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
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 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
95 // using an immediate type:
96 if (cln::cl_immediate_p(i))
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 // i.e. satisfies cl_immediate_p(), we save space and dereferences by
110 // using an immediate type:
111 if (cln::cl_immediate_p(i))
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 // make it safe by adding explicit sign
164 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
166 std::string::size_type delim;
168 // chop ss into terms from left to right
170 bool imaginary = false;
171 delim = ss.find_first_of(std::string("+-"),1);
172 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
173 if ((delim != std::string::npos) && (ss.at(delim-1) == 'E'))
174 delim = ss.find_first_of(std::string("+-"),delim+1);
175 term = ss.substr(0,delim);
176 if (delim != std::string::npos)
177 ss = ss.substr(delim);
178 // is the term imaginary?
179 if (term.find("I") != std::string::npos) {
181 term = term.replace(term.find("I"),1,"");
183 if (term.find("*") != std::string::npos)
184 term = term.replace(term.find("*"),1,"");
185 // correct for trivial +/-I without explicit factor on I:
186 if (term.size() == 1)
190 if (term.find(".") != std::string::npos) {
191 // CLN's short type cl_SF is not very useful within the GiNaC
192 // framework where we are mainly interested in the arbitrary
193 // precision type cl_LF. Hence we go straight to the construction
194 // of generic floats. In order to create them we have to convert
195 // our own floating point notation used for output and construction
196 // from char * to CLN's generic notation:
197 // 3.14 --> 3.14e0_<Digits>
198 // 31.4E-1 --> 31.4e-1_<Digits>
200 // No exponent marker? Let's add a trivial one.
201 if (term.find("E") == std::string::npos)
204 term = term.replace(term.find("E"),1,"e");
205 // append _<Digits> to term
206 term += "_" + ToString((unsigned)Digits);
207 // construct float using cln::cl_F(const char *) ctor.
209 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
211 ctorval = ctorval + cln::cl_F(term.c_str());
213 // not a floating point number...
215 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
217 ctorval = ctorval + cln::cl_R(term.c_str());
219 } while(delim != std::string::npos);
221 setflag(status_flags::evaluated | status_flags::expanded);
225 /** Ctor from CLN types. This is for the initiated user or internal use
227 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
230 setflag(status_flags::evaluated | status_flags::expanded);
237 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
239 cln::cl_N ctorval = 0;
241 // Read number as string
243 if (n.find_string("number", str)) {
244 std::istringstream s(str);
245 cln::cl_idecoded_float re, im;
249 case 'R': // Integer-decoded real number
250 s >> re.sign >> re.mantissa >> re.exponent;
251 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
253 case 'C': // Integer-decoded complex number
254 s >> re.sign >> re.mantissa >> re.exponent;
255 s >> im.sign >> im.mantissa >> im.exponent;
256 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
257 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
259 default: // Ordinary number
266 setflag(status_flags::evaluated | status_flags::expanded);
269 void numeric::archive(archive_node &n) const
271 inherited::archive(n);
273 // Write number as string
274 std::ostringstream s;
275 if (this->is_crational())
276 s << cln::the<cln::cl_N>(value);
278 // Non-rational numbers are written in an integer-decoded format
279 // to preserve the precision
280 if (this->is_real()) {
281 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
283 s << re.sign << " " << re.mantissa << " " << re.exponent;
285 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
286 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
288 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
289 s << im.sign << " " << im.mantissa << " " << im.exponent;
292 n.add_string("number", s.str());
295 DEFAULT_UNARCHIVE(numeric)
298 // functions overriding virtual functions from base classes
301 /** Helper function to print a real number in a nicer way than is CLN's
302 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
303 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
304 * long as it only uses cl_LF and no other floating point types that we might
305 * want to visibly distinguish from cl_LF.
307 * @see numeric::print() */
308 static void print_real_number(const print_context & c, const cln::cl_R &x)
310 cln::cl_print_flags ourflags;
311 if (cln::instanceof(x, cln::cl_RA_ring)) {
312 // case 1: integer or rational
313 if (cln::instanceof(x, cln::cl_I_ring) ||
314 !is_a<print_latex>(c)) {
315 cln::print_real(c.s, ourflags, x);
316 } else { // rational output in LaTeX context
318 cln::print_real(c.s, ourflags, cln::numerator(cln::the<cln::cl_RA>(x)));
320 cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
325 // make CLN believe this number has default_float_format, so it prints
326 // 'E' as exponent marker instead of 'L':
327 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
328 cln::print_real(c.s, ourflags, x);
332 /** This method adds to the output so it blends more consistently together
333 * with the other routines and produces something compatible to ginsh input.
335 * @see print_real_number() */
336 void numeric::print(const print_context & c, unsigned level) const
338 if (is_a<print_tree>(c)) {
340 c.s << std::string(level, ' ') << cln::the<cln::cl_N>(value)
341 << " (" << class_name() << ")"
342 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
345 } else if (is_a<print_csrc>(c)) {
347 std::ios::fmtflags oldflags = c.s.flags();
348 c.s.setf(std::ios::scientific);
349 if (this->is_rational() && !this->is_integer()) {
350 if (compare(_num0) > 0) {
352 if (is_a<print_csrc_cl_N>(c))
353 c.s << "cln::cl_F(\"" << numer().evalf() << "\")";
355 c.s << numer().to_double();
358 if (is_a<print_csrc_cl_N>(c))
359 c.s << "cln::cl_F(\"" << -numer().evalf() << "\")";
361 c.s << -numer().to_double();
364 if (is_a<print_csrc_cl_N>(c))
365 c.s << "cln::cl_F(\"" << denom().evalf() << "\")";
367 c.s << denom().to_double();
370 if (is_a<print_csrc_cl_N>(c))
371 c.s << "cln::cl_F(\"" << evalf() << "\")";
378 const std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
379 const std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
380 const std::string imag_sym = is_a<print_latex>(c) ? "i" : "I";
381 const std::string mul_sym = is_a<print_latex>(c) ? " " : "*";
382 const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
383 const cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
385 // case 1, real: x or -x
386 if ((precedence() <= level) && (!this->is_nonneg_integer())) {
388 print_real_number(c, r);
391 print_real_number(c, r);
395 // case 2, imaginary: y*I or -y*I
396 if ((precedence() <= level) && (i < 0)) {
398 c.s << par_open+imag_sym+par_close;
401 print_real_number(c, i);
402 c.s << mul_sym+imag_sym+par_close;
409 c.s << "-" << imag_sym;
411 print_real_number(c, i);
412 c.s << mul_sym+imag_sym;
417 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
418 if (precedence() <= level)
420 print_real_number(c, r);
425 print_real_number(c, i);
426 c.s << mul_sym+imag_sym;
433 print_real_number(c, i);
434 c.s << mul_sym+imag_sym;
437 if (precedence() <= level)
444 bool numeric::info(unsigned inf) const
447 case info_flags::numeric:
448 case info_flags::polynomial:
449 case info_flags::rational_function:
451 case info_flags::real:
453 case info_flags::rational:
454 case info_flags::rational_polynomial:
455 return is_rational();
456 case info_flags::crational:
457 case info_flags::crational_polynomial:
458 return is_crational();
459 case info_flags::integer:
460 case info_flags::integer_polynomial:
462 case info_flags::cinteger:
463 case info_flags::cinteger_polynomial:
464 return is_cinteger();
465 case info_flags::positive:
466 return is_positive();
467 case info_flags::negative:
468 return is_negative();
469 case info_flags::nonnegative:
470 return !is_negative();
471 case info_flags::posint:
472 return is_pos_integer();
473 case info_flags::negint:
474 return is_integer() && is_negative();
475 case info_flags::nonnegint:
476 return is_nonneg_integer();
477 case info_flags::even:
479 case info_flags::odd:
481 case info_flags::prime:
483 case info_flags::algebraic:
489 /** Disassemble real part and imaginary part to scan for the occurrence of a
490 * single number. Also handles the imaginary unit. It ignores the sign on
491 * both this and the argument, which may lead to what might appear as funny
492 * results: (2+I).has(-2) -> true. But this is consistent, since we also
493 * would like to have (-2+I).has(2) -> true and we want to think about the
494 * sign as a multiplicative factor. */
495 bool numeric::has(const ex &other) const
497 if (!is_ex_exactly_of_type(other, numeric))
499 const numeric &o = ex_to<numeric>(other);
500 if (this->is_equal(o) || this->is_equal(-o))
502 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
503 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
504 this->real().is_equal(-o) || this->imag().is_equal(-o));
506 if (o.is_equal(I)) // e.g scan for I in 42*I
507 return !this->is_real();
508 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
509 return (this->real().has(o*I) || this->imag().has(o*I) ||
510 this->real().has(-o*I) || this->imag().has(-o*I));
516 /** Evaluation of numbers doesn't do anything at all. */
517 ex numeric::eval(int level) const
519 // Warning: if this is ever gonna do something, the ex ctors from all kinds
520 // of numbers should be checking for status_flags::evaluated.
525 /** Cast numeric into a floating-point object. For example exact numeric(1) is
526 * returned as a 1.0000000000000000000000 and so on according to how Digits is
527 * currently set. In case the object already was a floating point number the
528 * precision is trimmed to match the currently set default.
530 * @param level ignored, only needed for overriding basic::evalf.
531 * @return an ex-handle to a numeric. */
532 ex numeric::evalf(int level) const
534 // level can safely be discarded for numeric objects.
535 return numeric(cln::cl_float(1.0, cln::default_float_format) *
536 (cln::the<cln::cl_N>(value)));
541 int numeric::compare_same_type(const basic &other) const
543 GINAC_ASSERT(is_exactly_a<numeric>(other));
544 const numeric &o = static_cast<const numeric &>(other);
546 return this->compare(o);
550 bool numeric::is_equal_same_type(const basic &other) const
552 GINAC_ASSERT(is_exactly_a<numeric>(other));
553 const numeric &o = static_cast<const numeric &>(other);
555 return this->is_equal(o);
559 unsigned numeric::calchash(void) const
561 // Use CLN's hashcode. Warning: It depends only on the number's value, not
562 // its type or precision (i.e. a true equivalence relation on numbers). As
563 // a consequence, 3 and 3.0 share the same hashvalue.
564 setflag(status_flags::hash_calculated);
565 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
570 // new virtual functions which can be overridden by derived classes
576 // non-virtual functions in this class
581 /** Numerical addition method. Adds argument to *this and returns result as
582 * a numeric object. */
583 const numeric numeric::add(const numeric &other) const
585 // Efficiency shortcut: trap the neutral element by pointer.
588 else if (&other==_num0_p)
591 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
595 /** Numerical subtraction method. Subtracts argument from *this and returns
596 * result as a numeric object. */
597 const numeric numeric::sub(const numeric &other) const
599 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
603 /** Numerical multiplication method. Multiplies *this and argument and returns
604 * result as a numeric object. */
605 const numeric numeric::mul(const numeric &other) const
607 // Efficiency shortcut: trap the neutral element by pointer.
610 else if (&other==_num1_p)
613 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
617 /** Numerical division method. Divides *this by argument and returns result as
620 * @exception overflow_error (division by zero) */
621 const numeric numeric::div(const numeric &other) const
623 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
624 throw std::overflow_error("numeric::div(): division by zero");
625 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
629 /** Numerical exponentiation. Raises *this to the power given as argument and
630 * returns result as a numeric object. */
631 const numeric numeric::power(const numeric &other) const
633 // Efficiency shortcut: trap the neutral exponent by pointer.
637 if (cln::zerop(cln::the<cln::cl_N>(value))) {
638 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
639 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
640 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
641 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
642 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
643 throw std::overflow_error("numeric::eval(): division by zero");
647 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
651 const numeric &numeric::add_dyn(const numeric &other) const
653 // Efficiency shortcut: trap the neutral element by pointer.
656 else if (&other==_num0_p)
659 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
660 setflag(status_flags::dynallocated));
664 const numeric &numeric::sub_dyn(const numeric &other) const
666 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
667 setflag(status_flags::dynallocated));
671 const numeric &numeric::mul_dyn(const numeric &other) const
673 // Efficiency shortcut: trap the neutral element by pointer.
676 else if (&other==_num1_p)
679 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
680 setflag(status_flags::dynallocated));
684 const numeric &numeric::div_dyn(const numeric &other) const
686 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
687 throw std::overflow_error("division by zero");
688 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
689 setflag(status_flags::dynallocated));
693 const numeric &numeric::power_dyn(const numeric &other) const
695 // Efficiency shortcut: trap the neutral exponent by pointer.
699 if (cln::zerop(cln::the<cln::cl_N>(value))) {
700 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
701 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
702 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
703 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
704 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
705 throw std::overflow_error("numeric::eval(): division by zero");
709 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
710 setflag(status_flags::dynallocated));
714 const numeric &numeric::operator=(int i)
716 return operator=(numeric(i));
720 const numeric &numeric::operator=(unsigned int i)
722 return operator=(numeric(i));
726 const numeric &numeric::operator=(long i)
728 return operator=(numeric(i));
732 const numeric &numeric::operator=(unsigned long i)
734 return operator=(numeric(i));
738 const numeric &numeric::operator=(double d)
740 return operator=(numeric(d));
744 const numeric &numeric::operator=(const char * s)
746 return operator=(numeric(s));
750 /** Inverse of a number. */
751 const numeric numeric::inverse(void) const
753 if (cln::zerop(cln::the<cln::cl_N>(value)))
754 throw std::overflow_error("numeric::inverse(): division by zero");
755 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
759 /** Return the complex half-plane (left or right) in which the number lies.
760 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
761 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
763 * @see numeric::compare(const numeric &other) */
764 int numeric::csgn(void) const
766 if (cln::zerop(cln::the<cln::cl_N>(value)))
768 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
769 if (!cln::zerop(r)) {
775 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
783 /** This method establishes a canonical order on all numbers. For complex
784 * numbers this is not possible in a mathematically consistent way but we need
785 * to establish some order and it ought to be fast. So we simply define it
786 * to be compatible with our method csgn.
788 * @return csgn(*this-other)
789 * @see numeric::csgn(void) */
790 int numeric::compare(const numeric &other) const
792 // Comparing two real numbers?
793 if (cln::instanceof(value, cln::cl_R_ring) &&
794 cln::instanceof(other.value, cln::cl_R_ring))
795 // Yes, so just cln::compare them
796 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
798 // No, first cln::compare real parts...
799 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
802 // ...and then the imaginary parts.
803 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
808 bool numeric::is_equal(const numeric &other) const
810 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
814 /** True if object is zero. */
815 bool numeric::is_zero(void) const
817 return cln::zerop(cln::the<cln::cl_N>(value));
821 /** True if object is not complex and greater than zero. */
822 bool numeric::is_positive(void) const
825 return cln::plusp(cln::the<cln::cl_R>(value));
830 /** True if object is not complex and less than zero. */
831 bool numeric::is_negative(void) const
834 return cln::minusp(cln::the<cln::cl_R>(value));
839 /** True if object is a non-complex integer. */
840 bool numeric::is_integer(void) const
842 return cln::instanceof(value, cln::cl_I_ring);
846 /** True if object is an exact integer greater than zero. */
847 bool numeric::is_pos_integer(void) const
849 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
853 /** True if object is an exact integer greater or equal zero. */
854 bool numeric::is_nonneg_integer(void) const
856 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
860 /** True if object is an exact even integer. */
861 bool numeric::is_even(void) const
863 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
867 /** True if object is an exact odd integer. */
868 bool numeric::is_odd(void) const
870 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
874 /** Probabilistic primality test.
876 * @return true if object is exact integer and prime. */
877 bool numeric::is_prime(void) const
879 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
883 /** True if object is an exact rational number, may even be complex
884 * (denominator may be unity). */
885 bool numeric::is_rational(void) const
887 return cln::instanceof(value, cln::cl_RA_ring);
891 /** True if object is a real integer, rational or float (but not complex). */
892 bool numeric::is_real(void) const
894 return cln::instanceof(value, cln::cl_R_ring);
898 bool numeric::operator==(const numeric &other) const
900 return cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
904 bool numeric::operator!=(const numeric &other) const
906 return !cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
910 /** True if object is element of the domain of integers extended by I, i.e. is
911 * of the form a+b*I, where a and b are integers. */
912 bool numeric::is_cinteger(void) const
914 if (cln::instanceof(value, cln::cl_I_ring))
916 else if (!this->is_real()) { // complex case, handle n+m*I
917 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
918 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
925 /** True if object is an exact rational number, may even be complex
926 * (denominator may be unity). */
927 bool numeric::is_crational(void) const
929 if (cln::instanceof(value, cln::cl_RA_ring))
931 else if (!this->is_real()) { // complex case, handle Q(i):
932 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
933 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
940 /** Numerical comparison: less.
942 * @exception invalid_argument (complex inequality) */
943 bool numeric::operator<(const numeric &other) const
945 if (this->is_real() && other.is_real())
946 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
947 throw std::invalid_argument("numeric::operator<(): complex inequality");
951 /** Numerical comparison: less or equal.
953 * @exception invalid_argument (complex inequality) */
954 bool numeric::operator<=(const numeric &other) const
956 if (this->is_real() && other.is_real())
957 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
958 throw std::invalid_argument("numeric::operator<=(): complex inequality");
962 /** Numerical comparison: greater.
964 * @exception invalid_argument (complex inequality) */
965 bool numeric::operator>(const numeric &other) const
967 if (this->is_real() && other.is_real())
968 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
969 throw std::invalid_argument("numeric::operator>(): complex inequality");
973 /** Numerical comparison: greater or equal.
975 * @exception invalid_argument (complex inequality) */
976 bool numeric::operator>=(const numeric &other) const
978 if (this->is_real() && other.is_real())
979 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
980 throw std::invalid_argument("numeric::operator>=(): complex inequality");
984 /** Converts numeric types to machine's int. You should check with
985 * is_integer() if the number is really an integer before calling this method.
986 * You may also consider checking the range first. */
987 int numeric::to_int(void) const
989 GINAC_ASSERT(this->is_integer());
990 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
994 /** Converts numeric types to machine's long. You should check with
995 * is_integer() if the number is really an integer before calling this method.
996 * You may also consider checking the range first. */
997 long numeric::to_long(void) const
999 GINAC_ASSERT(this->is_integer());
1000 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1004 /** Converts numeric types to machine's double. You should check with is_real()
1005 * if the number is really not complex before calling this method. */
1006 double numeric::to_double(void) const
1008 GINAC_ASSERT(this->is_real());
1009 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1013 /** Returns a new CLN object of type cl_N, representing the value of *this.
1014 * This method may be used when mixing GiNaC and CLN in one project.
1016 cln::cl_N numeric::to_cl_N(void) const
1018 return cln::cl_N(cln::the<cln::cl_N>(value));
1022 /** Real part of a number. */
1023 const numeric numeric::real(void) const
1025 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1029 /** Imaginary part of a number. */
1030 const numeric numeric::imag(void) const
1032 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1036 /** Numerator. Computes the numerator of rational numbers, rationalized
1037 * numerator of complex if real and imaginary part are both rational numbers
1038 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1040 const numeric numeric::numer(void) const
1042 if (this->is_integer())
1043 return numeric(*this);
1045 else if (cln::instanceof(value, cln::cl_RA_ring))
1046 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1048 else if (!this->is_real()) { // complex case, handle Q(i):
1049 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1050 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1051 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1052 return numeric(*this);
1053 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1054 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1055 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1056 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1057 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1058 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1059 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1060 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1063 // at least one float encountered
1064 return numeric(*this);
1068 /** Denominator. Computes the denominator of rational numbers, common integer
1069 * denominator of complex if real and imaginary part are both rational numbers
1070 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1071 const numeric numeric::denom(void) const
1073 if (this->is_integer())
1076 if (cln::instanceof(value, cln::cl_RA_ring))
1077 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1079 if (!this->is_real()) { // complex case, handle Q(i):
1080 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1081 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1082 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1084 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1085 return numeric(cln::denominator(i));
1086 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1087 return numeric(cln::denominator(r));
1088 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1089 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1091 // at least one float encountered
1096 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1097 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1098 * 2^(n-1) <= x < 2^n.
1100 * @return number of bits (excluding sign) needed to represent that number
1101 * in two's complement if it is an integer, 0 otherwise. */
1102 int numeric::int_length(void) const
1104 if (this->is_integer())
1105 return cln::integer_length(cln::the<cln::cl_I>(value));
1114 /** Imaginary unit. This is not a constant but a numeric since we are
1115 * natively handing complex numbers anyways, so in each expression containing
1116 * an I it is automatically eval'ed away anyhow. */
1117 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1120 /** Exponential function.
1122 * @return arbitrary precision numerical exp(x). */
1123 const numeric exp(const numeric &x)
1125 return cln::exp(x.to_cl_N());
1129 /** Natural logarithm.
1131 * @param z complex number
1132 * @return arbitrary precision numerical log(x).
1133 * @exception pole_error("log(): logarithmic pole",0) */
1134 const numeric log(const numeric &z)
1137 throw pole_error("log(): logarithmic pole",0);
1138 return cln::log(z.to_cl_N());
1142 /** Numeric sine (trigonometric function).
1144 * @return arbitrary precision numerical sin(x). */
1145 const numeric sin(const numeric &x)
1147 return cln::sin(x.to_cl_N());
1151 /** Numeric cosine (trigonometric function).
1153 * @return arbitrary precision numerical cos(x). */
1154 const numeric cos(const numeric &x)
1156 return cln::cos(x.to_cl_N());
1160 /** Numeric tangent (trigonometric function).
1162 * @return arbitrary precision numerical tan(x). */
1163 const numeric tan(const numeric &x)
1165 return cln::tan(x.to_cl_N());
1169 /** Numeric inverse sine (trigonometric function).
1171 * @return arbitrary precision numerical asin(x). */
1172 const numeric asin(const numeric &x)
1174 return cln::asin(x.to_cl_N());
1178 /** Numeric inverse cosine (trigonometric function).
1180 * @return arbitrary precision numerical acos(x). */
1181 const numeric acos(const numeric &x)
1183 return cln::acos(x.to_cl_N());
1189 * @param z complex number
1191 * @exception pole_error("atan(): logarithmic pole",0) */
1192 const numeric atan(const numeric &x)
1195 x.real().is_zero() &&
1196 abs(x.imag()).is_equal(_num1))
1197 throw pole_error("atan(): logarithmic pole",0);
1198 return cln::atan(x.to_cl_N());
1204 * @param x real number
1205 * @param y real number
1206 * @return atan(y/x) */
1207 const numeric atan(const numeric &y, const numeric &x)
1209 if (x.is_real() && y.is_real())
1210 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1211 cln::the<cln::cl_R>(y.to_cl_N()));
1213 throw std::invalid_argument("atan(): complex argument");
1217 /** Numeric hyperbolic sine (trigonometric function).
1219 * @return arbitrary precision numerical sinh(x). */
1220 const numeric sinh(const numeric &x)
1222 return cln::sinh(x.to_cl_N());
1226 /** Numeric hyperbolic cosine (trigonometric function).
1228 * @return arbitrary precision numerical cosh(x). */
1229 const numeric cosh(const numeric &x)
1231 return cln::cosh(x.to_cl_N());
1235 /** Numeric hyperbolic tangent (trigonometric function).
1237 * @return arbitrary precision numerical tanh(x). */
1238 const numeric tanh(const numeric &x)
1240 return cln::tanh(x.to_cl_N());
1244 /** Numeric inverse hyperbolic sine (trigonometric function).
1246 * @return arbitrary precision numerical asinh(x). */
1247 const numeric asinh(const numeric &x)
1249 return cln::asinh(x.to_cl_N());
1253 /** Numeric inverse hyperbolic cosine (trigonometric function).
1255 * @return arbitrary precision numerical acosh(x). */
1256 const numeric acosh(const numeric &x)
1258 return cln::acosh(x.to_cl_N());
1262 /** Numeric inverse hyperbolic tangent (trigonometric function).
1264 * @return arbitrary precision numerical atanh(x). */
1265 const numeric atanh(const numeric &x)
1267 return cln::atanh(x.to_cl_N());
1271 /*static cln::cl_N Li2_series(const ::cl_N &x,
1272 const ::float_format_t &prec)
1274 // Note: argument must be in the unit circle
1275 // This is very inefficient unless we have fast floating point Bernoulli
1276 // numbers implemented!
1277 cln::cl_N c1 = -cln::log(1-x);
1279 // hard-wire the first two Bernoulli numbers
1280 cln::cl_N acc = c1 - cln::square(c1)/4;
1282 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1283 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1285 c1 = cln::square(c1);
1289 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1290 // 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));
1293 } while (acc != acc+aug);
1297 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1298 * circle) using a power series. */
1299 static cln::cl_N Li2_series(const cln::cl_N &x,
1300 const cln::float_format_t &prec)
1302 // Note: argument must be in the unit circle
1304 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1309 den = den + i; // 1, 4, 9, 16, ...
1313 } while (acc != acc+aug);
1317 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1318 static cln::cl_N Li2_projection(const cln::cl_N &x,
1319 const cln::float_format_t &prec)
1321 const cln::cl_R re = cln::realpart(x);
1322 const cln::cl_R im = cln::imagpart(x);
1323 if (re > cln::cl_F(".5"))
1324 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1326 - Li2_series(1-x, prec)
1327 - cln::log(x)*cln::log(1-x));
1328 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1329 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1330 return(- cln::square(cln::log(1-x))/2
1331 - Li2_series(x/(x-1), prec));
1332 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1333 // Li2(x^2)/2 - Li2(-x)
1334 return(Li2_projection(cln::square(x), prec)/2
1335 - Li2_projection(-x, prec));
1336 return Li2_series(x, prec);
1339 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1340 * the branch cut lies along the positive real axis, starting at 1 and
1341 * continuous with quadrant IV.
1343 * @return arbitrary precision numerical Li2(x). */
1344 const numeric Li2(const numeric &x)
1349 // what is the desired float format?
1350 // first guess: default format
1351 cln::float_format_t prec = cln::default_float_format;
1352 const cln::cl_N value = x.to_cl_N();
1353 // second guess: the argument's format
1354 if (!x.real().is_rational())
1355 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1356 else if (!x.imag().is_rational())
1357 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1359 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1360 return cln::zeta(2, prec);
1362 if (cln::abs(value) > 1)
1363 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1364 return(- cln::square(cln::log(-value))/2
1365 - cln::zeta(2, prec)
1366 - Li2_projection(cln::recip(value), prec));
1368 return Li2_projection(x.to_cl_N(), prec);
1372 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1373 * integer arguments. */
1374 const numeric zeta(const numeric &x)
1376 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1377 // only knows about integer arguments and zeta(3).evalf() automatically
1378 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1379 // being an exact zero for CLN, which can be tested and then we can just
1380 // pass the number casted to an int:
1382 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1383 if (cln::zerop(x.to_cl_N()-aux))
1384 return cln::zeta(aux);
1390 /** The Gamma function.
1391 * This is only a stub! */
1392 const numeric lgamma(const numeric &x)
1396 const numeric tgamma(const numeric &x)
1402 /** The psi function (aka polygamma function).
1403 * This is only a stub! */
1404 const numeric psi(const numeric &x)
1410 /** The psi functions (aka polygamma functions).
1411 * This is only a stub! */
1412 const numeric psi(const numeric &n, const numeric &x)
1418 /** Factorial combinatorial function.
1420 * @param n integer argument >= 0
1421 * @exception range_error (argument must be integer >= 0) */
1422 const numeric factorial(const numeric &n)
1424 if (!n.is_nonneg_integer())
1425 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1426 return numeric(cln::factorial(n.to_int()));
1430 /** The double factorial combinatorial function. (Scarcely used, but still
1431 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1433 * @param n integer argument >= -1
1434 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1435 * @exception range_error (argument must be integer >= -1) */
1436 const numeric doublefactorial(const numeric &n)
1438 if (n.is_equal(_num_1))
1441 if (!n.is_nonneg_integer())
1442 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1444 return numeric(cln::doublefactorial(n.to_int()));
1448 /** The Binomial coefficients. It computes the binomial coefficients. For
1449 * integer n and k and positive n this is the number of ways of choosing k
1450 * objects from n distinct objects. If n is negative, the formula
1451 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1452 const numeric binomial(const numeric &n, const numeric &k)
1454 if (n.is_integer() && k.is_integer()) {
1455 if (n.is_nonneg_integer()) {
1456 if (k.compare(n)!=1 && k.compare(_num0)!=-1)
1457 return numeric(cln::binomial(n.to_int(),k.to_int()));
1461 return _num_1.power(k)*binomial(k-n-_num1,k);
1465 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1466 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1470 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1471 * in the expansion of the function x/(e^x-1).
1473 * @return the nth Bernoulli number (a rational number).
1474 * @exception range_error (argument must be integer >= 0) */
1475 const numeric bernoulli(const numeric &nn)
1477 if (!nn.is_integer() || nn.is_negative())
1478 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1482 // The Bernoulli numbers are rational numbers that may be computed using
1485 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1487 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1488 // previous ones, the computation is necessarily very expensive. There are
1489 // several other ways of computing them, a particularly good one being
1493 // for (unsigned i=0; i<n; i++) {
1494 // c = exquo(c*(i-n),(i+2));
1495 // Bern = Bern + c*s/(i+2);
1496 // s = s + expt_pos(cl_I(i+2),n);
1500 // But if somebody works with the n'th Bernoulli number she is likely to
1501 // also need all previous Bernoulli numbers. So we need a complete remember
1502 // table and above divide and conquer algorithm is not suited to build one
1503 // up. The code below is adapted from Pari's function bernvec().
1505 // (There is an interesting relation with the tangent polynomials described
1506 // in `Concrete Mathematics', which leads to a program twice as fast as our
1507 // implementation below, but it requires storing one such polynomial in
1508 // addition to the remember table. This doubles the memory footprint so
1509 // we don't use it.)
1511 // the special cases not covered by the algorithm below
1512 if (nn.is_equal(_num1))
1517 // store nonvanishing Bernoulli numbers here
1518 static std::vector< cln::cl_RA > results;
1519 static int highest_result = 0;
1520 // algorithm not applicable to B(0), so just store it
1521 if (results.empty())
1522 results.push_back(cln::cl_RA(1));
1524 int n = nn.to_long();
1525 for (int i=highest_result; i<n/2; ++i) {
1531 for (int j=i; j>0; --j) {
1532 B = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
1538 B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
1539 results.push_back(B);
1542 return results[n/2];
1546 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1547 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1549 * @param n an integer
1550 * @return the nth Fibonacci number F(n) (an integer number)
1551 * @exception range_error (argument must be an integer) */
1552 const numeric fibonacci(const numeric &n)
1554 if (!n.is_integer())
1555 throw std::range_error("numeric::fibonacci(): argument must be integer");
1558 // The following addition formula holds:
1560 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1562 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1563 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1565 // Replace m by m+1:
1566 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1567 // Now put in m = n, to get
1568 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1569 // F(2n+1) = F(n)^2 + F(n+1)^2
1571 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1574 if (n.is_negative())
1576 return -fibonacci(-n);
1578 return fibonacci(-n);
1582 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1583 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1584 // Since a squaring is cheaper than a multiplication, better use
1585 // three squarings instead of one multiplication and two squarings.
1586 cln::cl_I u2 = cln::square(u);
1587 cln::cl_I v2 = cln::square(v);
1588 if (cln::logbitp(bit-1, m)) {
1589 v = cln::square(u + v) - u2;
1592 u = v2 - cln::square(v - u);
1597 // Here we don't use the squaring formula because one multiplication
1598 // is cheaper than two squarings.
1599 return u * ((v << 1) - u);
1601 return cln::square(u) + cln::square(v);
1605 /** Absolute value. */
1606 const numeric abs(const numeric& x)
1608 return cln::abs(x.to_cl_N());
1612 /** Modulus (in positive representation).
1613 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1614 * sign of a or is zero. This is different from Maple's modp, where the sign
1615 * of b is ignored. It is in agreement with Mathematica's Mod.
1617 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1618 * integer, 0 otherwise. */
1619 const numeric mod(const numeric &a, const numeric &b)
1621 if (a.is_integer() && b.is_integer())
1622 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1623 cln::the<cln::cl_I>(b.to_cl_N()));
1629 /** Modulus (in symmetric representation).
1630 * Equivalent to Maple's mods.
1632 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1633 const numeric smod(const numeric &a, const numeric &b)
1635 if (a.is_integer() && b.is_integer()) {
1636 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1637 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1638 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1644 /** Numeric integer remainder.
1645 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1646 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1647 * sign of a or is zero.
1649 * @return remainder of a/b if both are integer, 0 otherwise. */
1650 const numeric irem(const numeric &a, const numeric &b)
1652 if (a.is_integer() && b.is_integer())
1653 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1654 cln::the<cln::cl_I>(b.to_cl_N()));
1660 /** Numeric integer remainder.
1661 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1662 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1663 * and irem(a,b) has the sign of a or is zero.
1665 * @return remainder of a/b and quotient stored in q if both are integer,
1667 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1669 if (a.is_integer() && b.is_integer()) {
1670 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1671 cln::the<cln::cl_I>(b.to_cl_N()));
1672 q = rem_quo.quotient;
1673 return rem_quo.remainder;
1681 /** Numeric integer quotient.
1682 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1684 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1685 const numeric iquo(const numeric &a, const numeric &b)
1687 if (a.is_integer() && b.is_integer())
1688 return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1689 cln::the<cln::cl_I>(b.to_cl_N()));
1695 /** Numeric integer quotient.
1696 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1697 * r == a - iquo(a,b,r)*b.
1699 * @return truncated quotient of a/b and remainder stored in r if both are
1700 * integer, 0 otherwise. */
1701 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1703 if (a.is_integer() && b.is_integer()) {
1704 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1705 cln::the<cln::cl_I>(b.to_cl_N()));
1706 r = rem_quo.remainder;
1707 return rem_quo.quotient;
1715 /** Greatest Common Divisor.
1717 * @return The GCD of two numbers if both are integer, a numerical 1
1718 * if they are not. */
1719 const numeric gcd(const numeric &a, const numeric &b)
1721 if (a.is_integer() && b.is_integer())
1722 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1723 cln::the<cln::cl_I>(b.to_cl_N()));
1729 /** Least Common Multiple.
1731 * @return The LCM of two numbers if both are integer, the product of those
1732 * two numbers if they are not. */
1733 const numeric lcm(const numeric &a, const numeric &b)
1735 if (a.is_integer() && b.is_integer())
1736 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1737 cln::the<cln::cl_I>(b.to_cl_N()));
1743 /** Numeric square root.
1744 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1745 * should return integer 2.
1747 * @param z numeric argument
1748 * @return square root of z. Branch cut along negative real axis, the negative
1749 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1750 * where imag(z)>0. */
1751 const numeric sqrt(const numeric &z)
1753 return cln::sqrt(z.to_cl_N());
1757 /** Integer numeric square root. */
1758 const numeric isqrt(const numeric &x)
1760 if (x.is_integer()) {
1762 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1769 /** Floating point evaluation of Archimedes' constant Pi. */
1772 return numeric(cln::pi(cln::default_float_format));
1776 /** Floating point evaluation of Euler's constant gamma. */
1779 return numeric(cln::eulerconst(cln::default_float_format));
1783 /** Floating point evaluation of Catalan's constant. */
1784 ex CatalanEvalf(void)
1786 return numeric(cln::catalanconst(cln::default_float_format));
1790 /** _numeric_digits default ctor, checking for singleton invariance. */
1791 _numeric_digits::_numeric_digits()
1794 // It initializes to 17 digits, because in CLN float_format(17) turns out
1795 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1796 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1798 throw(std::runtime_error("I told you not to do instantiate me!"));
1800 cln::default_float_format = cln::float_format(17);
1804 /** Assign a native long to global Digits object. */
1805 _numeric_digits& _numeric_digits::operator=(long prec)
1808 cln::default_float_format = cln::float_format(prec);
1813 /** Convert global Digits object to native type long. */
1814 _numeric_digits::operator long()
1816 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1817 return (long)digits;
1821 /** Append global Digits object to ostream. */
1822 void _numeric_digits::print(std::ostream &os) const
1828 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1835 // static member variables
1840 bool _numeric_digits::too_late = false;
1843 /** Accuracy in decimal digits. Only object of this type! Can be set using
1844 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1845 _numeric_digits Digits;
1847 } // namespace GiNaC