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);
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);
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 numeric::numeric(const char *s) : basic(TINFO_numeric)
224 { // MISSING: treatment of complex numbers
225 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
227 value = new cl_LF(s);
231 setflag(status_flags::evaluated|
232 status_flags::hash_calculated);
235 /** Ctor from CLN types. This is for the initiated user or internal use
237 numeric::numeric(const cl_N & z) : basic(TINFO_numeric)
239 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
242 setflag(status_flags::evaluated|
243 status_flags::hash_calculated);
250 /** Construct object from archive_node. */
251 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
253 debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT);
256 // Read number as string
258 if (n.find_string("number", str)) {
260 istringstream s(str);
262 istrstream s(str.c_str(), str.size() + 1);
264 cl_idecoded_float re, im;
268 case 'R': // Integer-decoded real number
269 s >> re.sign >> re.mantissa >> re.exponent;
270 *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent);
272 case 'C': // Integer-decoded complex number
273 s >> re.sign >> re.mantissa >> re.exponent;
274 s >> im.sign >> im.mantissa >> im.exponent;
275 *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent),
276 im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent));
278 default: // Ordinary number
285 setflag(status_flags::evaluated|
286 status_flags::hash_calculated);
289 /** Unarchive the object. */
290 ex numeric::unarchive(const archive_node &n, const lst &sym_lst)
292 return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated);
295 /** Archive the object. */
296 void numeric::archive(archive_node &n) const
298 inherited::archive(n);
300 // Write number as string
305 ostrstream s(buf, 1024);
307 if (this->is_crational())
310 // Non-rational numbers are written in an integer-decoded format
311 // to preserve the precision
312 if (this->is_real()) {
313 cl_idecoded_float re = integer_decode_float(The(cl_F)(*value));
315 s << re.sign << " " << re.mantissa << " " << re.exponent;
317 cl_idecoded_float re = integer_decode_float(The(cl_F)(::realpart(*value)));
318 cl_idecoded_float im = integer_decode_float(The(cl_F)(::imagpart(*value)));
320 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
321 s << im.sign << " " << im.mantissa << " " << im.exponent;
325 n.add_string("number", s.str());
329 n.add_string("number", str);
334 // functions overriding virtual functions from bases classes
339 basic * numeric::duplicate() const
341 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
342 return new numeric(*this);
346 /** Helper function to print a real number in a nicer way than is CLN's
347 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
348 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
349 * long as it only uses cl_LF and no other floating point types.
351 * @see numeric::print() */
352 void print_real_number(ostream & os, const cl_R & num)
354 cl_print_flags ourflags;
355 if (::instanceof(num, ::cl_RA_ring)) {
356 // case 1: integer or rational, nothing special to do:
357 ::print_real(os, ourflags, num);
360 // make CLN believe this number has default_float_format, so it prints
361 // 'E' as exponent marker instead of 'L':
362 ourflags.default_float_format = ::cl_float_format(The(cl_F)(num));
363 ::print_real(os, ourflags, num);
368 /** This method adds to the output so it blends more consistently together
369 * with the other routines and produces something compatible to ginsh input.
371 * @see print_real_number() */
372 void numeric::print(ostream & os, unsigned upper_precedence) const
374 debugmsg("numeric print", LOGLEVEL_PRINT);
375 if (this->is_real()) {
376 // case 1, real: x or -x
377 if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) {
379 print_real_number(os, The(cl_R)(*value));
382 print_real_number(os, The(cl_R)(*value));
385 // case 2, imaginary: y*I or -y*I
386 if (::realpart(*value) == 0) {
387 if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) {
388 if (::imagpart(*value) == -1) {
392 print_real_number(os, The(cl_R)(::imagpart(*value)));
396 if (::imagpart(*value) == 1) {
399 if (::imagpart (*value) == -1) {
402 print_real_number(os, The(cl_R)(::imagpart(*value)));
408 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
409 if (precedence <= upper_precedence)
411 print_real_number(os, The(cl_R)(::realpart(*value)));
412 if (::imagpart(*value) < 0) {
413 if (::imagpart(*value) == -1) {
416 print_real_number(os, The(cl_R)(::imagpart(*value)));
420 if (::imagpart(*value) == 1) {
424 print_real_number(os, The(cl_R)(::imagpart(*value)));
428 if (precedence <= upper_precedence)
435 void numeric::printraw(ostream & os) const
437 // The method printraw doesn't do much, it simply uses CLN's operator<<()
438 // for output, which is ugly but reliable. e.g: 2+2i
439 debugmsg("numeric printraw", LOGLEVEL_PRINT);
440 os << "numeric(" << *value << ")";
444 void numeric::printtree(ostream & os, unsigned indent) const
446 debugmsg("numeric printtree", LOGLEVEL_PRINT);
447 os << string(indent,' ') << *value
449 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
450 << ", flags=" << flags << endl;
454 void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
456 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
457 ios::fmtflags oldflags = os.flags();
458 os.setf(ios::scientific);
459 if (this->is_rational() && !this->is_integer()) {
460 if (compare(_num0()) > 0) {
462 if (type == csrc_types::ctype_cl_N)
463 os << "cl_F(\"" << numer().evalf() << "\")";
465 os << numer().to_double();
468 if (type == csrc_types::ctype_cl_N)
469 os << "cl_F(\"" << -numer().evalf() << "\")";
471 os << -numer().to_double();
474 if (type == csrc_types::ctype_cl_N)
475 os << "cl_F(\"" << denom().evalf() << "\")";
477 os << denom().to_double();
480 if (type == csrc_types::ctype_cl_N)
481 os << "cl_F(\"" << evalf() << "\")";
489 bool numeric::info(unsigned inf) const
492 case info_flags::numeric:
493 case info_flags::polynomial:
494 case info_flags::rational_function:
496 case info_flags::real:
498 case info_flags::rational:
499 case info_flags::rational_polynomial:
500 return is_rational();
501 case info_flags::crational:
502 case info_flags::crational_polynomial:
503 return is_crational();
504 case info_flags::integer:
505 case info_flags::integer_polynomial:
507 case info_flags::cinteger:
508 case info_flags::cinteger_polynomial:
509 return is_cinteger();
510 case info_flags::positive:
511 return is_positive();
512 case info_flags::negative:
513 return is_negative();
514 case info_flags::nonnegative:
515 return !is_negative();
516 case info_flags::posint:
517 return is_pos_integer();
518 case info_flags::negint:
519 return is_integer() && is_negative();
520 case info_flags::nonnegint:
521 return is_nonneg_integer();
522 case info_flags::even:
524 case info_flags::odd:
526 case info_flags::prime:
528 case info_flags::algebraic:
534 /** Disassemble real part and imaginary part to scan for the occurrence of a
535 * single number. Also handles the imaginary unit. It ignores the sign on
536 * both this and the argument, which may lead to what might appear as funny
537 * results: (2+I).has(-2) -> true. But this is consistent, since we also
538 * would like to have (-2+I).has(2) -> true and we want to think about the
539 * sign as a multiplicative factor. */
540 bool numeric::has(const ex & other) const
542 if (!is_exactly_of_type(*other.bp, numeric))
544 const numeric & o = static_cast<numeric &>(const_cast<basic &>(*other.bp));
545 if (this->is_equal(o) || this->is_equal(-o))
547 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
548 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
549 this->real().is_equal(-o) || this->imag().is_equal(-o));
551 if (o.is_equal(I)) // e.g scan for I in 42*I
552 return !this->is_real();
553 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
554 return (this->real().has(o*I) || this->imag().has(o*I) ||
555 this->real().has(-o*I) || this->imag().has(-o*I));
561 /** Evaluation of numbers doesn't do anything at all. */
562 ex numeric::eval(int level) const
564 // Warning: if this is ever gonna do something, the ex ctors from all kinds
565 // of numbers should be checking for status_flags::evaluated.
570 /** Cast numeric into a floating-point object. For example exact numeric(1) is
571 * returned as a 1.0000000000000000000000 and so on according to how Digits is
574 * @param level ignored, but needed for overriding basic::evalf.
575 * @return an ex-handle to a numeric. */
576 ex numeric::evalf(int level) const
578 // level can safely be discarded for numeric objects.
579 return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN
584 /** Implementation of ex::diff() for a numeric. It always returns 0.
587 ex numeric::derivative(const symbol & s) const
593 int numeric::compare_same_type(const basic & other) const
595 GINAC_ASSERT(is_exactly_of_type(other, numeric));
596 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
598 if (*value == *o.value) {
606 bool numeric::is_equal_same_type(const basic & other) const
608 GINAC_ASSERT(is_exactly_of_type(other,numeric));
609 const numeric *o = static_cast<const numeric *>(&other);
611 return this->is_equal(*o);
614 unsigned numeric::calchash(void) const
616 return (hashvalue=cl_equal_hashcode(*value) | 0x80000000U);
618 cout << *value << "->" << hashvalue << endl;
619 hashvalue=HASHVALUE_NUMERIC+1000U;
620 return HASHVALUE_NUMERIC+1000U;
625 unsigned numeric::calchash(void) const
627 double d=to_double();
633 return 0x88000000U+s*unsigned(d/0x07FF0000);
639 // new virtual functions which can be overridden by derived classes
645 // non-virtual functions in this class
650 /** Numerical addition method. Adds argument to *this and returns result as
651 * a new numeric object. */
652 numeric numeric::add(const numeric & other) const
654 return numeric((*value)+(*other.value));
657 /** Numerical subtraction method. Subtracts argument from *this and returns
658 * result as a new numeric object. */
659 numeric numeric::sub(const numeric & other) const
661 return numeric((*value)-(*other.value));
664 /** Numerical multiplication method. Multiplies *this and argument and returns
665 * result as a new numeric object. */
666 numeric numeric::mul(const numeric & other) const
668 static const numeric * _num1p=&_num1();
671 } else if (&other==_num1p) {
674 return numeric((*value)*(*other.value));
677 /** Numerical division method. Divides *this by argument and returns result as
678 * a new numeric object.
680 * @exception overflow_error (division by zero) */
681 numeric numeric::div(const numeric & other) const
683 if (::zerop(*other.value))
684 throw (std::overflow_error("division by zero"));
685 return numeric((*value)/(*other.value));
688 numeric numeric::power(const numeric & other) const
690 static const numeric * _num1p = &_num1();
693 if (::zerop(*value)) {
694 if (::zerop(*other.value))
695 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
696 else if (::zerop(::realpart(*other.value)))
697 throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
698 else if (::minusp(::realpart(*other.value)))
699 throw (std::overflow_error("numeric::eval(): division by zero"));
703 return numeric(::expt(*value,*other.value));
706 /** Inverse of a number. */
707 numeric numeric::inverse(void) const
709 return numeric(::recip(*value)); // -> CLN
712 const numeric & numeric::add_dyn(const numeric & other) const
714 return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
715 setflag(status_flags::dynallocated));
718 const numeric & numeric::sub_dyn(const numeric & other) const
720 return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
721 setflag(status_flags::dynallocated));
724 const numeric & numeric::mul_dyn(const numeric & other) const
726 static const numeric * _num1p=&_num1();
729 } else if (&other==_num1p) {
732 return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
733 setflag(status_flags::dynallocated));
736 const numeric & numeric::div_dyn(const numeric & other) const
738 if (::zerop(*other.value))
739 throw (std::overflow_error("division by zero"));
740 return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
741 setflag(status_flags::dynallocated));
744 const numeric & numeric::power_dyn(const numeric & other) const
746 static const numeric * _num1p=&_num1();
749 if (::zerop(*value)) {
750 if (::zerop(*other.value))
751 throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
752 else if (::zerop(::realpart(*other.value)))
753 throw (std::domain_error("numeric::eval(): pow(0,I) is undefined"));
754 else if (::minusp(::realpart(*other.value)))
755 throw (std::overflow_error("numeric::eval(): division by zero"));
759 return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
760 setflag(status_flags::dynallocated));
763 const numeric & numeric::operator=(int i)
765 return operator=(numeric(i));
768 const numeric & numeric::operator=(unsigned int i)
770 return operator=(numeric(i));
773 const numeric & numeric::operator=(long i)
775 return operator=(numeric(i));
778 const numeric & numeric::operator=(unsigned long i)
780 return operator=(numeric(i));
783 const numeric & numeric::operator=(double d)
785 return operator=(numeric(d));
788 const numeric & numeric::operator=(const char * s)
790 return operator=(numeric(s));
793 /** Return the complex half-plane (left or right) in which the number lies.
794 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
795 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
797 * @see numeric::compare(const numeric & other) */
798 int numeric::csgn(void) const
802 if (!::zerop(::realpart(*value))) {
803 if (::plusp(::realpart(*value)))
808 if (::plusp(::imagpart(*value)))
815 /** This method establishes a canonical order on all numbers. For complex
816 * numbers this is not possible in a mathematically consistent way but we need
817 * to establish some order and it ought to be fast. So we simply define it
818 * to be compatible with our method csgn.
820 * @return csgn(*this-other)
821 * @see numeric::csgn(void) */
822 int numeric::compare(const numeric & other) const
824 // Comparing two real numbers?
825 if (this->is_real() && other.is_real())
826 // Yes, just compare them
827 return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
829 // No, first compare real parts
830 cl_signean real_cmp = ::cl_compare(::realpart(*value), ::realpart(*other.value));
834 return ::cl_compare(::imagpart(*value), ::imagpart(*other.value));
838 bool numeric::is_equal(const numeric & other) const
840 return (*value == *other.value);
843 /** True if object is zero. */
844 bool numeric::is_zero(void) const
846 return ::zerop(*value); // -> CLN
849 /** True if object is not complex and greater than zero. */
850 bool numeric::is_positive(void) const
853 return ::plusp(The(cl_R)(*value)); // -> CLN
857 /** True if object is not complex and less than zero. */
858 bool numeric::is_negative(void) const
861 return ::minusp(The(cl_R)(*value)); // -> CLN
865 /** True if object is a non-complex integer. */
866 bool numeric::is_integer(void) const
868 return ::instanceof(*value, ::cl_I_ring); // -> CLN
871 /** True if object is an exact integer greater than zero. */
872 bool numeric::is_pos_integer(void) const
874 return (this->is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN
877 /** True if object is an exact integer greater or equal zero. */
878 bool numeric::is_nonneg_integer(void) const
880 return (this->is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN
883 /** True if object is an exact even integer. */
884 bool numeric::is_even(void) const
886 return (this->is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN
889 /** True if object is an exact odd integer. */
890 bool numeric::is_odd(void) const
892 return (this->is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN
895 /** Probabilistic primality test.
897 * @return true if object is exact integer and prime. */
898 bool numeric::is_prime(void) const
900 return (this->is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN
903 /** True if object is an exact rational number, may even be complex
904 * (denominator may be unity). */
905 bool numeric::is_rational(void) const
907 return ::instanceof(*value, ::cl_RA_ring); // -> CLN
910 /** True if object is a real integer, rational or float (but not complex). */
911 bool numeric::is_real(void) const
913 return ::instanceof(*value, ::cl_R_ring); // -> CLN
916 bool numeric::operator==(const numeric & other) const
918 return (*value == *other.value); // -> CLN
921 bool numeric::operator!=(const numeric & other) const
923 return (*value != *other.value); // -> CLN
926 /** True if object is element of the domain of integers extended by I, i.e. is
927 * of the form a+b*I, where a and b are integers. */
928 bool numeric::is_cinteger(void) const
930 if (::instanceof(*value, ::cl_I_ring))
932 else if (!this->is_real()) { // complex case, handle n+m*I
933 if (::instanceof(::realpart(*value), ::cl_I_ring) &&
934 ::instanceof(::imagpart(*value), ::cl_I_ring))
940 /** True if object is an exact rational number, may even be complex
941 * (denominator may be unity). */
942 bool numeric::is_crational(void) const
944 if (::instanceof(*value, ::cl_RA_ring))
946 else if (!this->is_real()) { // complex case, handle Q(i):
947 if (::instanceof(::realpart(*value), ::cl_RA_ring) &&
948 ::instanceof(::imagpart(*value), ::cl_RA_ring))
954 /** Numerical comparison: less.
956 * @exception invalid_argument (complex inequality) */
957 bool numeric::operator<(const numeric & other) const
959 if (this->is_real() && other.is_real())
960 return (The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
961 throw (std::invalid_argument("numeric::operator<(): complex inequality"));
962 return false; // make compiler shut up
965 /** Numerical comparison: less or equal.
967 * @exception invalid_argument (complex inequality) */
968 bool numeric::operator<=(const numeric & other) const
970 if (this->is_real() && other.is_real())
971 return (The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
972 throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
973 return false; // make compiler shut up
976 /** Numerical comparison: greater.
978 * @exception invalid_argument (complex inequality) */
979 bool numeric::operator>(const numeric & other) const
981 if (this->is_real() && other.is_real())
982 return (The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
983 throw (std::invalid_argument("numeric::operator>(): complex inequality"));
984 return false; // make compiler shut up
987 /** Numerical comparison: greater or equal.
989 * @exception invalid_argument (complex inequality) */
990 bool numeric::operator>=(const numeric & other) const
992 if (this->is_real() && other.is_real())
993 return (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 /** Converts numeric types to machine's int. You should check with
999 * is_integer() if the number is really an integer before calling this method.
1000 * You may also consider checking the range first. */
1001 int numeric::to_int(void) const
1003 GINAC_ASSERT(this->is_integer());
1004 return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN
1007 /** Converts numeric types to machine's long. You should check with
1008 * is_integer() if the number is really an integer before calling this method.
1009 * You may also consider checking the range first. */
1010 long numeric::to_long(void) const
1012 GINAC_ASSERT(this->is_integer());
1013 return ::cl_I_to_long(The(cl_I)(*value)); // -> CLN
1016 /** Converts numeric types to machine's double. You should check with is_real()
1017 * if the number is really not complex before calling this method. */
1018 double numeric::to_double(void) const
1020 GINAC_ASSERT(this->is_real());
1021 return ::cl_double_approx(::realpart(*value)); // -> CLN
1024 /** Real part of a number. */
1025 const numeric numeric::real(void) const
1027 return numeric(::realpart(*value)); // -> CLN
1030 /** Imaginary part of a number. */
1031 const numeric numeric::imag(void) const
1033 return numeric(::imagpart(*value)); // -> CLN
1037 // Unfortunately, CLN did not provide an official way to access the numerator
1038 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
1039 // one finds how it works internally in src/rational/cl_RA.h:
1040 struct cl_heap_ratio : cl_heap {
1045 inline cl_heap_ratio* TheRatio (const cl_N& obj)
1046 { return (cl_heap_ratio*)(obj.pointer); }
1047 #endif // ndef SANE_LINKER
1049 /** Numerator. Computes the numerator of rational numbers, rationalized
1050 * numerator of complex if real and imaginary part are both rational numbers
1051 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1053 const numeric numeric::numer(void) const
1055 if (this->is_integer()) {
1056 return numeric(*this);
1059 else if (::instanceof(*value, ::cl_RA_ring)) {
1060 return numeric(::numerator(The(cl_RA)(*value)));
1062 else if (!this->is_real()) { // complex case, handle Q(i):
1063 cl_R r = ::realpart(*value);
1064 cl_R i = ::imagpart(*value);
1065 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1066 return numeric(*this);
1067 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1068 return numeric(::complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i))));
1069 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1070 return numeric(::complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r))));
1071 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring)) {
1072 cl_I s = ::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)));
1073 return numeric(::complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))),
1074 ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i))))));
1078 else if (instanceof(*value, ::cl_RA_ring)) {
1079 return numeric(TheRatio(*value)->numerator);
1081 else if (!this->is_real()) { // complex case, handle Q(i):
1082 cl_R r = ::realpart(*value);
1083 cl_R i = ::imagpart(*value);
1084 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1085 return numeric(*this);
1086 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1087 return numeric(::complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
1088 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1089 return numeric(::complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
1090 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring)) {
1091 cl_I s = ::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
1092 return numeric(::complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
1093 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
1096 #endif // def SANE_LINKER
1097 // at least one float encountered
1098 return numeric(*this);
1101 /** Denominator. Computes the denominator of rational numbers, common integer
1102 * denominator of complex if real and imaginary part are both rational numbers
1103 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1104 const numeric numeric::denom(void) const
1106 if (this->is_integer()) {
1110 if (instanceof(*value, ::cl_RA_ring)) {
1111 return numeric(::denominator(The(cl_RA)(*value)));
1113 if (!this->is_real()) { // complex case, handle Q(i):
1114 cl_R r = ::realpart(*value);
1115 cl_R i = ::imagpart(*value);
1116 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_I_ring))
1118 if (::instanceof(r, ::cl_I_ring) && ::instanceof(i, ::cl_RA_ring))
1119 return numeric(::denominator(The(cl_RA)(i)));
1120 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_I_ring))
1121 return numeric(::denominator(The(cl_RA)(r)));
1122 if (::instanceof(r, ::cl_RA_ring) && ::instanceof(i, ::cl_RA_ring))
1123 return numeric(::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
1126 if (instanceof(*value, ::cl_RA_ring)) {
1127 return numeric(TheRatio(*value)->denominator);
1129 if (!this->is_real()) { // complex case, handle Q(i):
1130 cl_R r = ::realpart(*value);
1131 cl_R i = ::imagpart(*value);
1132 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_I_ring))
1134 if (instanceof(r, ::cl_I_ring) && instanceof(i, ::cl_RA_ring))
1135 return numeric(TheRatio(i)->denominator);
1136 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_I_ring))
1137 return numeric(TheRatio(r)->denominator);
1138 if (instanceof(r, ::cl_RA_ring) && instanceof(i, ::cl_RA_ring))
1139 return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
1141 #endif // def SANE_LINKER
1142 // at least one float encountered
1146 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1147 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1148 * 2^(n-1) <= x < 2^n.
1150 * @return number of bits (excluding sign) needed to represent that number
1151 * in two's complement if it is an integer, 0 otherwise. */
1152 int numeric::int_length(void) const
1154 if (this->is_integer())
1155 return ::integer_length(The(cl_I)(*value)); // -> CLN
1162 // static member variables
1167 unsigned numeric::precedence = 30;
1173 const numeric some_numeric;
1174 const type_info & typeid_numeric=typeid(some_numeric);
1175 /** Imaginary unit. This is not a constant but a numeric since we are
1176 * natively handing complex numbers anyways. */
1177 const numeric I = numeric(::complex(cl_I(0),cl_I(1)));
1180 /** Exponential function.
1182 * @return arbitrary precision numerical exp(x). */
1183 const numeric exp(const numeric & x)
1185 return ::exp(*x.value); // -> CLN
1189 /** Natural logarithm.
1191 * @param z complex number
1192 * @return arbitrary precision numerical log(x).
1193 * @exception overflow_error (logarithmic singularity) */
1194 const numeric log(const numeric & z)
1197 throw (std::overflow_error("log(): logarithmic singularity"));
1198 return ::log(*z.value); // -> CLN
1202 /** Numeric sine (trigonometric function).
1204 * @return arbitrary precision numerical sin(x). */
1205 const numeric sin(const numeric & x)
1207 return ::sin(*x.value); // -> CLN
1211 /** Numeric cosine (trigonometric function).
1213 * @return arbitrary precision numerical cos(x). */
1214 const numeric cos(const numeric & x)
1216 return ::cos(*x.value); // -> CLN
1220 /** Numeric tangent (trigonometric function).
1222 * @return arbitrary precision numerical tan(x). */
1223 const numeric tan(const numeric & x)
1225 return ::tan(*x.value); // -> CLN
1229 /** Numeric inverse sine (trigonometric function).
1231 * @return arbitrary precision numerical asin(x). */
1232 const numeric asin(const numeric & x)
1234 return ::asin(*x.value); // -> CLN
1238 /** Numeric inverse cosine (trigonometric function).
1240 * @return arbitrary precision numerical acos(x). */
1241 const numeric acos(const numeric & x)
1243 return ::acos(*x.value); // -> CLN
1249 * @param z complex number
1251 * @exception overflow_error (logarithmic singularity) */
1252 const numeric atan(const numeric & x)
1255 x.real().is_zero() &&
1256 !abs(x.imag()).is_equal(_num1()))
1257 throw (std::overflow_error("atan(): logarithmic singularity"));
1258 return ::atan(*x.value); // -> CLN
1264 * @param x real number
1265 * @param y real number
1266 * @return atan(y/x) */
1267 const numeric atan(const numeric & y, const numeric & x)
1269 if (x.is_real() && y.is_real())
1270 return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
1272 throw (std::invalid_argument("numeric::atan(): complex argument"));
1276 /** Numeric hyperbolic sine (trigonometric function).
1278 * @return arbitrary precision numerical sinh(x). */
1279 const numeric sinh(const numeric & x)
1281 return ::sinh(*x.value); // -> CLN
1285 /** Numeric hyperbolic cosine (trigonometric function).
1287 * @return arbitrary precision numerical cosh(x). */
1288 const numeric cosh(const numeric & x)
1290 return ::cosh(*x.value); // -> CLN
1294 /** Numeric hyperbolic tangent (trigonometric function).
1296 * @return arbitrary precision numerical tanh(x). */
1297 const numeric tanh(const numeric & x)
1299 return ::tanh(*x.value); // -> CLN
1303 /** Numeric inverse hyperbolic sine (trigonometric function).
1305 * @return arbitrary precision numerical asinh(x). */
1306 const numeric asinh(const numeric & x)
1308 return ::asinh(*x.value); // -> CLN
1312 /** Numeric inverse hyperbolic cosine (trigonometric function).
1314 * @return arbitrary precision numerical acosh(x). */
1315 const numeric acosh(const numeric & x)
1317 return ::acosh(*x.value); // -> CLN
1321 /** Numeric inverse hyperbolic tangent (trigonometric function).
1323 * @return arbitrary precision numerical atanh(x). */
1324 const numeric atanh(const numeric & x)
1326 return ::atanh(*x.value); // -> CLN
1330 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1331 * integer arguments. */
1332 const numeric zeta(const numeric & x)
1334 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1335 // only knows about integer arguments and zeta(3).evalf() automatically
1336 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1337 // being an exact zero for CLN, which can be tested and then we can just
1338 // pass the number casted to an int:
1340 int aux = (int)(::cl_double_approx(::realpart(*x.value)));
1341 if (zerop(*x.value-aux))
1342 return ::cl_zeta(aux); // -> CLN
1344 clog << "zeta(" << x
1345 << "): Does anybody know good way to calculate this numerically?"
1351 /** The Gamma function.
1352 * This is only a stub! */
1353 const numeric lgamma(const numeric & x)
1355 clog << "lgamma(" << x
1356 << "): Does anybody know good way to calculate this numerically?"
1360 const numeric tgamma(const numeric & x)
1362 clog << "tgamma(" << x
1363 << "): Does anybody know good way to calculate this numerically?"
1369 /** The psi function (aka polygamma function).
1370 * This is only a stub! */
1371 const numeric psi(const numeric & x)
1374 << "): Does anybody know good way to calculate this numerically?"
1380 /** The psi functions (aka polygamma functions).
1381 * This is only a stub! */
1382 const numeric psi(const numeric & n, const numeric & x)
1384 clog << "psi(" << n << "," << x
1385 << "): Does anybody know good way to calculate this numerically?"
1391 /** Factorial combinatorial function.
1393 * @param n integer argument >= 0
1394 * @exception range_error (argument must be integer >= 0) */
1395 const numeric factorial(const numeric & n)
1397 if (!n.is_nonneg_integer())
1398 throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
1399 return numeric(::factorial(n.to_int())); // -> CLN
1403 /** The double factorial combinatorial function. (Scarcely used, but still
1404 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1406 * @param n integer argument >= -1
1407 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1408 * @exception range_error (argument must be integer >= -1) */
1409 const numeric doublefactorial(const numeric & n)
1411 if (n == numeric(-1)) {
1414 if (!n.is_nonneg_integer()) {
1415 throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
1417 return numeric(::doublefactorial(n.to_int())); // -> CLN
1421 /** The Binomial coefficients. It computes the binomial coefficients. For
1422 * integer n and k and positive n this is the number of ways of choosing k
1423 * objects from n distinct objects. If n is negative, the formula
1424 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1425 const numeric binomial(const numeric & n, const numeric & k)
1427 if (n.is_integer() && k.is_integer()) {
1428 if (n.is_nonneg_integer()) {
1429 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1430 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1434 return _num_1().power(k)*binomial(k-n-_num1(),k);
1438 // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit
1439 throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
1443 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1444 * in the expansion of the function x/(e^x-1).
1446 * @return the nth Bernoulli number (a rational number).
1447 * @exception range_error (argument must be integer >= 0) */
1448 const numeric bernoulli(const numeric & nn)
1450 if (!nn.is_integer() || nn.is_negative())
1451 throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
1454 if (!nn.compare(_num1()))
1455 return numeric(-1,2);
1458 // Until somebody has the blues and comes up with a much better idea and
1459 // codes it (preferably in CLN) we make this a remembering function which
1460 // computes its results using the defining formula
1461 // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
1463 // Be warned, though: the Bernoulli numbers are computationally very
1464 // expensive anyhow and you shouldn't expect miracles to happen.
1465 static vector<numeric> results;
1466 static int highest_result = -1;
1467 int n = nn.sub(_num2()).div(_num2()).to_int();
1468 if (n <= highest_result)
1470 if (results.capacity() < (unsigned)(n+1))
1471 results.reserve(n+1);
1473 numeric tmp; // used to store the sum
1474 for (int i=highest_result+1; i<=n; ++i) {
1475 // the first two elements:
1476 tmp = numeric(-2*i-1,2);
1477 // accumulate the remaining elements:
1478 for (int j=0; j<i; ++j)
1479 tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
1480 // divide by -(nn+1) and store result:
1481 results.push_back(-tmp/numeric(2*i+3));
1488 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1489 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1491 * @param n an integer
1492 * @return the nth Fibonacci number F(n) (an integer number)
1493 * @exception range_error (argument must be an integer) */
1494 const numeric fibonacci(const numeric & n)
1496 if (!n.is_integer())
1497 throw (std::range_error("numeric::fibonacci(): argument must be integer"));
1498 // The following addition formula holds:
1499 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1500 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1501 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1503 // Replace m by m+1:
1504 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1505 // Now put in m = n, to get
1506 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1507 // F(2n+1) = F(n)^2 + F(n+1)^2
1509 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1512 if (n.is_negative())
1514 return -fibonacci(-n);
1516 return fibonacci(-n);
1520 cl_I m = The(cl_I)(*n.value) >> 1L; // floor(n/2);
1521 for (uintL bit=::integer_length(m); bit>0; --bit) {
1522 // Since a squaring is cheaper than a multiplication, better use
1523 // three squarings instead of one multiplication and two squarings.
1524 cl_I u2 = ::square(u);
1525 cl_I v2 = ::square(v);
1526 if (::logbitp(bit-1, m)) {
1527 v = ::square(u + v) - u2;
1530 u = v2 - ::square(v - u);
1535 // Here we don't use the squaring formula because one multiplication
1536 // is cheaper than two squarings.
1537 return u * ((v << 1) - u);
1539 return ::square(u) + ::square(v);
1543 /** Absolute value. */
1544 numeric abs(const numeric & x)
1546 return ::abs(*x.value); // -> CLN
1550 /** Modulus (in positive representation).
1551 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1552 * sign of a or is zero. This is different from Maple's modp, where the sign
1553 * of b is ignored. It is in agreement with Mathematica's Mod.
1555 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1556 * integer, 0 otherwise. */
1557 numeric mod(const numeric & a, const numeric & b)
1559 if (a.is_integer() && b.is_integer())
1560 return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1562 return _num0(); // Throw?
1566 /** Modulus (in symmetric representation).
1567 * Equivalent to Maple's mods.
1569 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1570 numeric smod(const numeric & a, const numeric & b)
1572 if (a.is_integer() && b.is_integer()) {
1573 cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1;
1574 return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
1576 return _num0(); // Throw?
1580 /** Numeric integer remainder.
1581 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1582 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1583 * sign of a or is zero.
1585 * @return remainder of a/b if both are integer, 0 otherwise. */
1586 numeric irem(const numeric & a, const numeric & b)
1588 if (a.is_integer() && b.is_integer())
1589 return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1591 return _num0(); // Throw?
1595 /** Numeric integer remainder.
1596 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1597 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1598 * and irem(a,b) has the sign of a or is zero.
1600 * @return remainder of a/b and quotient stored in q if both are integer,
1602 numeric irem(const numeric & a, const numeric & b, numeric & q)
1604 if (a.is_integer() && b.is_integer()) { // -> CLN
1605 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1606 q = rem_quo.quotient;
1607 return rem_quo.remainder;
1611 return _num0(); // Throw?
1616 /** Numeric integer quotient.
1617 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1619 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1620 numeric iquo(const numeric & a, const numeric & b)
1622 if (a.is_integer() && b.is_integer())
1623 return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1625 return _num0(); // Throw?
1629 /** Numeric integer quotient.
1630 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1631 * r == a - iquo(a,b,r)*b.
1633 * @return truncated quotient of a/b and remainder stored in r if both are
1634 * integer, 0 otherwise. */
1635 numeric iquo(const numeric & a, const numeric & b, numeric & r)
1637 if (a.is_integer() && b.is_integer()) { // -> CLN
1638 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1639 r = rem_quo.remainder;
1640 return rem_quo.quotient;
1643 return _num0(); // Throw?
1648 /** Numeric square root.
1649 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1650 * should return integer 2.
1652 * @param z numeric argument
1653 * @return square root of z. Branch cut along negative real axis, the negative
1654 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1655 * where imag(z)>0. */
1656 numeric sqrt(const numeric & z)
1658 return ::sqrt(*z.value); // -> CLN
1662 /** Integer numeric square root. */
1663 numeric isqrt(const numeric & x)
1665 if (x.is_integer()) {
1667 ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
1670 return _num0(); // Throw?
1674 /** Greatest Common Divisor.
1676 * @return The GCD of two numbers if both are integer, a numerical 1
1677 * if they are not. */
1678 numeric gcd(const numeric & a, const numeric & b)
1680 if (a.is_integer() && b.is_integer())
1681 return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1687 /** Least Common Multiple.
1689 * @return The LCM of two numbers if both are integer, the product of those
1690 * two numbers if they are not. */
1691 numeric lcm(const numeric & a, const numeric & b)
1693 if (a.is_integer() && b.is_integer())
1694 return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1696 return *a.value * *b.value;
1700 /** Floating point evaluation of Archimedes' constant Pi. */
1703 return numeric(::cl_pi(cl_default_float_format)); // -> CLN
1707 /** Floating point evaluation of Euler's constant gamma. */
1710 return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN
1714 /** Floating point evaluation of Catalan's constant. */
1715 ex CatalanEvalf(void)
1717 return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN
1721 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1722 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1723 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1724 _numeric_digits::_numeric_digits()
1729 cl_default_float_format = ::cl_float_format(17);
1733 _numeric_digits& _numeric_digits::operator=(long prec)
1736 cl_default_float_format = ::cl_float_format(prec);
1741 _numeric_digits::operator long()
1743 return (long)digits;
1747 void _numeric_digits::print(ostream & os) const
1749 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1754 ostream& operator<<(ostream& os, const _numeric_digits & e)
1761 // static member variables
1766 bool _numeric_digits::too_late = false;
1769 /** Accuracy in decimal digits. Only object of this type! Can be set using
1770 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1771 _numeric_digits Digits;
1773 #ifndef NO_NAMESPACE_GINAC
1774 } // namespace GiNaC
1775 #endif // ndef NO_NAMESPACE_GINAC