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) 1999-2004 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
37 #include "operators.h"
42 // CLN should pollute the global namespace as little as possible. Hence, we
43 // include most of it here and include only the part needed for properly
44 // declaring cln::cl_number in numeric.h. This can only be safely done in
45 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
46 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
48 #include <cln/output.h>
49 #include <cln/integer_io.h>
50 #include <cln/integer_ring.h>
51 #include <cln/rational_io.h>
52 #include <cln/rational_ring.h>
53 #include <cln/lfloat_class.h>
54 #include <cln/lfloat_io.h>
55 #include <cln/real_io.h>
56 #include <cln/real_ring.h>
57 #include <cln/complex_io.h>
58 #include <cln/complex_ring.h>
59 #include <cln/numtheory.h>
63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(numeric, basic,
64 print_func<print_context>(&numeric::do_print).
65 print_func<print_latex>(&numeric::do_print_latex).
66 print_func<print_csrc>(&numeric::do_print_csrc).
67 print_func<print_csrc_cl_N>(&numeric::do_print_csrc_cl_N).
68 print_func<print_tree>(&numeric::do_print_tree).
69 print_func<print_python_repr>(&numeric::do_print_python_repr))
72 // default constructor
75 /** default ctor. Numerically it initializes to an integer zero. */
76 numeric::numeric() : basic(TINFO_numeric)
79 setflag(status_flags::evaluated | status_flags::expanded);
88 numeric::numeric(int i) : basic(TINFO_numeric)
90 // Not the whole int-range is available if we don't cast to long
91 // first. This is due to the behaviour of the cl_I-ctor, which
92 // emphasizes efficiency. However, if the integer is small enough
93 // we save space and dereferences by using an immediate type.
94 // (C.f. <cln/object.h>)
95 if (i < (1L << (cl_value_len-1)) && i >= -(1L << (cl_value_len-1)))
98 value = cln::cl_I(static_cast<long>(i));
99 setflag(status_flags::evaluated | status_flags::expanded);
103 numeric::numeric(unsigned int i) : basic(TINFO_numeric)
105 // Not the whole uint-range is available if we don't cast to ulong
106 // first. This is due to the behaviour of the cl_I-ctor, which
107 // emphasizes efficiency. However, if the integer is small enough
108 // we save space and dereferences by using an immediate type.
109 // (C.f. <cln/object.h>)
110 if (i < (1U << (cl_value_len-1)))
111 value = cln::cl_I(i);
113 value = cln::cl_I(static_cast<unsigned long>(i));
114 setflag(status_flags::evaluated | status_flags::expanded);
118 numeric::numeric(long i) : basic(TINFO_numeric)
120 value = cln::cl_I(i);
121 setflag(status_flags::evaluated | status_flags::expanded);
125 numeric::numeric(unsigned long i) : basic(TINFO_numeric)
127 value = cln::cl_I(i);
128 setflag(status_flags::evaluated | status_flags::expanded);
132 /** Constructor for rational numerics a/b.
134 * @exception overflow_error (division by zero) */
135 numeric::numeric(long numer, long denom) : basic(TINFO_numeric)
138 throw std::overflow_error("division by zero");
139 value = cln::cl_I(numer) / cln::cl_I(denom);
140 setflag(status_flags::evaluated | status_flags::expanded);
144 numeric::numeric(double d) : basic(TINFO_numeric)
146 // We really want to explicitly use the type cl_LF instead of the
147 // more general cl_F, since that would give us a cl_DF only which
148 // will not be promoted to cl_LF if overflow occurs:
149 value = cln::cl_float(d, cln::default_float_format);
150 setflag(status_flags::evaluated | status_flags::expanded);
154 /** ctor from C-style string. It also accepts complex numbers in GiNaC
155 * notation like "2+5*I". */
156 numeric::numeric(const char *s) : basic(TINFO_numeric)
158 cln::cl_N ctorval = 0;
159 // parse complex numbers (functional but not completely safe, unfortunately
160 // std::string does not understand regexpese):
161 // ss should represent a simple sum like 2+5*I
163 std::string::size_type delim;
165 // make this implementation safe by adding explicit sign
166 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
169 // We use 'E' as exponent marker in the output, but some people insist on
170 // writing 'e' at input, so let's substitute them right at the beginning:
171 while ((delim = ss.find("e"))!=std::string::npos)
172 ss.replace(delim,1,"E");
176 // chop ss into terms from left to right
178 bool imaginary = false;
179 delim = ss.find_first_of(std::string("+-"),1);
180 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
181 if (delim!=std::string::npos && ss.at(delim-1)=='E')
182 delim = ss.find_first_of(std::string("+-"),delim+1);
183 term = ss.substr(0,delim);
184 if (delim!=std::string::npos)
185 ss = ss.substr(delim);
186 // is the term imaginary?
187 if (term.find("I")!=std::string::npos) {
189 term.erase(term.find("I"),1);
191 if (term.find("*")!=std::string::npos)
192 term.erase(term.find("*"),1);
193 // correct for trivial +/-I without explicit factor on I:
198 if (term.find('.')!=std::string::npos || term.find('E')!=std::string::npos) {
199 // CLN's short type cl_SF is not very useful within the GiNaC
200 // framework where we are mainly interested in the arbitrary
201 // precision type cl_LF. Hence we go straight to the construction
202 // of generic floats. In order to create them we have to convert
203 // our own floating point notation used for output and construction
204 // from char * to CLN's generic notation:
205 // 3.14 --> 3.14e0_<Digits>
206 // 31.4E-1 --> 31.4e-1_<Digits>
208 // No exponent marker? Let's add a trivial one.
209 if (term.find("E")==std::string::npos)
212 term = term.replace(term.find("E"),1,"e");
213 // append _<Digits> to term
214 term += "_" + ToString((unsigned)Digits);
215 // construct float using cln::cl_F(const char *) ctor.
217 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
219 ctorval = ctorval + cln::cl_F(term.c_str());
221 // this is not a floating point number...
223 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
225 ctorval = ctorval + cln::cl_R(term.c_str());
227 } while (delim != std::string::npos);
229 setflag(status_flags::evaluated | status_flags::expanded);
233 /** Ctor from CLN types. This is for the initiated user or internal use
235 numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric)
238 setflag(status_flags::evaluated | status_flags::expanded);
245 numeric::numeric(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
247 cln::cl_N ctorval = 0;
249 // Read number as string
251 if (n.find_string("number", str)) {
252 std::istringstream s(str);
253 cln::cl_idecoded_float re, im;
257 case 'R': // Integer-decoded real number
258 s >> re.sign >> re.mantissa >> re.exponent;
259 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
261 case 'C': // Integer-decoded complex number
262 s >> re.sign >> re.mantissa >> re.exponent;
263 s >> im.sign >> im.mantissa >> im.exponent;
264 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
265 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
267 default: // Ordinary number
274 setflag(status_flags::evaluated | status_flags::expanded);
277 void numeric::archive(archive_node &n) const
279 inherited::archive(n);
281 // Write number as string
282 std::ostringstream s;
283 if (this->is_crational())
286 // Non-rational numbers are written in an integer-decoded format
287 // to preserve the precision
288 if (this->is_real()) {
289 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
291 s << re.sign << " " << re.mantissa << " " << re.exponent;
293 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
294 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
296 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
297 s << im.sign << " " << im.mantissa << " " << im.exponent;
300 n.add_string("number", s.str());
303 DEFAULT_UNARCHIVE(numeric)
306 // functions overriding virtual functions from base classes
309 /** Helper function to print a real number in a nicer way than is CLN's
310 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
311 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
312 * long as it only uses cl_LF and no other floating point types that we might
313 * want to visibly distinguish from cl_LF.
315 * @see numeric::print() */
316 static void print_real_number(const print_context & c, const cln::cl_R & x)
318 cln::cl_print_flags ourflags;
319 if (cln::instanceof(x, cln::cl_RA_ring)) {
320 // case 1: integer or rational
321 if (cln::instanceof(x, cln::cl_I_ring) ||
322 !is_a<print_latex>(c)) {
323 cln::print_real(c.s, ourflags, x);
324 } else { // rational output in LaTeX context
328 cln::print_real(c.s, ourflags, cln::abs(cln::numerator(cln::the<cln::cl_RA>(x))));
330 cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
335 // make CLN believe this number has default_float_format, so it prints
336 // 'E' as exponent marker instead of 'L':
337 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
338 cln::print_real(c.s, ourflags, x);
342 /** Helper function to print integer number in C++ source format.
344 * @see numeric::print() */
345 static void print_integer_csrc(const print_context & c, const cln::cl_I & x)
347 // Print small numbers in compact float format, but larger numbers in
349 const int max_cln_int = 536870911; // 2^29-1
350 if (x >= cln::cl_I(-max_cln_int) && x <= cln::cl_I(max_cln_int))
351 c.s << cln::cl_I_to_int(x) << ".0";
353 c.s << cln::double_approx(x);
356 /** Helper function to print real number in C++ source format.
358 * @see numeric::print() */
359 static void print_real_csrc(const print_context & c, const cln::cl_R & x)
361 if (cln::instanceof(x, cln::cl_I_ring)) {
364 print_integer_csrc(c, cln::the<cln::cl_I>(x));
366 } else if (cln::instanceof(x, cln::cl_RA_ring)) {
369 const cln::cl_I numer = cln::numerator(cln::the<cln::cl_RA>(x));
370 const cln::cl_I denom = cln::denominator(cln::the<cln::cl_RA>(x));
371 if (cln::plusp(x) > 0) {
373 print_integer_csrc(c, numer);
376 print_integer_csrc(c, -numer);
379 print_integer_csrc(c, denom);
385 c.s << cln::double_approx(x);
389 /** Helper function to print real number in C++ source format using cl_N types.
391 * @see numeric::print() */
392 static void print_real_cl_N(const print_context & c, const cln::cl_R & x)
394 if (cln::instanceof(x, cln::cl_I_ring)) {
397 c.s << "cln::cl_I(\"";
398 print_real_number(c, x);
401 } else if (cln::instanceof(x, cln::cl_RA_ring)) {
404 cln::cl_print_flags ourflags;
405 c.s << "cln::cl_RA(\"";
406 cln::print_rational(c.s, ourflags, cln::the<cln::cl_RA>(x));
412 c.s << "cln::cl_F(\"";
413 print_real_number(c, cln::cl_float(1.0, cln::default_float_format) * x);
414 c.s << "_" << Digits << "\")";
418 void numeric::print_numeric(const print_context & c, const char *par_open, const char *par_close, const char *imag_sym, const char *mul_sym, unsigned level) const
420 const cln::cl_R r = cln::realpart(value);
421 const cln::cl_R i = cln::imagpart(value);
425 // case 1, real: x or -x
426 if ((precedence() <= level) && (!this->is_nonneg_integer())) {
428 print_real_number(c, r);
431 print_real_number(c, r);
437 // case 2, imaginary: y*I or -y*I
441 if (precedence()<=level)
444 c.s << "-" << imag_sym;
446 print_real_number(c, i);
447 c.s << mul_sym << imag_sym;
449 if (precedence()<=level)
455 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
456 if (precedence() <= level)
458 print_real_number(c, r);
461 c.s << "-" << imag_sym;
463 print_real_number(c, i);
464 c.s << mul_sym << imag_sym;
468 c.s << "+" << imag_sym;
471 print_real_number(c, i);
472 c.s << mul_sym << imag_sym;
475 if (precedence() <= level)
481 void numeric::do_print(const print_context & c, unsigned level) const
483 print_numeric(c, "(", ")", "I", "*", level);
486 void numeric::do_print_latex(const print_latex & c, unsigned level) const
488 print_numeric(c, "{(", ")}", "i", " ", level);
491 void numeric::do_print_csrc(const print_csrc & c, unsigned level) const
493 std::ios::fmtflags oldflags = c.s.flags();
494 c.s.setf(std::ios::scientific);
495 int oldprec = c.s.precision();
498 if (is_a<print_csrc_double>(c))
499 c.s.precision(std::numeric_limits<double>::digits10 + 1);
501 c.s.precision(std::numeric_limits<float>::digits10 + 1);
503 if (this->is_real()) {
506 print_real_csrc(c, cln::the<cln::cl_R>(value));
511 c.s << "std::complex<";
512 if (is_a<print_csrc_double>(c))
517 print_real_csrc(c, cln::realpart(value));
519 print_real_csrc(c, cln::imagpart(value));
524 c.s.precision(oldprec);
527 void numeric::do_print_csrc_cl_N(const print_csrc_cl_N & c, unsigned level) const
529 if (this->is_real()) {
532 print_real_cl_N(c, cln::the<cln::cl_R>(value));
537 c.s << "cln::complex(";
538 print_real_cl_N(c, cln::realpart(value));
540 print_real_cl_N(c, cln::imagpart(value));
545 void numeric::do_print_tree(const print_tree & c, unsigned level) const
547 c.s << std::string(level, ' ') << value
548 << " (" << class_name() << ")" << " @" << this
549 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
553 void numeric::do_print_python_repr(const print_python_repr & c, unsigned level) const
555 c.s << class_name() << "('";
556 print_numeric(c, "(", ")", "I", "*", level);
560 bool numeric::info(unsigned inf) const
563 case info_flags::numeric:
564 case info_flags::polynomial:
565 case info_flags::rational_function:
567 case info_flags::real:
569 case info_flags::rational:
570 case info_flags::rational_polynomial:
571 return is_rational();
572 case info_flags::crational:
573 case info_flags::crational_polynomial:
574 return is_crational();
575 case info_flags::integer:
576 case info_flags::integer_polynomial:
578 case info_flags::cinteger:
579 case info_flags::cinteger_polynomial:
580 return is_cinteger();
581 case info_flags::positive:
582 return is_positive();
583 case info_flags::negative:
584 return is_negative();
585 case info_flags::nonnegative:
586 return !is_negative();
587 case info_flags::posint:
588 return is_pos_integer();
589 case info_flags::negint:
590 return is_integer() && is_negative();
591 case info_flags::nonnegint:
592 return is_nonneg_integer();
593 case info_flags::even:
595 case info_flags::odd:
597 case info_flags::prime:
599 case info_flags::algebraic:
605 int numeric::degree(const ex & s) const
610 int numeric::ldegree(const ex & s) const
615 ex numeric::coeff(const ex & s, int n) const
617 return n==0 ? *this : _ex0;
620 /** Disassemble real part and imaginary part to scan for the occurrence of a
621 * single number. Also handles the imaginary unit. It ignores the sign on
622 * both this and the argument, which may lead to what might appear as funny
623 * results: (2+I).has(-2) -> true. But this is consistent, since we also
624 * would like to have (-2+I).has(2) -> true and we want to think about the
625 * sign as a multiplicative factor. */
626 bool numeric::has(const ex &other) const
628 if (!is_exactly_a<numeric>(other))
630 const numeric &o = ex_to<numeric>(other);
631 if (this->is_equal(o) || this->is_equal(-o))
633 if (o.imag().is_zero()) // e.g. scan for 3 in -3*I
634 return (this->real().is_equal(o) || this->imag().is_equal(o) ||
635 this->real().is_equal(-o) || this->imag().is_equal(-o));
637 if (o.is_equal(I)) // e.g scan for I in 42*I
638 return !this->is_real();
639 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
640 return (this->real().has(o*I) || this->imag().has(o*I) ||
641 this->real().has(-o*I) || this->imag().has(-o*I));
647 /** Evaluation of numbers doesn't do anything at all. */
648 ex numeric::eval(int level) const
650 // Warning: if this is ever gonna do something, the ex ctors from all kinds
651 // of numbers should be checking for status_flags::evaluated.
656 /** Cast numeric into a floating-point object. For example exact numeric(1) is
657 * returned as a 1.0000000000000000000000 and so on according to how Digits is
658 * currently set. In case the object already was a floating point number the
659 * precision is trimmed to match the currently set default.
661 * @param level ignored, only needed for overriding basic::evalf.
662 * @return an ex-handle to a numeric. */
663 ex numeric::evalf(int level) const
665 // level can safely be discarded for numeric objects.
666 return numeric(cln::cl_float(1.0, cln::default_float_format) * value);
669 ex numeric::conjugate() const
674 return numeric(cln::conjugate(this->value));
679 int numeric::compare_same_type(const basic &other) const
681 GINAC_ASSERT(is_exactly_a<numeric>(other));
682 const numeric &o = static_cast<const numeric &>(other);
684 return this->compare(o);
688 bool numeric::is_equal_same_type(const basic &other) const
690 GINAC_ASSERT(is_exactly_a<numeric>(other));
691 const numeric &o = static_cast<const numeric &>(other);
693 return this->is_equal(o);
697 unsigned numeric::calchash() const
699 // Base computation of hashvalue on CLN's hashcode. Note: That depends
700 // only on the number's value, not its type or precision (i.e. a true
701 // equivalence relation on numbers). As a consequence, 3 and 3.0 share
702 // the same hashvalue. That shouldn't really matter, though.
703 setflag(status_flags::hash_calculated);
704 hashvalue = golden_ratio_hash(cln::equal_hashcode(value));
710 // new virtual functions which can be overridden by derived classes
716 // non-virtual functions in this class
721 /** Numerical addition method. Adds argument to *this and returns result as
722 * a numeric object. */
723 const numeric numeric::add(const numeric &other) const
725 return numeric(value + other.value);
729 /** Numerical subtraction method. Subtracts argument from *this and returns
730 * result as a numeric object. */
731 const numeric numeric::sub(const numeric &other) const
733 return numeric(value - other.value);
737 /** Numerical multiplication method. Multiplies *this and argument and returns
738 * result as a numeric object. */
739 const numeric numeric::mul(const numeric &other) const
741 return numeric(value * other.value);
745 /** Numerical division method. Divides *this by argument and returns result as
748 * @exception overflow_error (division by zero) */
749 const numeric numeric::div(const numeric &other) const
751 if (cln::zerop(other.value))
752 throw std::overflow_error("numeric::div(): division by zero");
753 return numeric(value / other.value);
757 /** Numerical exponentiation. Raises *this to the power given as argument and
758 * returns result as a numeric object. */
759 const numeric numeric::power(const numeric &other) const
761 // Shortcut for efficiency and numeric stability (as in 1.0 exponent):
762 // trap the neutral exponent.
763 if (&other==_num1_p || cln::equal(other.value,_num1.value))
766 if (cln::zerop(value)) {
767 if (cln::zerop(other.value))
768 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
769 else if (cln::zerop(cln::realpart(other.value)))
770 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
771 else if (cln::minusp(cln::realpart(other.value)))
772 throw std::overflow_error("numeric::eval(): division by zero");
776 return numeric(cln::expt(value, other.value));
781 /** Numerical addition method. Adds argument to *this and returns result as
782 * a numeric object on the heap. Use internally only for direct wrapping into
783 * an ex object, where the result would end up on the heap anyways. */
784 const numeric &numeric::add_dyn(const numeric &other) const
786 // Efficiency shortcut: trap the neutral element by pointer. This hack
787 // is supposed to keep the number of distinct numeric objects low.
790 else if (&other==_num0_p)
793 return static_cast<const numeric &>((new numeric(value + other.value))->
794 setflag(status_flags::dynallocated));
798 /** Numerical subtraction method. Subtracts argument from *this and returns
799 * result as a numeric object on the heap. Use internally only for direct
800 * wrapping into an ex object, where the result would end up on the heap
802 const numeric &numeric::sub_dyn(const numeric &other) const
804 // Efficiency shortcut: trap the neutral exponent (first by pointer). This
805 // hack is supposed to keep the number of distinct numeric objects low.
806 if (&other==_num0_p || cln::zerop(other.value))
809 return static_cast<const numeric &>((new numeric(value - other.value))->
810 setflag(status_flags::dynallocated));
814 /** Numerical multiplication method. Multiplies *this and argument and returns
815 * result as a numeric object on the heap. Use internally only for direct
816 * wrapping into an ex object, where the result would end up on the heap
818 const numeric &numeric::mul_dyn(const numeric &other) const
820 // Efficiency shortcut: trap the neutral element by pointer. This hack
821 // is supposed to keep the number of distinct numeric objects low.
824 else if (&other==_num1_p)
827 return static_cast<const numeric &>((new numeric(value * other.value))->
828 setflag(status_flags::dynallocated));
832 /** Numerical division method. Divides *this by argument and returns result as
833 * a numeric object on the heap. Use internally only for direct wrapping
834 * into an ex object, where the result would end up on the heap
837 * @exception overflow_error (division by zero) */
838 const numeric &numeric::div_dyn(const numeric &other) const
840 // Efficiency shortcut: trap the neutral element by pointer. This hack
841 // is supposed to keep the number of distinct numeric objects low.
844 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
845 throw std::overflow_error("division by zero");
846 return static_cast<const numeric &>((new numeric(value / other.value))->
847 setflag(status_flags::dynallocated));
851 /** Numerical exponentiation. Raises *this to the power given as argument and
852 * returns result as a numeric object on the heap. Use internally only for
853 * direct wrapping into an ex object, where the result would end up on the
855 const numeric &numeric::power_dyn(const numeric &other) const
857 // Efficiency shortcut: trap the neutral exponent (first try by pointer, then
858 // try harder, since calls to cln::expt() below may return amazing results for
859 // floating point exponent 1.0).
860 if (&other==_num1_p || cln::equal(other.value, _num1.value))
863 if (cln::zerop(value)) {
864 if (cln::zerop(other.value))
865 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
866 else if (cln::zerop(cln::realpart(other.value)))
867 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
868 else if (cln::minusp(cln::realpart(other.value)))
869 throw std::overflow_error("numeric::eval(): division by zero");
873 return static_cast<const numeric &>((new numeric(cln::expt(value, other.value)))->
874 setflag(status_flags::dynallocated));
878 const numeric &numeric::operator=(int i)
880 return operator=(numeric(i));
884 const numeric &numeric::operator=(unsigned int i)
886 return operator=(numeric(i));
890 const numeric &numeric::operator=(long i)
892 return operator=(numeric(i));
896 const numeric &numeric::operator=(unsigned long i)
898 return operator=(numeric(i));
902 const numeric &numeric::operator=(double d)
904 return operator=(numeric(d));
908 const numeric &numeric::operator=(const char * s)
910 return operator=(numeric(s));
914 /** Inverse of a number. */
915 const numeric numeric::inverse() const
917 if (cln::zerop(value))
918 throw std::overflow_error("numeric::inverse(): division by zero");
919 return numeric(cln::recip(value));
923 /** Return the complex half-plane (left or right) in which the number lies.
924 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
925 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
927 * @see numeric::compare(const numeric &other) */
928 int numeric::csgn() const
930 if (cln::zerop(value))
932 cln::cl_R r = cln::realpart(value);
933 if (!cln::zerop(r)) {
939 if (cln::plusp(cln::imagpart(value)))
947 /** This method establishes a canonical order on all numbers. For complex
948 * numbers this is not possible in a mathematically consistent way but we need
949 * to establish some order and it ought to be fast. So we simply define it
950 * to be compatible with our method csgn.
952 * @return csgn(*this-other)
953 * @see numeric::csgn() */
954 int numeric::compare(const numeric &other) const
956 // Comparing two real numbers?
957 if (cln::instanceof(value, cln::cl_R_ring) &&
958 cln::instanceof(other.value, cln::cl_R_ring))
959 // Yes, so just cln::compare them
960 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
962 // No, first cln::compare real parts...
963 cl_signean real_cmp = cln::compare(cln::realpart(value), cln::realpart(other.value));
966 // ...and then the imaginary parts.
967 return cln::compare(cln::imagpart(value), cln::imagpart(other.value));
972 bool numeric::is_equal(const numeric &other) const
974 return cln::equal(value, other.value);
978 /** True if object is zero. */
979 bool numeric::is_zero() const
981 return cln::zerop(value);
985 /** True if object is not complex and greater than zero. */
986 bool numeric::is_positive() const
988 if (cln::instanceof(value, cln::cl_R_ring)) // real?
989 return cln::plusp(cln::the<cln::cl_R>(value));
994 /** True if object is not complex and less than zero. */
995 bool numeric::is_negative() const
997 if (cln::instanceof(value, cln::cl_R_ring)) // real?
998 return cln::minusp(cln::the<cln::cl_R>(value));
1003 /** True if object is a non-complex integer. */
1004 bool numeric::is_integer() const
1006 return cln::instanceof(value, cln::cl_I_ring);
1010 /** True if object is an exact integer greater than zero. */
1011 bool numeric::is_pos_integer() const
1013 return (cln::instanceof(value, cln::cl_I_ring) && cln::plusp(cln::the<cln::cl_I>(value)));
1017 /** True if object is an exact integer greater or equal zero. */
1018 bool numeric::is_nonneg_integer() const
1020 return (cln::instanceof(value, cln::cl_I_ring) && !cln::minusp(cln::the<cln::cl_I>(value)));
1024 /** True if object is an exact even integer. */
1025 bool numeric::is_even() const
1027 return (cln::instanceof(value, cln::cl_I_ring) && cln::evenp(cln::the<cln::cl_I>(value)));
1031 /** True if object is an exact odd integer. */
1032 bool numeric::is_odd() const
1034 return (cln::instanceof(value, cln::cl_I_ring) && cln::oddp(cln::the<cln::cl_I>(value)));
1038 /** Probabilistic primality test.
1040 * @return true if object is exact integer and prime. */
1041 bool numeric::is_prime() const
1043 return (cln::instanceof(value, cln::cl_I_ring) // integer?
1044 && cln::plusp(cln::the<cln::cl_I>(value)) // positive?
1045 && cln::isprobprime(cln::the<cln::cl_I>(value)));
1049 /** True if object is an exact rational number, may even be complex
1050 * (denominator may be unity). */
1051 bool numeric::is_rational() const
1053 return cln::instanceof(value, cln::cl_RA_ring);
1057 /** True if object is a real integer, rational or float (but not complex). */
1058 bool numeric::is_real() const
1060 return cln::instanceof(value, cln::cl_R_ring);
1064 bool numeric::operator==(const numeric &other) const
1066 return cln::equal(value, other.value);
1070 bool numeric::operator!=(const numeric &other) const
1072 return !cln::equal(value, other.value);
1076 /** True if object is element of the domain of integers extended by I, i.e. is
1077 * of the form a+b*I, where a and b are integers. */
1078 bool numeric::is_cinteger() const
1080 if (cln::instanceof(value, cln::cl_I_ring))
1082 else if (!this->is_real()) { // complex case, handle n+m*I
1083 if (cln::instanceof(cln::realpart(value), cln::cl_I_ring) &&
1084 cln::instanceof(cln::imagpart(value), cln::cl_I_ring))
1091 /** True if object is an exact rational number, may even be complex
1092 * (denominator may be unity). */
1093 bool numeric::is_crational() const
1095 if (cln::instanceof(value, cln::cl_RA_ring))
1097 else if (!this->is_real()) { // complex case, handle Q(i):
1098 if (cln::instanceof(cln::realpart(value), cln::cl_RA_ring) &&
1099 cln::instanceof(cln::imagpart(value), cln::cl_RA_ring))
1106 /** Numerical comparison: less.
1108 * @exception invalid_argument (complex inequality) */
1109 bool numeric::operator<(const numeric &other) const
1111 if (this->is_real() && other.is_real())
1112 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
1113 throw std::invalid_argument("numeric::operator<(): complex inequality");
1117 /** Numerical comparison: less or equal.
1119 * @exception invalid_argument (complex inequality) */
1120 bool numeric::operator<=(const numeric &other) const
1122 if (this->is_real() && other.is_real())
1123 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
1124 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1128 /** Numerical comparison: greater.
1130 * @exception invalid_argument (complex inequality) */
1131 bool numeric::operator>(const numeric &other) const
1133 if (this->is_real() && other.is_real())
1134 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1135 throw std::invalid_argument("numeric::operator>(): complex inequality");
1139 /** Numerical comparison: greater or equal.
1141 * @exception invalid_argument (complex inequality) */
1142 bool numeric::operator>=(const numeric &other) const
1144 if (this->is_real() && other.is_real())
1145 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1146 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1150 /** Converts numeric types to machine's int. You should check with
1151 * is_integer() if the number is really an integer before calling this method.
1152 * You may also consider checking the range first. */
1153 int numeric::to_int() const
1155 GINAC_ASSERT(this->is_integer());
1156 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1160 /** Converts numeric types to machine's long. You should check with
1161 * is_integer() if the number is really an integer before calling this method.
1162 * You may also consider checking the range first. */
1163 long numeric::to_long() const
1165 GINAC_ASSERT(this->is_integer());
1166 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1170 /** Converts numeric types to machine's double. You should check with is_real()
1171 * if the number is really not complex before calling this method. */
1172 double numeric::to_double() const
1174 GINAC_ASSERT(this->is_real());
1175 return cln::double_approx(cln::realpart(value));
1179 /** Returns a new CLN object of type cl_N, representing the value of *this.
1180 * This method may be used when mixing GiNaC and CLN in one project.
1182 cln::cl_N numeric::to_cl_N() const
1188 /** Real part of a number. */
1189 const numeric numeric::real() const
1191 return numeric(cln::realpart(value));
1195 /** Imaginary part of a number. */
1196 const numeric numeric::imag() const
1198 return numeric(cln::imagpart(value));
1202 /** Numerator. Computes the numerator of rational numbers, rationalized
1203 * numerator of complex if real and imaginary part are both rational numbers
1204 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1206 const numeric numeric::numer() const
1208 if (cln::instanceof(value, cln::cl_I_ring))
1209 return numeric(*this); // integer case
1211 else if (cln::instanceof(value, cln::cl_RA_ring))
1212 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1214 else if (!this->is_real()) { // complex case, handle Q(i):
1215 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(value));
1216 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(value));
1217 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1218 return numeric(*this);
1219 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1220 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1221 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1222 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1223 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1224 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1225 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1226 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1229 // at least one float encountered
1230 return numeric(*this);
1234 /** Denominator. Computes the denominator of rational numbers, common integer
1235 * denominator of complex if real and imaginary part are both rational numbers
1236 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1237 const numeric numeric::denom() const
1239 if (cln::instanceof(value, cln::cl_I_ring))
1240 return _num1; // integer case
1242 if (cln::instanceof(value, cln::cl_RA_ring))
1243 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1245 if (!this->is_real()) { // complex case, handle Q(i):
1246 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(value));
1247 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(value));
1248 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1250 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1251 return numeric(cln::denominator(i));
1252 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1253 return numeric(cln::denominator(r));
1254 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1255 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1257 // at least one float encountered
1262 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1263 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1264 * 2^(n-1) <= x < 2^n.
1266 * @return number of bits (excluding sign) needed to represent that number
1267 * in two's complement if it is an integer, 0 otherwise. */
1268 int numeric::int_length() const
1270 if (cln::instanceof(value, cln::cl_I_ring))
1271 return cln::integer_length(cln::the<cln::cl_I>(value));
1280 /** Imaginary unit. This is not a constant but a numeric since we are
1281 * natively handing complex numbers anyways, so in each expression containing
1282 * an I it is automatically eval'ed away anyhow. */
1283 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1286 /** Exponential function.
1288 * @return arbitrary precision numerical exp(x). */
1289 const numeric exp(const numeric &x)
1291 return cln::exp(x.to_cl_N());
1295 /** Natural logarithm.
1297 * @param x complex number
1298 * @return arbitrary precision numerical log(x).
1299 * @exception pole_error("log(): logarithmic pole",0) */
1300 const numeric log(const numeric &x)
1303 throw pole_error("log(): logarithmic pole",0);
1304 return cln::log(x.to_cl_N());
1308 /** Numeric sine (trigonometric function).
1310 * @return arbitrary precision numerical sin(x). */
1311 const numeric sin(const numeric &x)
1313 return cln::sin(x.to_cl_N());
1317 /** Numeric cosine (trigonometric function).
1319 * @return arbitrary precision numerical cos(x). */
1320 const numeric cos(const numeric &x)
1322 return cln::cos(x.to_cl_N());
1326 /** Numeric tangent (trigonometric function).
1328 * @return arbitrary precision numerical tan(x). */
1329 const numeric tan(const numeric &x)
1331 return cln::tan(x.to_cl_N());
1335 /** Numeric inverse sine (trigonometric function).
1337 * @return arbitrary precision numerical asin(x). */
1338 const numeric asin(const numeric &x)
1340 return cln::asin(x.to_cl_N());
1344 /** Numeric inverse cosine (trigonometric function).
1346 * @return arbitrary precision numerical acos(x). */
1347 const numeric acos(const numeric &x)
1349 return cln::acos(x.to_cl_N());
1355 * @param x complex number
1357 * @exception pole_error("atan(): logarithmic pole",0) */
1358 const numeric atan(const numeric &x)
1361 x.real().is_zero() &&
1362 abs(x.imag()).is_equal(_num1))
1363 throw pole_error("atan(): logarithmic pole",0);
1364 return cln::atan(x.to_cl_N());
1370 * @param x real number
1371 * @param y real number
1372 * @return atan(y/x) */
1373 const numeric atan(const numeric &y, const numeric &x)
1375 if (x.is_real() && y.is_real())
1376 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1377 cln::the<cln::cl_R>(y.to_cl_N()));
1379 throw std::invalid_argument("atan(): complex argument");
1383 /** Numeric hyperbolic sine (trigonometric function).
1385 * @return arbitrary precision numerical sinh(x). */
1386 const numeric sinh(const numeric &x)
1388 return cln::sinh(x.to_cl_N());
1392 /** Numeric hyperbolic cosine (trigonometric function).
1394 * @return arbitrary precision numerical cosh(x). */
1395 const numeric cosh(const numeric &x)
1397 return cln::cosh(x.to_cl_N());
1401 /** Numeric hyperbolic tangent (trigonometric function).
1403 * @return arbitrary precision numerical tanh(x). */
1404 const numeric tanh(const numeric &x)
1406 return cln::tanh(x.to_cl_N());
1410 /** Numeric inverse hyperbolic sine (trigonometric function).
1412 * @return arbitrary precision numerical asinh(x). */
1413 const numeric asinh(const numeric &x)
1415 return cln::asinh(x.to_cl_N());
1419 /** Numeric inverse hyperbolic cosine (trigonometric function).
1421 * @return arbitrary precision numerical acosh(x). */
1422 const numeric acosh(const numeric &x)
1424 return cln::acosh(x.to_cl_N());
1428 /** Numeric inverse hyperbolic tangent (trigonometric function).
1430 * @return arbitrary precision numerical atanh(x). */
1431 const numeric atanh(const numeric &x)
1433 return cln::atanh(x.to_cl_N());
1437 /*static cln::cl_N Li2_series(const ::cl_N &x,
1438 const ::float_format_t &prec)
1440 // Note: argument must be in the unit circle
1441 // This is very inefficient unless we have fast floating point Bernoulli
1442 // numbers implemented!
1443 cln::cl_N c1 = -cln::log(1-x);
1445 // hard-wire the first two Bernoulli numbers
1446 cln::cl_N acc = c1 - cln::square(c1)/4;
1448 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1449 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1451 c1 = cln::square(c1);
1455 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1456 // 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));
1459 } while (acc != acc+aug);
1463 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1464 * circle) using a power series. */
1465 static cln::cl_N Li2_series(const cln::cl_N &x,
1466 const cln::float_format_t &prec)
1468 // Note: argument must be in the unit circle
1470 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1475 den = den + i; // 1, 4, 9, 16, ...
1479 } while (acc != acc+aug);
1483 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1484 static cln::cl_N Li2_projection(const cln::cl_N &x,
1485 const cln::float_format_t &prec)
1487 const cln::cl_R re = cln::realpart(x);
1488 const cln::cl_R im = cln::imagpart(x);
1489 if (re > cln::cl_F(".5"))
1490 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1492 - Li2_series(1-x, prec)
1493 - cln::log(x)*cln::log(1-x));
1494 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1495 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1496 return(- cln::square(cln::log(1-x))/2
1497 - Li2_series(x/(x-1), prec));
1498 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1499 // Li2(x^2)/2 - Li2(-x)
1500 return(Li2_projection(cln::square(x), prec)/2
1501 - Li2_projection(-x, prec));
1502 return Li2_series(x, prec);
1505 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1506 * the branch cut lies along the positive real axis, starting at 1 and
1507 * continuous with quadrant IV.
1509 * @return arbitrary precision numerical Li2(x). */
1510 const numeric Li2(const numeric &x)
1515 // what is the desired float format?
1516 // first guess: default format
1517 cln::float_format_t prec = cln::default_float_format;
1518 const cln::cl_N value = x.to_cl_N();
1519 // second guess: the argument's format
1520 if (!x.real().is_rational())
1521 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1522 else if (!x.imag().is_rational())
1523 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1525 if (value==1) // may cause trouble with log(1-x)
1526 return cln::zeta(2, prec);
1528 if (cln::abs(value) > 1)
1529 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1530 return(- cln::square(cln::log(-value))/2
1531 - cln::zeta(2, prec)
1532 - Li2_projection(cln::recip(value), prec));
1534 return Li2_projection(x.to_cl_N(), prec);
1538 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1539 * integer arguments. */
1540 const numeric zeta(const numeric &x)
1542 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1543 // only knows about integer arguments and zeta(3).evalf() automatically
1544 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1545 // being an exact zero for CLN, which can be tested and then we can just
1546 // pass the number casted to an int:
1548 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1549 if (cln::zerop(x.to_cl_N()-aux))
1550 return cln::zeta(aux);
1556 /** The Gamma function.
1557 * This is only a stub! */
1558 const numeric lgamma(const numeric &x)
1562 const numeric tgamma(const numeric &x)
1568 /** The psi function (aka polygamma function).
1569 * This is only a stub! */
1570 const numeric psi(const numeric &x)
1576 /** The psi functions (aka polygamma functions).
1577 * This is only a stub! */
1578 const numeric psi(const numeric &n, const numeric &x)
1584 /** Factorial combinatorial function.
1586 * @param n integer argument >= 0
1587 * @exception range_error (argument must be integer >= 0) */
1588 const numeric factorial(const numeric &n)
1590 if (!n.is_nonneg_integer())
1591 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1592 return numeric(cln::factorial(n.to_int()));
1596 /** The double factorial combinatorial function. (Scarcely used, but still
1597 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1599 * @param n integer argument >= -1
1600 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1601 * @exception range_error (argument must be integer >= -1) */
1602 const numeric doublefactorial(const numeric &n)
1604 if (n.is_equal(_num_1))
1607 if (!n.is_nonneg_integer())
1608 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1610 return numeric(cln::doublefactorial(n.to_int()));
1614 /** The Binomial coefficients. It computes the binomial coefficients. For
1615 * integer n and k and positive n this is the number of ways of choosing k
1616 * objects from n distinct objects. If n is negative, the formula
1617 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1618 const numeric binomial(const numeric &n, const numeric &k)
1620 if (n.is_integer() && k.is_integer()) {
1621 if (n.is_nonneg_integer()) {
1622 if (k.compare(n)!=1 && k.compare(_num0)!=-1)
1623 return numeric(cln::binomial(n.to_int(),k.to_int()));
1627 return _num_1.power(k)*binomial(k-n-_num1,k);
1631 // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
1632 throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
1636 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1637 * in the expansion of the function x/(e^x-1).
1639 * @return the nth Bernoulli number (a rational number).
1640 * @exception range_error (argument must be integer >= 0) */
1641 const numeric bernoulli(const numeric &nn)
1643 if (!nn.is_integer() || nn.is_negative())
1644 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1648 // The Bernoulli numbers are rational numbers that may be computed using
1651 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1653 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1654 // previous ones, the computation is necessarily very expensive. There are
1655 // several other ways of computing them, a particularly good one being
1659 // for (unsigned i=0; i<n; i++) {
1660 // c = exquo(c*(i-n),(i+2));
1661 // Bern = Bern + c*s/(i+2);
1662 // s = s + expt_pos(cl_I(i+2),n);
1666 // But if somebody works with the n'th Bernoulli number she is likely to
1667 // also need all previous Bernoulli numbers. So we need a complete remember
1668 // table and above divide and conquer algorithm is not suited to build one
1669 // up. The formula below accomplishes this. It is a modification of the
1670 // defining formula above but the computation of the binomial coefficients
1671 // is carried along in an inline fashion. It also honors the fact that
1672 // B_n is zero when n is odd and greater than 1.
1674 // (There is an interesting relation with the tangent polynomials described
1675 // in `Concrete Mathematics', which leads to a program a little faster as
1676 // our implementation below, but it requires storing one such polynomial in
1677 // addition to the remember table. This doubles the memory footprint so
1678 // we don't use it.)
1680 const unsigned n = nn.to_int();
1682 // the special cases not covered by the algorithm below
1684 return (n==1) ? _num_1_2 : _num0;
1688 // store nonvanishing Bernoulli numbers here
1689 static std::vector< cln::cl_RA > results;
1690 static unsigned next_r = 0;
1692 // algorithm not applicable to B(2), so just store it
1694 results.push_back(cln::recip(cln::cl_RA(6)));
1698 return results[n/2-1];
1700 results.reserve(n/2);
1701 for (unsigned p=next_r; p<=n; p+=2) {
1702 cln::cl_I c = 1; // seed for binonmial coefficients
1703 cln::cl_RA b = cln::cl_RA(1-p)/2;
1704 const unsigned p3 = p+3;
1705 const unsigned pm = p-2;
1707 // test if intermediate unsigned int can be represented by immediate
1708 // objects by CLN (i.e. < 2^29 for 32 Bit machines, see <cln/object.h>)
1709 if (p < (1UL<<cl_value_len/2)) {
1710 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1711 c = cln::exquo(c * ((p3-i) * p_2), (i-1)*k);
1712 b = b + c*results[k-1];
1715 for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
1716 c = cln::exquo((c * (p3-i)) * p_2, cln::cl_I(i-1)*k);
1717 b = b + c*results[k-1];
1720 results.push_back(-b/(p+1));
1723 return results[n/2-1];
1727 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1728 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1730 * @param n an integer
1731 * @return the nth Fibonacci number F(n) (an integer number)
1732 * @exception range_error (argument must be an integer) */
1733 const numeric fibonacci(const numeric &n)
1735 if (!n.is_integer())
1736 throw std::range_error("numeric::fibonacci(): argument must be integer");
1739 // The following addition formula holds:
1741 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1743 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1744 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1746 // Replace m by m+1:
1747 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1748 // Now put in m = n, to get
1749 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1750 // F(2n+1) = F(n)^2 + F(n+1)^2
1752 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1755 if (n.is_negative())
1757 return -fibonacci(-n);
1759 return fibonacci(-n);
1763 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1764 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1765 // Since a squaring is cheaper than a multiplication, better use
1766 // three squarings instead of one multiplication and two squarings.
1767 cln::cl_I u2 = cln::square(u);
1768 cln::cl_I v2 = cln::square(v);
1769 if (cln::logbitp(bit-1, m)) {
1770 v = cln::square(u + v) - u2;
1773 u = v2 - cln::square(v - u);
1778 // Here we don't use the squaring formula because one multiplication
1779 // is cheaper than two squarings.
1780 return u * ((v << 1) - u);
1782 return cln::square(u) + cln::square(v);
1786 /** Absolute value. */
1787 const numeric abs(const numeric& x)
1789 return cln::abs(x.to_cl_N());
1793 /** Modulus (in positive representation).
1794 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1795 * sign of a or is zero. This is different from Maple's modp, where the sign
1796 * of b is ignored. It is in agreement with Mathematica's Mod.
1798 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1799 * integer, 0 otherwise. */
1800 const numeric mod(const numeric &a, const numeric &b)
1802 if (a.is_integer() && b.is_integer())
1803 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1804 cln::the<cln::cl_I>(b.to_cl_N()));
1810 /** Modulus (in symmetric representation).
1811 * Equivalent to Maple's mods.
1813 * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */
1814 const numeric smod(const numeric &a, const numeric &b)
1816 if (a.is_integer() && b.is_integer()) {
1817 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1818 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1819 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1825 /** Numeric integer remainder.
1826 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1827 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1828 * sign of a or is zero.
1830 * @return remainder of a/b if both are integer, 0 otherwise.
1831 * @exception overflow_error (division by zero) if b is zero. */
1832 const numeric irem(const numeric &a, const numeric &b)
1835 throw std::overflow_error("numeric::irem(): division by zero");
1836 if (a.is_integer() && b.is_integer())
1837 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1838 cln::the<cln::cl_I>(b.to_cl_N()));
1844 /** Numeric integer remainder.
1845 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1846 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1847 * and irem(a,b) has the sign of a or is zero.
1849 * @return remainder of a/b and quotient stored in q if both are integer,
1851 * @exception overflow_error (division by zero) if b is zero. */
1852 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1855 throw std::overflow_error("numeric::irem(): division by zero");
1856 if (a.is_integer() && b.is_integer()) {
1857 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1858 cln::the<cln::cl_I>(b.to_cl_N()));
1859 q = rem_quo.quotient;
1860 return rem_quo.remainder;
1868 /** Numeric integer quotient.
1869 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1871 * @return truncated quotient of a/b if both are integer, 0 otherwise.
1872 * @exception overflow_error (division by zero) if b is zero. */
1873 const numeric iquo(const numeric &a, const numeric &b)
1876 throw std::overflow_error("numeric::iquo(): division by zero");
1877 if (a.is_integer() && b.is_integer())
1878 return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1879 cln::the<cln::cl_I>(b.to_cl_N()));
1885 /** Numeric integer quotient.
1886 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1887 * r == a - iquo(a,b,r)*b.
1889 * @return truncated quotient of a/b and remainder stored in r if both are
1890 * integer, 0 otherwise.
1891 * @exception overflow_error (division by zero) if b is zero. */
1892 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1895 throw std::overflow_error("numeric::iquo(): division by zero");
1896 if (a.is_integer() && b.is_integer()) {
1897 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1898 cln::the<cln::cl_I>(b.to_cl_N()));
1899 r = rem_quo.remainder;
1900 return rem_quo.quotient;
1908 /** Greatest Common Divisor.
1910 * @return The GCD of two numbers if both are integer, a numerical 1
1911 * if they are not. */
1912 const numeric gcd(const numeric &a, const numeric &b)
1914 if (a.is_integer() && b.is_integer())
1915 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1916 cln::the<cln::cl_I>(b.to_cl_N()));
1922 /** Least Common Multiple.
1924 * @return The LCM of two numbers if both are integer, the product of those
1925 * two numbers if they are not. */
1926 const numeric lcm(const numeric &a, const numeric &b)
1928 if (a.is_integer() && b.is_integer())
1929 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1930 cln::the<cln::cl_I>(b.to_cl_N()));
1936 /** Numeric square root.
1937 * If possible, sqrt(x) should respect squares of exact numbers, i.e. sqrt(4)
1938 * should return integer 2.
1940 * @param x numeric argument
1941 * @return square root of x. Branch cut along negative real axis, the negative
1942 * real axis itself where imag(x)==0 and real(x)<0 belongs to the upper part
1943 * where imag(x)>0. */
1944 const numeric sqrt(const numeric &x)
1946 return cln::sqrt(x.to_cl_N());
1950 /** Integer numeric square root. */
1951 const numeric isqrt(const numeric &x)
1953 if (x.is_integer()) {
1955 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
1962 /** Floating point evaluation of Archimedes' constant Pi. */
1965 return numeric(cln::pi(cln::default_float_format));
1969 /** Floating point evaluation of Euler's constant gamma. */
1972 return numeric(cln::eulerconst(cln::default_float_format));
1976 /** Floating point evaluation of Catalan's constant. */
1979 return numeric(cln::catalanconst(cln::default_float_format));
1983 /** _numeric_digits default ctor, checking for singleton invariance. */
1984 _numeric_digits::_numeric_digits()
1987 // It initializes to 17 digits, because in CLN float_format(17) turns out
1988 // to be 61 (<64) while float_format(18)=65. The reason is we want to
1989 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
1991 throw(std::runtime_error("I told you not to do instantiate me!"));
1993 cln::default_float_format = cln::float_format(17);
1997 /** Assign a native long to global Digits object. */
1998 _numeric_digits& _numeric_digits::operator=(long prec)
2001 cln::default_float_format = cln::float_format(prec);
2006 /** Convert global Digits object to native type long. */
2007 _numeric_digits::operator long()
2009 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
2010 return (long)digits;
2014 /** Append global Digits object to ostream. */
2015 void _numeric_digits::print(std::ostream &os) const
2021 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
2028 // static member variables
2033 bool _numeric_digits::too_late = false;
2036 /** Accuracy in decimal digits. Only object of this type! Can be set using
2037 * assignment from C++ unsigned ints and evaluated like any built-in type. */
2038 _numeric_digits Digits;
2040 } // namespace GiNaC