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) 1999-2000 Johannes Gutenberg University Mainz, Germany
12 * This program is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU General Public License as published by
14 * the Free Software Foundation; either version 2 of the License, or
15 * (at your option) any later version.
17 * This program is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU General Public License for more details.
22 * You should have received a copy of the GNU General Public License
23 * along with this program; if not, write to the Free Software
24 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
33 #if defined(HAVE_SSTREAM)
35 #elif defined(HAVE_STRSTREAM)
38 #error Need either sstream or strstream
47 // CLN should not pollute the global namespace, hence we include it here
48 // instead of in some header file where it would propagate to other parts.
49 // Also, we only need a subset of CLN, so we don't include the complete cln.h:
51 #include <cln/cl_output.h>
52 #include <cln/cl_integer_io.h>
53 #include <cln/cl_integer_ring.h>
54 #include <cln/cl_rational_io.h>
55 #include <cln/cl_rational_ring.h>
56 #include <cln/cl_lfloat_class.h>
57 #include <cln/cl_lfloat_io.h>
58 #include <cln/cl_real_io.h>
59 #include <cln/cl_real_ring.h>
60 #include <cln/cl_complex_io.h>
61 #include <cln/cl_complex_ring.h>
62 #include <cln/cl_numtheory.h>
63 #else // def HAVE_CLN_CLN_H
64 #include <cl_output.h>
65 #include <cl_integer_io.h>
66 #include <cl_integer_ring.h>
67 #include <cl_rational_io.h>
68 #include <cl_rational_ring.h>
69 #include <cl_lfloat_class.h>
70 #include <cl_lfloat_io.h>
71 #include <cl_real_io.h>
72 #include <cl_real_ring.h>
73 #include <cl_complex_io.h>
74 #include <cl_complex_ring.h>
75 #include <cl_numtheory.h>
76 #endif // def HAVE_CLN_CLN_H
78 #ifndef NO_NAMESPACE_GINAC
80 #endif // ndef NO_NAMESPACE_GINAC
82 // linker has no problems finding text symbols for numerator or denominator
85 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
88 // default constructor, destructor, copy constructor assignment
89 // operator and helpers
94 /** default ctor. Numerically it initializes to an integer zero. */
95 numeric::numeric() : basic(TINFO_numeric)
97 debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
101 setflag(status_flags::evaluated |
102 status_flags::expanded |
103 status_flags::hash_calculated);
108 debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
112 numeric::numeric(const numeric & other)
114 debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
118 const numeric & numeric::operator=(const numeric & other)
120 debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
121 if (this != &other) {
130 void numeric::copy(const numeric & other)
133 value = new ::cl_N(*other.value);
136 void numeric::destroy(bool call_parent)
139 if (call_parent) basic::destroy(call_parent);
143 // other constructors
148 numeric::numeric(int i) : basic(TINFO_numeric)
150 debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
151 // Not the whole int-range is available if we don't cast to long
152 // first. This is due to the behaviour of the cl_I-ctor, which
153 // emphasizes efficiency:
154 value = new ::cl_I((long) i);
156 setflag(status_flags::evaluated|
157 status_flags::hash_calculated);
161 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
163 debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
164 // Not the whole uint-range is available if we don't cast to ulong
165 // first. This is due to the behaviour of the cl_I-ctor, which
166 // emphasizes efficiency:
167 value = new ::cl_I((unsigned long)i);
169 setflag(status_flags::evaluated|
170 status_flags::hash_calculated);
174 numeric::numeric(long i) : basic(TINFO_numeric)
176 debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
177 value = new ::cl_I(i);
179 setflag(status_flags::evaluated|
180 status_flags::hash_calculated);
184 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
186 debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
187 value = new ::cl_I(i);
189 setflag(status_flags::evaluated|
190 status_flags::hash_calculated);
193 /** Ctor for rational numerics a/b.
195 * @exception overflow_error (division by zero) */
196 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
198 debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
200 throw (std::overflow_error("division by zero"));
201 value = new ::cl_I(numer);
202 *value = *value / ::cl_I(denom);
204 setflag(status_flags::evaluated|
205 status_flags::hash_calculated);
209 numeric::numeric(double d) : basic(TINFO_numeric)
211 debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
212 // We really want to explicitly use the type cl_LF instead of the
213 // more general cl_F, since that would give us a cl_DF only which
214 // will not be promoted to cl_LF if overflow occurs:
216 *value = cl_float(d, cl_default_float_format);
218 setflag(status_flags::evaluated|
219 status_flags::hash_calculated);
223 /** ctor from C-style string. It also accepts complex numbers in GiNaC
224 * notation like "2+5*I". */
225 numeric::numeric(const char *s) : basic(TINFO_numeric)
227 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
228 value = new ::cl_N(0);
229 // parse complex numbers (functional but not completely safe, unfortunately
230 // std::string does not understand regexpese):
231 // ss should represent a simple sum like 2+5*I
233 // make it safe by adding explicit sign
234 if (ss.at(0) != '+' && ss.at(0) != '-')
236 std::string::size_type delim;
238 // chop ss into terms from left to right
240 bool imaginary = false;
241 delim = ss.find_first_of(std::string("+-"),1);
242 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
243 if (delim != std::string::npos &&
244 ss.at(delim-1) == 'E')
245 delim = ss.find_first_of(std::string("+-"),delim+1);
246 term = ss.substr(0,delim);
247 if (delim != std::string::npos)
248 ss = ss.substr(delim);
249 // is the term imaginary?
250 if (term.find("I") != std::string::npos) {
252 term = term.replace(term.find("I"),1,"");
254 if (term.find("*") != std::string::npos)
255 term = term.replace(term.find("*"),1,"");
256 // correct for trivial +/-I without explicit factor on I:
257 if (term.size() == 1)
261 const char *cs = term.c_str();
262 // CLN's short types are not useful within the GiNaC framework, hence
263 // we go straight to the construction of a long float. Simply using
264 // cl_N(s) would require us to use add a CLN exponent mark, otherwise
265 // we would not be save from over-/underflows.
268 *value = *value + ::complex(cl_I(0),::cl_LF(cs));
270 *value = *value + ::cl_LF(cs);
273 *value = *value + ::complex(cl_I(0),::cl_R(cs));
275 *value = *value + ::cl_R(cs);
276 } while(delim != std::string::npos);
278 setflag(status_flags::evaluated|
279 status_flags::hash_calculated);
282 /** Ctor from CLN types. This is for the initiated user or internal use
284 numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
286 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
287 value = new ::cl_N(z);
289 setflag(status_flags::evaluated|
290 status_flags::hash_calculated);
297 /** Construct object from archive_node. */
298 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
300 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
303 // Read number as string
305 if (n.find_string("number", str)) {
307 std::istringstream s(str);
309 std::istrstream s(str.c_str(), str.size() + 1);
311 ::cl_idecoded_float re, im;
315 case 'R': // Integer-decoded real number
316 s >> re.sign >> re.mantissa >> re.exponent;
317 *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
319 case 'C': // Integer-decoded complex number
320 s >> re.sign >> re.mantissa >> re.exponent;
321 s >> im.sign >> im.mantissa >> im.exponent;
322 *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
323 im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
325 default: // Ordinary number
332 setflag(status_flags::evaluated|
333 status_flags::hash_calculated);
336 /** Unarchive the object. */
337 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
339 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
342 /** Archive the object. */
343 void numeric::archive(archive_node &n) const
345 inherited::archive(n);
347 // Write number as string
349 std::ostringstream s;
352 std::ostrstream s(buf, 1024);
354 if (this->is_crational())
357 // Non-rational numbers are written in an integer-decoded format
358 // to preserve the precision
359 if (this->is_real()) {
360 cl_idecoded_float re = integer_decode_float(The(::cl_F)(*value));
362 s << re.sign << " " << re.mantissa << " " << re.exponent;
364 cl_idecoded_float re = integer_decode_float(The(::cl_F)(::realpart(*value)));
365 cl_idecoded_float im = integer_decode_float(The(::cl_F)(::imagpart(*value)));
367 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
368 s << im.sign << " " << im.mantissa << " " << im.exponent;
372 n.add_string("number", s.str());
375 std::string str(buf);
376 n.add_string("number", str);
381 // functions overriding virtual functions from bases classes
386 basic * numeric::duplicate() const
388 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
389 return new numeric(*this);
393 /** Helper function to print a real number in a nicer way than is CLN's
394 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
395 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
396 * long as it only uses cl_LF and no other floating point types.
398 * @see numeric::print() */
399 static void print_real_number(ostream & os, const cl_R & num)
401 cl_print_flags ourflags;
402 if (::instanceof(num, ::cl_RA_ring)) {
403 // case 1: integer or rational, nothing special to do:
404 ::print_real(os, ourflags, num);
407 // make CLN believe this number has default_float_format, so it prints
408 // 'E' as exponent marker instead of 'L':
409 ourflags.default_float_format = ::cl_float_format(The(::cl_F)(num));
410 ::print_real(os, ourflags, num);
415 /** This method adds to the output so it blends more consistently together
416 * with the other routines and produces something compatible to ginsh input.
418 * @see print_real_number() */
419 void numeric::print(ostream & os, unsigned upper_precedence) const
421 debugmsg("numeric print", LOGLEVEL_PRINT);
422 if (this->is_real()) {
423 // case 1, real: x or -x
424 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
426 print_real_number(os, The(::cl_R)(*value));
429 print_real_number(os, The(::cl_R)(*value));
432 // case 2, imaginary: y*I or -y*I
433 if (::realpart(*value) == 0) {
434 if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
435 if (::imagpart(*value) == -1) {
439 print_real_number(os, The(::cl_R)(::imagpart(*value)));
443 if (::imagpart(*value) == 1) {
446 if (::imagpart (*value) == -1) {
449 print_real_number(os, The(::cl_R)(::imagpart(*value)));
455 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
456 if (precedence <= upper_precedence)
458 print_real_number(os, The(::cl_R)(::realpart(*value)));
459 if (::imagpart(*value) < 0) {
460 if (::imagpart(*value) == -1) {
463 print_real_number(os, The(::cl_R)(::imagpart(*value)));
467 if (::imagpart(*value) == 1) {
471 print_real_number(os, The(::cl_R)(::imagpart(*value)));
475 if (precedence <= upper_precedence)
482 void numeric::printraw(ostream & os) const
484 // The method printraw doesn't do much, it simply uses CLN's operator<<()
485 // for output, which is ugly but reliable. e.g: 2+2i
486 debugmsg("numeric printraw", LOGLEVEL_PRINT);
487 os << "numeric(" << *value << ")";
491 void numeric::printtree(ostream & os, unsigned indent) const
493 debugmsg("numeric printtree", LOGLEVEL_PRINT);
494 os << std::string(indent,' ') << *value
496 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
497 << ", flags=" << flags << endl;
501 void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
503 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
504 ios::fmtflags oldflags = os.flags();
505 os.setf(ios::scientific);
506 if (this->is_rational() && !this->is_integer()) {
507 if (compare(_num0()) > 0) {
509 if (type == csrc_types::ctype_cl_N)
510 os << "cl_F(\"" << numer().evalf() << "\")";
512 os << numer().to_double();
515 if (type == csrc_types::ctype_cl_N)
516 os << "cl_F(\"" << -numer().evalf() << "\")";
518 os << -numer().to_double();
521 if (type == csrc_types::ctype_cl_N)
522 os << "cl_F(\"" << denom().evalf() << "\")";
524 os << denom().to_double();
527 if (type == csrc_types::ctype_cl_N)
528 os << "cl_F(\"" << evalf() << "\")";
536 bool numeric::info(unsigned inf) const
539 case info_flags::numeric:
540 case info_flags::polynomial:
541 case info_flags::rational_function:
543 case info_flags::real:
545 case info_flags::rational:
546 case info_flags::rational_polynomial:
547 return is_rational();
548 case info_flags::crational:
549 case info_flags::crational_polynomial:
550 return is_crational();
551 case info_flags::integer:
552 case info_flags::integer_polynomial:
554 case info_flags::cinteger:
555 case info_flags::cinteger_polynomial:
556 return is_cinteger();
557 case info_flags::positive:
558 return is_positive();
559 case info_flags::negative:
560 return is_negative();
561 case info_flags::nonnegative:
562 return !is_negative();
563 case info_flags::posint:
564 return is_pos_integer();
565 case info_flags::negint:
566 return is_integer() && is_negative();
567 case info_flags::nonnegint:
568 return is_nonneg_integer();
569 case info_flags::even:
571 case info_flags::odd:
573 case info_flags::prime:
575 case info_flags::algebraic:
581 /** Disassemble real part and imaginary part to scan for the occurrence of a
582 * single number. Also handles the imaginary unit. It ignores the sign on
583 * both this and the argument, which may lead to what might appear as funny
584 * results: (2+I).has(-2) -> true. But this is consistent, since we also
585 * would like to have (-2+I).has(2) -> true and we want to think about the
586 * sign as a multiplicative factor. */
587 bool numeric::has(const ex & other) const
589 if (!is_exactly_of_type(*other.bp, numeric))
591 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
592 if (this->is_equal(o) || this->is_equal(-o))
594 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
595 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
596 this->real().is_equal(-o) || this->imag().is_equal(-o));
598 if (o.is_equal(I)) // e.g scan for I in 42*I
599 return !this->is_real();
600 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
601 return (this->real().has(o*I) || this->imag().has(o*I) ||
602 this->real().has(-o*I) || this->imag().has(-o*I));
608 /** Evaluation of numbers doesn't do anything at all. */
609 ex numeric::eval(int level) const
611 // Warning: if this is ever gonna do something, the ex ctors from all kinds
612 // of numbers should be checking for status_flags::evaluated.
617 /** Cast numeric into a floating-point object. For example exact numeric(1) is
618 * returned as a 1.0000000000000000000000 and so on according to how Digits is
621 * @param level ignored, but needed for overriding basic::evalf.
622 * @return an ex-handle to a numeric. */
623 ex numeric::evalf(int level) const
625 // level can safely be discarded for numeric objects.
626 return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
631 /** Implementation of ex::diff() for a numeric. It always returns 0.
634 ex numeric::derivative(const symbol & s) const
640 int numeric::compare_same_type(const basic & other) const
642 GINAC_ASSERT(is_exactly_of_type(other, numeric));
643 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
645 if (*value == *o.value) {
653 bool numeric::is_equal_same_type(const basic & other) const
655 GINAC_ASSERT(is_exactly_of_type(other,numeric));
656 const numeric *o = static_cast<const numeric *>(&other);
658 return this->is_equal(*o);
662 unsigned numeric::calchash(void) const
664 // Use CLN's hashcode. Warning: It depends only on the number's value, not
665 // its type or precision (i.e. a true equivalence relation on numbers). As
666 // a consequence, 3 and 3.0 share the same hashvalue.
667 return (hashvalue = cl_equal_hashcode(*value) | 0x80000000U);
672 // new virtual functions which can be overridden by derived classes
678 // non-virtual functions in this class
683 /** Numerical addition method. Adds argument to *this and returns result as
684 * a new numeric object. */
685 numeric numeric::add(const numeric & other) const
687 return numeric((*value)+(*other.value));
690 /** Numerical subtraction method. Subtracts argument from *this and returns
691 * result as a new numeric object. */
692 numeric numeric::sub(const numeric & other) const
694 return numeric((*value)-(*other.value));
697 /** Numerical multiplication method. Multiplies *this and argument and returns
698 * result as a new numeric object. */
699 numeric numeric::mul(const numeric & other) const
701 static const numeric * _num1p=&_num1();
704 } else if (&other==_num1p) {
707 return numeric((*value)*(*other.value));
710 /** Numerical division method. Divides *this by argument and returns result as
711 * a new numeric object.
713 * @exception overflow_error (division by zero) */
714 numeric numeric::div(const numeric & other) const
716 if (::zerop(*other.value))
717 throw (std::overflow_error("division by zero"));
718 return numeric((*value)/(*other.value));
721 numeric numeric::power(const numeric & other) const
723 static const numeric * _num1p = &_num1();
726 if (::zerop(*value)) {
727 if (::zerop(*other.value))
728 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
729 else if (::zerop(::realpart(*other.value)))
730 throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
731 else if (::minusp(::realpart(*other.value)))
732 throw (std::overflow_error("numeric::eval(): division by zero"));
736 return numeric(::expt(*value,*other.value));
739 /** Inverse of a number. */
740 numeric numeric::inverse(void) const
742 return numeric(::recip(*value)); // -> CLN
745 const numeric & numeric::add_dyn(const numeric & other) const
747 return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
748 setflag(status_flags::dynallocated));
751 const numeric & numeric::sub_dyn(const numeric & other) const
753 return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
754 setflag(status_flags::dynallocated));
757 const numeric & numeric::mul_dyn(const numeric & other) const
759 static const numeric * _num1p=&_num1();
762 } else if (&other==_num1p) {
765 return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
766 setflag(status_flags::dynallocated));
769 const numeric & numeric::div_dyn(const numeric & other) const
771 if (::zerop(*other.value))
772 throw (std::overflow_error("division by zero"));
773 return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
774 setflag(status_flags::dynallocated));
777 const numeric & numeric::power_dyn(const numeric & other) const
779 static const numeric * _num1p=&_num1();
782 if (::zerop(*value)) {
783 if (::zerop(*other.value))
784 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
785 else if (::zerop(::realpart(*other.value)))
786 throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
787 else if (::minusp(::realpart(*other.value)))
788 throw (std::overflow_error("numeric::eval(): division by zero"));
792 return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
793 setflag(status_flags::dynallocated));
796 const numeric & numeric::operator=(int i)
798 return operator=(numeric(i));
801 const numeric & numeric::operator=(unsigned int i)
803 return operator=(numeric(i));
806 const numeric & numeric::operator=(long i)
808 return operator=(numeric(i));
811 const numeric & numeric::operator=(unsigned long i)
813 return operator=(numeric(i));
816 const numeric & numeric::operator=(double d)
818 return operator=(numeric(d));
821 const numeric & numeric::operator=(const char * s)
823 return operator=(numeric(s));
826 /** Return the complex half-plane (left or right) in which the number lies.
827 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
828 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
830 * @see numeric::compare(const numeric & other) */
831 int numeric::csgn(void) const
835 if (!::zerop(::realpart(*value))) {
836 if (::plusp(::realpart(*value)))
841 if (::plusp(::imagpart(*value)))
848 /** This method establishes a canonical order on all numbers. For complex
849 * numbers this is not possible in a mathematically consistent way but we need
850 * to establish some order and it ought to be fast. So we simply define it
851 * to be compatible with our method csgn.
853 * @return csgn(*this-other)
854 * @see numeric::csgn(void) */
855 int numeric::compare(const numeric & other) const
857 // Comparing two real numbers?
858 if (this->is_real() && other.is_real())
859 // Yes, just compare them
860 return ::cl_compare(The(::cl_R)(*value), The(::cl_R)(*other.value));
862 // No, first compare real parts
863 cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
867 return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
871 bool numeric::is_equal(const numeric & other) const
873 return (*value == *other.value);
876 /** True if object is zero. */
877 bool numeric::is_zero(void) const
879 return ::zerop(*value); // -> CLN
882 /** True if object is not complex and greater than zero. */
883 bool numeric::is_positive(void) const
886 return ::plusp(The(::cl_R)(*value)); // -> CLN
890 /** True if object is not complex and less than zero. */
891 bool numeric::is_negative(void) const
894 return ::minusp(The(::cl_R)(*value)); // -> CLN
898 /** True if object is a non-complex integer. */
899 bool numeric::is_integer(void) const
901 return ::instanceof(*value, ::cl_I_ring); // -> CLN
904 /** True if object is an exact integer greater than zero. */
905 bool numeric::is_pos_integer(void) const
907 return (this->is_integer() && ::plusp(The(::cl_I)(*value))); // -> CLN
910 /** True if object is an exact integer greater or equal zero. */
911 bool numeric::is_nonneg_integer(void) const
913 return (this->is_integer() && !::minusp(The(::cl_I)(*value))); // -> CLN
916 /** True if object is an exact even integer. */
917 bool numeric::is_even(void) const
919 return (this->is_integer() && ::evenp(The(::cl_I)(*value))); // -> CLN
922 /** True if object is an exact odd integer. */
923 bool numeric::is_odd(void) const
925 return (this->is_integer() && ::oddp(The(::cl_I)(*value))); // -> CLN
928 /** Probabilistic primality test.
930 * @return true if object is exact integer and prime. */
931 bool numeric::is_prime(void) const
933 return (this->is_integer() && ::isprobprime(The(::cl_I)(*value))); // -> CLN
936 /** True if object is an exact rational number, may even be complex
937 * (denominator may be unity). */
938 bool numeric::is_rational(void) const
940 return ::instanceof(*value, ::cl_RA_ring); // -> CLN
943 /** True if object is a real integer, rational or float (but not complex). */
944 bool numeric::is_real(void) const
946 return ::instanceof(*value, ::cl_R_ring); // -> CLN
949 bool numeric::operator==(const numeric & other) const
951 return (*value == *other.value); // -> CLN
954 bool numeric::operator!=(const numeric & other) const
956 return (*value != *other.value); // -> CLN
959 /** True if object is element of the domain of integers extended by I, i.e. is
960 * of the form a+b*I, where a and b are integers. */
961 bool numeric::is_cinteger(void) const
963 if (::instanceof(*value, ::cl_I_ring))
965 else if (!this->is_real()) { // complex case, handle n+m*I
966 if (::instanceof(::realpart(*value), ::cl_I_ring) &&
967 ::instanceof(::imagpart(*value), ::cl_I_ring))
973 /** True if object is an exact rational number, may even be complex
974 * (denominator may be unity). */
975 bool numeric::is_crational(void) const
977 if (::instanceof(*value, ::cl_RA_ring))
979 else if (!this->is_real()) { // complex case, handle Q(i):
980 if (::instanceof(::realpart(*value), ::cl_RA_ring) &&
981 ::instanceof(::imagpart(*value), ::cl_RA_ring))
987 /** Numerical comparison: less.
989 * @exception invalid_argument (complex inequality) */
990 bool numeric::operator<(const numeric & other) const
992 if (this->is_real() && other.is_real())
993 return (The(::cl_R)(*value) < The(::cl_R)(*other.value)); // -> CLN
994 throw (std::invalid_argument("numeric::operator<(): complex inequality"));
995 return false; // make compiler shut up
998 /** Numerical comparison: less or equal.
1000 * @exception invalid_argument (complex inequality) */
1001 bool numeric::operator<=(const numeric & other) const
1003 if (this->is_real() && other.is_real())
1004 return (The(::cl_R)(*value) <= The(::cl_R)(*other.value)); // -> CLN
1005 throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
1006 return false; // make compiler shut up
1009 /** Numerical comparison: greater.
1011 * @exception invalid_argument (complex inequality) */
1012 bool numeric::operator>(const numeric & other) const
1014 if (this->is_real() && other.is_real())
1015 return (The(::cl_R)(*value) > The(::cl_R)(*other.value)); // -> CLN
1016 throw (std::invalid_argument("numeric::operator>(): complex inequality"));
1017 return false; // make compiler shut up
1020 /** Numerical comparison: greater or equal.
1022 * @exception invalid_argument (complex inequality) */
1023 bool numeric::operator>=(const numeric & other) const
1025 if (this->is_real() && other.is_real())
1026 return (The(::cl_R)(*value) >= The(::cl_R)(*other.value)); // -> CLN
1027 throw (std::invalid_argument("numeric::operator>=(): complex inequality"));
1028 return false; // make compiler shut up
1031 /** Converts numeric types to machine's int. You should check with
1032 * is_integer() if the number is really an integer before calling this method.
1033 * You may also consider checking the range first. */
1034 int numeric::to_int(void) const
1036 GINAC_ASSERT(this->is_integer());
1037 return ::cl_I_to_int(The(::cl_I)(*value)); // -> CLN
1040 /** Converts numeric types to machine's long. You should check with
1041 * is_integer() if the number is really an integer before calling this method.
1042 * You may also consider checking the range first. */
1043 long numeric::to_long(void) const
1045 GINAC_ASSERT(this->is_integer());
1046 return ::cl_I_to_long(The(::cl_I)(*value)); // -> CLN
1049 /** Converts numeric types to machine's double. You should check with is_real()
1050 * if the number is really not complex before calling this method. */
1051 double numeric::to_double(void) const
1053 GINAC_ASSERT(this->is_real());
1054 return ::cl_double_approx(::realpart(*value)); // -> CLN
1057 /** Real part of a number. */
1058 const numeric numeric::real(void) const
1060 return numeric(::realpart(*value)); // -> CLN
1063 /** Imaginary part of a number. */
1064 const numeric numeric::imag(void) const
1066 return numeric(::imagpart(*value)); // -> CLN
1070 // Unfortunately, CLN did not provide an official way to access the numerator
1071 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
1072 // one finds how it works internally in src/rational/cl_RA.h:
1073 struct cl_heap_ratio : cl_heap {
1078 inline cl_heap_ratio* TheRatio (const cl_N& obj)
1079 { return (cl_heap_ratio*)(obj.pointer); }
1080 #endif // ndef SANE_LINKER
1082 /** Numerator. Computes the numerator of rational numbers, rationalized
1083 * numerator of complex if real and imaginary part are both rational numbers
1084 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1086 const numeric numeric::numer(void) const
1088 if (this->is_integer()) {
1089 return numeric(*this);
1092 else if (::instanceof(*value, ::cl_RA_ring)) {
1093 return numeric(::numerator(The(::cl_RA)(*value)));
1095 else if (!this->is_real()) { // complex case, handle Q(i):
1096 cl_R r = ::realpart(*value);
1097 cl_R i = ::imagpart(*value);
1098 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1099 return numeric(*this);
1100 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1101 return numeric(::complex(r*::denominator(The(::cl_RA)(i)), ::numerator(The(::cl_RA)(i))));
1102 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1103 return numeric(::complex(::numerator(The(::cl_RA)(r)), i*::denominator(The(::cl_RA)(r))));
1104 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring)) {
1105 cl_I s = ::lcm(::denominator(The(::cl_RA)(r)), ::denominator(The(::cl_RA)(i)));
1106 return numeric(::complex(::numerator(The(::cl_RA)(r))*(exquo(s,::denominator(The(::cl_RA)(r)))),
1107 ::numerator(The(::cl_RA)(i))*(exquo(s,::denominator(The(::cl_RA)(i))))));
1111 else if (instanceof(*value, ::cl_RA_ring)) {
1112 return numeric(TheRatio(*value)->numerator);
1114 else if (!this->is_real()) { // complex case, handle Q(i):
1115 cl_R r = ::realpart(*value);
1116 cl_R i = ::imagpart(*value);
1117 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1118 return numeric(*this);
1119 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1120 return numeric(::complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
1121 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1122 return numeric(::complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
1123 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring)) {
1124 cl_I s = ::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
1125 return numeric(::complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
1126 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
1129 #endif // def SANE_LINKER
1130 // at least one float encountered
1131 return numeric(*this);
1134 /** Denominator. Computes the denominator of rational numbers, common integer
1135 * denominator of complex if real and imaginary part are both rational numbers
1136 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1137 const numeric numeric::denom(void) const
1139 if (this->is_integer()) {
1143 if (instanceof(*value, ::cl_RA_ring)) {
1144 return numeric(::denominator(The(::cl_RA)(*value)));
1146 if (!this->is_real()) { // complex case, handle Q(i):
1147 cl_R r = ::realpart(*value);
1148 cl_R i = ::imagpart(*value);
1149 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1151 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1152 return numeric(::denominator(The(::cl_RA)(i)));
1153 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1154 return numeric(::denominator(The(::cl_RA)(r)));
1155 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring))
1156 return numeric(::lcm(::denominator(The(::cl_RA)(r)), ::denominator(The(::cl_RA)(i))));
1159 if (instanceof(*value, ::cl_RA_ring)) {
1160 return numeric(TheRatio(*value)->denominator);
1162 if (!this->is_real()) { // complex case, handle Q(i):
1163 cl_R r = ::realpart(*value);
1164 cl_R i = ::imagpart(*value);
1165 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1167 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1168 return numeric(TheRatio(i)->denominator);
1169 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1170 return numeric(TheRatio(r)->denominator);
1171 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring))
1172 return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
1174 #endif // def SANE_LINKER
1175 // at least one float encountered
1179 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1180 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1181 * 2^(n-1) <= x < 2^n.
1183 * @return number of bits (excluding sign) needed to represent that number
1184 * in two's complement if it is an integer, 0 otherwise. */
1185 int numeric::int_length(void) const
1187 if (this->is_integer())
1188 return ::integer_length(The(::cl_I)(*value)); // -> CLN
1195 // static member variables
1200 unsigned numeric::precedence = 30;
1206 const numeric some_numeric;
1207 const type_info & typeid_numeric=typeid(some_numeric);
1208 /** Imaginary unit. This is not a constant but a numeric since we are
1209 * natively handing complex numbers anyways. */
1210 const numeric I = numeric(::complex(cl_I(0),cl_I(1)));
1213 /** Exponential function.
1215 * @return arbitrary precision numerical exp(x). */
1216 const numeric exp(const numeric & x)
1218 return ::exp(*x.value); // -> CLN
1222 /** Natural logarithm.
1224 * @param z complex number
1225 * @return arbitrary precision numerical log(x).
1226 * @exception overflow_error (logarithmic singularity) */
1227 const numeric log(const numeric & z)
1230 throw (std::overflow_error("log(): logarithmic singularity"));
1231 return ::log(*z.value); // -> CLN
1235 /** Numeric sine (trigonometric function).
1237 * @return arbitrary precision numerical sin(x). */
1238 const numeric sin(const numeric & x)
1240 return ::sin(*x.value); // -> CLN
1244 /** Numeric cosine (trigonometric function).
1246 * @return arbitrary precision numerical cos(x). */
1247 const numeric cos(const numeric & x)
1249 return ::cos(*x.value); // -> CLN
1253 /** Numeric tangent (trigonometric function).
1255 * @return arbitrary precision numerical tan(x). */
1256 const numeric tan(const numeric & x)
1258 return ::tan(*x.value); // -> CLN
1262 /** Numeric inverse sine (trigonometric function).
1264 * @return arbitrary precision numerical asin(x). */
1265 const numeric asin(const numeric & x)
1267 return ::asin(*x.value); // -> CLN
1271 /** Numeric inverse cosine (trigonometric function).
1273 * @return arbitrary precision numerical acos(x). */
1274 const numeric acos(const numeric & x)
1276 return ::acos(*x.value); // -> CLN
1282 * @param z complex number
1284 * @exception overflow_error (logarithmic singularity) */
1285 const numeric atan(const numeric & x)
1288 x.real().is_zero() &&
1289 !abs(x.imag()).is_equal(_num1()))
1290 throw (std::overflow_error("atan(): logarithmic singularity"));
1291 return ::atan(*x.value); // -> CLN
1297 * @param x real number
1298 * @param y real number
1299 * @return atan(y/x) */
1300 const numeric atan(const numeric & y, const numeric & x)
1302 if (x.is_real() && y.is_real())
1303 return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
1305 throw (std::invalid_argument("numeric::atan(): complex argument"));
1309 /** Numeric hyperbolic sine (trigonometric function).
1311 * @return arbitrary precision numerical sinh(x). */
1312 const numeric sinh(const numeric & x)
1314 return ::sinh(*x.value); // -> CLN
1318 /** Numeric hyperbolic cosine (trigonometric function).
1320 * @return arbitrary precision numerical cosh(x). */
1321 const numeric cosh(const numeric & x)
1323 return ::cosh(*x.value); // -> CLN
1327 /** Numeric hyperbolic tangent (trigonometric function).
1329 * @return arbitrary precision numerical tanh(x). */
1330 const numeric tanh(const numeric & x)
1332 return ::tanh(*x.value); // -> CLN
1336 /** Numeric inverse hyperbolic sine (trigonometric function).
1338 * @return arbitrary precision numerical asinh(x). */
1339 const numeric asinh(const numeric & x)
1341 return ::asinh(*x.value); // -> CLN
1345 /** Numeric inverse hyperbolic cosine (trigonometric function).
1347 * @return arbitrary precision numerical acosh(x). */
1348 const numeric acosh(const numeric & x)
1350 return ::acosh(*x.value); // -> CLN
1354 /** Numeric inverse hyperbolic tangent (trigonometric function).
1356 * @return arbitrary precision numerical atanh(x). */
1357 const numeric atanh(const numeric & x)
1359 return ::atanh(*x.value); // -> CLN
1363 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1364 * integer arguments. */
1365 const numeric zeta(const numeric & x)
1367 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1368 // only knows about integer arguments and zeta(3).evalf() automatically
1369 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1370 // being an exact zero for CLN, which can be tested and then we can just
1371 // pass the number casted to an int:
1373 int aux = (int)(::cl_double_approx(::realpart(*x.value)));
1374 if (zerop(*x.value-aux))
1375 return ::cl_zeta(aux); // -> CLN
1377 clog << "zeta(" << x
1378 << "): Does anybody know good way to calculate this numerically?"
1384 /** The Gamma function.
1385 * This is only a stub! */
1386 const numeric lgamma(const numeric & x)
1388 clog << "lgamma(" << x
1389 << "): Does anybody know good way to calculate this numerically?"
1393 const numeric tgamma(const numeric & x)
1395 clog << "tgamma(" << x
1396 << "): Does anybody know good way to calculate this numerically?"
1402 /** The psi function (aka polygamma function).
1403 * This is only a stub! */
1404 const numeric psi(const numeric & x)
1407 << "): Does anybody know good way to calculate this numerically?"
1413 /** The psi functions (aka polygamma functions).
1414 * This is only a stub! */
1415 const numeric psi(const numeric & n, const numeric & x)
1417 clog << "psi(" << n << "," << x
1418 << "): Does anybody know good way to calculate this numerically?"
1424 /** Factorial combinatorial function.
1426 * @param n integer argument >= 0
1427 * @exception range_error (argument must be integer >= 0) */
1428 const numeric factorial(const numeric & n)
1430 if (!n.is_nonneg_integer())
1431 throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
1432 return numeric(::factorial(n.to_int())); // -> CLN
1436 /** The double factorial combinatorial function. (Scarcely used, but still
1437 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1439 * @param n integer argument >= -1
1440 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1441 * @exception range_error (argument must be integer >= -1) */
1442 const numeric doublefactorial(const numeric & n)
1444 if (n == numeric(-1)) {
1447 if (!n.is_nonneg_integer()) {
1448 throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
1450 return numeric(::doublefactorial(n.to_int())); // -> CLN
1454 /** The Binomial coefficients. It computes the binomial coefficients. For
1455 * integer n and k and positive n this is the number of ways of choosing k
1456 * objects from n distinct objects. If n is negative, the formula
1457 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1458 const numeric binomial(const numeric & n, const numeric & k)
1460 if (n.is_integer() && k.is_integer()) {
1461 if (n.is_nonneg_integer()) {
1462 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1463 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1467 return _num_1().power(k)*binomial(k-n-_num1(),k);
1471 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1472 throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
1476 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1477 * in the expansion of the function x/(e^x-1).
1479 * @return the nth Bernoulli number (a rational number).
1480 * @exception range_error (argument must be integer >= 0) */
1481 const numeric bernoulli(const numeric & nn)
1483 if (!nn.is_integer() || nn.is_negative())
1484 throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
1488 // The Bernoulli numbers are rational numbers that may be computed using
1491 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1493 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1494 // previous ones, the computation is necessarily very expensive. There are
1495 // several other ways of computing them, a particularly good one being
1499 // for (unsigned i=0; i<n; i++) {
1500 // c = exquo(c*(i-n),(i+2));
1501 // Bern = Bern + c*s/(i+2);
1502 // s = s + expt_pos(cl_I(i+2),n);
1506 // But if somebody works with the n'th Bernoulli number she is likely to
1507 // also need all previous Bernoulli numbers. So we need a complete remember
1508 // table and above divide and conquer algorithm is not suited to build one
1509 // up. The code below is adapted from Pari's function bernvec().
1511 // (There is an interesting relation with the tangent polynomials described
1512 // in `Concrete Mathematics', which leads to a program twice as fast as our
1513 // implementation below, but it requires storing one such polynomial in
1514 // addition to the remember table. This doubles the memory footprint so
1515 // we don't use it.)
1517 // the special cases not covered by the algorithm below
1518 if (!nn.compare(_num1()))
1519 return numeric(-1,2);
1523 // store nonvanishing Bernoulli numbers here
1524 static vector< ::cl_RA > results;
1525 static int highest_result = 0;
1526 // algorithm not applicable to B(0), so just store it
1527 if (results.size()==0)
1528 results.push_back(::cl_RA(1));
1530 int n = nn.to_long();
1531 for (int i=highest_result; i<n/2; ++i) {
1537 for (int j=i; j>0; --j) {
1538 B = cl_I(n*m) * (B+results[j]) / (d1*d2);
1544 B = (1 - ((B+1)/(2*i+3))) / (cl_I(1)<<(2*i+2));
1545 results.push_back(B);
1548 return results[n/2];
1552 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1553 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1555 * @param n an integer
1556 * @return the nth Fibonacci number F(n) (an integer number)
1557 * @exception range_error (argument must be an integer) */
1558 const numeric fibonacci(const numeric & n)
1560 if (!n.is_integer())
1561 throw (std::range_error("numeric::fibonacci(): argument must be integer"));
1564 // This is based on an implementation that can be found in CLN's example
1565 // directory. There, it is done recursively, which may be more elegant
1566 // than our non-recursive implementation that has to resort to some bit-
1567 // fiddling. This is, however, a matter of taste.
1568 // The following addition formula holds:
1570 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1572 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1573 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1575 // Replace m by m+1:
1576 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1577 // Now put in m = n, to get
1578 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1579 // F(2n+1) = F(n)^2 + F(n+1)^2
1581 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1584 if (n.is_negative())
1586 return -fibonacci(-n);
1588 return fibonacci(-n);
1592 ::cl_I m = The(::cl_I)(*n.value) >> 1L; // floor(n/2);
1593 for (uintL bit=::integer_length(m); bit>0; --bit) {
1594 // Since a squaring is cheaper than a multiplication, better use
1595 // three squarings instead of one multiplication and two squarings.
1596 ::cl_I u2 = ::square(u);
1597 ::cl_I v2 = ::square(v);
1598 if (::logbitp(bit-1, m)) {
1599 v = ::square(u + v) - u2;
1602 u = v2 - ::square(v - u);
1607 // Here we don't use the squaring formula because one multiplication
1608 // is cheaper than two squarings.
1609 return u * ((v << 1) - u);
1611 return ::square(u) + ::square(v);
1615 /** Absolute value. */
1616 numeric abs(const numeric & x)
1618 return ::abs(*x.value); // -> CLN
1622 /** Modulus (in positive representation).
1623 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1624 * sign of a or is zero. This is different from Maple's modp, where the sign
1625 * of b is ignored. It is in agreement with Mathematica's Mod.
1627 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1628 * integer, 0 otherwise. */
1629 numeric mod(const numeric & a, const numeric & b)
1631 if (a.is_integer() && b.is_integer())
1632 return ::mod(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1634 return _num0(); // Throw?
1638 /** Modulus (in symmetric representation).
1639 * Equivalent to Maple's mods.
1641 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1642 numeric smod(const numeric & a, const numeric & b)
1644 if (a.is_integer() && b.is_integer()) {
1645 cl_I b2 = The(::cl_I)(ceiling1(The(::cl_I)(*b.value) / 2)) - 1;
1646 return ::mod(The(::cl_I)(*a.value) + b2, The(::cl_I)(*b.value)) - b2;
1648 return _num0(); // Throw?
1652 /** Numeric integer remainder.
1653 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1654 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1655 * sign of a or is zero.
1657 * @return remainder of a/b if both are integer, 0 otherwise. */
1658 numeric irem(const numeric & a, const numeric & b)
1660 if (a.is_integer() && b.is_integer())
1661 return ::rem(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1663 return _num0(); // Throw?
1667 /** Numeric integer remainder.
1668 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1669 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1670 * and irem(a,b) has the sign of a or is zero.
1672 * @return remainder of a/b and quotient stored in q if both are integer,
1674 numeric irem(const numeric & a, const numeric & b, numeric & q)
1676 if (a.is_integer() && b.is_integer()) { // -> CLN
1677 cl_I_div_t rem_quo = truncate2(The(::cl_I)(*a.value), The(::cl_I)(*b.value));
1678 q = rem_quo.quotient;
1679 return rem_quo.remainder;
1683 return _num0(); // Throw?
1688 /** Numeric integer quotient.
1689 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1691 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1692 numeric iquo(const numeric & a, const numeric & b)
1694 if (a.is_integer() && b.is_integer())
1695 return truncate1(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1697 return _num0(); // Throw?
1701 /** Numeric integer quotient.
1702 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1703 * r == a - iquo(a,b,r)*b.
1705 * @return truncated quotient of a/b and remainder stored in r if both are
1706 * integer, 0 otherwise. */
1707 numeric iquo(const numeric & a, const numeric & b, numeric & r)
1709 if (a.is_integer() && b.is_integer()) { // -> CLN
1710 cl_I_div_t rem_quo = truncate2(The(::cl_I)(*a.value), The(::cl_I)(*b.value));
1711 r = rem_quo.remainder;
1712 return rem_quo.quotient;
1715 return _num0(); // Throw?
1720 /** Numeric square root.
1721 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1722 * should return integer 2.
1724 * @param z numeric argument
1725 * @return square root of z. Branch cut along negative real axis, the negative
1726 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1727 * where imag(z)>0. */
1728 numeric sqrt(const numeric & z)
1730 return ::sqrt(*z.value); // -> CLN
1734 /** Integer numeric square root. */
1735 numeric isqrt(const numeric & x)
1737 if (x.is_integer()) {
1739 ::isqrt(The(::cl_I)(*x.value), &root); // -> CLN
1742 return _num0(); // Throw?
1746 /** Greatest Common Divisor.
1748 * @return The GCD of two numbers if both are integer, a numerical 1
1749 * if they are not. */
1750 numeric gcd(const numeric & a, const numeric & b)
1752 if (a.is_integer() && b.is_integer())
1753 return ::gcd(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1759 /** Least Common Multiple.
1761 * @return The LCM of two numbers if both are integer, the product of those
1762 * two numbers if they are not. */
1763 numeric lcm(const numeric & a, const numeric & b)
1765 if (a.is_integer() && b.is_integer())
1766 return ::lcm(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN
1768 return *a.value * *b.value;
1772 /** Floating point evaluation of Archimedes' constant Pi. */
1775 return numeric(::cl_pi(cl_default_float_format)); // -> CLN
1779 /** Floating point evaluation of Euler's constant gamma. */
1782 return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
1786 /** Floating point evaluation of Catalan's constant. */
1787 ex CatalanEvalf(void)
1789 return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
1793 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1794 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1795 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1796 _numeric_digits::_numeric_digits()
1801 cl_default_float_format = ::cl_float_format(17);
1805 _numeric_digits& _numeric_digits::operator=(long prec)
1808 cl_default_float_format = ::cl_float_format(prec);
1813 _numeric_digits::operator long()
1815 return (long)digits;
1819 void _numeric_digits::print(ostream & os) const
1821 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1826 ostream& operator<<(ostream& os, const _numeric_digits & e)
1833 // static member variables
1838 bool _numeric_digits::too_late = false;
1841 /** Accuracy in decimal digits. Only object of this type! Can be set using
1842 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1843 _numeric_digits Digits;
1845 #ifndef NO_NAMESPACE_GINAC
1846 } // namespace GiNaC
1847 #endif // ndef NO_NAMESPACE_GINAC