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 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
35 // CLN should not pollute the global namespace, hence we include it here
36 // instead of in some header file where it would propagate to other parts:
43 #ifndef NO_GINAC_NAMESPACE
45 #endif // ndef NO_GINAC_NAMESPACE
47 // linker has no problems finding text symbols for numerator or denominator
51 // default constructor, destructor, copy constructor assignment
52 // operator and helpers
57 /** default ctor. Numerically it initializes to an integer zero. */
58 numeric::numeric() : basic(TINFO_numeric)
60 debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT);
64 setflag(status_flags::evaluated|
65 status_flags::hash_calculated);
70 debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT);
74 numeric::numeric(numeric const & other)
76 debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT);
80 numeric const & numeric::operator=(numeric const & other)
82 debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT);
92 void numeric::copy(numeric const & other)
95 value = new cl_N(*other.value);
98 void numeric::destroy(bool call_parent)
101 if (call_parent) basic::destroy(call_parent);
105 // other constructors
110 numeric::numeric(int i) : basic(TINFO_numeric)
112 debugmsg("numeric constructor from int",LOGLEVEL_CONSTRUCT);
113 // Not the whole int-range is available if we don't cast to long
114 // first. This is due to the behaviour of the cl_I-ctor, which
115 // emphasizes efficiency:
116 value = new cl_I((long) i);
118 setflag(status_flags::evaluated|
119 status_flags::hash_calculated);
122 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
124 debugmsg("numeric constructor from uint",LOGLEVEL_CONSTRUCT);
125 // Not the whole uint-range is available if we don't cast to ulong
126 // first. This is due to the behaviour of the cl_I-ctor, which
127 // emphasizes efficiency:
128 value = new cl_I((unsigned long)i);
130 setflag(status_flags::evaluated|
131 status_flags::hash_calculated);
134 numeric::numeric(long i) : basic(TINFO_numeric)
136 debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT);
139 setflag(status_flags::evaluated|
140 status_flags::hash_calculated);
143 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
145 debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT);
148 setflag(status_flags::evaluated|
149 status_flags::hash_calculated);
152 /** Ctor for rational numerics a/b.
154 * @exception overflow_error (division by zero) */
155 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
157 debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
159 throw (std::overflow_error("division by zero"));
160 value = new cl_I(numer);
161 *value = *value / cl_I(denom);
163 setflag(status_flags::evaluated|
164 status_flags::hash_calculated);
167 numeric::numeric(double d) : basic(TINFO_numeric)
169 debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT);
170 // We really want to explicitly use the type cl_LF instead of the
171 // more general cl_F, since that would give us a cl_DF only which
172 // will not be promoted to cl_LF if overflow occurs:
174 *value = cl_float(d, cl_default_float_format);
176 setflag(status_flags::evaluated|
177 status_flags::hash_calculated);
180 numeric::numeric(char const *s) : basic(TINFO_numeric)
181 { // MISSING: treatment of complex and ints and rationals.
182 debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT);
184 value = new cl_LF(s);
188 setflag(status_flags::evaluated|
189 status_flags::hash_calculated);
192 /** Ctor from CLN types. This is for the initiated user or internal use
194 numeric::numeric(cl_N const & z) : basic(TINFO_numeric)
196 debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT);
199 setflag(status_flags::evaluated|
200 status_flags::hash_calculated);
204 // functions overriding virtual functions from bases classes
209 basic * numeric::duplicate() const
211 debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE);
212 return new numeric(*this);
215 // The method printraw doesn't do much, it simply uses CLN's operator<<() for
216 // output, which is ugly but reliable. Examples:
218 void numeric::printraw(ostream & os) const
220 debugmsg("numeric printraw", LOGLEVEL_PRINT);
221 os << "numeric(" << *value << ")";
224 // The method print adds to the output so it blends more consistently together
225 // with the other routines and produces something compatible to Maple input.
226 void numeric::print(ostream & os, unsigned upper_precedence) const
228 debugmsg("numeric print", LOGLEVEL_PRINT);
230 // case 1, real: x or -x
231 if ((precedence<=upper_precedence) && (!is_pos_integer())) {
232 os << "(" << *value << ")";
237 // case 2, imaginary: y*I or -y*I
238 if (realpart(*value) == 0) {
239 if ((precedence<=upper_precedence) && (imagpart(*value) < 0)) {
240 if (imagpart(*value) == -1) {
243 os << "(" << imagpart(*value) << "*I)";
246 if (imagpart(*value) == 1) {
249 if (imagpart (*value) == -1) {
252 os << imagpart(*value) << "*I";
257 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
258 if (precedence <= upper_precedence) os << "(";
259 os << realpart(*value);
260 if (imagpart(*value) < 0) {
261 if (imagpart(*value) == -1) {
264 os << imagpart(*value) << "*I";
267 if (imagpart(*value) == 1) {
270 os << "+" << imagpart(*value) << "*I";
273 if (precedence <= upper_precedence) os << ")";
278 void numeric::printtree(ostream & os, unsigned indent) const
280 debugmsg("numeric printtree", LOGLEVEL_PRINT);
281 os << string(indent,' ') << *value
283 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
284 << ", flags=" << flags << endl;
287 void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
289 debugmsg("numeric print csrc", LOGLEVEL_PRINT);
290 ios::fmtflags oldflags = os.flags();
291 os.setf(ios::scientific);
292 if (is_rational() && !is_integer()) {
293 if (compare(numZERO()) > 0) {
295 if (type == csrc_types::ctype_cl_N)
296 os << "cl_F(\"" << numer().evalf() << "\")";
298 os << numer().to_double();
301 if (type == csrc_types::ctype_cl_N)
302 os << "cl_F(\"" << -numer().evalf() << "\")";
304 os << -numer().to_double();
307 if (type == csrc_types::ctype_cl_N)
308 os << "cl_F(\"" << denom().evalf() << "\")";
310 os << denom().to_double();
313 if (type == csrc_types::ctype_cl_N)
314 os << "cl_F(\"" << evalf() << "\")";
321 bool numeric::info(unsigned inf) const
324 case info_flags::numeric:
325 case info_flags::polynomial:
326 case info_flags::rational_function:
328 case info_flags::real:
330 case info_flags::rational:
331 case info_flags::rational_polynomial:
332 return is_rational();
333 case info_flags::crational:
334 case info_flags::crational_polynomial:
335 return is_crational();
336 case info_flags::integer:
337 case info_flags::integer_polynomial:
339 case info_flags::cinteger:
340 case info_flags::cinteger_polynomial:
341 return is_cinteger();
342 case info_flags::positive:
343 return is_positive();
344 case info_flags::negative:
345 return is_negative();
346 case info_flags::nonnegative:
347 return compare(numZERO())>=0;
348 case info_flags::posint:
349 return is_pos_integer();
350 case info_flags::negint:
351 return is_integer() && (compare(numZERO())<0);
352 case info_flags::nonnegint:
353 return is_nonneg_integer();
354 case info_flags::even:
356 case info_flags::odd:
358 case info_flags::prime:
364 /** Cast numeric into a floating-point object. For example exact numeric(1) is
365 * returned as a 1.0000000000000000000000 and so on according to how Digits is
368 * @param level ignored, but needed for overriding basic::evalf.
369 * @return an ex-handle to a numeric. */
370 ex numeric::evalf(int level) const
372 // level can safely be discarded for numeric objects.
373 return numeric(cl_float(1.0, cl_default_float_format) * (*value)); // -> CLN
378 int numeric::compare_same_type(basic const & other) const
380 GINAC_ASSERT(is_exactly_of_type(other, numeric));
381 numeric const & o = static_cast<numeric &>(const_cast<basic &>(other));
383 if (*value == *o.value) {
390 bool numeric::is_equal_same_type(basic const & other) const
392 GINAC_ASSERT(is_exactly_of_type(other,numeric));
393 numeric const *o = static_cast<numeric const *>(&other);
399 unsigned numeric::calchash(void) const
401 double d=to_double();
407 return 0x88000000U+s*unsigned(d/0x07FF0000);
413 // new virtual functions which can be overridden by derived classes
419 // non-virtual functions in this class
424 /** Numerical addition method. Adds argument to *this and returns result as
425 * a new numeric object. */
426 numeric numeric::add(numeric const & other) const
428 return numeric((*value)+(*other.value));
431 /** Numerical subtraction method. Subtracts argument from *this and returns
432 * result as a new numeric object. */
433 numeric numeric::sub(numeric const & other) const
435 return numeric((*value)-(*other.value));
438 /** Numerical multiplication method. Multiplies *this and argument and returns
439 * result as a new numeric object. */
440 numeric numeric::mul(numeric const & other) const
442 static const numeric * numONEp=&numONE();
445 } else if (&other==numONEp) {
448 return numeric((*value)*(*other.value));
451 /** Numerical division method. Divides *this by argument and returns result as
452 * a new numeric object.
454 * @exception overflow_error (division by zero) */
455 numeric numeric::div(numeric const & other) const
457 if (::zerop(*other.value))
458 throw (std::overflow_error("division by zero"));
459 return numeric((*value)/(*other.value));
462 numeric numeric::power(numeric const & other) const
464 static const numeric * numONEp=&numONE();
467 if (::zerop(*value) && other.is_real() && ::minusp(realpart(*other.value)))
468 throw (std::overflow_error("division by zero"));
469 return numeric(::expt(*value,*other.value));
472 /** Inverse of a number. */
473 numeric numeric::inverse(void) const
475 return numeric(::recip(*value)); // -> CLN
478 numeric const & numeric::add_dyn(numeric const & other) const
480 return static_cast<numeric const &>((new numeric((*value)+(*other.value)))->
481 setflag(status_flags::dynallocated));
484 numeric const & numeric::sub_dyn(numeric const & other) const
486 return static_cast<numeric const &>((new numeric((*value)-(*other.value)))->
487 setflag(status_flags::dynallocated));
490 numeric const & numeric::mul_dyn(numeric const & other) const
492 static const numeric * numONEp=&numONE();
495 } else if (&other==numONEp) {
498 return static_cast<numeric const &>((new numeric((*value)*(*other.value)))->
499 setflag(status_flags::dynallocated));
502 numeric const & numeric::div_dyn(numeric const & other) const
504 if (::zerop(*other.value))
505 throw (std::overflow_error("division by zero"));
506 return static_cast<numeric const &>((new numeric((*value)/(*other.value)))->
507 setflag(status_flags::dynallocated));
510 numeric const & numeric::power_dyn(numeric const & other) const
512 static const numeric * numONEp=&numONE();
515 if (::zerop(*value) && other.is_real() && ::minusp(realpart(*other.value)))
516 throw (std::overflow_error("division by zero"));
517 return static_cast<numeric const &>((new numeric(::expt(*value,*other.value)))->
518 setflag(status_flags::dynallocated));
521 numeric const & numeric::operator=(int i)
523 return operator=(numeric(i));
526 numeric const & numeric::operator=(unsigned int i)
528 return operator=(numeric(i));
531 numeric const & numeric::operator=(long i)
533 return operator=(numeric(i));
536 numeric const & numeric::operator=(unsigned long i)
538 return operator=(numeric(i));
541 numeric const & numeric::operator=(double d)
543 return operator=(numeric(d));
546 numeric const & numeric::operator=(char const * s)
548 return operator=(numeric(s));
551 /** Return the complex half-plane (left or right) in which the number lies.
552 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
553 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
555 * @see numeric::compare(numeric const & other) */
556 int numeric::csgn(void) const
560 if (!::zerop(realpart(*value))) {
561 if (::plusp(realpart(*value)))
566 if (::plusp(imagpart(*value)))
573 /** This method establishes a canonical order on all numbers. For complex
574 * numbers this is not possible in a mathematically consistent way but we need
575 * to establish some order and it ought to be fast. So we simply define it
576 * to be compatible with our method csgn.
578 * @return csgn(*this-other)
579 * @see numeric::csgn(void) */
580 int numeric::compare(numeric const & other) const
582 // Comparing two real numbers?
583 if (is_real() && other.is_real())
584 // Yes, just compare them
585 return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value));
587 // No, first compare real parts
588 cl_signean real_cmp = ::cl_compare(realpart(*value), realpart(*other.value));
592 return ::cl_compare(imagpart(*value), imagpart(*other.value));
596 bool numeric::is_equal(numeric const & other) const
598 return (*value == *other.value);
601 /** True if object is zero. */
602 bool numeric::is_zero(void) const
604 return ::zerop(*value); // -> CLN
607 /** True if object is not complex and greater than zero. */
608 bool numeric::is_positive(void) const
611 return ::plusp(The(cl_R)(*value)); // -> CLN
615 /** True if object is not complex and less than zero. */
616 bool numeric::is_negative(void) const
619 return ::minusp(The(cl_R)(*value)); // -> CLN
623 /** True if object is a non-complex integer. */
624 bool numeric::is_integer(void) const
626 return ::instanceof(*value, cl_I_ring); // -> CLN
629 /** True if object is an exact integer greater than zero. */
630 bool numeric::is_pos_integer(void) const
632 return (is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN
635 /** True if object is an exact integer greater or equal zero. */
636 bool numeric::is_nonneg_integer(void) const
638 return (is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN
641 /** True if object is an exact even integer. */
642 bool numeric::is_even(void) const
644 return (is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN
647 /** True if object is an exact odd integer. */
648 bool numeric::is_odd(void) const
650 return (is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN
653 /** Probabilistic primality test.
655 * @return true if object is exact integer and prime. */
656 bool numeric::is_prime(void) const
658 return (is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN
661 /** True if object is an exact rational number, may even be complex
662 * (denominator may be unity). */
663 bool numeric::is_rational(void) const
665 return ::instanceof(*value, cl_RA_ring); // -> CLN
668 /** True if object is a real integer, rational or float (but not complex). */
669 bool numeric::is_real(void) const
671 return ::instanceof(*value, cl_R_ring); // -> CLN
674 bool numeric::operator==(numeric const & other) const
676 return (*value == *other.value); // -> CLN
679 bool numeric::operator!=(numeric const & other) const
681 return (*value != *other.value); // -> CLN
684 /** True if object is element of the domain of integers extended by I, i.e. is
685 * of the form a+b*I, where a and b are integers. */
686 bool numeric::is_cinteger(void) const
688 if (::instanceof(*value, cl_I_ring))
690 else if (!is_real()) { // complex case, handle n+m*I
691 if (::instanceof(realpart(*value), cl_I_ring) &&
692 ::instanceof(imagpart(*value), cl_I_ring))
698 /** True if object is an exact rational number, may even be complex
699 * (denominator may be unity). */
700 bool numeric::is_crational(void) const
702 if (::instanceof(*value, cl_RA_ring))
704 else if (!is_real()) { // complex case, handle Q(i):
705 if (::instanceof(realpart(*value), cl_RA_ring) &&
706 ::instanceof(imagpart(*value), cl_RA_ring))
712 /** Numerical comparison: less.
714 * @exception invalid_argument (complex inequality) */
715 bool numeric::operator<(numeric const & other) const
717 if (is_real() && other.is_real())
718 return (bool)(The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
719 throw (std::invalid_argument("numeric::operator<(): complex inequality"));
720 return false; // make compiler shut up
723 /** Numerical comparison: less or equal.
725 * @exception invalid_argument (complex inequality) */
726 bool numeric::operator<=(numeric const & other) const
728 if (is_real() && other.is_real())
729 return (bool)(The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
730 throw (std::invalid_argument("numeric::operator<=(): complex inequality"));
731 return false; // make compiler shut up
734 /** Numerical comparison: greater.
736 * @exception invalid_argument (complex inequality) */
737 bool numeric::operator>(numeric const & other) const
739 if (is_real() && other.is_real())
740 return (bool)(The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
741 throw (std::invalid_argument("numeric::operator>(): complex inequality"));
742 return false; // make compiler shut up
745 /** Numerical comparison: greater or equal.
747 * @exception invalid_argument (complex inequality) */
748 bool numeric::operator>=(numeric const & other) const
750 if (is_real() && other.is_real())
751 return (bool)(The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
752 throw (std::invalid_argument("numeric::operator>=(): complex inequality"));
753 return false; // make compiler shut up
756 /** Converts numeric types to machine's int. You should check with is_integer()
757 * if the number is really an integer before calling this method. */
758 int numeric::to_int(void) const
760 GINAC_ASSERT(is_integer());
761 return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN
764 /** Converts numeric types to machine's double. You should check with is_real()
765 * if the number is really not complex before calling this method. */
766 double numeric::to_double(void) const
768 GINAC_ASSERT(is_real());
769 return ::cl_double_approx(realpart(*value)); // -> CLN
772 /** Real part of a number. */
773 numeric numeric::real(void) const
775 return numeric(::realpart(*value)); // -> CLN
778 /** Imaginary part of a number. */
779 numeric numeric::imag(void) const
781 return numeric(::imagpart(*value)); // -> CLN
785 // Unfortunately, CLN did not provide an official way to access the numerator
786 // or denominator of a rational number (cl_RA). Doing some excavations in CLN
787 // one finds how it works internally in src/rational/cl_RA.h:
788 struct cl_heap_ratio : cl_heap {
793 inline cl_heap_ratio* TheRatio (const cl_N& obj)
794 { return (cl_heap_ratio*)(obj.pointer); }
795 #endif // ndef SANE_LINKER
797 /** Numerator. Computes the numerator of rational numbers, rationalized
798 * numerator of complex if real and imaginary part are both rational numbers
799 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
801 numeric numeric::numer(void) const
804 return numeric(*this);
807 else if (::instanceof(*value, cl_RA_ring)) {
808 return numeric(::numerator(The(cl_RA)(*value)));
810 else if (!is_real()) { // complex case, handle Q(i):
811 cl_R r = ::realpart(*value);
812 cl_R i = ::imagpart(*value);
813 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
814 return numeric(*this);
815 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
816 return numeric(complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i))));
817 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
818 return numeric(complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r))));
819 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring)) {
820 cl_I s = lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)));
821 return numeric(complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))),
822 ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i))))));
826 else if (instanceof(*value, cl_RA_ring)) {
827 return numeric(TheRatio(*value)->numerator);
829 else if (!is_real()) { // complex case, handle Q(i):
830 cl_R r = realpart(*value);
831 cl_R i = imagpart(*value);
832 if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
833 return numeric(*this);
834 if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
835 return numeric(complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator));
836 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
837 return numeric(complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator));
838 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) {
839 cl_I s = lcm(TheRatio(r)->denominator, TheRatio(i)->denominator);
840 return numeric(complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)),
841 TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator))));
844 #endif // def SANE_LINKER
845 // at least one float encountered
846 return numeric(*this);
849 /** Denominator. Computes the denominator of rational numbers, common integer
850 * denominator of complex if real and imaginary part are both rational numbers
851 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
852 numeric numeric::denom(void) const
858 if (instanceof(*value, cl_RA_ring)) {
859 return numeric(::denominator(The(cl_RA)(*value)));
861 if (!is_real()) { // complex case, handle Q(i):
862 cl_R r = realpart(*value);
863 cl_R i = imagpart(*value);
864 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring))
866 if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring))
867 return numeric(::denominator(The(cl_RA)(i)));
868 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring))
869 return numeric(::denominator(The(cl_RA)(r)));
870 if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring))
871 return numeric(lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
874 if (instanceof(*value, cl_RA_ring)) {
875 return numeric(TheRatio(*value)->denominator);
877 if (!is_real()) { // complex case, handle Q(i):
878 cl_R r = realpart(*value);
879 cl_R i = imagpart(*value);
880 if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring))
882 if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring))
883 return numeric(TheRatio(i)->denominator);
884 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring))
885 return numeric(TheRatio(r)->denominator);
886 if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring))
887 return numeric(lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
889 #endif // def SANE_LINKER
890 // at least one float encountered
894 /** Size in binary notation. For integers, this is the smallest n >= 0 such
895 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
896 * 2^(n-1) <= x < 2^n.
898 * @return number of bits (excluding sign) needed to represent that number
899 * in two's complement if it is an integer, 0 otherwise. */
900 int numeric::int_length(void) const
903 return ::integer_length(The(cl_I)(*value)); // -> CLN
910 // static member variables
915 unsigned numeric::precedence = 30;
921 const numeric some_numeric;
922 type_info const & typeid_numeric=typeid(some_numeric);
923 /** Imaginary unit. This is not a constant but a numeric since we are
924 * natively handing complex numbers anyways. */
925 const numeric I = numeric(complex(cl_I(0),cl_I(1)));
931 numeric const & numZERO(void)
933 const static ex eZERO = ex((new numeric(0))->setflag(status_flags::dynallocated));
934 const static numeric * nZERO = static_cast<const numeric *>(eZERO.bp);
938 numeric const & numONE(void)
940 const static ex eONE = ex((new numeric(1))->setflag(status_flags::dynallocated));
941 const static numeric * nONE = static_cast<const numeric *>(eONE.bp);
945 numeric const & numTWO(void)
947 const static ex eTWO = ex((new numeric(2))->setflag(status_flags::dynallocated));
948 const static numeric * nTWO = static_cast<const numeric *>(eTWO.bp);
952 numeric const & numTHREE(void)
954 const static ex eTHREE = ex((new numeric(3))->setflag(status_flags::dynallocated));
955 const static numeric * nTHREE = static_cast<const numeric *>(eTHREE.bp);
959 numeric const & numMINUSONE(void)
961 const static ex eMINUSONE = ex((new numeric(-1))->setflag(status_flags::dynallocated));
962 const static numeric * nMINUSONE = static_cast<const numeric *>(eMINUSONE.bp);
966 numeric const & numHALF(void)
968 const static ex eHALF = ex((new numeric(1, 2))->setflag(status_flags::dynallocated));
969 const static numeric * nHALF = static_cast<const numeric *>(eHALF.bp);
973 /** Exponential function.
975 * @return arbitrary precision numerical exp(x). */
976 numeric exp(numeric const & x)
978 return ::exp(*x.value); // -> CLN
981 /** Natural logarithm.
983 * @param z complex number
984 * @return arbitrary precision numerical log(x).
985 * @exception overflow_error (logarithmic singularity) */
986 numeric log(numeric const & z)
989 throw (std::overflow_error("log(): logarithmic singularity"));
990 return ::log(*z.value); // -> CLN
993 /** Numeric sine (trigonometric function).
995 * @return arbitrary precision numerical sin(x). */
996 numeric sin(numeric const & x)
998 return ::sin(*x.value); // -> CLN
1001 /** Numeric cosine (trigonometric function).
1003 * @return arbitrary precision numerical cos(x). */
1004 numeric cos(numeric const & x)
1006 return ::cos(*x.value); // -> CLN
1009 /** Numeric tangent (trigonometric function).
1011 * @return arbitrary precision numerical tan(x). */
1012 numeric tan(numeric const & x)
1014 return ::tan(*x.value); // -> CLN
1017 /** Numeric inverse sine (trigonometric function).
1019 * @return arbitrary precision numerical asin(x). */
1020 numeric asin(numeric const & x)
1022 return ::asin(*x.value); // -> CLN
1025 /** Numeric inverse cosine (trigonometric function).
1027 * @return arbitrary precision numerical acos(x). */
1028 numeric acos(numeric const & x)
1030 return ::acos(*x.value); // -> CLN
1035 * @param z complex number
1037 * @exception overflow_error (logarithmic singularity) */
1038 numeric atan(numeric const & x)
1041 x.real().is_zero() &&
1042 !abs(x.imag()).is_equal(numONE()))
1043 throw (std::overflow_error("atan(): logarithmic singularity"));
1044 return ::atan(*x.value); // -> CLN
1049 * @param x real number
1050 * @param y real number
1051 * @return atan(y/x) */
1052 numeric atan(numeric const & y, numeric const & x)
1054 if (x.is_real() && y.is_real())
1055 return ::atan(realpart(*x.value), realpart(*y.value)); // -> CLN
1057 throw (std::invalid_argument("numeric::atan(): complex argument"));
1060 /** Numeric hyperbolic sine (trigonometric function).
1062 * @return arbitrary precision numerical sinh(x). */
1063 numeric sinh(numeric const & x)
1065 return ::sinh(*x.value); // -> CLN
1068 /** Numeric hyperbolic cosine (trigonometric function).
1070 * @return arbitrary precision numerical cosh(x). */
1071 numeric cosh(numeric const & x)
1073 return ::cosh(*x.value); // -> CLN
1076 /** Numeric hyperbolic tangent (trigonometric function).
1078 * @return arbitrary precision numerical tanh(x). */
1079 numeric tanh(numeric const & x)
1081 return ::tanh(*x.value); // -> CLN
1084 /** Numeric inverse hyperbolic sine (trigonometric function).
1086 * @return arbitrary precision numerical asinh(x). */
1087 numeric asinh(numeric const & x)
1089 return ::asinh(*x.value); // -> CLN
1092 /** Numeric inverse hyperbolic cosine (trigonometric function).
1094 * @return arbitrary precision numerical acosh(x). */
1095 numeric acosh(numeric const & x)
1097 return ::acosh(*x.value); // -> CLN
1100 /** Numeric inverse hyperbolic tangent (trigonometric function).
1102 * @return arbitrary precision numerical atanh(x). */
1103 numeric atanh(numeric const & x)
1105 return ::atanh(*x.value); // -> CLN
1108 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1109 * integer arguments. */
1110 numeric zeta(numeric const & x)
1112 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1113 // only knows about integer arguments and zeta(3).evalf() automatically
1114 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1115 // being an exact zero for CLN, which can be tested and then we can just
1116 // pass the number casted to an int:
1118 int aux = (int)(::cl_double_approx(realpart(*x.value)));
1119 if (zerop(*x.value-aux))
1120 return ::cl_zeta(aux); // -> CLN
1122 clog << "zeta(" << x
1123 << "): Does anybody know good way to calculate this numerically?"
1128 /** The gamma function.
1129 * This is only a stub! */
1130 numeric gamma(numeric const & x)
1132 clog << "gamma(" << x
1133 << "): Does anybody know good way to calculate this numerically?"
1138 /** The psi function (aka polygamma function).
1139 * This is only a stub! */
1140 numeric psi(numeric const & x)
1143 << "): Does anybody know good way to calculate this numerically?"
1148 /** The psi functions (aka polygamma functions).
1149 * This is only a stub! */
1150 numeric psi(numeric const & n, numeric const & x)
1152 clog << "psi(" << n << "," << x
1153 << "): Does anybody know good way to calculate this numerically?"
1158 /** Factorial combinatorial function.
1160 * @exception range_error (argument must be integer >= 0) */
1161 numeric factorial(numeric const & nn)
1163 if (!nn.is_nonneg_integer())
1164 throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
1165 return numeric(::factorial(nn.to_int())); // -> CLN
1168 /** The double factorial combinatorial function. (Scarcely used, but still
1169 * useful in cases, like for exact results of Gamma(n+1/2) for instance.)
1171 * @param n integer argument >= -1
1172 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == 1 == (-1)!!
1173 * @exception range_error (argument must be integer >= -1) */
1174 numeric doublefactorial(numeric const & nn)
1176 // META-NOTE: The whole shit here will become obsolete and may be moved
1177 // out once CLN learns about double factorial, which should be as soon as
1180 // We store the results separately for even and odd arguments. This has
1181 // the advantage that we don't have to compute any even result at all if
1182 // the function is always called with odd arguments and vice versa. There
1183 // is no tradeoff involved in this, it is guaranteed to save time as well
1184 // as memory. (If this is not enough justification consider the Gamma
1185 // function of half integer arguments: it only needs odd doublefactorials.)
1186 static vector<numeric> evenresults;
1187 static int highest_evenresult = -1;
1188 static vector<numeric> oddresults;
1189 static int highest_oddresult = -1;
1191 if (nn == numeric(-1)) {
1194 if (!nn.is_nonneg_integer()) {
1195 throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
1198 int n = nn.div(numTWO()).to_int();
1199 if (n <= highest_evenresult) {
1200 return evenresults[n];
1202 if (evenresults.capacity() < (unsigned)(n+1)) {
1203 evenresults.reserve(n+1);
1205 if (highest_evenresult < 0) {
1206 evenresults.push_back(numONE());
1207 highest_evenresult=0;
1209 for (int i=highest_evenresult+1; i<=n; i++) {
1210 evenresults.push_back(numeric(evenresults[i-1].mul(numeric(i*2))));
1212 highest_evenresult=n;
1213 return evenresults[n];
1215 int n = nn.sub(numONE()).div(numTWO()).to_int();
1216 if (n <= highest_oddresult) {
1217 return oddresults[n];
1219 if (oddresults.capacity() < (unsigned)n) {
1220 oddresults.reserve(n+1);
1222 if (highest_oddresult < 0) {
1223 oddresults.push_back(numONE());
1224 highest_oddresult=0;
1226 for (int i=highest_oddresult+1; i<=n; i++) {
1227 oddresults.push_back(numeric(oddresults[i-1].mul(numeric(i*2+1))));
1229 highest_oddresult=n;
1230 return oddresults[n];
1234 /** The Binomial coefficients. It computes the binomial coefficients. For
1235 * integer n and k and positive n this is the number of ways of choosing k
1236 * objects from n distinct objects. If n is negative, the formula
1237 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1238 numeric binomial(numeric const & n, numeric const & k)
1240 if (n.is_integer() && k.is_integer()) {
1241 if (n.is_nonneg_integer()) {
1242 if (k.compare(n)!=1 && k.compare(numZERO())!=-1)
1243 return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN
1247 return numMINUSONE().power(k)*binomial(k-n-numONE(),k);
1251 // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit
1252 throw (std::range_error("numeric::binomial(): don´t know how to evaluate that."));
1255 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1256 * in the expansion of the function x/(e^x-1).
1258 * @return the nth Bernoulli number (a rational number).
1259 * @exception range_error (argument must be integer >= 0) */
1260 numeric bernoulli(numeric const & nn)
1262 if (!nn.is_integer() || nn.is_negative())
1263 throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0"));
1266 if (!nn.compare(numONE()))
1267 return numeric(-1,2);
1270 // Until somebody has the Blues and comes up with a much better idea and
1271 // codes it (preferably in CLN) we make this a remembering function which
1272 // computes its results using the formula
1273 // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k))
1275 static vector<numeric> results;
1276 static int highest_result = -1;
1277 int n = nn.sub(numTWO()).div(numTWO()).to_int();
1278 if (n <= highest_result)
1280 if (results.capacity() < (unsigned)(n+1))
1281 results.reserve(n+1);
1283 numeric tmp; // used to store the sum
1284 for (int i=highest_result+1; i<=n; ++i) {
1285 // the first two elements:
1286 tmp = numeric(-2*i-1,2);
1287 // accumulate the remaining elements:
1288 for (int j=0; j<i; ++j)
1289 tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
1290 // divide by -(nn+1) and store result:
1291 results.push_back(-tmp/numeric(2*i+3));
1297 /** Absolute value. */
1298 numeric abs(numeric const & x)
1300 return ::abs(*x.value); // -> CLN
1303 /** Modulus (in positive representation).
1304 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1305 * sign of a or is zero. This is different from Maple's modp, where the sign
1306 * of b is ignored. It is in agreement with Mathematica's Mod.
1308 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1309 * integer, 0 otherwise. */
1310 numeric mod(numeric const & a, numeric const & b)
1312 if (a.is_integer() && b.is_integer())
1313 return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1315 return numZERO(); // Throw?
1318 /** Modulus (in symmetric representation).
1319 * Equivalent to Maple's mods.
1321 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1322 numeric smod(numeric const & a, numeric const & b)
1324 if (a.is_integer() && b.is_integer()) {
1325 cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1;
1326 return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2;
1328 return numZERO(); // Throw?
1331 /** Numeric integer remainder.
1332 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1333 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1334 * sign of a or is zero.
1336 * @return remainder of a/b if both are integer, 0 otherwise. */
1337 numeric irem(numeric const & a, numeric const & b)
1339 if (a.is_integer() && b.is_integer())
1340 return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1342 return numZERO(); // Throw?
1345 /** Numeric integer remainder.
1346 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1347 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1348 * and irem(a,b) has the sign of a or is zero.
1350 * @return remainder of a/b and quotient stored in q if both are integer,
1352 numeric irem(numeric const & a, numeric const & b, numeric & q)
1354 if (a.is_integer() && b.is_integer()) { // -> CLN
1355 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1356 q = rem_quo.quotient;
1357 return rem_quo.remainder;
1361 return numZERO(); // Throw?
1365 /** Numeric integer quotient.
1366 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1368 * @return truncated quotient of a/b if both are integer, 0 otherwise. */
1369 numeric iquo(numeric const & a, numeric const & b)
1371 if (a.is_integer() && b.is_integer())
1372 return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1374 return numZERO(); // Throw?
1377 /** Numeric integer quotient.
1378 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1379 * r == a - iquo(a,b,r)*b.
1381 * @return truncated quotient of a/b and remainder stored in r if both are
1382 * integer, 0 otherwise. */
1383 numeric iquo(numeric const & a, numeric const & b, numeric & r)
1385 if (a.is_integer() && b.is_integer()) { // -> CLN
1386 cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value));
1387 r = rem_quo.remainder;
1388 return rem_quo.quotient;
1391 return numZERO(); // Throw?
1395 /** Numeric square root.
1396 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1397 * should return integer 2.
1399 * @param z numeric argument
1400 * @return square root of z. Branch cut along negative real axis, the negative
1401 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1402 * where imag(z)>0. */
1403 numeric sqrt(numeric const & z)
1405 return ::sqrt(*z.value); // -> CLN
1408 /** Integer numeric square root. */
1409 numeric isqrt(numeric const & x)
1411 if (x.is_integer()) {
1413 ::isqrt(The(cl_I)(*x.value), &root); // -> CLN
1416 return numZERO(); // Throw?
1419 /** Greatest Common Divisor.
1421 * @return The GCD of two numbers if both are integer, a numerical 1
1422 * if they are not. */
1423 numeric gcd(numeric const & a, numeric const & b)
1425 if (a.is_integer() && b.is_integer())
1426 return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1431 /** Least Common Multiple.
1433 * @return The LCM of two numbers if both are integer, the product of those
1434 * two numbers if they are not. */
1435 numeric lcm(numeric const & a, numeric const & b)
1437 if (a.is_integer() && b.is_integer())
1438 return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN
1440 return *a.value * *b.value;
1445 return numeric(cl_pi(cl_default_float_format)); // -> CLN
1448 ex EulerGammaEvalf(void)
1450 return numeric(cl_eulerconst(cl_default_float_format)); // -> CLN
1453 ex CatalanEvalf(void)
1455 return numeric(cl_catalanconst(cl_default_float_format)); // -> CLN
1458 // It initializes to 17 digits, because in CLN cl_float_format(17) turns out to
1459 // be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead
1460 // of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary.
1461 _numeric_digits::_numeric_digits()
1466 cl_default_float_format = cl_float_format(17);
1469 _numeric_digits& _numeric_digits::operator=(long prec)
1472 cl_default_float_format = cl_float_format(prec);
1476 _numeric_digits::operator long()
1478 return (long)digits;
1481 void _numeric_digits::print(ostream & os) const
1483 debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
1487 ostream& operator<<(ostream& os, _numeric_digits const & e)
1494 // static member variables
1499 bool _numeric_digits::too_late = false;
1501 /** Accuracy in decimal digits. Only object of this type! Can be set using
1502 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1503 _numeric_digits Digits;
1505 #ifndef NO_GINAC_NAMESPACE
1506 } // namespace GiNaC
1507 #endif // ndef NO_GINAC_NAMESPACE