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) 1999-2003 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
41 // CLN should pollute the global namespace as little as possible. Hence, we
42 // include most of it here and include only the part needed for properly
43 // declaring cln::cl_number in numeric.h. This can only be safely done in
44 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
45 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
47 #include <cln/output.h>
48 #include <cln/integer_io.h>
49 #include <cln/integer_ring.h>
50 #include <cln/rational_io.h>
51 #include <cln/rational_ring.h>
52 #include <cln/lfloat_class.h>
53 #include <cln/lfloat_io.h>
54 #include <cln/real_io.h>
55 #include <cln/real_ring.h>
56 #include <cln/complex_io.h>
57 #include <cln/complex_ring.h>
58 #include <cln/numtheory.h>
62 GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic)
65 // default ctor, dtor, copy ctor, assignment operator and helpers
68 /** default ctor. Numerically it initializes to an integer zero. */
69 numeric::numeric() : basic(TINFO_numeric)
72 setflag(status_flags::evaluated | status_flags::expanded);
75 void numeric::copy(const numeric &other)
77 inherited::copy(other);
81 DEFAULT_DESTROY(numeric)
89 numeric::numeric(int i) : basic(TINFO_numeric)
91 // Not the whole int-range is available if we don't cast to long
92 // first. This is due to the behaviour of the cl_I-ctor, which
93 // emphasizes efficiency. However, if the integer is small enough
94 // we save space and dereferences by using an immediate type.
95 // (C.f. <cln/object.h>)
96 if (i < (1L << (cl_value_len-1)) && i >= -(1L << (cl_value_len-1)))
99 value = cln::cl_I(static_cast<long>(i));
100 setflag(status_flags::evaluated | status_flags::expanded);
104 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
106 // Not the whole uint-range is available if we don't cast to ulong
107 // first. This is due to the behaviour of the cl_I-ctor, which
108 // emphasizes efficiency. However, if the integer is small enough
109 // we save space and dereferences by using an immediate type.
110 // (C.f. <cln/object.h>)
111 if (i < (1U << (cl_value_len-1)))
112 value = cln::cl_I(i);
114 value = cln::cl_I(static_cast<unsigned long>(i));
115 setflag(status_flags::evaluated | status_flags::expanded);
119 numeric::numeric(long i) : basic(TINFO_numeric)
121 value = cln::cl_I(i);
122 setflag(status_flags::evaluated | status_flags::expanded);
126 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
128 value = cln::cl_I(i);
129 setflag(status_flags::evaluated | status_flags::expanded);
133 /** Constructor for rational numerics a/b.
135 * @exception overflow_error (division by zero) */
136 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
139 throw std::overflow_error("division by zero");
140 value = cln::cl_I(numer) / cln::cl_I(denom);
141 setflag(status_flags::evaluated | status_flags::expanded);
145 numeric::numeric(double d) : basic(TINFO_numeric)
147 // We really want to explicitly use the type cl_LF instead of the
148 // more general cl_F, since that would give us a cl_DF only which
149 // will not be promoted to cl_LF if overflow occurs:
150 value = cln::cl_float(d, cln::default_float_format);
151 setflag(status_flags::evaluated | status_flags::expanded);
155 /** ctor from C-style string. It also accepts complex numbers in GiNaC
156 * notation like "2+5*I". */
157 numeric::numeric(const char *s) : basic(TINFO_numeric)
159 cln::cl_N ctorval = 0;
160 // parse complex numbers (functional but not completely safe, unfortunately
161 // std::string does not understand regexpese):
162 // ss should represent a simple sum like 2+5*I
164 std::string::size_type delim;
166 // make this implementation safe by adding explicit sign
167 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
170 // We use 'E' as exponent marker in the output, but some people insist on
171 // writing 'e' at input, so let's substitute them right at the beginning:
172 while ((delim = ss.find("e"))!=std::string::npos)
173 ss.replace(delim,1,"E");
177 // chop ss into terms from left to right
179 bool imaginary = false;
180 delim = ss.find_first_of(std::string("+-"),1);
181 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
182 if (delim!=std::string::npos && ss.at(delim-1)=='E')
183 delim = ss.find_first_of(std::string("+-"),delim+1);
184 term = ss.substr(0,delim);
185 if (delim!=std::string::npos)
186 ss = ss.substr(delim);
187 // is the term imaginary?
188 if (term.find("I")!=std::string::npos) {
190 term.erase(term.find("I"),1);
192 if (term.find("*")!=std::string::npos)
193 term.erase(term.find("*"),1);
194 // correct for trivial +/-I without explicit factor on I:
199 if (term.find('.')!=std::string::npos || term.find('E')!=std::string::npos) {
200 // CLN's short type cl_SF is not very useful within the GiNaC
201 // framework where we are mainly interested in the arbitrary
202 // precision type cl_LF. Hence we go straight to the construction
203 // of generic floats. In order to create them we have to convert
204 // our own floating point notation used for output and construction
205 // from char * to CLN's generic notation:
206 // 3.14 --> 3.14e0_<Digits>
207 // 31.4E-1 --> 31.4e-1_<Digits>
209 // No exponent marker? Let's add a trivial one.
210 if (term.find("E")==std::string::npos)
213 term = term.replace(term.find("E"),1,"e");
214 // append _<Digits> to term
215 term += "_" + ToString((unsigned)Digits);
216 // construct float using cln::cl_F(const char *) ctor.
218 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
220 ctorval = ctorval + cln::cl_F(term.c_str());
222 // this is not a floating point number...
224 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
226 ctorval = ctorval + cln::cl_R(term.c_str());
228 } while (delim != std::string::npos);
230 setflag(status_flags::evaluated | status_flags::expanded);
234 /** Ctor from CLN types. This is for the initiated user or internal use
236 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
239 setflag(status_flags::evaluated | status_flags::expanded);
246 numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
248 cln::cl_N ctorval = 0;
250 // Read number as string
252 if (n.find_string("number", str)) {
253 std::istringstream s(str);
254 cln::cl_idecoded_float re, im;
258 case 'R': // Integer-decoded real number
259 s >> re.sign >> re.mantissa >> re.exponent;
260 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
262 case 'C': // Integer-decoded complex number
263 s >> re.sign >> re.mantissa >> re.exponent;
264 s >> im.sign >> im.mantissa >> im.exponent;
265 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
266 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
268 default: // Ordinary number
275 setflag(status_flags::evaluated | status_flags::expanded);
278 void numeric::archive(archive_node &n) const
280 inherited::archive(n);
282 // Write number as string
283 std::ostringstream s;
284 if (this->is_crational())
285 s << cln::the<cln::cl_N>(value);
287 // Non-rational numbers are written in an integer-decoded format
288 // to preserve the precision
289 if (this->is_real()) {
290 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
292 s << re.sign << " " << re.mantissa << " " << re.exponent;
294 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
295 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
297 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
298 s << im.sign << " " << im.mantissa << " " << im.exponent;
301 n.add_string("number", s.str());
304 DEFAULT_UNARCHIVE(numeric)
307 // functions overriding virtual functions from base classes
310 /** Helper function to print a real number in a nicer way than is CLN's
311 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
312 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
313 * long as it only uses cl_LF and no other floating point types that we might
314 * want to visibly distinguish from cl_LF.
316 * @see numeric::print() */
317 static void print_real_number(const print_context & c, const cln::cl_R & x)
319 cln::cl_print_flags ourflags;
320 if (cln::instanceof(x, cln::cl_RA_ring)) {
321 // case 1: integer or rational
322 if (cln::instanceof(x, cln::cl_I_ring) ||
323 !is_a<print_latex>(c)) {
324 cln::print_real(c.s, ourflags, x);
325 } else { // rational output in LaTeX context
329 cln::print_real(c.s, ourflags, cln::abs(cln::numerator(cln::the<cln::cl_RA>(x))));
331 cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
336 // make CLN believe this number has default_float_format, so it prints
337 // 'E' as exponent marker instead of 'L':
338 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
339 cln::print_real(c.s, ourflags, x);
343 /** Helper function to print integer number in C++ source format.
345 * @see numeric::print() */
346 static void print_integer_csrc(const print_context & c, const cln::cl_I & x)
348 // Print small numbers in compact float format, but larger numbers in
350 const int max_cln_int = 536870911; // 2^29-1
351 if (x >= cln::cl_I(-max_cln_int) && x <= cln::cl_I(max_cln_int))
352 c.s << cln::cl_I_to_int(x) << ".0";
354 c.s << cln::double_approx(x);
357 /** Helper function to print real number in C++ source format.
359 * @see numeric::print() */
360 static void print_real_csrc(const print_context & c, const cln::cl_R & x)
362 if (cln::instanceof(x, cln::cl_I_ring)) {
365 print_integer_csrc(c, cln::the<cln::cl_I>(x));
367 } else if (cln::instanceof(x, cln::cl_RA_ring)) {
370 const cln::cl_I numer = cln::numerator(cln::the<cln::cl_RA>(x));
371 const cln::cl_I denom = cln::denominator(cln::the<cln::cl_RA>(x));
372 if (cln::plusp(x) > 0) {
374 print_integer_csrc(c, numer);
377 print_integer_csrc(c, -numer);
380 print_integer_csrc(c, denom);
386 c.s << cln::double_approx(x);
390 /** Helper function to print real number in C++ source format using cl_N types.
392 * @see numeric::print() */
393 static void print_real_cl_N(const print_context & c, const cln::cl_R & x)
395 if (cln::instanceof(x, cln::cl_I_ring)) {
398 c.s << "cln::cl_I(\"";
399 print_real_number(c, x);
402 } else if (cln::instanceof(x, cln::cl_RA_ring)) {
405 cln::cl_print_flags ourflags;
406 c.s << "cln::cl_RA(\"";
407 cln::print_rational(c.s, ourflags, cln::the<cln::cl_RA>(x));
413 c.s << "cln::cl_F(\"";
414 print_real_number(c, cln::cl_float(1.0, cln::default_float_format) * x);
415 c.s << "_" << Digits << "\")";
419 /** This method adds to the output so it blends more consistently together
420 * with the other routines and produces something compatible to ginsh input.
422 * @see print_real_number() */
423 void numeric::print(const print_context & c, unsigned level) const
425 if (is_a<print_tree>(c)) {
427 c.s << std::string(level, ' ') << cln::the<cln::cl_N>(value)
428 << " (" << class_name() << ")"
429 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
432 } else if (is_a<print_csrc_cl_N>(c)) {
435 if (this->is_real()) {
438 print_real_cl_N(c, cln::the<cln::cl_R>(value));
443 c.s << "cln::complex(";
444 print_real_cl_N(c, cln::realpart(cln::the<cln::cl_N>(value)));
446 print_real_cl_N(c, cln::imagpart(cln::the<cln::cl_N>(value)));
450 } else if (is_a<print_csrc>(c)) {
453 std::ios::fmtflags oldflags = c.s.flags();
454 c.s.setf(std::ios::scientific);
455 int oldprec = c.s.precision();
458 if (is_a<print_csrc_double>(c))
463 if (this->is_real()) {
466 print_real_csrc(c, cln::the<cln::cl_R>(value));
471 c.s << "std::complex<";
472 if (is_a<print_csrc_double>(c))
477 print_real_csrc(c, cln::realpart(cln::the<cln::cl_N>(value)));
479 print_real_csrc(c, cln::imagpart(cln::the<cln::cl_N>(value)));
484 c.s.precision(oldprec);
488 const std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
489 const std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
490 const std::string imag_sym = is_a<print_latex>(c) ? "i" : "I";
491 const std::string mul_sym = is_a<print_latex>(c) ? " " : "*";
492 const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
493 const cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
495 if (is_a<print_python_repr>(c))
496 c.s << class_name() << "('";
498 // case 1, real: x or -x
499 if ((precedence() <= level) && (!this->is_nonneg_integer())) {
501 print_real_number(c, r);
504 print_real_number(c, r);
508 // case 2, imaginary: y*I or -y*I
512 if (precedence()<=level)
515 c.s << "-" << imag_sym;
517 print_real_number(c, i);
518 c.s << mul_sym+imag_sym;
520 if (precedence()<=level)
524 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
525 if (precedence() <= level)
527 print_real_number(c, r);
532 print_real_number(c, i);
533 c.s << mul_sym+imag_sym;
540 print_real_number(c, i);
541 c.s << mul_sym+imag_sym;
544 if (precedence() <= level)
548 if (is_a<print_python_repr>(c))
553 bool numeric::info(unsigned inf) const
556 case info_flags::numeric:
557 case info_flags::polynomial:
558 case info_flags::rational_function:
560 case info_flags::real:
562 case info_flags::rational:
563 case info_flags::rational_polynomial:
564 return is_rational();
565 case info_flags::crational:
566 case info_flags::crational_polynomial:
567 return is_crational();
568 case info_flags::integer:
569 case info_flags::integer_polynomial:
571 case info_flags::cinteger:
572 case info_flags::cinteger_polynomial:
573 return is_cinteger();
574 case info_flags::positive:
575 return is_positive();
576 case info_flags::negative:
577 return is_negative();
578 case info_flags::nonnegative:
579 return !is_negative();
580 case info_flags::posint:
581 return is_pos_integer();
582 case info_flags::negint:
583 return is_integer() && is_negative();
584 case info_flags::nonnegint:
585 return is_nonneg_integer();
586 case info_flags::even:
588 case info_flags::odd:
590 case info_flags::prime:
592 case info_flags::algebraic:
598 int numeric::degree(const ex & s) const
603 int numeric::ldegree(const ex & s) const
608 ex numeric::coeff(const ex & s, int n) const
610 return n==0 ? *this : _ex0;
613 /** Disassemble real part and imaginary part to scan for the occurrence of a
614 * single number. Also handles the imaginary unit. It ignores the sign on
615 * both this and the argument, which may lead to what might appear as funny
616 * results: (2+I).has(-2) -> true. But this is consistent, since we also
617 * would like to have (-2+I).has(2) -> true and we want to think about the
618 * sign as a multiplicative factor. */
619 bool numeric::has(const ex &other) const
621 if (!is_ex_exactly_of_type(other, numeric))
623 const numeric &o = ex_to<numeric>(other);
624 if (this->is_equal(o) || this->is_equal(-o))
626 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
627 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
628 this->real().is_equal(-o) || this->imag().is_equal(-o));
630 if (o.is_equal(I)) // e.g scan for I in 42*I
631 return !this->is_real();
632 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
633 return (this->real().has(o*I) || this->imag().has(o*I) ||
634 this->real().has(-o*I) || this->imag().has(-o*I));
640 /** Evaluation of numbers doesn't do anything at all. */
641 ex numeric::eval(int level) const
643 // Warning: if this is ever gonna do something, the ex ctors from all kinds
644 // of numbers should be checking for status_flags::evaluated.
649 /** Cast numeric into a floating-point object. For example exact numeric(1) is
650 * returned as a 1.0000000000000000000000 and so on according to how Digits is
651 * currently set. In case the object already was a floating point number the
652 * precision is trimmed to match the currently set default.
654 * @param level ignored, only needed for overriding basic::evalf.
655 * @return an ex-handle to a numeric. */
656 ex numeric::evalf(int level) const
658 // level can safely be discarded for numeric objects.
659 return numeric(cln::cl_float(1.0, cln::default_float_format) *
660 (cln::the<cln::cl_N>(value)));
665 int numeric::compare_same_type(const basic &other) const
667 GINAC_ASSERT(is_exactly_a<numeric>(other));
668 const numeric &o = static_cast<const numeric &>(other);
670 return this->compare(o);
674 bool numeric::is_equal_same_type(const basic &other) const
676 GINAC_ASSERT(is_exactly_a<numeric>(other));
677 const numeric &o = static_cast<const numeric &>(other);
679 return this->is_equal(o);
683 unsigned numeric::calchash(void) const
685 // Use CLN's hashcode. Warning: It depends only on the number's value, not
686 // its type or precision (i.e. a true equivalence relation on numbers). As
687 // a consequence, 3 and 3.0 share the same hashvalue.
688 setflag(status_flags::hash_calculated);
689 return (hashvalue = cln::equal_hashcode(cln::the<cln::cl_N>(value)) | 0x80000000U);
694 // new virtual functions which can be overridden by derived classes
700 // non-virtual functions in this class
705 /** Numerical addition method. Adds argument to *this and returns result as
706 * a numeric object. */
707 const numeric numeric::add(const numeric &other) const
709 // Efficiency shortcut: trap the neutral element by pointer.
712 else if (&other==_num0_p)
715 return numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value));
719 /** Numerical subtraction method. Subtracts argument from *this and returns
720 * result as a numeric object. */
721 const numeric numeric::sub(const numeric &other) const
723 return numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value));
727 /** Numerical multiplication method. Multiplies *this and argument and returns
728 * result as a numeric object. */
729 const numeric numeric::mul(const numeric &other) const
731 // Efficiency shortcut: trap the neutral element by pointer.
734 else if (&other==_num1_p)
737 return numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value));
741 /** Numerical division method. Divides *this by argument and returns result as
744 * @exception overflow_error (division by zero) */
745 const numeric numeric::div(const numeric &other) const
747 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
748 throw std::overflow_error("numeric::div(): division by zero");
749 return numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value));
753 /** Numerical exponentiation. Raises *this to the power given as argument and
754 * returns result as a numeric object. */
755 const numeric numeric::power(const numeric &other) const
757 // Efficiency shortcut: trap the neutral exponent by pointer.
761 if (cln::zerop(cln::the<cln::cl_N>(value))) {
762 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
763 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
764 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
765 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
766 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
767 throw std::overflow_error("numeric::eval(): division by zero");
771 return numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value)));
775 const numeric &numeric::add_dyn(const numeric &other) const
777 // Efficiency shortcut: trap the neutral element by pointer.
780 else if (&other==_num0_p)
783 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)+cln::the<cln::cl_N>(other.value)))->
784 setflag(status_flags::dynallocated));
788 const numeric &numeric::sub_dyn(const numeric &other) const
790 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)-cln::the<cln::cl_N>(other.value)))->
791 setflag(status_flags::dynallocated));
795 const numeric &numeric::mul_dyn(const numeric &other) const
797 // Efficiency shortcut: trap the neutral element by pointer.
800 else if (&other==_num1_p)
803 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)*cln::the<cln::cl_N>(other.value)))->
804 setflag(status_flags::dynallocated));
808 const numeric &numeric::div_dyn(const numeric &other) const
810 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
811 throw std::overflow_error("division by zero");
812 return static_cast<const numeric &>((new numeric(cln::the<cln::cl_N>(value)/cln::the<cln::cl_N>(other.value)))->
813 setflag(status_flags::dynallocated));
817 const numeric &numeric::power_dyn(const numeric &other) const
819 // Efficiency shortcut: trap the neutral exponent by pointer.
823 if (cln::zerop(cln::the<cln::cl_N>(value))) {
824 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
825 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
826 else if (cln::zerop(cln::realpart(cln::the<cln::cl_N>(other.value))))
827 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
828 else if (cln::minusp(cln::realpart(cln::the<cln::cl_N>(other.value))))
829 throw std::overflow_error("numeric::eval(): division by zero");
833 return static_cast<const numeric &>((new numeric(cln::expt(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value))))->
834 setflag(status_flags::dynallocated));
838 const numeric &numeric::operator=(int i)
840 return operator=(numeric(i));
844 const numeric &numeric::operator=(unsigned int i)
846 return operator=(numeric(i));
850 const numeric &numeric::operator=(long i)
852 return operator=(numeric(i));
856 const numeric &numeric::operator=(unsigned long i)
858 return operator=(numeric(i));
862 const numeric &numeric::operator=(double d)
864 return operator=(numeric(d));
868 const numeric &numeric::operator=(const char * s)
870 return operator=(numeric(s));
874 /** Inverse of a number. */
875 const numeric numeric::inverse(void) const
877 if (cln::zerop(cln::the<cln::cl_N>(value)))
878 throw std::overflow_error("numeric::inverse(): division by zero");
879 return numeric(cln::recip(cln::the<cln::cl_N>(value)));
883 /** Return the complex half-plane (left or right) in which the number lies.
884 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
885 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
887 * @see numeric::compare(const numeric &other) */
888 int numeric::csgn(void) const
890 if (cln::zerop(cln::the<cln::cl_N>(value)))
892 cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
893 if (!cln::zerop(r)) {
899 if (cln::plusp(cln::imagpart(cln::the<cln::cl_N>(value))))
907 /** This method establishes a canonical order on all numbers. For complex
908 * numbers this is not possible in a mathematically consistent way but we need
909 * to establish some order and it ought to be fast. So we simply define it
910 * to be compatible with our method csgn.
912 * @return csgn(*this-other)
913 * @see numeric::csgn(void) */
914 int numeric::compare(const numeric &other) const
916 // Comparing two real numbers?
917 if (cln::instanceof(value, cln::cl_R_ring) &&
918 cln::instanceof(other.value, cln::cl_R_ring))
919 // Yes, so just cln::compare them
920 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
922 // No, first cln::compare real parts...
923 cl_signean real_cmp = cln::compare(cln::realpart(cln::the<cln::cl_N>(value)), cln::realpart(cln::the<cln::cl_N>(other.value)));
926 // ...and then the imaginary parts.
927 return cln::compare(cln::imagpart(cln::the<cln::cl_N>(value)), cln::imagpart(cln::the<cln::cl_N>(other.value)));
932 bool numeric::is_equal(const numeric &other) const
934 return cln::equal(cln::the<cln::cl_N>(value),cln::the<cln::cl_N>(other.value));
938 /** True if object is zero. */
939 bool numeric::is_zero(void) const
941 return cln::zerop(cln::the<cln::cl_N>(value));
945 /** True if object is not complex and greater than zero. */
946 bool numeric::is_positive(void) const
949 return cln::plusp(cln::the<cln::cl_R>(value));
954 /** True if object is not complex and less than zero. */
955 bool numeric::is_negative(void) const
958 return cln::minusp(cln::the<cln::cl_R>(value));
963 /** True if object is a non-complex integer. */
964 bool numeric::is_integer(void) const
966 return cln::instanceof(value, cln::cl_I_ring);
970 /** True if object is an exact integer greater than zero. */
971 bool numeric::is_pos_integer(void) const
973 return (this->is_integer() && cln::plusp(cln::the<cln::cl_I>(value)));
977 /** True if object is an exact integer greater or equal zero. */
978 bool numeric::is_nonneg_integer(void) const
980 return (this->is_integer() && !cln::minusp(cln::the<cln::cl_I>(value)));
984 /** True if object is an exact even integer. */
985 bool numeric::is_even(void) const
987 return (this->is_integer() && cln::evenp(cln::the<cln::cl_I>(value)));
991 /** True if object is an exact odd integer. */
992 bool numeric::is_odd(void) const
994 return (this->is_integer() && cln::oddp(cln::the<cln::cl_I>(value)));
998 /** Probabilistic primality test.
1000 * @return true if object is exact integer and prime. */
1001 bool numeric::is_prime(void) const
1003 return (this->is_integer() && cln::isprobprime(cln::the<cln::cl_I>(value)));
1007 /** True if object is an exact rational number, may even be complex
1008 * (denominator may be unity). */
1009 bool numeric::is_rational(void) const
1011 return cln::instanceof(value, cln::cl_RA_ring);
1015 /** True if object is a real integer, rational or float (but not complex). */
1016 bool numeric::is_real(void) const
1018 return cln::instanceof(value, cln::cl_R_ring);
1022 bool numeric::operator==(const numeric &other) const
1024 return cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
1028 bool numeric::operator!=(const numeric &other) const
1030 return !cln::equal(cln::the<cln::cl_N>(value), cln::the<cln::cl_N>(other.value));
1034 /** True if object is element of the domain of integers extended by I, i.e. is
1035 * of the form a+b*I, where a and b are integers. */
1036 bool numeric::is_cinteger(void) const
1038 if (cln::instanceof(value, cln::cl_I_ring))
1040 else if (!this->is_real()) { // complex case, handle n+m*I
1041 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring) &&
1042 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_I_ring))
1049 /** True if object is an exact rational number, may even be complex
1050 * (denominator may be unity). */
1051 bool numeric::is_crational(void) const
1053 if (cln::instanceof(value, cln::cl_RA_ring))
1055 else if (!this->is_real()) { // complex case, handle Q(i):
1056 if (cln::instanceof(cln::realpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring) &&
1057 cln::instanceof(cln::imagpart(cln::the<cln::cl_N>(value)), cln::cl_RA_ring))
1064 /** Numerical comparison: less.
1066 * @exception invalid_argument (complex inequality) */
1067 bool numeric::operator<(const numeric &other) const
1069 if (this->is_real() && other.is_real())
1070 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
1071 throw std::invalid_argument("numeric::operator<(): complex inequality");
1075 /** Numerical comparison: less or equal.
1077 * @exception invalid_argument (complex inequality) */
1078 bool numeric::operator<=(const numeric &other) const
1080 if (this->is_real() && other.is_real())
1081 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
1082 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1086 /** Numerical comparison: greater.
1088 * @exception invalid_argument (complex inequality) */
1089 bool numeric::operator>(const numeric &other) const
1091 if (this->is_real() && other.is_real())
1092 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1093 throw std::invalid_argument("numeric::operator>(): complex inequality");
1097 /** Numerical comparison: greater or equal.
1099 * @exception invalid_argument (complex inequality) */
1100 bool numeric::operator>=(const numeric &other) const
1102 if (this->is_real() && other.is_real())
1103 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1104 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1108 /** Converts numeric types to machine's int. You should check with
1109 * is_integer() if the number is really an integer before calling this method.
1110 * You may also consider checking the range first. */
1111 int numeric::to_int(void) const
1113 GINAC_ASSERT(this->is_integer());
1114 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1118 /** Converts numeric types to machine's long. You should check with
1119 * is_integer() if the number is really an integer before calling this method.
1120 * You may also consider checking the range first. */
1121 long numeric::to_long(void) const
1123 GINAC_ASSERT(this->is_integer());
1124 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1128 /** Converts numeric types to machine's double. You should check with is_real()
1129 * if the number is really not complex before calling this method. */
1130 double numeric::to_double(void) const
1132 GINAC_ASSERT(this->is_real());
1133 return cln::double_approx(cln::realpart(cln::the<cln::cl_N>(value)));
1137 /** Returns a new CLN object of type cl_N, representing the value of *this.
1138 * This method may be used when mixing GiNaC and CLN in one project.
1140 cln::cl_N numeric::to_cl_N(void) const
1142 return cln::cl_N(cln::the<cln::cl_N>(value));
1146 /** Real part of a number. */
1147 const numeric numeric::real(void) const
1149 return numeric(cln::realpart(cln::the<cln::cl_N>(value)));
1153 /** Imaginary part of a number. */
1154 const numeric numeric::imag(void) const
1156 return numeric(cln::imagpart(cln::the<cln::cl_N>(value)));
1160 /** Numerator. Computes the numerator of rational numbers, rationalized
1161 * numerator of complex if real and imaginary part are both rational numbers
1162 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1164 const numeric numeric::numer(void) const
1166 if (this->is_integer())
1167 return numeric(*this);
1169 else if (cln::instanceof(value, cln::cl_RA_ring))
1170 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1172 else if (!this->is_real()) { // complex case, handle Q(i):
1173 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1174 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1175 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1176 return numeric(*this);
1177 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1178 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1179 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1180 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1181 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1182 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1183 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1184 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1187 // at least one float encountered
1188 return numeric(*this);
1192 /** Denominator. Computes the denominator of rational numbers, common integer
1193 * denominator of complex if real and imaginary part are both rational numbers
1194 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1195 const numeric numeric::denom(void) const
1197 if (this->is_integer())
1200 if (cln::instanceof(value, cln::cl_RA_ring))
1201 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1203 if (!this->is_real()) { // complex case, handle Q(i):
1204 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(cln::the<cln::cl_N>(value)));
1205 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(cln::the<cln::cl_N>(value)));
1206 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1208 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1209 return numeric(cln::denominator(i));
1210 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1211 return numeric(cln::denominator(r));
1212 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1213 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1215 // at least one float encountered
1220 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1221 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1222 * 2^(n-1) <= x < 2^n.
1224 * @return number of bits (excluding sign) needed to represent that number
1225 * in two's complement if it is an integer, 0 otherwise. */
1226 int numeric::int_length(void) const
1228 if (this->is_integer())
1229 return cln::integer_length(cln::the<cln::cl_I>(value));
1238 /** Imaginary unit. This is not a constant but a numeric since we are
1239 * natively handing complex numbers anyways, so in each expression containing
1240 * an I it is automatically eval'ed away anyhow. */
1241 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1244 /** Exponential function.
1246 * @return arbitrary precision numerical exp(x). */
1247 const numeric exp(const numeric &x)
1249 return cln::exp(x.to_cl_N());
1253 /** Natural logarithm.
1255 * @param z complex number
1256 * @return arbitrary precision numerical log(x).
1257 * @exception pole_error("log(): logarithmic pole",0) */
1258 const numeric log(const numeric &z)
1261 throw pole_error("log(): logarithmic pole",0);
1262 return cln::log(z.to_cl_N());
1266 /** Numeric sine (trigonometric function).
1268 * @return arbitrary precision numerical sin(x). */
1269 const numeric sin(const numeric &x)
1271 return cln::sin(x.to_cl_N());
1275 /** Numeric cosine (trigonometric function).
1277 * @return arbitrary precision numerical cos(x). */
1278 const numeric cos(const numeric &x)
1280 return cln::cos(x.to_cl_N());
1284 /** Numeric tangent (trigonometric function).
1286 * @return arbitrary precision numerical tan(x). */
1287 const numeric tan(const numeric &x)
1289 return cln::tan(x.to_cl_N());
1293 /** Numeric inverse sine (trigonometric function).
1295 * @return arbitrary precision numerical asin(x). */
1296 const numeric asin(const numeric &x)
1298 return cln::asin(x.to_cl_N());
1302 /** Numeric inverse cosine (trigonometric function).
1304 * @return arbitrary precision numerical acos(x). */
1305 const numeric acos(const numeric &x)
1307 return cln::acos(x.to_cl_N());
1313 * @param z complex number
1315 * @exception pole_error("atan(): logarithmic pole",0) */
1316 const numeric atan(const numeric &x)
1319 x.real().is_zero() &&
1320 abs(x.imag()).is_equal(_num1))
1321 throw pole_error("atan(): logarithmic pole",0);
1322 return cln::atan(x.to_cl_N());
1328 * @param x real number
1329 * @param y real number
1330 * @return atan(y/x) */
1331 const numeric atan(const numeric &y, const numeric &x)
1333 if (x.is_real() && y.is_real())
1334 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1335 cln::the<cln::cl_R>(y.to_cl_N()));
1337 throw std::invalid_argument("atan(): complex argument");
1341 /** Numeric hyperbolic sine (trigonometric function).
1343 * @return arbitrary precision numerical sinh(x). */
1344 const numeric sinh(const numeric &x)
1346 return cln::sinh(x.to_cl_N());
1350 /** Numeric hyperbolic cosine (trigonometric function).
1352 * @return arbitrary precision numerical cosh(x). */
1353 const numeric cosh(const numeric &x)
1355 return cln::cosh(x.to_cl_N());
1359 /** Numeric hyperbolic tangent (trigonometric function).
1361 * @return arbitrary precision numerical tanh(x). */
1362 const numeric tanh(const numeric &x)
1364 return cln::tanh(x.to_cl_N());
1368 /** Numeric inverse hyperbolic sine (trigonometric function).
1370 * @return arbitrary precision numerical asinh(x). */
1371 const numeric asinh(const numeric &x)
1373 return cln::asinh(x.to_cl_N());
1377 /** Numeric inverse hyperbolic cosine (trigonometric function).
1379 * @return arbitrary precision numerical acosh(x). */
1380 const numeric acosh(const numeric &x)
1382 return cln::acosh(x.to_cl_N());
1386 /** Numeric inverse hyperbolic tangent (trigonometric function).
1388 * @return arbitrary precision numerical atanh(x). */
1389 const numeric atanh(const numeric &x)
1391 return cln::atanh(x.to_cl_N());
1395 /*static cln::cl_N Li2_series(const ::cl_N &x,
1396 const ::float_format_t &prec)
1398 // Note: argument must be in the unit circle
1399 // This is very inefficient unless we have fast floating point Bernoulli
1400 // numbers implemented!
1401 cln::cl_N c1 = -cln::log(1-x);
1403 // hard-wire the first two Bernoulli numbers
1404 cln::cl_N acc = c1 - cln::square(c1)/4;
1406 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1407 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1409 c1 = cln::square(c1);
1413 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1414 // aug = c2 * cln::cl_I(i%2 ? 1 : -1) / cln::cl_I(2*i+1) * cln::cl_zeta(2*i, prec) / piac / (cln::cl_I(1)<<(2*i-1));
1417 } while (acc != acc+aug);
1421 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1422 * circle) using a power series. */
1423 static cln::cl_N Li2_series(const cln::cl_N &x,
1424 const cln::float_format_t &prec)
1426 // Note: argument must be in the unit circle
1428 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1433 den = den + i; // 1, 4, 9, 16, ...
1437 } while (acc != acc+aug);
1441 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1442 static cln::cl_N Li2_projection(const cln::cl_N &x,
1443 const cln::float_format_t &prec)
1445 const cln::cl_R re = cln::realpart(x);
1446 const cln::cl_R im = cln::imagpart(x);
1447 if (re > cln::cl_F(".5"))
1448 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1450 - Li2_series(1-x, prec)
1451 - cln::log(x)*cln::log(1-x));
1452 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1453 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1454 return(- cln::square(cln::log(1-x))/2
1455 - Li2_series(x/(x-1), prec));
1456 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1457 // Li2(x^2)/2 - Li2(-x)
1458 return(Li2_projection(cln::square(x), prec)/2
1459 - Li2_projection(-x, prec));
1460 return Li2_series(x, prec);
1463 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1464 * the branch cut lies along the positive real axis, starting at 1 and
1465 * continuous with quadrant IV.
1467 * @return arbitrary precision numerical Li2(x). */
1468 const numeric Li2(const numeric &x)
1473 // what is the desired float format?
1474 // first guess: default format
1475 cln::float_format_t prec = cln::default_float_format;
1476 const cln::cl_N value = x.to_cl_N();
1477 // second guess: the argument's format
1478 if (!x.real().is_rational())
1479 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1480 else if (!x.imag().is_rational())
1481 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1483 if (cln::the<cln::cl_N>(value)==1) // may cause trouble with log(1-x)
1484 return cln::zeta(2, prec);
1486 if (cln::abs(value) > 1)
1487 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1488 return(- cln::square(cln::log(-value))/2
1489 - cln::zeta(2, prec)
1490 - Li2_projection(cln::recip(value), prec));
1492 return Li2_projection(x.to_cl_N(), prec);
1496 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1497 * integer arguments. */
1498 const numeric zeta(const numeric &x)
1500 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1501 // only knows about integer arguments and zeta(3).evalf() automatically
1502 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1503 // being an exact zero for CLN, which can be tested and then we can just
1504 // pass the number casted to an int:
1506 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1507 if (cln::zerop(x.to_cl_N()-aux))
1508 return cln::zeta(aux);
1514 /** The Gamma function.
1515 * This is only a stub! */
1516 const numeric lgamma(const numeric &x)
1520 const numeric tgamma(const numeric &x)
1526 /** The psi function (aka polygamma function).
1527 * This is only a stub! */
1528 const numeric psi(const numeric &x)
1534 /** The psi functions (aka polygamma functions).
1535 * This is only a stub! */
1536 const numeric psi(const numeric &n, const numeric &x)
1542 /** Factorial combinatorial function.
1544 * @param n integer argument >= 0
1545 * @exception range_error (argument must be integer >= 0) */
1546 const numeric factorial(const numeric &n)
1548 if (!n.is_nonneg_integer())
1549 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1550 return numeric(cln::factorial(n.to_int()));
1554 /** The double factorial combinatorial function. (Scarcely used, but still
1555 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1557 * @param n integer argument >= -1
1558 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1559 * @exception range_error (argument must be integer >= -1) */
1560 const numeric doublefactorial(const numeric &n)
1562 if (n.is_equal(_num_1))
1565 if (!n.is_nonneg_integer())
1566 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1568 return numeric(cln::doublefactorial(n.to_int()));
1572 /** The Binomial coefficients. It computes the binomial coefficients. For
1573 * integer n and k and positive n this is the number of ways of choosing k
1574 * objects from n distinct objects. If n is negative, the formula
1575 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1576 const numeric binomial(const numeric &n, const numeric &k)
1578 if (n.is_integer() && k.is_integer()) {
1579 if (n.is_nonneg_integer()) {
1580 if (k.compare(n)!=1 && k.compare(_num0)!=-1)
1581 return numeric(cln::binomial(n.to_int(),k.to_int()));
1585 return _num_1.power(k)*binomial(k-n-_num1,k);
1589 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1590 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1594 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1595 * in the expansion of the function x/(e^x-1).
1597 * @return the nth Bernoulli number (a rational number).
1598 * @exception range_error (argument must be integer >= 0) */
1599 const numeric bernoulli(const numeric &nn)
1601 if (!nn.is_integer() || nn.is_negative())
1602 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1606 // The Bernoulli numbers are rational numbers that may be computed using
1609 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1611 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1612 // previous ones, the computation is necessarily very expensive. There are
1613 // several other ways of computing them, a particularly good one being
1617 // for (unsigned i=0; i<n; i++) {
1618 // c = exquo(c*(i-n),(i+2));
1619 // Bern = Bern + c*s/(i+2);
1620 // s = s + expt_pos(cl_I(i+2),n);
1624 // But if somebody works with the n'th Bernoulli number she is likely to
1625 // also need all previous Bernoulli numbers. So we need a complete remember
1626 // table and above divide and conquer algorithm is not suited to build one
1627 // up. The formula below accomplishes this. It is a modification of the
1628 // defining formula above but the computation of the binomial coefficients
1629 // is carried along in an inline fashion. It also honors the fact that
1630 // B_n is zero when n is odd and greater than 1.
1632 // (There is an interesting relation with the tangent polynomials described
1633 // in `Concrete Mathematics', which leads to a program a little faster as
1634 // our implementation below, but it requires storing one such polynomial in
1635 // addition to the remember table. This doubles the memory footprint so
1636 // we don't use it.)
1638 const unsigned n = nn.to_int();
1640 // the special cases not covered by the algorithm below
1642 return (n==1) ? _num_1_2 : _num0;
1646 // store nonvanishing Bernoulli numbers here
1647 static std::vector< cln::cl_RA > results;
1648 static unsigned next_r = 0;
1650 // algorithm not applicable to B(2), so just store it
1652 results.push_back(cln::recip(cln::cl_RA(6)));
1656 return results[n/2-1];
1658 results.reserve(n/2);
1659 for (unsigned p=next_r; p<=n; p+=2) {
1660 cln::cl_I c = 1; // seed for binonmial coefficients
1661 cln::cl_RA b = cln::cl_RA(1-p)/2;
1662 const unsigned p3 = p+3;
1663 const unsigned pm = p-2;
1665 // test if intermediate unsigned int can be represented by immediate
1666 // objects by CLN (i.e. < 2^29 for 32 Bit machines, see <cln/object.h>)
1667 if (p < (1UL<<cl_value_len/2)) {
1668 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1669 c = cln::exquo(c * ((p3-i) * p_2), (i-1)*k);
1670 b = b + c*results[k-1];
1673 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1674 c = cln::exquo((c * (p3-i)) * p_2, cln::cl_I(i-1)*k);
1675 b = b + c*results[k-1];
1678 results.push_back(-b/(p+1));
1681 return results[n/2-1];
1685 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1686 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1688 * @param n an integer
1689 * @return the nth Fibonacci number F(n) (an integer number)
1690 * @exception range_error (argument must be an integer) */
1691 const numeric fibonacci(const numeric &n)
1693 if (!n.is_integer())
1694 throw std::range_error("numeric::fibonacci(): argument must be integer");
1697 // The following addition formula holds:
1699 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1701 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1702 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1704 // Replace m by m+1:
1705 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1706 // Now put in m = n, to get
1707 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1708 // F(2n+1) = F(n)^2 + F(n+1)^2
1710 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1713 if (n.is_negative())
1715 return -fibonacci(-n);
1717 return fibonacci(-n);
1721 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1722 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1723 // Since a squaring is cheaper than a multiplication, better use
1724 // three squarings instead of one multiplication and two squarings.
1725 cln::cl_I u2 = cln::square(u);
1726 cln::cl_I v2 = cln::square(v);
1727 if (cln::logbitp(bit-1, m)) {
1728 v = cln::square(u + v) - u2;
1731 u = v2 - cln::square(v - u);
1736 // Here we don't use the squaring formula because one multiplication
1737 // is cheaper than two squarings.
1738 return u * ((v << 1) - u);
1740 return cln::square(u) + cln::square(v);
1744 /** Absolute value. */
1745 const numeric abs(const numeric& x)
1747 return cln::abs(x.to_cl_N());
1751 /** Modulus (in positive representation).
1752 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1753 * sign of a or is zero. This is different from Maple's modp, where the sign
1754 * of b is ignored. It is in agreement with Mathematica's Mod.
1756 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1757 * integer, 0 otherwise. */
1758 const numeric mod(const numeric &a, const numeric &b)
1760 if (a.is_integer() && b.is_integer())
1761 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1762 cln::the<cln::cl_I>(b.to_cl_N()));
1768 /** Modulus (in symmetric representation).
1769 * Equivalent to Maple's mods.
1771 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1772 const numeric smod(const numeric &a, const numeric &b)
1774 if (a.is_integer() && b.is_integer()) {
1775 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1776 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1777 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1783 /** Numeric integer remainder.
1784 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1785 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1786 * sign of a or is zero.
1788 * @return remainder of a/b if both are integer, 0 otherwise.
1789 * @exception overflow_error (division by zero) if b is zero. */
1790 const numeric irem(const numeric &a, const numeric &b)
1793 throw std::overflow_error("numeric::irem(): division by zero");
1794 if (a.is_integer() && b.is_integer())
1795 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1796 cln::the<cln::cl_I>(b.to_cl_N()));
1802 /** Numeric integer remainder.
1803 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1804 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1805 * and irem(a,b) has the sign of a or is zero.
1807 * @return remainder of a/b and quotient stored in q if both are integer,
1809 * @exception overflow_error (division by zero) if b is zero. */
1810 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1813 throw std::overflow_error("numeric::irem(): division by zero");
1814 if (a.is_integer() && b.is_integer()) {
1815 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1816 cln::the<cln::cl_I>(b.to_cl_N()));
1817 q = rem_quo.quotient;
1818 return rem_quo.remainder;
1826 /** Numeric integer quotient.
1827 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1829 * @return truncated quotient of a/b if both are integer, 0 otherwise.
1830 * @exception overflow_error (division by zero) if b is zero. */
1831 const numeric iquo(const numeric &a, const numeric &b)
1834 throw std::overflow_error("numeric::iquo(): division by zero");
1835 if (a.is_integer() && b.is_integer())
1836 return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1837 cln::the<cln::cl_I>(b.to_cl_N()));
1843 /** Numeric integer quotient.
1844 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1845 * r == a - iquo(a,b,r)*b.
1847 * @return truncated quotient of a/b and remainder stored in r if both are
1848 * integer, 0 otherwise.
1849 * @exception overflow_error (division by zero) if b is zero. */
1850 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1853 throw std::overflow_error("numeric::iquo(): division by zero");
1854 if (a.is_integer() && b.is_integer()) {
1855 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1856 cln::the<cln::cl_I>(b.to_cl_N()));
1857 r = rem_quo.remainder;
1858 return rem_quo.quotient;
1866 /** Greatest Common Divisor.
1868 * @return The GCD of two numbers if both are integer, a numerical 1
1869 * if they are not. */
1870 const numeric gcd(const numeric &a, const numeric &b)
1872 if (a.is_integer() && b.is_integer())
1873 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1874 cln::the<cln::cl_I>(b.to_cl_N()));
1880 /** Least Common Multiple.
1882 * @return The LCM of two numbers if both are integer, the product of those
1883 * two numbers if they are not. */
1884 const numeric lcm(const numeric &a, const numeric &b)
1886 if (a.is_integer() && b.is_integer())
1887 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1888 cln::the<cln::cl_I>(b.to_cl_N()));
1894 /** Numeric square root.
1895 * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4)
1896 * should return integer 2.
1898 * @param z numeric argument
1899 * @return square root of z. Branch cut along negative real axis, the negative
1900 * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part
1901 * where imag(z)>0. */
1902 const numeric sqrt(const numeric &z)
1904 return cln::sqrt(z.to_cl_N());
1908 /** Integer numeric square root. */
1909 const numeric isqrt(const numeric &x)
1911 if (x.is_integer()) {
1913 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1920 /** Floating point evaluation of Archimedes' constant Pi. */
1923 return numeric(cln::pi(cln::default_float_format));
1927 /** Floating point evaluation of Euler's constant gamma. */
1930 return numeric(cln::eulerconst(cln::default_float_format));
1934 /** Floating point evaluation of Catalan's constant. */
1935 ex CatalanEvalf(void)
1937 return numeric(cln::catalanconst(cln::default_float_format));
1941 /** _numeric_digits default ctor, checking for singleton invariance. */
1942 _numeric_digits::_numeric_digits()
1945 // It initializes to 17 digits, because in CLN float_format(17) turns out
1946 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1947 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1949 throw(std::runtime_error("I told you not to do instantiate me!"));
1951 cln::default_float_format = cln::float_format(17);
1955 /** Assign a native long to global Digits object. */
1956 _numeric_digits& _numeric_digits::operator=(long prec)
1959 cln::default_float_format = cln::float_format(prec);
1964 /** Convert global Digits object to native type long. */
1965 _numeric_digits::operator long()
1967 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
1968 return (long)digits;
1972 /** Append global Digits object to ostream. */
1973 void _numeric_digits::print(std::ostream &os) const
1979 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
1986 // static member variables
1991 bool _numeric_digits::too_late = false;
1994 /** Accuracy in decimal digits. Only object of this type! Can be set using
1995 * assignment from C++ unsigned ints and evaluated like any built-in type. */
1996 _numeric_digits Digits;
1998 } // namespace GiNaC