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_GINAC_NAMESPACE
78 #endif // ndef NO_GINAC_NAMESPACE
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(cl_N const & 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
325 // Non-rational numbers are written in an integer-decoded format
326 // to preserve the precision
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);
347 // Non-rational numbers are written in an integer-decoded format
348 // to preserve the precision
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);
385 // case 1, real: x or -x
386 if ((precedence<=upper_precedence) && (!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 << ")";
441 void numeric::printtree(ostream & os, unsigned indent) const
443 debugmsg("numeric printtree", LOGLEVEL_PRINT);
444 os << string(indent,' ') << *value
446 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
447 << ", flags=" << flags << endl;
450 void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
452 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
453 ios::fmtflags oldflags = os.flags();
454 os.setf(ios::scientific);
455 if (is_rational() && !is_integer()) {
456 if (compare(_num0()) > 0) {
458 if (type == csrc_types::ctype_cl_N)
459 os << "cl_F(\"" << numer().evalf() << "\")";
461 os << numer().to_double();
464 if (type == csrc_types::ctype_cl_N)
465 os << "cl_F(\"" << -numer().evalf() << "\")";
467 os << -numer().to_double();
470 if (type == csrc_types::ctype_cl_N)
471 os << "cl_F(\"" << denom().evalf() << "\")";
473 os << denom().to_double();
476 if (type == csrc_types::ctype_cl_N)
477 os << "cl_F(\"" << evalf() << "\")";
484 bool numeric::info(unsigned inf) const
487 case info_flags::numeric:
488 case info_flags::polynomial:
489 case info_flags::rational_function:
491 case info_flags::real:
493 case info_flags::rational:
494 case info_flags::rational_polynomial:
495 return is_rational();
496 case info_flags::crational:
497 case info_flags::crational_polynomial:
498 return is_crational();
499 case info_flags::integer:
500 case info_flags::integer_polynomial:
502 case info_flags::cinteger:
503 case info_flags::cinteger_polynomial:
504 return is_cinteger();
505 case info_flags::positive:
506 return is_positive();
507 case info_flags::negative:
508 return is_negative();
509 case info_flags::nonnegative:
510 return compare(_num0())>=0;
511 case info_flags::posint:
512 return is_pos_integer();
513 case info_flags::negint:
514 return is_integer() && (compare(_num0())<0);
515 case info_flags::nonnegint:
516 return is_nonneg_integer();
517 case info_flags::even:
519 case info_flags::odd:
521 case info_flags::prime:
527 /** Cast numeric into a floating-point object. For example exact numeric(1) is
528 * returned as a 1.0000000000000000000000 and so on according to how Digits is
531 * @param level ignored, but needed for overriding basic::evalf.
532 * @return an ex-handle to a numeric. */
533 ex numeric::evalf(int level) const
535 // level can safely be discarded for numeric objects.
536 return numeric(cl_float(1.0, cl_default_float_format) * (*value)); // -> CLN
541 int numeric::compare_same_type(const basic & other) const
543 GINAC_ASSERT(is_exactly_of_type(other, numeric));
544 const numeric & o = static_cast<numeric &>(const_cast<basic &>(other));
546 if (*value == *o.value) {
553 bool numeric::is_equal_same_type(const basic & other) const
555 GINAC_ASSERT(is_exactly_of_type(other,numeric));
556 const numeric *o = static_cast<const numeric *>(&other);
562 unsigned numeric::calchash(void) const
564 double d=to_double();
570 return 0x88000000U+s*unsigned(d/0x07FF0000);
576 // new virtual functions which can be overridden by derived classes
582 // non-virtual functions in this class
587 /** Numerical addition method. Adds argument to *this and returns result as
588 * a new numeric object. */
589 numeric numeric::add(const numeric & other) const
591 return numeric((*value)+(*other.value));
594 /** Numerical subtraction method. Subtracts argument from *this and returns
595 * result as a new numeric object. */
596 numeric numeric::sub(const numeric & other) const
598 return numeric((*value)-(*other.value));
601 /** Numerical multiplication method. Multiplies *this and argument and returns
602 * result as a new numeric object. */
603 numeric numeric::mul(const numeric & other) const
605 static const numeric * _num1p=&_num1();
608 } else if (&other==_num1p) {
611 return numeric((*value)*(*other.value));
614 /** Numerical division method. Divides *this by argument and returns result as
615 * a new numeric object.
617 * @exception overflow_error (division by zero) */
618 numeric numeric::div(const numeric & other) const
620 if (::zerop(*other.value))
621 throw (std::overflow_error("division by zero"));
622 return numeric((*value)/(*other.value));
625 numeric numeric::power(const numeric & other) const
627 static const numeric * _num1p=&_num1();
630 if (::zerop(*value) && other.is_real() && ::minusp(realpart(*other.value)))
631 throw (std::overflow_error("division by zero"));
632 return numeric(::expt(*value,*other.value));
635 /** Inverse of a number. */
636 numeric numeric::inverse(void) const
638 return numeric(::recip(*value)); // -> CLN
641 const numeric & numeric::add_dyn(const numeric & other) const
643 return static_cast<const numeric &>((new numeric((*value)+(*other.value)))->
644 setflag(status_flags::dynallocated));
647 const numeric & numeric::sub_dyn(const numeric & other) const
649 return static_cast<const numeric &>((new numeric((*value)-(*other.value)))->
650 setflag(status_flags::dynallocated));
653 const numeric & numeric::mul_dyn(const numeric & other) const
655 static const numeric * _num1p=&_num1();
658 } else if (&other==_num1p) {
661 return static_cast<const numeric &>((new numeric((*value)*(*other.value)))->
662 setflag(status_flags::dynallocated));
665 const numeric & numeric::div_dyn(const numeric & other) const
667 if (::zerop(*other.value))
668 throw (std::overflow_error("division by zero"));
669 return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
670 setflag(status_flags::dynallocated));
673 const numeric & numeric::power_dyn(const numeric & other) const
675 static const numeric * _num1p=&_num1();
678 if (::zerop(*value) && other.is_real() && ::minusp(realpart(*other.value)))
679 throw (std::overflow_error("division by zero"));
680 return static_cast<const numeric &>((new numeric(::expt(*value,*other.value)))->
681 setflag(status_flags::dynallocated));
684 const numeric & numeric::operator=(int i)
686 return operator=(numeric(i));
689 const numeric & numeric::operator=(unsigned int i)
691 return operator=(numeric(i));
694 const numeric & numeric::operator=(long i)
696 return operator=(numeric(i));
699 const numeric & numeric::operator=(unsigned long i)
701 return operator=(numeric(i));
704 const numeric & numeric::operator=(double d)
706 return operator=(numeric(d));
709 const numeric & numeric::operator=(const char * s)
711 return operator=(numeric(s));
714 /** Return the complex half-plane (left or right) in which the number lies.
715 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
716 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
718 * @see numeric::compare(const numeric & other) */
719 int numeric::csgn(void) const
723 if (!::zerop(realpart(*value))) {
724 if (::plusp(realpart(*value)))
729 if (::plusp(imagpart(*value)))
736 /** This method establishes a canonical order on all numbers. For complex
737 * numbers this is not possible in a mathematically consistent way but we need
738 * to establish some order and it ought to be fast. So we simply define it
739 * to be compatible with our method csgn.
741 * @return csgn(*this-other)
742 * @see numeric::csgn(void) */
743 int numeric::compare(const numeric & other) const
745 // Comparing two real numbers?
746 if (is_real() && other.is_real())
747 // Yes, just compare them
748 return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
750 // No, first compare real parts
751 cl_signean real_cmp = ::cl_compare(realpart(*value), realpart(*other.value));
755 return ::cl_compare(imagpart(*value), imagpart(*other.value));
759 bool numeric::is_equal(const numeric & other) const
761 return (*value == *other.value);
764 /** True if object is zero. */
765 bool numeric::is_zero(void) const
767 return ::zerop(*value); // -> CLN
770 /** True if object is not complex and greater than zero. */
771 bool numeric::is_positive(void) const
774 return ::plusp(The(cl_R)(*value)); // -> CLN
778 /** True if object is not complex and less than zero. */
779 bool numeric::is_negative(void) const
782 return ::minusp(The(cl_R)(*value)); // -> CLN
786 /** True if object is a non-complex integer. */
787 bool numeric::is_integer(void) const
789 return ::instanceof(*value, cl_I_ring); // -> CLN
792 /** True if object is an exact integer greater than zero. */
793 bool numeric::is_pos_integer(void) const
795 return (is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN
798 /** True if object is an exact integer greater or equal zero. */
799 bool numeric::is_nonneg_integer(void) const
801 return (is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN
804 /** True if object is an exact even integer. */
805 bool numeric::is_even(void) const
807 return (is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN
810 /** True if object is an exact odd integer. */
811 bool numeric::is_odd(void) const
813 return (is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN
816 /** Probabilistic primality test.
818 * @return true if object is exact integer and prime. */
819 bool numeric::is_prime(void) const
821 return (is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN
824 /** True if object is an exact rational number, may even be complex
825 * (denominator may be unity). */
826 bool numeric::is_rational(void) const
828 return ::instanceof(*value, cl_RA_ring); // -> CLN
831 /** True if object is a real integer, rational or float (but not complex). */
832 bool numeric::is_real(void) const
834 return ::instanceof(*value, cl_R_ring); // -> CLN
837 bool numeric::operator==(const numeric & other) const
839 return (*value == *other.value); // -> CLN
842 bool numeric::operator!=(const numeric & other) const
844 return (*value != *other.value); // -> CLN
847 /** True if object is element of the domain of integers extended by I, i.e. is
848 * of the form a+b*I, where a and b are integers. */
849 bool numeric::is_cinteger(void) const
851 if (::instanceof(*value, cl_I_ring))
853 else if (!is_real()) { // complex case, handle n+m*I
854 if (::instanceof(realpart(*value), cl_I_ring) &&
855 ::instanceof(imagpart(*value), cl_I_ring))
861 /** True if object is an exact rational number, may even be complex
862 * (denominator may be unity). */
863 bool numeric::is_crational(void) const
865 if (::instanceof(*value, cl_RA_ring))
867 else if (!is_real()) { // complex case, handle Q(i):
868 if (::instanceof(realpart(*value), cl_RA_ring) &&
869 ::instanceof(imagpart(*value), cl_RA_ring))
875 /** Numerical comparison: less.
877 * @exception invalid_argument (complex inequality) */
878 bool numeric::operator<(const numeric & other) const
880 if (is_real() && other.is_real())
881 return (bool)(The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
882 throw (std::invalid_argument("numeric::operator<(): complex inequality"));
883 return false; // make compiler shut up
886 /** Numerical comparison: less or equal.
888 * @exception invalid_argument (complex inequality) */
889 bool numeric::operator<=(const numeric & other) const
891 if (is_real() && other.is_real())
892 return (bool)(The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
893 throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
894 return false; // make compiler shut up
897 /** Numerical comparison: greater.
899 * @exception invalid_argument (complex inequality) */
900 bool numeric::operator>(const numeric & other) const
902 if (is_real() && other.is_real())
903 return (bool)(The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
904 throw (std::invalid_argument("numeric::operator>(): complex inequality"));
905 return false; // make compiler shut up
908 /** Numerical comparison: greater or equal.
910 * @exception invalid_argument (complex inequality) */
911 bool numeric::operator>=(const numeric & other) const
913 if (is_real() && other.is_real())
914 return (bool)(The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
915 throw (std::invalid_argument("numeric::operator>=(): complex inequality"));
916 return false; // make compiler shut up
919 /** Converts numeric types to machine's int. You should check with
920 * is_integer() if the number is really an integer before calling this method.
921 * You may also consider checking the range first. */
922 int numeric::to_int(void) const
924 GINAC_ASSERT(is_integer());
925 return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN
928 /** Converts numeric types to machine's long. You should check with
929 * is_integer() if the number is really an integer before calling this method.
930 * You may also consider checking the range first. */
931 long numeric::to_long(void) const
933 GINAC_ASSERT(is_integer());
934 return ::cl_I_to_long(The(cl_I)(*value)); // -> CLN
937 /** Converts numeric types to machine's double. You should check with is_real()
938 * if the number is really not complex before calling this method. */
939 double numeric::to_double(void) const
941 GINAC_ASSERT(is_real());
942 return ::cl_double_approx(realpart(*value)); // -> CLN
945 /** Real part of a number. */
946 numeric numeric::real(void) const
948 return numeric(::realpart(*value)); // -> CLN
951 /** Imaginary part of a number. */
952 numeric numeric::imag(void) const
954 return numeric(::imagpart(*value)); // -> CLN
958 // Unfortunately, CLN did not provide an official way to access the numerator
959 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
960 // one finds how it works internally in src/rational/cl_RA.h:
961 struct cl_heap_ratio : cl_heap {
966 inline cl_heap_ratio* TheRatio (const cl_N& obj)
967 { return (cl_heap_ratio*)(obj.pointer); }
968 #endif // ndef SANE_LINKER
970 /** Numerator. Computes the numerator of rational numbers, rationalized
971 * numerator of complex if real and imaginary part are both rational numbers
972 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
974 numeric numeric::numer(void) const
977 return numeric(*this);
980 else if (::instanceof(*value, cl_RA_ring)) {
981 return numeric(::numerator(The(cl_RA)(*value)));
983 else if (!is_real()) { // complex case, handle Q(i):
984 cl_R r = ::realpart(*value);
985 cl_R i = ::imagpart(*value);
986 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
987 return numeric(*this);
988 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
989 return numeric(complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i))));
990 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
991 return numeric(complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r))));
992 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring)) {
993 cl_I s = lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)));
994 return numeric(complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))),
995 ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i))))));
999 else if (instanceof(*value, cl_RA_ring)) {
1000 return numeric(TheRatio(*value)->numerator);
1002 else if (!is_real()) { // complex case, handle Q(i):
1003 cl_R r = realpart(*value);
1004 cl_R i = imagpart(*value);
1005 if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
1006 return numeric(*this);
1007 if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
1008 return numeric(complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
1009 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
1010 return numeric(complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
1011 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) {
1012 cl_I s = lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
1013 return numeric(complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
1014 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
1017 #endif // def SANE_LINKER
1018 // at least one float encountered
1019 return numeric(*this);
1022 /** Denominator. Computes the denominator of rational numbers, common integer
1023 * denominator of complex if real and imaginary part are both rational numbers
1024 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1025 numeric numeric::denom(void) const
1031 if (instanceof(*value, cl_RA_ring)) {
1032 return numeric(::denominator(The(cl_RA)(*value)));
1034 if (!is_real()) { // complex case, handle Q(i):
1035 cl_R r = realpart(*value);
1036 cl_R i = imagpart(*value);
1037 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
1039 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
1040 return numeric(::denominator(The(cl_RA)(i)));
1041 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
1042 return numeric(::denominator(The(cl_RA)(r)));
1043 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring))
1044 return numeric(lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
1047 if (instanceof(*value, cl_RA_ring)) {
1048 return numeric(TheRatio(*value)->denominator);
1050 if (!is_real()) { // complex case, handle Q(i):
1051 cl_R r = realpart(*value);
1052 cl_R i = imagpart(*value);
1053 if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
1055 if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
1056 return numeric(TheRatio(i)->denominator);
1057 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
1058 return numeric(TheRatio(r)->denominator);
1059 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring))
1060 return numeric(lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
1062 #endif // def SANE_LINKER
1063 // at least one float encountered
1067 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1068 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1069 * 2^(n-1) <= x < 2^n.
1071 * @return number of bits (excluding sign) needed to represent that number
1072 * in two's complement if it is an integer, 0 otherwise. */
1073 int numeric::int_length(void) const
1076 return ::integer_length(The(cl_I)(*value)); // -> CLN
1083 // static member variables
1088 unsigned numeric::precedence = 30;
1094 const numeric some_numeric;
1095 const type_info & typeid_numeric=typeid(some_numeric);
1096 /** Imaginary unit. This is not a constant but a numeric since we are
1097 * natively handing complex numbers anyways. */
1098 const numeric I = numeric(complex(cl_I(0),cl_I(1)));
1101 /** Exponential function.
1103 * @return arbitrary precision numerical exp(x). */
1104 const numeric exp(const numeric & x)
1106 return ::exp(*x.value); // -> CLN
1110 /** Natural logarithm.
1112 * @param z complex number
1113 * @return arbitrary precision numerical log(x).
1114 * @exception overflow_error (logarithmic singularity) */
1115 const numeric log(const numeric & z)
1118 throw (std::overflow_error("log(): logarithmic singularity"));
1119 return ::log(*z.value); // -> CLN
1123 /** Numeric sine (trigonometric function).
1125 * @return arbitrary precision numerical sin(x). */
1126 const numeric sin(const numeric & x)
1128 return ::sin(*x.value); // -> CLN
1132 /** Numeric cosine (trigonometric function).
1134 * @return arbitrary precision numerical cos(x). */
1135 const numeric cos(const numeric & x)
1137 return ::cos(*x.value); // -> CLN
1141 /** Numeric tangent (trigonometric function).
1143 * @return arbitrary precision numerical tan(x). */
1144 const numeric tan(const numeric & x)
1146 return ::tan(*x.value); // -> CLN
1150 /** Numeric inverse sine (trigonometric function).
1152 * @return arbitrary precision numerical asin(x). */
1153 const numeric asin(const numeric & x)
1155 return ::asin(*x.value); // -> CLN
1159 /** Numeric inverse cosine (trigonometric function).
1161 * @return arbitrary precision numerical acos(x). */
1162 const numeric acos(const numeric & x)
1164 return ::acos(*x.value); // -> CLN
1170 * @param z complex number
1172 * @exception overflow_error (logarithmic singularity) */
1173 const numeric atan(const numeric & x)
1176 x.real().is_zero() &&
1177 !abs(x.imag()).is_equal(_num1()))
1178 throw (std::overflow_error("atan(): logarithmic singularity"));
1179 return ::atan(*x.value); // -> CLN
1185 * @param x real number
1186 * @param y real number
1187 * @return atan(y/x) */
1188 const numeric atan(const numeric & y, const numeric & x)
1190 if (x.is_real() && y.is_real())
1191 return ::atan(realpart(*x.value), realpart(*y.value)); // -> CLN
1193 throw (std::invalid_argument("numeric::atan(): complex argument"));
1197 /** Numeric hyperbolic sine (trigonometric function).
1199 * @return arbitrary precision numerical sinh(x). */
1200 const numeric sinh(const numeric & x)
1202 return ::sinh(*x.value); // -> CLN
1206 /** Numeric hyperbolic cosine (trigonometric function).
1208 * @return arbitrary precision numerical cosh(x). */
1209 const numeric cosh(const numeric & x)
1211 return ::cosh(*x.value); // -> CLN
1215 /** Numeric hyperbolic tangent (trigonometric function).
1217 * @return arbitrary precision numerical tanh(x). */
1218 const numeric tanh(const numeric & x)
1220 return ::tanh(*x.value); // -> CLN
1224 /** Numeric inverse hyperbolic sine (trigonometric function).
1226 * @return arbitrary precision numerical asinh(x). */
1227 const numeric asinh(const numeric & x)
1229 return ::asinh(*x.value); // -> CLN
1233 /** Numeric inverse hyperbolic cosine (trigonometric function).
1235 * @return arbitrary precision numerical acosh(x). */
1236 const numeric acosh(const numeric & x)
1238 return ::acosh(*x.value); // -> CLN
1242 /** Numeric inverse hyperbolic tangent (trigonometric function).
1244 * @return arbitrary precision numerical atanh(x). */
1245 const numeric atanh(const numeric & x)
1247 return ::atanh(*x.value); // -> CLN
1251 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1252 * integer arguments. */
1253 const numeric zeta(const numeric & x)
1255 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1256 // only knows about integer arguments and zeta(3).evalf() automatically
1257 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1258 // being an exact zero for CLN, which can be tested and then we can just
1259 // pass the number casted to an int:
1261 int aux = (int)(::cl_double_approx(realpart(*x.value)));
1262 if (zerop(*x.value-aux))
1263 return ::cl_zeta(aux); // -> CLN
1265 clog << "zeta(" << x
1266 << "): Does anybody know good way to calculate this numerically?"
1272 /** The gamma function.
1273 * This is only a stub! */
1274 const numeric gamma(const numeric & x)
1276 clog << "gamma(" << x
1277 << "): Does anybody know good way to calculate this numerically?"
1283 /** The psi function (aka polygamma function).
1284 * This is only a stub! */
1285 const numeric psi(const numeric & x)
1288 << "): Does anybody know good way to calculate this numerically?"
1294 /** The psi functions (aka polygamma functions).
1295 * This is only a stub! */
1296 const numeric psi(const numeric & n, const numeric & x)
1298 clog << "psi(" << n << "," << x
1299 << "): Does anybody know good way to calculate this numerically?"
1305 /** Factorial combinatorial function.
1307 * @param n integer argument >= 0
1308 * @exception range_error (argument must be integer >= 0) */
1309 const numeric factorial(const numeric & n)
1311 if (!n.is_nonneg_integer())
1312 throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
1313 return numeric(::factorial(n.to_int())); // -> CLN
1317 /** The double factorial combinatorial function. (Scarcely used, but still
1318 * useful in cases, like for exact results of Gamma(n+1/2) for instance.)
1320 * @param n integer argument >= -1
1321 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1322 * @exception range_error (argument must be integer >= -1) */
1323 const numeric doublefactorial(const numeric & n)
1325 if (n == numeric(-1)) {
1328 if (!n.is_nonneg_integer()) {
1329 throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
1331 return numeric(::doublefactorial(n.to_int())); // -> CLN
1335 /** The Binomial coefficients. It computes the binomial coefficients. For
1336 * integer n and k and positive n this is the number of ways of choosing k
1337 * objects from n distinct objects. If n is negative, the formula
1338 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1339 const numeric binomial(const numeric & n, const numeric & k)
1341 if (n.is_integer() && k.is_integer()) {
1342 if (n.is_nonneg_integer()) {
1343 if (k.compare(n)!=1 && k.compare(_num0())!=-1)
1344 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1348 return _num_1().power(k)*binomial(k-n-_num1(),k);
1352 // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit
1353 throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
1357 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1358 * in the expansion of the function x/(e^x-1).
1360 * @return the nth Bernoulli number (a rational number).
1361 * @exception range_error (argument must be integer >= 0) */
1362 const numeric bernoulli(const numeric & nn)
1364 if (!nn.is_integer() || nn.is_negative())
1365 throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
1368 if (!nn.compare(_num1()))
1369 return numeric(-1,2);
1372 // Until somebody has the Blues and comes up with a much better idea and
1373 // codes it (preferably in CLN) we make this a remembering function which
1374 // computes its results using the formula
1375 // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
1377 static vector<numeric> results;
1378 static int highest_result = -1;
1379 int n = nn.sub(_num2()).div(_num2()).to_int();
1380 if (n <= highest_result)
1382 if (results.capacity() < (unsigned)(n+1))
1383 results.reserve(n+1);
1385 numeric tmp; // used to store the sum
1386 for (int i=highest_result+1; i<=n; ++i) {
1387 // the first two elements:
1388 tmp = numeric(-2*i-1,2);
1389 // accumulate the remaining elements:
1390 for (int j=0; j<i; ++j)
1391 tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
1392 // divide by -(nn+1) and store result:
1393 results.push_back(-tmp/numeric(2*i+3));
1400 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1401 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1403 * @param n an integer
1404 * @return the nth Fibonacci number F(n) (an integer number)
1405 * @exception range_error (argument must be an integer) */
1406 const numeric fibonacci(const numeric & n)
1408 if (!n.is_integer()) {
1409 throw (std::range_error("numeric::fibonacci(): argument must be integer"));
1411 // For positive arguments compute the nearest integer to
1412 // ((1+sqrt(5))/2)^n/sqrt(5). For negative arguments, apply an additional
1413 // sign. Note that we are falling back to longs, but this should suffice
1416 const long nn = ::abs(n.to_double());
1417 if (n.is_negative() && n.is_even())
1420 // Need a precision of ((1+sqrt(5))/2)^-n.
1421 cl_float_format_t prec = ::cl_float_format((int)(0.208987641*nn+5));
1422 cl_R sqrt5 = ::sqrt(::cl_float(5,prec));
1423 cl_R phi = (1+sqrt5)/2;
1424 return numeric(::round1(::expt(phi,nn)/sqrt5)*sig);
1428 /** Absolute value. */
1429 numeric abs(const numeric & x)
1431 return ::abs(*x.value); // -> CLN
1435 /** Modulus (in positive representation).
1436 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1437 * sign of a or is zero. This is different from Maple's modp, where the sign
1438 * of b is ignored. It is in agreement with Mathematica's Mod.
1440 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1441 * integer, 0 otherwise. */
1442 numeric mod(const numeric & a, const numeric & b)
1444 if (a.is_integer() && b.is_integer())
1445 return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1447 return _num0(); // Throw?
1451 /** Modulus (in symmetric representation).
1452 * Equivalent to Maple's mods.
1454 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1455 numeric smod(const numeric & a, const numeric & b)
1457 // FIXME: Should this become a member function?
1458 if (a.is_integer() && b.is_integer()) {
1459 cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1;
1460 return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
1462 return _num0(); // Throw?
1466 /** Numeric integer remainder.
1467 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1468 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1469 * sign of a or is zero.
1471 * @return remainder of a/b if both are integer, 0 otherwise. */
1472 numeric irem(const numeric & a, const numeric & b)
1474 if (a.is_integer() && b.is_integer())
1475 return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1477 return _num0(); // Throw?
1481 /** Numeric integer remainder.
1482 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1483 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1484 * and irem(a,b) has the sign of a or is zero.
1486 * @return remainder of a/b and quotient stored in q if both are integer,
1488 numeric irem(const numeric & a, const numeric & b, numeric & q)
1490 if (a.is_integer() && b.is_integer()) { // -> CLN
1491 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1492 q = rem_quo.quotient;
1493 return rem_quo.remainder;
1497 return _num0(); // Throw?
1502 /** Numeric integer quotient.
1503 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1505 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1506 numeric iquo(const numeric & a, const numeric & b)
1508 if (a.is_integer() && b.is_integer())
1509 return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1511 return _num0(); // Throw?
1515 /** Numeric integer quotient.
1516 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1517 * r == a - iquo(a,b,r)*b.
1519 * @return truncated quotient of a/b and remainder stored in r if both are
1520 * integer, 0 otherwise. */
1521 numeric iquo(const numeric & a, const numeric & b, numeric & r)
1523 if (a.is_integer() && b.is_integer()) { // -> CLN
1524 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1525 r = rem_quo.remainder;
1526 return rem_quo.quotient;
1529 return _num0(); // Throw?
1534 /** Numeric square root.
1535 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1536 * should return integer 2.
1538 * @param z numeric argument
1539 * @return square root of z. Branch cut along negative real axis, the negative
1540 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1541 * where imag(z)>0. */
1542 numeric sqrt(const numeric & z)
1544 return ::sqrt(*z.value); // -> CLN
1548 /** Integer numeric square root. */
1549 numeric isqrt(const numeric & x)
1551 if (x.is_integer()) {
1553 ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
1556 return _num0(); // Throw?
1560 /** Greatest Common Divisor.
1562 * @return The GCD of two numbers if both are integer, a numerical 1
1563 * if they are not. */
1564 numeric gcd(const numeric & a, const numeric & b)
1566 if (a.is_integer() && b.is_integer())
1567 return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1573 /** Least Common Multiple.
1575 * @return The LCM of two numbers if both are integer, the product of those
1576 * two numbers if they are not. */
1577 numeric lcm(const numeric & a, const numeric & b)
1579 if (a.is_integer() && b.is_integer())
1580 return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1582 return *a.value * *b.value;
1586 /** Floating point evaluation of Archimedes' constant Pi. */
1589 return numeric(cl_pi(cl_default_float_format)); // -> CLN
1593 /** Floating point evaluation of Euler's constant Gamma. */
1594 ex EulerGammaEvalf(void)
1596 return numeric(cl_eulerconst(cl_default_float_format)); // -> CLN
1600 /** Floating point evaluation of Catalan's constant. */
1601 ex CatalanEvalf(void)
1603 return numeric(cl_catalanconst(cl_default_float_format)); // -> CLN
1607 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1608 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1609 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1610 _numeric_digits::_numeric_digits()
1615 cl_default_float_format = cl_float_format(17);
1619 _numeric_digits& _numeric_digits::operator=(long prec)
1622 cl_default_float_format = cl_float_format(prec);
1627 _numeric_digits::operator long()
1629 return (long)digits;
1633 void _numeric_digits::print(ostream & os) const
1635 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1640 ostream& operator<<(ostream& os, const _numeric_digits & e)
1647 // static member variables
1652 bool _numeric_digits::too_late = false;
1654 /** Accuracy in decimal digits. Only object of this type! Can be set using
1655 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1656 _numeric_digits Digits;
1658 #ifndef NO_GINAC_NAMESPACE
1659 } // namespace GiNaC
1660 #endif // ndef NO_GINAC_NAMESPACE