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_integer_io.h>
52 #include <cln/cl_integer_ring.h>
53 #include <cln/cl_rational_io.h>
54 #include <cln/cl_rational_ring.h>
55 #include <cln/cl_lfloat_class.h>
56 #include <cln/cl_lfloat_io.h>
57 #include <cln/cl_real_io.h>
58 #include <cln/cl_real_ring.h>
59 #include <cln/cl_complex_io.h>
60 #include <cln/cl_complex_ring.h>
61 #include <cln/cl_numtheory.h>
62 #else // def HAVE_CLN_CLN_H
63 #include <cl_integer_io.h>
64 #include <cl_integer_ring.h>
65 #include <cl_rational_io.h>
66 #include <cl_rational_ring.h>
67 #include <cl_lfloat_class.h>
68 #include <cl_lfloat_io.h>
69 #include <cl_real_io.h>
70 #include <cl_real_ring.h>
71 #include <cl_complex_io.h>
72 #include <cl_complex_ring.h>
73 #include <cl_numtheory.h>
74 #endif // def HAVE_CLN_CLN_H
76 #ifndef NO_NAMESPACE_GINAC
78 #endif // ndef NO_NAMESPACE_GINAC
80 // linker has no problems finding text symbols for numerator or denominator
83 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
86 // default constructor, destructor, copy constructor assignment
87 // operator and helpers
92 /** default ctor. Numerically it initializes to an integer zero. */
93 numeric::numeric() : basic(TINFO_numeric)
95 debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
99 setflag(status_flags::evaluated|
100 status_flags::hash_calculated);
105 debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
109 numeric::numeric(const numeric & other)
111 debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
115 const numeric & numeric::operator=(const numeric & other)
117 debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
118 if (this != &other) {
127 void numeric::copy(const numeric & other)
130 value = new cl_N(*other.value);
133 void numeric::destroy(bool call_parent)
136 if (call_parent) basic::destroy(call_parent);
140 // other constructors
145 numeric::numeric(int i) : basic(TINFO_numeric)
147 debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
148 // Not the whole int-range is available if we don't cast to long
149 // first. This is due to the behaviour of the cl_I-ctor, which
150 // emphasizes efficiency:
151 value = new cl_I((long) i);
153 setflag(status_flags::evaluated|
154 status_flags::hash_calculated);
158 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
160 debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
161 // Not the whole uint-range is available if we don't cast to ulong
162 // first. This is due to the behaviour of the cl_I-ctor, which
163 // emphasizes efficiency:
164 value = new cl_I((unsigned long)i);
166 setflag(status_flags::evaluated|
167 status_flags::hash_calculated);
171 numeric::numeric(long i) : basic(TINFO_numeric)
173 debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
176 setflag(status_flags::evaluated|
177 status_flags::hash_calculated);
181 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
183 debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
186 setflag(status_flags::evaluated|
187 status_flags::hash_calculated);
190 /** Ctor for rational numerics a/b.
192 * @exception overflow_error (division by zero) */
193 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
195 debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
197 throw (std::overflow_error("division by zero"));
198 value = new cl_I(numer);
199 *value = *value / cl_I(denom);
201 setflag(status_flags::evaluated|
202 status_flags::hash_calculated);
206 numeric::numeric(double d) : basic(TINFO_numeric)
208 debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
209 // We really want to explicitly use the type cl_LF instead of the
210 // more general cl_F, since that would give us a cl_DF only which
211 // will not be promoted to cl_LF if overflow occurs:
213 *value = cl_float(d, cl_default_float_format);
215 setflag(status_flags::evaluated|
216 status_flags::hash_calculated);
220 numeric::numeric(const char *s) : basic(TINFO_numeric)
221 { // MISSING: treatment of complex and ints and rationals.
222 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
224 value = new cl_LF(s);
228 setflag(status_flags::evaluated|
229 status_flags::hash_calculated);
232 /** Ctor from CLN types. This is for the initiated user or internal use
234 numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
236 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
239 setflag(status_flags::evaluated|
240 status_flags::hash_calculated);
247 /** Construct object from archive_node. */
248 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
250 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
253 // Read number as string
255 if (n.find_string("number", str)) {
256 istringstream s(str);
257 cl_idecoded_float re, im;
261 case 'N': // Ordinary number
262 case 'R': // Integer-decoded real number
263 s >> re.sign >> re.mantissa >> re.exponent;
264 *value = re.sign * re.mantissa * expt(cl_float(2.0, cl_default_float_format), re.exponent);
266 case 'C': // Integer-decoded complex number
267 s >> re.sign >> re.mantissa >> re.exponent;
268 s >> im.sign >> im.mantissa >> im.exponent;
269 *value = complex(re.sign * re.mantissa * expt(cl_float(2.0, cl_default_float_format), re.exponent),
270 im.sign * im.mantissa * expt(cl_float(2.0, cl_default_float_format), im.exponent));
272 default: // Ordinary number
279 // Read number as string
281 if (n.find_string("number", str)) {
282 istrstream f(str.c_str(), str.size() + 1);
283 cl_idecoded_float re, im;
287 case 'R': // Integer-decoded real number
288 f >> re.sign >> re.mantissa >> re.exponent;
289 *value = re.sign * re.mantissa * expt(cl_float(2.0, cl_default_float_format), re.exponent);
291 case 'C': // Integer-decoded complex number
292 f >> re.sign >> re.mantissa >> re.exponent;
293 f >> im.sign >> im.mantissa >> im.exponent;
294 *value = complex(re.sign * re.mantissa * expt(cl_float(2.0, cl_default_float_format), re.exponent),
295 im.sign * im.mantissa * expt(cl_float(2.0, cl_default_float_format), im.exponent));
297 default: // Ordinary number
305 setflag(status_flags::evaluated|
306 status_flags::hash_calculated);
309 /** Unarchive the object. */
310 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
312 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
315 /** Archive the object. */
316 void numeric::archive(archive_node &n) const
318 inherited::archive(n);
320 // Write number as string
322 if (this->is_crational())
325 // Non-rational numbers are written in an integer-decoded format
326 // to preserve the precision
327 if (this->is_real()) {
328 cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
330 s << re.sign << " " << re.mantissa << " " << re.exponent;
332 cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
333 cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
335 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
336 s << im.sign << " " << im.mantissa << " " << im.exponent;
339 n.add_string("number", s.str());
341 // Write number as string
343 ostrstream f(buf, 1024);
344 if (this->is_crational())
347 // Non-rational numbers are written in an integer-decoded format
348 // to preserve the precision
349 if (this->is_real()) {
350 cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
352 f << re.sign << " " << re.mantissa << " " << re.exponent << ends;
354 cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
355 cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
357 f << re.sign << " " << re.mantissa << " " << re.exponent << " ";
358 f << im.sign << " " << im.mantissa << " " << im.exponent << ends;
362 n.add_string("number", str);
367 // functions overriding virtual functions from bases classes
372 basic * numeric::duplicate() const
374 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
375 return new numeric(*this);
378 void numeric::print(ostream & os, unsigned upper_precedence) const
380 // The method print adds to the output so it blends more consistently
381 // together with the other routines and produces something compatible to
383 debugmsg("numeric print", LOGLEVEL_PRINT);
384 if (this->is_real()) {
385 // case 1, real: x or -x
386 if ((precedence<=upper_precedence) && (!this->is_pos_integer())) {
387 os << "(" << *value << ")";
392 // case 2, imaginary: y*I or -y*I
393 if (::realpart(*value) == 0) {
394 if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
395 if (::imagpart(*value) == -1) {
398 os << "(" << ::imagpart(*value) << "*I)";
401 if (::imagpart(*value) == 1) {
404 if (::imagpart (*value) == -1) {
407 os << ::imagpart(*value) << "*I";
412 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
413 if (precedence <= upper_precedence) os << "(";
414 os << ::realpart(*value);
415 if (::imagpart(*value) < 0) {
416 if (::imagpart(*value) == -1) {
419 os << ::imagpart(*value) << "*I";
422 if (::imagpart(*value) == 1) {
425 os << "+" << ::imagpart(*value) << "*I";
428 if (precedence <= upper_precedence) os << ")";
434 void numeric::printraw(ostream & os) const
436 // The method printraw doesn't do much, it simply uses CLN's operator<<()
437 // for output, which is ugly but reliable. e.g: 2+2i
438 debugmsg("numeric printraw", LOGLEVEL_PRINT);
439 os << "numeric(" << *value << ")";
443 void numeric::printtree(ostream & os, unsigned indent) const
445 debugmsg("numeric printtree", LOGLEVEL_PRINT);
446 os << string(indent,' ') << *value
448 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
449 << ", flags=" << flags << endl;
453 void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
455 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
456 ios::fmtflags oldflags = os.flags();
457 os.setf(ios::scientific);
458 if (this->is_rational() && !this->is_integer()) {
459 if (compare(_num0()) > 0) {
461 if (type == csrc_types::ctype_cl_N)
462 os << "cl_F(\"" << numer().evalf() << "\")";
464 os << numer().to_double();
467 if (type == csrc_types::ctype_cl_N)
468 os << "cl_F(\"" << -numer().evalf() << "\")";
470 os << -numer().to_double();
473 if (type == csrc_types::ctype_cl_N)
474 os << "cl_F(\"" << denom().evalf() << "\")";
476 os << denom().to_double();
479 if (type == csrc_types::ctype_cl_N)
480 os << "cl_F(\"" << evalf() << "\")";
488 bool numeric::info(unsigned inf) const
491 case info_flags::numeric:
492 case info_flags::polynomial:
493 case info_flags::rational_function:
495 case info_flags::real:
497 case info_flags::rational:
498 case info_flags::rational_polynomial:
499 return is_rational();
500 case info_flags::crational:
501 case info_flags::crational_polynomial:
502 return is_crational();
503 case info_flags::integer:
504 case info_flags::integer_polynomial:
506 case info_flags::cinteger:
507 case info_flags::cinteger_polynomial:
508 return is_cinteger();
509 case info_flags::positive:
510 return is_positive();
511 case info_flags::negative:
512 return is_negative();
513 case info_flags::nonnegative:
514 return compare(_num0())>=0;
515 case info_flags::posint:
516 return is_pos_integer();
517 case info_flags::negint:
518 return is_integer() && (compare(_num0())<0);
519 case info_flags::nonnegint:
520 return is_nonneg_integer();
521 case info_flags::even:
523 case info_flags::odd:
525 case info_flags::prime:
531 /** Disassemble real part and imaginary part to scan for the occurrence of a
532 * single number. Also handles the imaginary unit. */
533 bool numeric::has(const ex & other) const
535 if (!is_exactly_of_type(*other.bp, numeric))
537 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
538 if (this->is_equal(o))
540 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
541 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
542 this->real().is_equal(-o) || this->imag().is_equal(-o));
544 if (o.is_equal(I)) // e.g scan for I in 42*I
545 return !this->is_real();
546 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
547 return (this->real().has(o*I) || this->imag().has(o*I) ||
548 this->real().has(-o*I) || this->imag().has(-o*I));
554 /** Evaluation of numbers doesn't do anything. */
555 ex numeric::eval(int level) const
557 // Warning: if this is ever gonna do something, the ex ctors from all kinds
558 // of numbers should be checking for status_flags::evaluated.
563 /** Cast numeric into a floating-point object. For example exact numeric(1) is
564 * returned as a 1.0000000000000000000000 and so on according to how Digits is
567 * @param level ignored, but needed for overriding basic::evalf.
568 * @return an ex-handle to a numeric. */
569 ex numeric::evalf(int level) const
571 // level can safely be discarded for numeric objects.
572 return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
577 /** Implementation of ex::diff() for a numeric. It always returns 0.
580 ex numeric::derivative(const symbol & s) const
586 int numeric::compare_same_type(const basic & other) const
588 GINAC_ASSERT(is_exactly_of_type(other, numeric));
589 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
591 if (*value == *o.value) {
599 bool numeric::is_equal_same_type(const basic & other) const
601 GINAC_ASSERT(is_exactly_of_type(other,numeric));
602 const numeric *o = static_cast<const numeric *>(&other);
604 return this->is_equal(*o);
608 unsigned numeric::calchash(void) const
610 double d=to_double();
616 return 0x88000000U+s*unsigned(d/0x07FF0000);
622 // new virtual functions which can be overridden by derived classes
628 // non-virtual functions in this class
633 /** Numerical addition method. Adds argument to *this and returns result as
634 * a new numeric object. */
635 numeric numeric::add(const numeric & other) const
637 return numeric((*value)+(*other.value));
640 /** Numerical subtraction method. Subtracts argument from *this and returns
641 * result as a new numeric object. */
642 numeric numeric::sub(const numeric & other) const
644 return numeric((*value)-(*other.value));
647 /** Numerical multiplication method. Multiplies *this and argument and returns
648 * result as a new numeric object. */
649 numeric numeric::mul(const numeric & other) const
651 static const numeric * _num1p=&_num1();
654 } else if (&other==_num1p) {
657 return numeric((*value)*(*other.value));
660 /** Numerical division method. Divides *this by argument and returns result as
661 * a new numeric object.
663 * @exception overflow_error (division by zero) */
664 numeric numeric::div(const numeric & other) const
666 if (::zerop(*other.value))
667 throw (std::overflow_error("division by zero"));
668 return numeric((*value)/(*other.value));
671 numeric numeric::power(const numeric & other) const
673 static const numeric * _num1p=&_num1();
676 if (::zerop(*value)) {
677 if (::zerop(*other.value))
678 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
679 else if (other.is_real() && !::plusp(::realpart(*other.value)))
680 throw (std::overflow_error("numeric::eval(): division by zero"));
684 return numeric(::expt(*value,*other.value));
687 /** Inverse of a number. */
688 numeric numeric::inverse(void) const
690 return numeric(::recip(*value)); // -> CLN
693 const numeric & numeric::add_dyn(const numeric & other) const
695 return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
696 setflag(status_flags::dynallocated));
699 const numeric & numeric::sub_dyn(const numeric & other) const
701 return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
702 setflag(status_flags::dynallocated));
705 const numeric & numeric::mul_dyn(const numeric & other) const
707 static const numeric * _num1p=&_num1();
710 } else if (&other==_num1p) {
713 return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
714 setflag(status_flags::dynallocated));
717 const numeric & numeric::div_dyn(const numeric & other) const
719 if (::zerop(*other.value))
720 throw (std::overflow_error("division by zero"));
721 return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
722 setflag(status_flags::dynallocated));
725 const numeric & numeric::power_dyn(const numeric & other) const
727 static const numeric * _num1p=&_num1();
730 if (::zerop(*value)) {
731 if (::zerop(*other.value))
732 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
733 else if (other.is_real() && !::plusp(::realpart(*other.value)))
734 throw (std::overflow_error("numeric::eval(): division by zero"));
738 return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
739 setflag(status_flags::dynallocated));
742 const numeric & numeric::operator=(int i)
744 return operator=(numeric(i));
747 const numeric & numeric::operator=(unsigned int i)
749 return operator=(numeric(i));
752 const numeric & numeric::operator=(long i)
754 return operator=(numeric(i));
757 const numeric & numeric::operator=(unsigned long i)
759 return operator=(numeric(i));
762 const numeric & numeric::operator=(double d)
764 return operator=(numeric(d));
767 const numeric & numeric::operator=(const char * s)
769 return operator=(numeric(s));
772 /** Return the complex half-plane (left or right) in which the number lies.
773 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
774 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
776 * @see numeric::compare(const numeric & other) */
777 int numeric::csgn(void) const
781 if (!::zerop(::realpart(*value))) {
782 if (::plusp(::realpart(*value)))
787 if (::plusp(::imagpart(*value)))
794 /** This method establishes a canonical order on all numbers. For complex
795 * numbers this is not possible in a mathematically consistent way but we need
796 * to establish some order and it ought to be fast. So we simply define it
797 * to be compatible with our method csgn.
799 * @return csgn(*this-other)
800 * @see numeric::csgn(void) */
801 int numeric::compare(const numeric & other) const
803 // Comparing two real numbers?
804 if (this->is_real() && other.is_real())
805 // Yes, just compare them
806 return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
808 // No, first compare real parts
809 cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
813 return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
817 bool numeric::is_equal(const numeric & other) const
819 return (*value == *other.value);
822 /** True if object is zero. */
823 bool numeric::is_zero(void) const
825 return ::zerop(*value); // -> CLN
828 /** True if object is not complex and greater than zero. */
829 bool numeric::is_positive(void) const
832 return ::plusp(The(cl_R)(*value)); // -> CLN
836 /** True if object is not complex and less than zero. */
837 bool numeric::is_negative(void) const
840 return ::minusp(The(cl_R)(*value)); // -> CLN
844 /** True if object is a non-complex integer. */
845 bool numeric::is_integer(void) const
847 return ::instanceof(*value, cl_I_ring); // -> CLN
850 /** True if object is an exact integer greater than zero. */
851 bool numeric::is_pos_integer(void) const
853 return (this->is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN
856 /** True if object is an exact integer greater or equal zero. */
857 bool numeric::is_nonneg_integer(void) const
859 return (this->is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN
862 /** True if object is an exact even integer. */
863 bool numeric::is_even(void) const
865 return (this->is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN
868 /** True if object is an exact odd integer. */
869 bool numeric::is_odd(void) const
871 return (this->is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN
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() && ::isprobprime(The(cl_I)(*value))); // -> CLN
882 /** True if object is an exact rational number, may even be complex
883 * (denominator may be unity). */
884 bool numeric::is_rational(void) const
886 return ::instanceof(*value, cl_RA_ring); // -> CLN
889 /** True if object is a real integer, rational or float (but not complex). */
890 bool numeric::is_real(void) const
892 return ::instanceof(*value, cl_R_ring); // -> CLN
895 bool numeric::operator==(const numeric & other) const
897 return (*value == *other.value); // -> CLN
900 bool numeric::operator!=(const numeric & other) const
902 return (*value != *other.value); // -> CLN
905 /** True if object is element of the domain of integers extended by I, i.e. is
906 * of the form a+b*I, where a and b are integers. */
907 bool numeric::is_cinteger(void) const
909 if (::instanceof(*value, cl_I_ring))
911 else if (!this->is_real()) { // complex case, handle n+m*I
912 if (::instanceof(::realpart(*value), cl_I_ring) &&
913 ::instanceof(::imagpart(*value), cl_I_ring))
919 /** True if object is an exact rational number, may even be complex
920 * (denominator may be unity). */
921 bool numeric::is_crational(void) const
923 if (::instanceof(*value, cl_RA_ring))
925 else if (!this->is_real()) { // complex case, handle Q(i):
926 if (::instanceof(::realpart(*value), cl_RA_ring) &&
927 ::instanceof(::imagpart(*value), cl_RA_ring))
933 /** Numerical comparison: less.
935 * @exception invalid_argument (complex inequality) */
936 bool numeric::operator<(const numeric & other) const
938 if (this->is_real() && other.is_real())
939 return (bool)(The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
940 throw (std::invalid_argument("numeric::operator<(): complex inequality"));
941 return false; // make compiler shut up
944 /** Numerical comparison: less or equal.
946 * @exception invalid_argument (complex inequality) */
947 bool numeric::operator<=(const numeric & other) const
949 if (this->is_real() && other.is_real())
950 return (bool)(The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
951 throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
952 return false; // make compiler shut up
955 /** Numerical comparison: greater.
957 * @exception invalid_argument (complex inequality) */
958 bool numeric::operator>(const numeric & other) const
960 if (this->is_real() && other.is_real())
961 return (bool)(The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
962 throw (std::invalid_argument("numeric::operator>(): complex inequality"));
963 return false; // make compiler shut up
966 /** Numerical comparison: greater or equal.
968 * @exception invalid_argument (complex inequality) */
969 bool numeric::operator>=(const numeric & other) const
971 if (this->is_real() && other.is_real())
972 return (bool)(The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
973 throw (std::invalid_argument("numeric::operator>=(): complex inequality"));
974 return false; // make compiler shut up
977 /** Converts numeric types to machine's int. You should check with
978 * is_integer() if the number is really an integer before calling this method.
979 * You may also consider checking the range first. */
980 int numeric::to_int(void) const
982 GINAC_ASSERT(this->is_integer());
983 return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN
986 /** Converts numeric types to machine's long. You should check with
987 * is_integer() if the number is really an integer before calling this method.
988 * You may also consider checking the range first. */
989 long numeric::to_long(void) const
991 GINAC_ASSERT(this->is_integer());
992 return ::cl_I_to_long(The(cl_I)(*value)); // -> CLN
995 /** Converts numeric types to machine's double. You should check with is_real()
996 * if the number is really not complex before calling this method. */
997 double numeric::to_double(void) const
999 GINAC_ASSERT(this->is_real());
1000 return ::cl_double_approx(::realpart(*value)); // -> CLN
1003 /** Real part of a number. */
1004 numeric numeric::real(void) const
1006 return numeric(::realpart(*value)); // -> CLN
1009 /** Imaginary part of a number. */
1010 numeric numeric::imag(void) const
1012 return numeric(::imagpart(*value)); // -> CLN
1016 // Unfortunately, CLN did not provide an official way to access the numerator
1017 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
1018 // one finds how it works internally in src/rational/cl_RA.h:
1019 struct cl_heap_ratio : cl_heap {
1024 inline cl_heap_ratio* TheRatio (const cl_N& obj)
1025 { return (cl_heap_ratio*)(obj.pointer); }
1026 #endif // ndef SANE_LINKER
1028 /** Numerator. Computes the numerator of rational numbers, rationalized
1029 * numerator of complex if real and imaginary part are both rational numbers
1030 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1032 numeric numeric::numer(void) const
1034 if (this->is_integer()) {
1035 return numeric(*this);
1038 else if (::instanceof(*value, cl_RA_ring)) {
1039 return numeric(::numerator(The(cl_RA)(*value)));
1041 else if (!this->is_real()) { // complex case, handle Q(i):
1042 cl_R r = ::realpart(*value);
1043 cl_R i = ::imagpart(*value);
1044 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
1045 return numeric(*this);
1046 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
1047 return numeric(complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i))));
1048 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
1049 return numeric(complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r))));
1050 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring)) {
1051 cl_I s = lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)));
1052 return numeric(complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))),
1053 ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i))))));
1057 else if (instanceof(*value, cl_RA_ring)) {
1058 return numeric(TheRatio(*value)->numerator);
1060 else if (!this->is_real()) { // complex case, handle Q(i):
1061 cl_R r = ::realpart(*value);
1062 cl_R i = ::imagpart(*value);
1063 if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
1064 return numeric(*this);
1065 if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
1066 return numeric(complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
1067 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
1068 return numeric(complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
1069 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) {
1070 cl_I s = lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
1071 return numeric(complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
1072 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
1075 #endif // def SANE_LINKER
1076 // at least one float encountered
1077 return numeric(*this);
1080 /** Denominator. Computes the denominator of rational numbers, common integer
1081 * denominator of complex if real and imaginary part are both rational numbers
1082 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1083 numeric numeric::denom(void) const
1085 if (this->is_integer()) {
1089 if (instanceof(*value, cl_RA_ring)) {
1090 return numeric(::denominator(The(cl_RA)(*value)));
1092 if (!this->is_real()) { // complex case, handle Q(i):
1093 cl_R r = ::realpart(*value);
1094 cl_R i = ::imagpart(*value);
1095 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
1097 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
1098 return numeric(::denominator(The(cl_RA)(i)));
1099 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
1100 return numeric(::denominator(The(cl_RA)(r)));
1101 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring))
1102 return numeric(lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
1105 if (instanceof(*value, cl_RA_ring)) {
1106 return numeric(TheRatio(*value)->denominator);
1108 if (!this->is_real()) { // complex case, handle Q(i):
1109 cl_R r = ::realpart(*value);
1110 cl_R i = ::imagpart(*value);
1111 if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
1113 if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
1114 return numeric(TheRatio(i)->denominator);
1115 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
1116 return numeric(TheRatio(r)->denominator);
1117 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring))
1118 return numeric(lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
1120 #endif // def SANE_LINKER
1121 // at least one float encountered
1125 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1126 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1127 * 2^(n-1) <= x < 2^n.
1129 * @return number of bits (excluding sign) needed to represent that number
1130 * in two's complement if it is an integer, 0 otherwise. */
1131 int numeric::int_length(void) const
1133 if (this->is_integer())
1134 return ::integer_length(The(cl_I)(*value)); // -> CLN
1141 // static member variables
1146 unsigned numeric::precedence = 30;
1152 const numeric some_numeric;
1153 const type_info & typeid_numeric=typeid(some_numeric);
1154 /** Imaginary unit. This is not a constant but a numeric since we are
1155 * natively handing complex numbers anyways. */
1156 const numeric I = numeric(complex(cl_I(0),cl_I(1)));
1159 /** Exponential function.
1161 * @return arbitrary precision numerical exp(x). */
1162 const numeric exp(const numeric & x)
1164 return ::exp(*x.value); // -> CLN
1168 /** Natural logarithm.
1170 * @param z complex number
1171 * @return arbitrary precision numerical log(x).
1172 * @exception overflow_error (logarithmic singularity) */
1173 const numeric log(const numeric & z)
1176 throw (std::overflow_error("log(): logarithmic singularity"));
1177 return ::log(*z.value); // -> CLN
1181 /** Numeric sine (trigonometric function).
1183 * @return arbitrary precision numerical sin(x). */
1184 const numeric sin(const numeric & x)
1186 return ::sin(*x.value); // -> CLN
1190 /** Numeric cosine (trigonometric function).
1192 * @return arbitrary precision numerical cos(x). */
1193 const numeric cos(const numeric & x)
1195 return ::cos(*x.value); // -> CLN
1199 /** Numeric tangent (trigonometric function).
1201 * @return arbitrary precision numerical tan(x). */
1202 const numeric tan(const numeric & x)
1204 return ::tan(*x.value); // -> CLN
1208 /** Numeric inverse sine (trigonometric function).
1210 * @return arbitrary precision numerical asin(x). */
1211 const numeric asin(const numeric & x)
1213 return ::asin(*x.value); // -> CLN
1217 /** Numeric inverse cosine (trigonometric function).
1219 * @return arbitrary precision numerical acos(x). */
1220 const numeric acos(const numeric & x)
1222 return ::acos(*x.value); // -> CLN
1228 * @param z complex number
1230 * @exception overflow_error (logarithmic singularity) */
1231 const numeric atan(const numeric & x)
1234 x.real().is_zero() &&
1235 !abs(x.imag()).is_equal(_num1()))
1236 throw (std::overflow_error("atan(): logarithmic singularity"));
1237 return ::atan(*x.value); // -> CLN
1243 * @param x real number
1244 * @param y real number
1245 * @return atan(y/x) */
1246 const numeric atan(const numeric & y, const numeric & x)
1248 if (x.is_real() && y.is_real())
1249 return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
1251 throw (std::invalid_argument("numeric::atan(): complex argument"));
1255 /** Numeric hyperbolic sine (trigonometric function).
1257 * @return arbitrary precision numerical sinh(x). */
1258 const numeric sinh(const numeric & x)
1260 return ::sinh(*x.value); // -> CLN
1264 /** Numeric hyperbolic cosine (trigonometric function).
1266 * @return arbitrary precision numerical cosh(x). */
1267 const numeric cosh(const numeric & x)
1269 return ::cosh(*x.value); // -> CLN
1273 /** Numeric hyperbolic tangent (trigonometric function).
1275 * @return arbitrary precision numerical tanh(x). */
1276 const numeric tanh(const numeric & x)
1278 return ::tanh(*x.value); // -> CLN
1282 /** Numeric inverse hyperbolic sine (trigonometric function).
1284 * @return arbitrary precision numerical asinh(x). */
1285 const numeric asinh(const numeric & x)
1287 return ::asinh(*x.value); // -> CLN
1291 /** Numeric inverse hyperbolic cosine (trigonometric function).
1293 * @return arbitrary precision numerical acosh(x). */
1294 const numeric acosh(const numeric & x)
1296 return ::acosh(*x.value); // -> CLN
1300 /** Numeric inverse hyperbolic tangent (trigonometric function).
1302 * @return arbitrary precision numerical atanh(x). */
1303 const numeric atanh(const numeric & x)
1305 return ::atanh(*x.value); // -> CLN
1309 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1310 * integer arguments. */
1311 const numeric zeta(const numeric & x)
1313 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1314 // only knows about integer arguments and zeta(3).evalf() automatically
1315 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1316 // being an exact zero for CLN, which can be tested and then we can just
1317 // pass the number casted to an int:
1319 int aux = (int)(::cl_double_approx(::realpart(*x.value)));
1320 if (zerop(*x.value-aux))
1321 return ::cl_zeta(aux); // -> CLN
1323 clog << "zeta(" << x
1324 << "): Does anybody know good way to calculate this numerically?"
1330 /** The gamma function.
1331 * This is only a stub! */
1332 const numeric gamma(const numeric & x)
1334 clog << "gamma(" << x
1335 << "): Does anybody know good way to calculate this numerically?"
1341 /** The psi function (aka polygamma function).
1342 * This is only a stub! */
1343 const numeric psi(const numeric & x)
1346 << "): Does anybody know good way to calculate this numerically?"
1352 /** The psi functions (aka polygamma functions).
1353 * This is only a stub! */
1354 const numeric psi(const numeric & n, const numeric & x)
1356 clog << "psi(" << n << "," << x
1357 << "): Does anybody know good way to calculate this numerically?"
1363 /** Factorial combinatorial function.
1365 * @param n integer argument >= 0
1366 * @exception range_error (argument must be integer >= 0) */
1367 const numeric factorial(const numeric & n)
1369 if (!n.is_nonneg_integer())
1370 throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
1371 return numeric(::factorial(n.to_int())); // -> CLN
1375 /** The double factorial combinatorial function. (Scarcely used, but still
1376 * useful in cases, like for exact results of Gamma(n+1/2) for instance.)
1378 * @param n integer argument >= -1
1379 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1380 * @exception range_error (argument must be integer >= -1) */
1381 const numeric doublefactorial(const numeric & n)
1383 if (n == numeric(-1)) {
1386 if (!n.is_nonneg_integer()) {
1387 throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
1389 return numeric(::doublefactorial(n.to_int())); // -> CLN
1393 /** The Binomial coefficients. It computes the binomial coefficients. For
1394 * integer n and k and positive n this is the number of ways of choosing k
1395 * objects from n distinct objects. If n is negative, the formula
1396 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1397 const numeric binomial(const numeric & n, const numeric & k)
1399 if (n.is_integer() && k.is_integer()) {
1400 if (n.is_nonneg_integer()) {
1401 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1402 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1406 return _num_1().power(k)*binomial(k-n-_num1(),k);
1410 // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit
1411 throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
1415 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1416 * in the expansion of the function x/(e^x-1).
1418 * @return the nth Bernoulli number (a rational number).
1419 * @exception range_error (argument must be integer >= 0) */
1420 const numeric bernoulli(const numeric & nn)
1422 if (!nn.is_integer() || nn.is_negative())
1423 throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
1426 if (!nn.compare(_num1()))
1427 return numeric(-1,2);
1430 // Until somebody has the Blues and comes up with a much better idea and
1431 // codes it (preferably in CLN) we make this a remembering function which
1432 // computes its results using the formula
1433 // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
1435 static vector<numeric> results;
1436 static int highest_result = -1;
1437 int n = nn.sub(_num2()).div(_num2()).to_int();
1438 if (n <= highest_result)
1440 if (results.capacity() < (unsigned)(n+1))
1441 results.reserve(n+1);
1443 numeric tmp; // used to store the sum
1444 for (int i=highest_result+1; i<=n; ++i) {
1445 // the first two elements:
1446 tmp = numeric(-2*i-1,2);
1447 // accumulate the remaining elements:
1448 for (int j=0; j<i; ++j)
1449 tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
1450 // divide by -(nn+1) and store result:
1451 results.push_back(-tmp/numeric(2*i+3));
1458 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1459 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1461 * @param n an integer
1462 * @return the nth Fibonacci number F(n) (an integer number)
1463 * @exception range_error (argument must be an integer) */
1464 const numeric fibonacci(const numeric & n)
1466 if (!n.is_integer()) {
1467 throw (std::range_error("numeric::fibonacci(): argument must be integer"));
1469 // For positive arguments compute the nearest integer to
1470 // ((1+sqrt(5))/2)^n/sqrt(5). For negative arguments, apply an additional
1471 // sign. Note that we are falling back to longs, but this should suffice
1474 const long nn = ::abs(n.to_double());
1475 if (n.is_negative() && n.is_even())
1478 // Need a precision of ((1+sqrt(5))/2)^-n.
1479 cl_float_format_t prec = ::cl_float_format((int)(0.208987641*nn+5));
1480 cl_R sqrt5 = ::sqrt(::cl_float(5,prec));
1481 cl_R phi = (1+sqrt5)/2;
1482 return numeric(::round1(::expt(phi,nn)/sqrt5)*sig);
1486 /** Absolute value. */
1487 numeric abs(const numeric & x)
1489 return ::abs(*x.value); // -> CLN
1493 /** Modulus (in positive representation).
1494 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1495 * sign of a or is zero. This is different from Maple's modp, where the sign
1496 * of b is ignored. It is in agreement with Mathematica's Mod.
1498 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1499 * integer, 0 otherwise. */
1500 numeric mod(const numeric & a, const numeric & b)
1502 if (a.is_integer() && b.is_integer())
1503 return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1505 return _num0(); // Throw?
1509 /** Modulus (in symmetric representation).
1510 * Equivalent to Maple's mods.
1512 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1513 numeric smod(const numeric & a, const numeric & b)
1515 if (a.is_integer() && b.is_integer()) {
1516 cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1;
1517 return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
1519 return _num0(); // Throw?
1523 /** Numeric integer remainder.
1524 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1525 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1526 * sign of a or is zero.
1528 * @return remainder of a/b if both are integer, 0 otherwise. */
1529 numeric irem(const numeric & a, const numeric & b)
1531 if (a.is_integer() && b.is_integer())
1532 return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1534 return _num0(); // Throw?
1538 /** Numeric integer remainder.
1539 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1540 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1541 * and irem(a,b) has the sign of a or is zero.
1543 * @return remainder of a/b and quotient stored in q if both are integer,
1545 numeric irem(const numeric & a, const numeric & b, numeric & q)
1547 if (a.is_integer() && b.is_integer()) { // -> CLN
1548 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1549 q = rem_quo.quotient;
1550 return rem_quo.remainder;
1554 return _num0(); // Throw?
1559 /** Numeric integer quotient.
1560 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1562 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1563 numeric iquo(const numeric & a, const numeric & b)
1565 if (a.is_integer() && b.is_integer())
1566 return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1568 return _num0(); // Throw?
1572 /** Numeric integer quotient.
1573 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1574 * r == a - iquo(a,b,r)*b.
1576 * @return truncated quotient of a/b and remainder stored in r if both are
1577 * integer, 0 otherwise. */
1578 numeric iquo(const numeric & a, const numeric & b, numeric & r)
1580 if (a.is_integer() && b.is_integer()) { // -> CLN
1581 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1582 r = rem_quo.remainder;
1583 return rem_quo.quotient;
1586 return _num0(); // Throw?
1591 /** Numeric square root.
1592 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1593 * should return integer 2.
1595 * @param z numeric argument
1596 * @return square root of z. Branch cut along negative real axis, the negative
1597 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1598 * where imag(z)>0. */
1599 numeric sqrt(const numeric & z)
1601 return ::sqrt(*z.value); // -> CLN
1605 /** Integer numeric square root. */
1606 numeric isqrt(const numeric & x)
1608 if (x.is_integer()) {
1610 ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
1613 return _num0(); // Throw?
1617 /** Greatest Common Divisor.
1619 * @return The GCD of two numbers if both are integer, a numerical 1
1620 * if they are not. */
1621 numeric gcd(const numeric & a, const numeric & b)
1623 if (a.is_integer() && b.is_integer())
1624 return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1630 /** Least Common Multiple.
1632 * @return The LCM of two numbers if both are integer, the product of those
1633 * two numbers if they are not. */
1634 numeric lcm(const numeric & a, const numeric & b)
1636 if (a.is_integer() && b.is_integer())
1637 return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1639 return *a.value * *b.value;
1643 /** Floating point evaluation of Archimedes' constant Pi. */
1646 return numeric(::cl_pi(cl_default_float_format)); // -> CLN
1650 /** Floating point evaluation of Euler's constant Gamma. */
1651 ex EulerGammaEvalf(void)
1653 return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
1657 /** Floating point evaluation of Catalan's constant. */
1658 ex CatalanEvalf(void)
1660 return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
1664 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1665 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1666 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1667 _numeric_digits::_numeric_digits()
1672 cl_default_float_format = ::cl_float_format(17);
1676 _numeric_digits& _numeric_digits::operator=(long prec)
1679 cl_default_float_format = ::cl_float_format(prec);
1684 _numeric_digits::operator long()
1686 return (long)digits;
1690 void _numeric_digits::print(ostream & os) const
1692 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1697 ostream& operator<<(ostream& os, const _numeric_digits & e)
1704 // static member variables
1709 bool _numeric_digits::too_late = false;
1712 /** Accuracy in decimal digits. Only object of this type! Can be set using
1713 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1714 _numeric_digits Digits;
1716 #ifndef NO_NAMESPACE_GINAC
1717 } // namespace GiNaC
1718 #endif // ndef NO_NAMESPACE_GINAC