(const numeric &other) const
979 if (this->is_real() && other.is_real())
980 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
981 throw std::invalid_argument("numeric::operator<(): complex inequality");
985 /** Numerical comparison: less or equal.
987 * @exception invalid_argument (complex inequality) */
988 bool numeric::operator<=(const numeric &other) const
990 if (this->is_real() && other.is_real())
991 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
992 throw std::invalid_argument("numeric::operator<=(): complex inequality");
996 /** Numerical comparison: greater.
998 * @exception invalid_argument (complex inequality) */
999 bool numeric::operator>(const numeric &other) const
1001 if (this->is_real() && other.is_real())
1002 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1003 throw std::invalid_argument("numeric::operator>(): complex inequality");
1007 /** Numerical comparison: greater or equal.
1009 * @exception invalid_argument (complex inequality) */
1010 bool numeric::operator>=(const numeric &other) const
1012 if (this->is_real() && other.is_real())
1013 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1014 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1018 /** Converts numeric types to machine's int. You should check with
1019 * is_integer() if the number is really an integer before calling this method.
1020 * You may also consider checking the range first. */
1021 int numeric::to_int(void) const
1023 GINAC_ASSERT(this->is_integer());
1024 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1028 /** Converts numeric types to machine's long. You should check with
1029 * is_integer() if the number is really an integer before calling this method.
1030 * You may also consider checking the range first. */
1031 long numeric::to_long(void) const
1033 GINAC_ASSERT(this->is_integer());
1034 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1038 /** Converts numeric types to machine's double. You should check with is_real()
1039 * if the number is really not complex before calling this method. */
1040 double numeric::to_double(void) const
1042 GINAC_ASSERT(this->is_real());
1043 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1047 /** Returns a new CLN object of type cl_N, representing the value of *this.
1048 * This method may be used when mixing GiNaC and CLN in one project.
1050 cln::cl_N numeric::to_cl_N(void) const
1052 return cln::cl_N(cln::the<cln::cl_N>(value));
1056 /** Real part of a number. */
1057 const numeric numeric::real(void) const
1059 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1063 /** Imaginary part of a number. */
1064 const numeric numeric::imag(void) const
1066 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1070 /** Numerator. Computes the numerator of rational numbers, rationalized
1071 * numerator of complex if real and imaginary part are both rational numbers
1072 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1074 const numeric numeric::numer(void) const
1076 if (this->is_integer())
1077 return numeric(*this);
1079 else if (cln::instanceof(value, cln::cl_RA_ring))
1080 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1082 else if (!this->is_real()) { // complex case, handle Q(i):
1083 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1084 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1085 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1086 return numeric(*this);
1087 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1088 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1089 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1090 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1091 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1092 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1093 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1094 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1097 // at least one float encountered
1098 return numeric(*this);
1102 /** Denominator. Computes the denominator of rational numbers, common integer
1103 * denominator of complex if real and imaginary part are both rational numbers
1104 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1105 const numeric numeric::denom(void) const
1107 if (this->is_integer())
1110 if (cln::instanceof(value, cln::cl_RA_ring))
1111 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1113 if (!this->is_real()) { // complex case, handle Q(i):
1114 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1115 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1116 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1118 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1119 return numeric(cln::denominator(i));
1120 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1121 return numeric(cln::denominator(r));
1122 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1123 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1125 // at least one float encountered
1130 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1131 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1132 * 2^(n-1) <= x < 2^n.
1134 * @return number of bits (excluding sign) needed to represent that number
1135 * in two's complement if it is an integer, 0 otherwise. */
1136 int numeric::int_length(void) const
1138 if (this->is_integer())
1139 return cln::integer_length(cln::the<cln::cl_I>(value));
1148 /** Imaginary unit. This is not a constant but a numeric since we are
1149 * natively handing complex numbers anyways, so in each expression containing
1150 * an I it is automatically eval'ed away anyhow. */
1151 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1154 /** Exponential function.
1156 * @return arbitrary precision numerical exp(x). */
1157 const numeric exp(const numeric &x)
1159 return cln::exp(x.to_cl_N());
1163 /** Natural logarithm.
1165 * @param z complex number
1166 * @return arbitrary precision numerical log(x).
1167 * @exception pole_error("log(): logarithmic pole",0) */
1168 const numeric log(const numeric &z)
1171 throw pole_error("log(): logarithmic pole",0);
1172 return cln::log(z.to_cl_N());
1176 /** Numeric sine (trigonometric function).
1178 * @return arbitrary precision numerical sin(x). */
1179 const numeric sin(const numeric &x)
1181 return cln::sin(x.to_cl_N());
1185 /** Numeric cosine (trigonometric function).
1187 * @return arbitrary precision numerical cos(x). */
1188 const numeric cos(const numeric &x)
1190 return cln::cos(x.to_cl_N());
1194 /** Numeric tangent (trigonometric function).
1196 * @return arbitrary precision numerical tan(x). */
1197 const numeric tan(const numeric &x)
1199 return cln::tan(x.to_cl_N());
1203 /** Numeric inverse sine (trigonometric function).
1205 * @return arbitrary precision numerical asin(x). */
1206 const numeric asin(const numeric &x)
1208 return cln::asin(x.to_cl_N());
1212 /** Numeric inverse cosine (trigonometric function).
1214 * @return arbitrary precision numerical acos(x). */
1215 const numeric acos(const numeric &x)
1217 return cln::acos(x.to_cl_N());
1223 * @param z complex number
1225 * @exception pole_error("atan(): logarithmic pole",0) */
1226 const numeric atan(const numeric &x)
1229 x.real().is_zero() &&
1230 abs(x.imag()).is_equal(_num1))
1231 throw pole_error("atan(): logarithmic pole",0);
1232 return cln::atan(x.to_cl_N());
1238 * @param x real number
1239 * @param y real number
1240 * @return atan(y/x) */
1241 const numeric atan(const numeric &y, const numeric &x)
1243 if (x.is_real() && y.is_real())
1244 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1245 cln::the<cln::cl_R>(y.to_cl_N()));
1247 throw std::invalid_argument("atan(): complex argument");
1251 /** Numeric hyperbolic sine (trigonometric function).
1253 * @return arbitrary precision numerical sinh(x). */
1254 const numeric sinh(const numeric &x)
1256 return cln::sinh(x.to_cl_N());
1260 /** Numeric hyperbolic cosine (trigonometric function).
1262 * @return arbitrary precision numerical cosh(x). */
1263 const numeric cosh(const numeric &x)
1265 return cln::cosh(x.to_cl_N());
1269 /** Numeric hyperbolic tangent (trigonometric function).
1271 * @return arbitrary precision numerical tanh(x). */
1272 const numeric tanh(const numeric &x)
1274 return cln::tanh(x.to_cl_N());
1278 /** Numeric inverse hyperbolic sine (trigonometric function).
1280 * @return arbitrary precision numerical asinh(x). */
1281 const numeric asinh(const numeric &x)
1283 return cln::asinh(x.to_cl_N());
1287 /** Numeric inverse hyperbolic cosine (trigonometric function).
1289 * @return arbitrary precision numerical acosh(x). */
1290 const numeric acosh(const numeric &x)
1292 return cln::acosh(x.to_cl_N());
1296 /** Numeric inverse hyperbolic tangent (trigonometric function).
1298 * @return arbitrary precision numerical atanh(x). */
1299 const numeric atanh(const numeric &x)
1301 return cln::atanh(x.to_cl_N());
1305 /*static cln::cl_N Li2_series(const ::cl_N &x,
1306 const ::float_format_t &prec)
1308 // Note: argument must be in the unit circle
1309 // This is very inefficient unless we have fast floating point Bernoulli
1310 // numbers implemented!
1311 cln::cl_N c1 = -cln::log(1-x);
1313 // hard-wire the first two Bernoulli numbers
1314 cln::cl_N acc = c1 - cln::square(c1)/4;
1316 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1317 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1319 c1 = cln::square(c1);
1323 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1324 // 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));
1327 } while (acc != acc+aug);
1331 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1332 * circle) using a power series. */
1333 static cln::cl_N Li2_series(const cln::cl_N &x,
1334 const cln::float_format_t &prec)
1336 // Note: argument must be in the unit circle
1338 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1343 den = den + i; // 1, 4, 9, 16, ...
1347 } while (acc != acc+aug);
1351 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1352 static cln::cl_N Li2_projection(const cln::cl_N &x,
1353 const cln::float_format_t &prec)
1355 const cln::cl_R re = cln::realpart(x);
1356 const cln::cl_R im = cln::imagpart(x);
1357 if (re > cln::cl_F(".5"))
1358 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1360 - Li2_series(1-x, prec)
1361 - cln::log(x)*cln::log(1-x));
1362 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1363 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1364 return(- cln::square(cln::log(1-x))/2
1365 - Li2_series(x/(x-1), prec));
1366 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1367 // Li2(x^2)/2 - Li2(-x)
1368 return(Li2_projection(cln::square(x), prec)/2
1369 - Li2_projection(-x, prec));
1370 return Li2_series(x, prec);
1373 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1374 * the branch cut lies along the positive real axis, starting at 1 and
1375 * continuous with quadrant IV.
1377 * @return arbitrary precision numerical Li2(x). */
1378 const numeric Li2(const numeric &x)
1383 // what is the desired float format?
1384 // first guess: default format
1385 cln::float_format_t prec = cln::default_float_format;
1386 const cln::cl_N value = x.to_cl_N();
1387 // second guess: the argument's format
1388 if (!x.real().is_rational())
1389 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1390 else if (!x.imag().is_rational())
1391 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1393 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1394 return cln::zeta(2, prec);
1396 if (cln::abs(value) > 1)
1397 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1398 return(- cln::square(cln::log(-value))/2
1399 - cln::zeta(2, prec)
1400 - Li2_projection(cln::recip(value), prec));
1402 return Li2_projection(x.to_cl_N(), prec);
1406 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1407 * integer arguments. */
1408 const numeric zeta(const numeric &x)
1410 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1411 // only knows about integer arguments and zeta(3).evalf() automatically
1412 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1413 // being an exact zero for CLN, which can be tested and then we can just
1414 // pass the number casted to an int:
1416 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1417 if (cln::zerop(x.to_cl_N()-aux))
1418 return cln::zeta(aux);
1424 /** The Gamma function.
1425 * This is only a stub! */
1426 const numeric lgamma(const numeric &x)
1430 const numeric tgamma(const numeric &x)
1436 /** The psi function (aka polygamma function).
1437 * This is only a stub! */
1438 const numeric psi(const numeric &x)
1444 /** The psi functions (aka polygamma functions).
1445 * This is only a stub! */
1446 const numeric psi(const numeric &n, const numeric &x)
1452 /** Factorial combinatorial function.
1454 * @param n integer argument >= 0
1455 * @exception range_error (argument must be integer >= 0) */
1456 const numeric factorial(const numeric &n)
1458 if (!n.is_nonneg_integer())
1459 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1460 return numeric(cln::factorial(n.to_int()));
1464 /** The double factorial combinatorial function. (Scarcely used, but still
1465 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1467 * @param n integer argument >= -1
1468 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1469 * @exception range_error (argument must be integer >= -1) */
1470 const numeric doublefactorial(const numeric &n)
1472 if (n.is_equal(_num_1))
1475 if (!n.is_nonneg_integer())
1476 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1478 return numeric(cln::doublefactorial(n.to_int()));
1482 /** The Binomial coefficients. It computes the binomial coefficients. For
1483 * integer n and k and positive n this is the number of ways of choosing k
1484 * objects from n distinct objects. If n is negative, the formula
1485 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1486 const numeric binomial(const numeric &n, const numeric &k)
1488 if (n.is_integer() && k.is_integer()) {
1489 if (n.is_nonneg_integer()) {
1490 if (k.compare(n)!=1 && k.compare(_num0)!=-1)
1491 return numeric(cln::binomial(n.to_int(),k.to_int()));
1495 return _num_1.power(k)*binomial(k-n-_num1,k);
1499 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1500 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1504 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1505 * in the expansion of the function x/(e^x-1).
1507 * @return the nth Bernoulli number (a rational number).
1508 * @exception range_error (argument must be integer >= 0) */
1509 const numeric bernoulli(const numeric &nn)
1511 if (!nn.is_integer() || nn.is_negative())
1512 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1516 // The Bernoulli numbers are rational numbers that may be computed using
1519 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1521 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1522 // previous ones, the computation is necessarily very expensive. There are
1523 // several other ways of computing them, a particularly good one being
1527 // for (unsigned i=0; i<n; i++) {
1528 // c = exquo(c*(i-n),(i+2));
1529 // Bern = Bern + c*s/(i+2);
1530 // s = s + expt_pos(cl_I(i+2),n);
1534 // But if somebody works with the n'th Bernoulli number she is likely to
1535 // also need all previous Bernoulli numbers. So we need a complete remember
1536 // table and above divide and conquer algorithm is not suited to build one
1537 // up. The formula below accomplishes this. It is a modification of the
1538 // defining formula above but the computation of the binomial coefficients
1539 // is carried along in an inline fashion. It also honors the fact that
1540 // B_n is zero when n is odd and greater than 1.
1542 // (There is an interesting relation with the tangent polynomials described
1543 // in `Concrete Mathematics', which leads to a program a little faster as
1544 // our implementation below, but it requires storing one such polynomial in
1545 // addition to the remember table. This doubles the memory footprint so
1546 // we don't use it.)
1548 const unsigned n = nn.to_int();
1550 // the special cases not covered by the algorithm below
1552 return (n==1) ? _num_1_2 : _num0;
1556 // store nonvanishing Bernoulli numbers here
1557 static std::vector< cln::cl_RA > results;
1558 static unsigned next_r = 0;
1560 // algorithm not applicable to B(2), so just store it
1562 results.push_back(cln::recip(cln::cl_RA(6)));
1566 return results[n/2-1];
1568 results.reserve(n/2);
1569 for (unsigned p=next_r; p<=n; p+=2) {
1570 cln::cl_I c = 1; // seed for binonmial coefficients
1571 cln::cl_RA b = cln::cl_RA(1-p)/2;
1572 const unsigned p3 = p+3;
1573 const unsigned pm = p-2;
1575 // test if intermediate unsigned int can be represented by immediate
1576 // objects by CLN (i.e. < 2^29 for 32 Bit machines, see <cln/object.h>)
1577 if (p < (1UL<<cl_value_len/2)) {
1578 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1579 c = cln::exquo(c * ((p3-i) * p_2), (i-1)*k);
1580 b = b + c*results[k-1];
1583 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1584 c = cln::exquo((c * (p3-i)) * p_2, cln::cl_I(i-1)*k);
1585 b = b + c*results[k-1];
1588 results.push_back(-b/(p+1));
1591 return results[n/2-1];
1595 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1596 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1598 * @param n an integer
1599 * @return the nth Fibonacci number F(n) (an integer number)
1600 * @exception range_error (argument must be an integer) */
1601 const numeric fibonacci(const numeric &n)
1603 if (!n.is_integer())
1604 throw std::range_error("numeric::fibonacci(): argument must be integer");
1607 // The following addition formula holds:
1609 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1611 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1612 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1614 // Replace m by m+1:
1615 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1616 // Now put in m = n, to get
1617 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1618 // F(2n+1) = F(n)^2 + F(n+1)^2
1620 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1623 if (n.is_negative())
1625 return -fibonacci(-n);
1627 return fibonacci(-n);
1631 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1632 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1633 // Since a squaring is cheaper than a multiplication, better use
1634 // three squarings instead of one multiplication and two squarings.
1635 cln::cl_I u2 = cln::square(u);
1636 cln::cl_I v2 = cln::square(v);
1637 if (cln::logbitp(bit-1, m)) {
1638 v = cln::square(u + v) - u2;
1641 u = v2 - cln::square(v - u);
1646 // Here we don't use the squaring formula because one multiplication
1647 // is cheaper than two squarings.
1648 return u * ((v << 1) - u);
1650 return cln::square(u) + cln::square(v);
1654 /** Absolute value. */
1655 const numeric abs(const numeric& x)
1657 return cln::abs(x.to_cl_N());
1661 /** Modulus (in positive representation).
1662 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1663 * sign of a or is zero. This is different from Maple's modp, where the sign
1664 * of b is ignored. It is in agreement with Mathematica's Mod.
1666 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1667 * integer, 0 otherwise. */
1668 const numeric mod(const numeric &a, const numeric &b)
1670 if (a.is_integer() && b.is_integer())
1671 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1672 cln::the<cln::cl_I>(b.to_cl_N()));
1678 /** Modulus (in symmetric representation).
1679 * Equivalent to Maple's mods.
1681 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1682 const numeric smod(const numeric &a, const numeric &b)
1684 if (a.is_integer() && b.is_integer()) {
1685 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1686 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1687 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1693 /** Numeric integer remainder.
1694 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1695 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1696 * sign of a or is zero.
1698 * @return remainder of a/b if both are integer, 0 otherwise.
1699 * @exception overflow_error (division by zero) if b is zero. */
1700 const numeric irem(const numeric &a, const numeric &b)
1703 throw std::overflow_error("numeric::irem(): division by zero");
1704 if (a.is_integer() && b.is_integer())
1705 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1706 cln::the<cln::cl_I>(b.to_cl_N()));
1712 /** Numeric integer remainder.
1713 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1714 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1715 * and irem(a,b) has the sign of a or is zero.
1717 * @return remainder of a/b and quotient stored in q if both are integer,
1719 * @exception overflow_error (division by zero) if b is zero. */
1720 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1723 throw std::overflow_error("numeric::irem(): division by zero");
1724 if (a.is_integer() && b.is_integer()) {
1725 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1726 cln::the<cln::cl_I>(b.to_cl_N()));
1727 q = rem_quo.quotient;
1728 return rem_quo.remainder;
1736 /** Numeric integer quotient.
1737 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1739 * @return truncated quotient of a/b if both are integer, 0 otherwise.
1740 * @exception overflow_error (division by zero) if b is zero. */
1741 const numeric iquo(const numeric &a, const numeric &b)
1744 throw std::overflow_error("numeric::iquo(): division by zero");
1745 if (a.is_integer() && b.is_integer())
1746 return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1747 cln::the<cln::cl_I>(b.to_cl_N()));
1753 /** Numeric integer quotient.
1754 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1755 * r == a - iquo(a,b,r)*b.
1757 * @return truncated quotient of a/b and remainder stored in r if both are
1758 * integer, 0 otherwise.
1759 * @exception overflow_error (division by zero) if b is zero. */
1760 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1763 throw std::overflow_error("numeric::iquo(): division by zero");
1764 if (a.is_integer() && b.is_integer()) {
1765 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1766 cln::the<cln::cl_I>(b.to_cl_N()));
1767 r = rem_quo.remainder;
1768 return rem_quo.quotient;
1776 /** Greatest Common Divisor.
1778 * @return The GCD of two numbers if both are integer, a numerical 1
1779 * if they are not. */
1780 const numeric gcd(const numeric &a, const numeric &b)
1782 if (a.is_integer() && b.is_integer())
1783 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1784 cln::the<cln::cl_I>(b.to_cl_N()));
1790 /** Least Common Multiple.
1792 * @return The LCM of two numbers if both are integer, the product of those
1793 * two numbers if they are not. */
1794 const numeric lcm(const numeric &a, const numeric &b)
1796 if (a.is_integer() && b.is_integer())
1797 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1798 cln::the<cln::cl_I>(b.to_cl_N()));
1804 /** Numeric square root.
1805 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1806 * should return integer 2.
1808 * @param z numeric argument
1809 * @return square root of z. Branch cut along negative real axis, the negative
1810 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1811 * where imag(z)>0. */
1812 const numeric sqrt(const numeric &z)
1814 return cln::sqrt(z.to_cl_N());
1818 /** Integer numeric square root. */
1819 const numeric isqrt(const numeric &x)
1821 if (x.is_integer()) {
1823 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1830 /** Floating point evaluation of Archimedes' constant Pi. */
1833 return numeric(cln::pi(cln::default_float_format));
1837 /** Floating point evaluation of Euler's constant gamma. */
1840 return numeric(cln::eulerconst(cln::default_float_format));
1844 /** Floating point evaluation of Catalan's constant. */
1845 ex CatalanEvalf(void)
1847 return numeric(cln::catalanconst(cln::default_float_format));
1851 /** _numeric_digits default ctor, checking for singleton invariance. */
1852 _numeric_digits::_numeric_digits()
1855 // It initializes to 17 digits, because in CLN float_format(17) turns out
1856 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1857 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1859 throw(std::runtime_error("I told you not to do instantiate me!"));
1861 cln::default_float_format = cln::float_format(17);
1865 /** Assign a native long to global Digits object. */
1866 _numeric_digits& _numeric_digits::operator=(long prec)
1869 cln::default_float_format = cln::float_format(prec);
1874 /** Convert global Digits object to native type long. */
1875 _numeric_digits::operator long()
1877 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1878 return (long)digits;
1882 /** Append global Digits object to ostream. */
1883 void _numeric_digits::print(std::ostream &os) const
1889 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1896 // static member variables
1901 bool _numeric_digits::too_late = false;
1904 /** Accuracy in decimal digits. Only object of this type! Can be set using
1905 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1906 _numeric_digits Digits;
1908 } // namespace GiNaC