X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnumeric.cpp;h=022993d03d2ba29ec1e19c34ce191143bb656533;hp=4937448515915c2c629959570a5b839bccdab5f7;hb=b9f93aaeedcdcc014e1aa28d1edb0e920c527f66;hpb=4afb5bbe2c0b0a60928120a042997ba7d89e8f5c diff --git a/ginac/numeric.cpp b/ginac/numeric.cpp index 49374485..022993d0 100644 --- a/ginac/numeric.cpp +++ b/ginac/numeric.cpp @@ -7,7 +7,7 @@ * of special functions or implement the interface to the bignum package. */ /* - * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -24,82 +24,59 @@ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ +#include "config.h" + #include #include +#include +#include +#include #include "numeric.h" #include "ex.h" -#include "config.h" -#include "debugmsg.h" +#include "operators.h" +#include "archive.h" +#include "tostring.h" #include "utils.h" -// CLN should not pollute the global namespace, hence we include it here -// instead of in some header file where it would propagate to other parts: -#ifdef HAVE_CLN_CLN_H -#include -#else -#include -#endif +// CLN should pollute the global namespace as little as possible. Hence, we +// include most of it here and include only the part needed for properly +// declaring cln::cl_number in numeric.h. This can only be safely done in +// namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a +// subset of CLN, so we don't include the complete but only the +// essential stuff: +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include -#ifndef NO_GINAC_NAMESPACE namespace GiNaC { -#endif // ndef NO_GINAC_NAMESPACE -// linker has no problems finding text symbols for numerator or denominator -//#define SANE_LINKER +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(numeric, basic, + print_func(&numeric::do_print). + print_func(&numeric::do_print_latex). + print_func(&numeric::do_print_csrc). + print_func(&numeric::do_print_csrc_cl_N). + print_func(&numeric::do_print_tree). + print_func(&numeric::do_print_python_repr)) ////////// -// default constructor, destructor, copy constructor assignment -// operator and helpers +// default constructor ////////// -// public - /** default ctor. Numerically it initializes to an integer zero. */ numeric::numeric() : basic(TINFO_numeric) { - debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT); - value = new cl_N; - *value=cl_I(0); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); -} - -numeric::~numeric() -{ - debugmsg("numeric destructor" ,LOGLEVEL_DESTRUCT); - destroy(0); -} - -numeric::numeric(const numeric & other) -{ - debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT); - copy(other); -} - -const numeric & numeric::operator=(const numeric & other) -{ - debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT); - if (this != &other) { - destroy(1); - copy(other); - } - return *this; -} - -// protected - -void numeric::copy(const numeric & other) -{ - basic::copy(other); - value = new cl_N(*other.value); -} - -void numeric::destroy(bool call_parent) -{ - delete value; - if (call_parent) basic::destroy(call_parent); + value = cln::cl_I(0); + setflag(status_flags::evaluated | status_flags::expanded); } ////////// @@ -110,304 +87,623 @@ void numeric::destroy(bool call_parent) numeric::numeric(int i) : basic(TINFO_numeric) { - debugmsg("const numericructor from int",LOGLEVEL_CONSTRUCT); - // Not the whole int-range is available if we don't cast to long - // first. This is due to the behaviour of the cl_I-ctor, which - // emphasizes efficiency: - value = new cl_I((long) i); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); + // Not the whole int-range is available if we don't cast to long + // first. This is due to the behaviour of the cl_I-ctor, which + // emphasizes efficiency. However, if the integer is small enough + // we save space and dereferences by using an immediate type. + // (C.f. ) + if (i < (1L << (cl_value_len-1)) && i >= -(1L << (cl_value_len-1))) + value = cln::cl_I(i); + else + value = cln::cl_I(static_cast(i)); + setflag(status_flags::evaluated | status_flags::expanded); } + numeric::numeric(unsigned int i) : basic(TINFO_numeric) { - debugmsg("const numericructor from uint",LOGLEVEL_CONSTRUCT); - // Not the whole uint-range is available if we don't cast to ulong - // first. This is due to the behaviour of the cl_I-ctor, which - // emphasizes efficiency: - value = new cl_I((unsigned long)i); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); + // Not the whole uint-range is available if we don't cast to ulong + // first. This is due to the behaviour of the cl_I-ctor, which + // emphasizes efficiency. However, if the integer is small enough + // we save space and dereferences by using an immediate type. + // (C.f. ) + if (i < (1U << (cl_value_len-1))) + value = cln::cl_I(i); + else + value = cln::cl_I(static_cast(i)); + setflag(status_flags::evaluated | status_flags::expanded); } + numeric::numeric(long i) : basic(TINFO_numeric) { - debugmsg("const numericructor from long",LOGLEVEL_CONSTRUCT); - value = new cl_I(i); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); + value = cln::cl_I(i); + setflag(status_flags::evaluated | status_flags::expanded); } + numeric::numeric(unsigned long i) : basic(TINFO_numeric) { - debugmsg("const numericructor from ulong",LOGLEVEL_CONSTRUCT); - value = new cl_I(i); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); + value = cln::cl_I(i); + setflag(status_flags::evaluated | status_flags::expanded); } -/** Ctor for rational numerics a/b. + +/** Constructor for rational numerics a/b. * * @exception overflow_error (division by zero) */ numeric::numeric(long numer, long denom) : basic(TINFO_numeric) { - debugmsg("const numericructor from long/long",LOGLEVEL_CONSTRUCT); - if (!denom) - throw (std::overflow_error("division by zero")); - value = new cl_I(numer); - *value = *value / cl_I(denom); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); + if (!denom) + throw std::overflow_error("division by zero"); + value = cln::cl_I(numer) / cln::cl_I(denom); + setflag(status_flags::evaluated | status_flags::expanded); } + numeric::numeric(double d) : basic(TINFO_numeric) { - debugmsg("const numericructor from double",LOGLEVEL_CONSTRUCT); - // We really want to explicitly use the type cl_LF instead of the - // more general cl_F, since that would give us a cl_DF only which - // will not be promoted to cl_LF if overflow occurs: - value = new cl_N; - *value = cl_float(d, cl_default_float_format); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); -} - -numeric::numeric(char const *s) : basic(TINFO_numeric) -{ // MISSING: treatment of complex and ints and rationals. - debugmsg("const numericructor from string",LOGLEVEL_CONSTRUCT); - if (strchr(s, '.')) - value = new cl_LF(s); - else - value = new cl_I(s); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); + // We really want to explicitly use the type cl_LF instead of the + // more general cl_F, since that would give us a cl_DF only which + // will not be promoted to cl_LF if overflow occurs: + value = cln::cl_float(d, cln::default_float_format); + setflag(status_flags::evaluated | status_flags::expanded); +} + + +/** ctor from C-style string. It also accepts complex numbers in GiNaC + * notation like "2+5*I". */ +numeric::numeric(const char *s) : basic(TINFO_numeric) +{ + cln::cl_N ctorval = 0; + // parse complex numbers (functional but not completely safe, unfortunately + // std::string does not understand regexpese): + // ss should represent a simple sum like 2+5*I + std::string ss = s; + std::string::size_type delim; + + // make this implementation safe by adding explicit sign + if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#') + ss = '+' + ss; + + // We use 'E' as exponent marker in the output, but some people insist on + // writing 'e' at input, so let's substitute them right at the beginning: + while ((delim = ss.find("e"))!=std::string::npos) + ss.replace(delim,1,"E"); + + // main parser loop: + do { + // chop ss into terms from left to right + std::string term; + bool imaginary = false; + delim = ss.find_first_of(std::string("+-"),1); + // Do we have an exponent marker like "31.415E-1"? If so, hop on! + if (delim!=std::string::npos && ss.at(delim-1)=='E') + delim = ss.find_first_of(std::string("+-"),delim+1); + term = ss.substr(0,delim); + if (delim!=std::string::npos) + ss = ss.substr(delim); + // is the term imaginary? + if (term.find("I")!=std::string::npos) { + // erase 'I': + term.erase(term.find("I"),1); + // erase '*': + if (term.find("*")!=std::string::npos) + term.erase(term.find("*"),1); + // correct for trivial +/-I without explicit factor on I: + if (term.size()==1) + term += '1'; + imaginary = true; + } + if (term.find('.')!=std::string::npos || term.find('E')!=std::string::npos) { + // CLN's short type cl_SF is not very useful within the GiNaC + // framework where we are mainly interested in the arbitrary + // precision type cl_LF. Hence we go straight to the construction + // of generic floats. In order to create them we have to convert + // our own floating point notation used for output and construction + // from char * to CLN's generic notation: + // 3.14 --> 3.14e0_ + // 31.4E-1 --> 31.4e-1_ + // and s on. + // No exponent marker? Let's add a trivial one. + if (term.find("E")==std::string::npos) + term += "E0"; + // E to lower case + term = term.replace(term.find("E"),1,"e"); + // append _ to term + term += "_" + ToString((unsigned)Digits); + // construct float using cln::cl_F(const char *) ctor. + if (imaginary) + ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str())); + else + ctorval = ctorval + cln::cl_F(term.c_str()); + } else { + // this is not a floating point number... + if (imaginary) + ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str())); + else + ctorval = ctorval + cln::cl_R(term.c_str()); + } + } while (delim != std::string::npos); + value = ctorval; + setflag(status_flags::evaluated | status_flags::expanded); } + /** Ctor from CLN types. This is for the initiated user or internal use * only. */ -numeric::numeric(cl_N const & z) : basic(TINFO_numeric) +numeric::numeric(const cln::cl_N &z) : basic(TINFO_numeric) { - debugmsg("const numericructor from cl_N", LOGLEVEL_CONSTRUCT); - value = new cl_N(z); - calchash(); - setflag(status_flags::evaluated| - status_flags::hash_calculated); + value = z; + setflag(status_flags::evaluated | status_flags::expanded); } ////////// -// functions overriding virtual functions from bases classes +// archiving ////////// -// public +numeric::numeric(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst) +{ + cln::cl_N ctorval = 0; + + // Read number as string + std::string str; + if (n.find_string("number", str)) { + std::istringstream s(str); + cln::cl_idecoded_float re, im; + char c; + s.get(c); + switch (c) { + case 'R': // Integer-decoded real number + s >> re.sign >> re.mantissa >> re.exponent; + ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent); + break; + case 'C': // Integer-decoded complex number + s >> re.sign >> re.mantissa >> re.exponent; + s >> im.sign >> im.mantissa >> im.exponent; + ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent), + im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent)); + break; + default: // Ordinary number + s.putback(c); + s >> ctorval; + break; + } + } + value = ctorval; + setflag(status_flags::evaluated | status_flags::expanded); +} + +void numeric::archive(archive_node &n) const +{ + inherited::archive(n); + + // Write number as string + std::ostringstream s; + if (this->is_crational()) + s << value; + else { + // Non-rational numbers are written in an integer-decoded format + // to preserve the precision + if (this->is_real()) { + cln::cl_idecoded_float re = cln::integer_decode_float(cln::the(value)); + s << "R"; + s << re.sign << " " << re.mantissa << " " << re.exponent; + } else { + cln::cl_idecoded_float re = cln::integer_decode_float(cln::the(cln::realpart(cln::the(value)))); + cln::cl_idecoded_float im = cln::integer_decode_float(cln::the(cln::imagpart(cln::the(value)))); + s << "C"; + s << re.sign << " " << re.mantissa << " " << re.exponent << " "; + s << im.sign << " " << im.mantissa << " " << im.exponent; + } + } + n.add_string("number", s.str()); +} + +DEFAULT_UNARCHIVE(numeric) + +////////// +// functions overriding virtual functions from base classes +////////// + +/** Helper function to print a real number in a nicer way than is CLN's + * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os + * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as + * long as it only uses cl_LF and no other floating point types that we might + * want to visibly distinguish from cl_LF. + * + * @see numeric::print() */ +static void print_real_number(const print_context & c, const cln::cl_R & x) +{ + cln::cl_print_flags ourflags; + if (cln::instanceof(x, cln::cl_RA_ring)) { + // case 1: integer or rational + if (cln::instanceof(x, cln::cl_I_ring) || + !is_a(c)) { + cln::print_real(c.s, ourflags, x); + } else { // rational output in LaTeX context + if (x < 0) + c.s << "-"; + c.s << "\\frac{"; + cln::print_real(c.s, ourflags, cln::abs(cln::numerator(cln::the(x)))); + c.s << "}{"; + cln::print_real(c.s, ourflags, cln::denominator(cln::the(x))); + c.s << '}'; + } + } else { + // case 2: float + // make CLN believe this number has default_float_format, so it prints + // 'E' as exponent marker instead of 'L': + ourflags.default_float_format = cln::float_format(cln::the(x)); + cln::print_real(c.s, ourflags, x); + } +} + +/** Helper function to print integer number in C++ source format. + * + * @see numeric::print() */ +static void print_integer_csrc(const print_context & c, const cln::cl_I & x) +{ + // Print small numbers in compact float format, but larger numbers in + // scientific format + const int max_cln_int = 536870911; // 2^29-1 + if (x >= cln::cl_I(-max_cln_int) && x <= cln::cl_I(max_cln_int)) + c.s << cln::cl_I_to_int(x) << ".0"; + else + c.s << cln::double_approx(x); +} + +/** Helper function to print real number in C++ source format. + * + * @see numeric::print() */ +static void print_real_csrc(const print_context & c, const cln::cl_R & x) +{ + if (cln::instanceof(x, cln::cl_I_ring)) { + + // Integer number + print_integer_csrc(c, cln::the(x)); + + } else if (cln::instanceof(x, cln::cl_RA_ring)) { + + // Rational number + const cln::cl_I numer = cln::numerator(cln::the(x)); + const cln::cl_I denom = cln::denominator(cln::the(x)); + if (cln::plusp(x) > 0) { + c.s << "("; + print_integer_csrc(c, numer); + } else { + c.s << "-("; + print_integer_csrc(c, -numer); + } + c.s << "/"; + print_integer_csrc(c, denom); + c.s << ")"; + + } else { + + // Anything else + c.s << cln::double_approx(x); + } +} + +/** Helper function to print real number in C++ source format using cl_N types. + * + * @see numeric::print() */ +static void print_real_cl_N(const print_context & c, const cln::cl_R & x) +{ + if (cln::instanceof(x, cln::cl_I_ring)) { + + // Integer number + c.s << "cln::cl_I(\""; + print_real_number(c, x); + c.s << "\")"; + + } else if (cln::instanceof(x, cln::cl_RA_ring)) { + + // Rational number + cln::cl_print_flags ourflags; + c.s << "cln::cl_RA(\""; + cln::print_rational(c.s, ourflags, cln::the(x)); + c.s << "\")"; + + } else { + + // Anything else + c.s << "cln::cl_F(\""; + print_real_number(c, cln::cl_float(1.0, cln::default_float_format) * x); + c.s << "_" << Digits << "\")"; + } +} + +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 +{ + const cln::cl_R r = cln::realpart(value); + const cln::cl_R i = cln::imagpart(value); + + if (cln::zerop(i)) { + + // case 1, real: x or -x + if ((precedence() <= level) && (!this->is_nonneg_integer())) { + c.s << par_open; + print_real_number(c, r); + c.s << par_close; + } else { + print_real_number(c, r); + } + + } else { + if (cln::zerop(r)) { + + // case 2, imaginary: y*I or -y*I + if (i == 1) + c.s << imag_sym; + else { + if (precedence()<=level) + c.s << par_open; + if (i == -1) + c.s << "-" << imag_sym; + else { + print_real_number(c, i); + c.s << mul_sym << imag_sym; + } + if (precedence()<=level) + c.s << par_close; + } + + } else { + + // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I + if (precedence() <= level) + c.s << par_open; + print_real_number(c, r); + if (i < 0) { + if (i == -1) { + c.s << "-" << imag_sym; + } else { + print_real_number(c, i); + c.s << mul_sym << imag_sym; + } + } else { + if (i == 1) { + c.s << "+" << imag_sym; + } else { + c.s << "+"; + print_real_number(c, i); + c.s << mul_sym << imag_sym; + } + } + if (precedence() <= level) + c.s << par_close; + } + } +} + +void numeric::do_print(const print_context & c, unsigned level) const +{ + print_numeric(c, "(", ")", "I", "*", level); +} + +void numeric::do_print_latex(const print_latex & c, unsigned level) const +{ + print_numeric(c, "{(", ")}", "i", " ", level); +} -basic * numeric::duplicate() const -{ - debugmsg("numeric duplicate", LOGLEVEL_DUPLICATE); - return new numeric(*this); -} - -void numeric::print(ostream & os, unsigned upper_precedence) const -{ - // The method print adds to the output so it blends more consistently - // together with the other routines and produces something compatible to - // ginsh input. - debugmsg("numeric print", LOGLEVEL_PRINT); - if (is_real()) { - // case 1, real: x or -x - if ((precedence<=upper_precedence) && (!is_pos_integer())) { - os << "(" << *value << ")"; - } else { - os << *value; - } - } else { - // case 2, imaginary: y*I or -y*I - if (realpart(*value) == 0) { - if ((precedence<=upper_precedence) && (imagpart(*value) < 0)) { - if (imagpart(*value) == -1) { - os << "(-I)"; - } else { - os << "(" << imagpart(*value) << "*I)"; - } - } else { - if (imagpart(*value) == 1) { - os << "I"; - } else { - if (imagpart (*value) == -1) { - os << "-I"; - } else { - os << imagpart(*value) << "*I"; - } - } - } - } else { - // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I - if (precedence <= upper_precedence) os << "("; - os << realpart(*value); - if (imagpart(*value) < 0) { - if (imagpart(*value) == -1) { - os << "-I"; - } else { - os << imagpart(*value) << "*I"; - } - } else { - if (imagpart(*value) == 1) { - os << "+I"; - } else { - os << "+" << imagpart(*value) << "*I"; - } - } - if (precedence <= upper_precedence) os << ")"; - } - } -} - - -void numeric::printraw(ostream & os) const -{ - // The method printraw doesn't do much, it simply uses CLN's operator<<() - // for output, which is ugly but reliable. e.g: 2+2i - debugmsg("numeric printraw", LOGLEVEL_PRINT); - os << "numeric(" << *value << ")"; -} -void numeric::printtree(ostream & os, unsigned indent) const -{ - debugmsg("numeric printtree", LOGLEVEL_PRINT); - os << string(indent,' ') << *value - << " (numeric): " - << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")" - << ", flags=" << flags << endl; -} - -void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const -{ - debugmsg("numeric print csrc", LOGLEVEL_PRINT); - ios::fmtflags oldflags = os.flags(); - os.setf(ios::scientific); - if (is_rational() && !is_integer()) { - if (compare(_num0()) > 0) { - os << "("; - if (type == csrc_types::ctype_cl_N) - os << "cl_F(\"" << numer().evalf() << "\")"; - else - os << numer().to_double(); - } else { - os << "-("; - if (type == csrc_types::ctype_cl_N) - os << "cl_F(\"" << -numer().evalf() << "\")"; - else - os << -numer().to_double(); - } - os << "/"; - if (type == csrc_types::ctype_cl_N) - os << "cl_F(\"" << denom().evalf() << "\")"; - else - os << denom().to_double(); - os << ")"; - } else { - if (type == csrc_types::ctype_cl_N) - os << "cl_F(\"" << evalf() << "\")"; - else - os << to_double(); - } - os.flags(oldflags); +void numeric::do_print_csrc(const print_csrc & c, unsigned level) const +{ + std::ios::fmtflags oldflags = c.s.flags(); + c.s.setf(std::ios::scientific); + int oldprec = c.s.precision(); + + // Set precision + if (is_a(c)) + c.s.precision(std::numeric_limits::digits10 + 1); + else + c.s.precision(std::numeric_limits::digits10 + 1); + + if (this->is_real()) { + + // Real number + print_real_csrc(c, cln::the(value)); + + } else { + + // Complex number + c.s << "std::complex<"; + if (is_a(c)) + c.s << "double>("; + else + c.s << "float>("; + + print_real_csrc(c, cln::realpart(value)); + c.s << ","; + print_real_csrc(c, cln::imagpart(value)); + c.s << ")"; + } + + c.s.flags(oldflags); + c.s.precision(oldprec); +} + +void numeric::do_print_csrc_cl_N(const print_csrc_cl_N & c, unsigned level) const +{ + if (this->is_real()) { + + // Real number + print_real_cl_N(c, cln::the(value)); + + } else { + + // Complex number + c.s << "cln::complex("; + print_real_cl_N(c, cln::realpart(value)); + c.s << ","; + print_real_cl_N(c, cln::imagpart(value)); + c.s << ")"; + } +} + +void numeric::do_print_tree(const print_tree & c, unsigned level) const +{ + c.s << std::string(level, ' ') << value + << " (" << class_name() << ")" << " @" << this + << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec + << std::endl; +} + +void numeric::do_print_python_repr(const print_python_repr & c, unsigned level) const +{ + c.s << class_name() << "('"; + print_numeric(c, "(", ")", "I", "*", level); + c.s << "')"; } bool numeric::info(unsigned inf) const { - switch (inf) { - case info_flags::numeric: - case info_flags::polynomial: - case info_flags::rational_function: - return true; - case info_flags::real: - return is_real(); - case info_flags::rational: - case info_flags::rational_polynomial: - return is_rational(); - case info_flags::crational: - case info_flags::crational_polynomial: - return is_crational(); - case info_flags::integer: - case info_flags::integer_polynomial: - return is_integer(); - case info_flags::cinteger: - case info_flags::cinteger_polynomial: - return is_cinteger(); - case info_flags::positive: - return is_positive(); - case info_flags::negative: - return is_negative(); - case info_flags::nonnegative: - return compare(_num0())>=0; - case info_flags::posint: - return is_pos_integer(); - case info_flags::negint: - return is_integer() && (compare(_num0())<0); - case info_flags::nonnegint: - return is_nonneg_integer(); - case info_flags::even: - return is_even(); - case info_flags::odd: - return is_odd(); - case info_flags::prime: - return is_prime(); - } - return false; + switch (inf) { + case info_flags::numeric: + case info_flags::polynomial: + case info_flags::rational_function: + return true; + case info_flags::real: + return is_real(); + case info_flags::rational: + case info_flags::rational_polynomial: + return is_rational(); + case info_flags::crational: + case info_flags::crational_polynomial: + return is_crational(); + case info_flags::integer: + case info_flags::integer_polynomial: + return is_integer(); + case info_flags::cinteger: + case info_flags::cinteger_polynomial: + return is_cinteger(); + case info_flags::positive: + return is_positive(); + case info_flags::negative: + return is_negative(); + case info_flags::nonnegative: + return !is_negative(); + case info_flags::posint: + return is_pos_integer(); + case info_flags::negint: + return is_integer() && is_negative(); + case info_flags::nonnegint: + return is_nonneg_integer(); + case info_flags::even: + return is_even(); + case info_flags::odd: + return is_odd(); + case info_flags::prime: + return is_prime(); + case info_flags::algebraic: + return !is_real(); + } + return false; +} + +int numeric::degree(const ex & s) const +{ + return 0; +} + +int numeric::ldegree(const ex & s) const +{ + return 0; +} + +ex numeric::coeff(const ex & s, int n) const +{ + return n==0 ? *this : _ex0; +} + +/** Disassemble real part and imaginary part to scan for the occurrence of a + * single number. Also handles the imaginary unit. It ignores the sign on + * both this and the argument, which may lead to what might appear as funny + * results: (2+I).has(-2) -> true. But this is consistent, since we also + * would like to have (-2+I).has(2) -> true and we want to think about the + * sign as a multiplicative factor. */ +bool numeric::has(const ex &other) const +{ + if (!is_exactly_a(other)) + return false; + const numeric &o = ex_to(other); + if (this->is_equal(o) || this->is_equal(-o)) + return true; + if (o.imag().is_zero()) // e.g. scan for 3 in -3*I + return (this->real().is_equal(o) || this->imag().is_equal(o) || + this->real().is_equal(-o) || this->imag().is_equal(-o)); + else { + if (o.is_equal(I)) // e.g scan for I in 42*I + return !this->is_real(); + if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1 + return (this->real().has(o*I) || this->imag().has(o*I) || + this->real().has(-o*I) || this->imag().has(-o*I)); + } + return false; +} + + +/** Evaluation of numbers doesn't do anything at all. */ +ex numeric::eval(int level) const +{ + // Warning: if this is ever gonna do something, the ex ctors from all kinds + // of numbers should be checking for status_flags::evaluated. + return this->hold(); } + /** Cast numeric into a floating-point object. For example exact numeric(1) is * returned as a 1.0000000000000000000000 and so on according to how Digits is - * currently set. + * currently set. In case the object already was a floating point number the + * precision is trimmed to match the currently set default. * - * @param level ignored, but needed for overriding basic::evalf. + * @param level ignored, only needed for overriding basic::evalf. * @return an ex-handle to a numeric. */ ex numeric::evalf(int level) const { - // level can safely be discarded for numeric objects. - return numeric(cl_float(1.0, cl_default_float_format) * (*value)); // -> CLN + // level can safely be discarded for numeric objects. + return numeric(cln::cl_float(1.0, cln::default_float_format) * value); } -// protected - -int numeric::compare_same_type(basic const & other) const +ex numeric::conjugate() const { - GINAC_ASSERT(is_exactly_of_type(other, numeric)); - const numeric & o = static_cast(const_cast(other)); + if (is_real()) { + return *this; + } + return numeric(cln::conjugate(this->value)); +} - if (*value == *o.value) { - return 0; - } +// protected - return compare(o); +int numeric::compare_same_type(const basic &other) const +{ + GINAC_ASSERT(is_exactly_a(other)); + const numeric &o = static_cast(other); + + return this->compare(o); } -bool numeric::is_equal_same_type(basic const & other) const + +bool numeric::is_equal_same_type(const basic &other) const { - GINAC_ASSERT(is_exactly_of_type(other,numeric)); - const numeric *o = static_cast(&other); - - return is_equal(*o); + GINAC_ASSERT(is_exactly_a(other)); + const numeric &o = static_cast(other); + + return this->is_equal(o); } -/* -unsigned numeric::calchash(void) const + +unsigned numeric::calchash() const { - double d=to_double(); - int s=d>0 ? 1 : -1; - d=fabs(d); - if (d>0x07FF0000) { - d=0x07FF0000; - } - return 0x88000000U+s*unsigned(d/0x07FF0000); + // Base computation of hashvalue on CLN's hashcode. Note: That depends + // only on the number's value, not its type or precision (i.e. a true + // equivalence relation on numbers). As a consequence, 3 and 3.0 share + // the same hashvalue. That shouldn't really matter, though. + setflag(status_flags::hash_calculated); + hashvalue = golden_ratio_hash(cln::equal_hashcode(value)); + return hashvalue; } -*/ ////////// @@ -423,838 +719,1077 @@ unsigned numeric::calchash(void) const // public /** Numerical addition method. Adds argument to *this and returns result as - * a new numeric object. */ -numeric numeric::add(const numeric & other) const + * a numeric object. */ +const numeric numeric::add(const numeric &other) const { - return numeric((*value)+(*other.value)); + return numeric(value + other.value); } + /** Numerical subtraction method. Subtracts argument from *this and returns - * result as a new numeric object. */ -numeric numeric::sub(const numeric & other) const + * result as a numeric object. */ +const numeric numeric::sub(const numeric &other) const { - return numeric((*value)-(*other.value)); + return numeric(value - other.value); } + /** Numerical multiplication method. Multiplies *this and argument and returns - * result as a new numeric object. */ -numeric numeric::mul(const numeric & other) const + * result as a numeric object. */ +const numeric numeric::mul(const numeric &other) const { - static const numeric * _num1p=&_num1(); - if (this==_num1p) { - return other; - } else if (&other==_num1p) { - return *this; - } - return numeric((*value)*(*other.value)); + return numeric(value * other.value); } + /** Numerical division method. Divides *this by argument and returns result as - * a new numeric object. + * a numeric object. * * @exception overflow_error (division by zero) */ -numeric numeric::div(const numeric & other) const +const numeric numeric::div(const numeric &other) const { - if (::zerop(*other.value)) - throw (std::overflow_error("division by zero")); - return numeric((*value)/(*other.value)); + if (cln::zerop(other.value)) + throw std::overflow_error("numeric::div(): division by zero"); + return numeric(value / other.value); } -numeric numeric::power(const numeric & other) const + +/** Numerical exponentiation. Raises *this to the power given as argument and + * returns result as a numeric object. */ +const numeric numeric::power(const numeric &other) const { - static const numeric * _num1p=&_num1(); - if (&other==_num1p) - return *this; - if (::zerop(*value) && other.is_real() && ::minusp(realpart(*other.value))) - throw (std::overflow_error("division by zero")); - return numeric(::expt(*value,*other.value)); + // Shortcut for efficiency and numeric stability (as in 1.0 exponent): + // trap the neutral exponent. + if (&other==_num1_p || cln::equal(other.value,_num1.value)) + return *this; + + if (cln::zerop(value)) { + if (cln::zerop(other.value)) + throw std::domain_error("numeric::eval(): pow(0,0) is undefined"); + else if (cln::zerop(cln::realpart(other.value))) + throw std::domain_error("numeric::eval(): pow(0,I) is undefined"); + else if (cln::minusp(cln::realpart(other.value))) + throw std::overflow_error("numeric::eval(): division by zero"); + else + return _num0; + } + return numeric(cln::expt(value, other.value)); } -/** Inverse of a number. */ -numeric numeric::inverse(void) const + + +/** Numerical addition method. Adds argument to *this and returns result as + * a numeric object on the heap. Use internally only for direct wrapping into + * an ex object, where the result would end up on the heap anyways. */ +const numeric &numeric::add_dyn(const numeric &other) const { - return numeric(::recip(*value)); // -> CLN + // Efficiency shortcut: trap the neutral element by pointer. This hack + // is supposed to keep the number of distinct numeric objects low. + if (this==_num0_p) + return other; + else if (&other==_num0_p) + return *this; + + return static_cast((new numeric(value + other.value))-> + setflag(status_flags::dynallocated)); } -const numeric & numeric::add_dyn(const numeric & other) const + +/** Numerical subtraction method. Subtracts argument from *this and returns + * result as a numeric object on the heap. Use internally only for direct + * wrapping into an ex object, where the result would end up on the heap + * anyways. */ +const numeric &numeric::sub_dyn(const numeric &other) const { - return static_cast((new numeric((*value)+(*other.value)))-> - setflag(status_flags::dynallocated)); + // Efficiency shortcut: trap the neutral exponent (first by pointer). This + // hack is supposed to keep the number of distinct numeric objects low. + if (&other==_num0_p || cln::zerop(other.value)) + return *this; + + return static_cast((new numeric(value - other.value))-> + setflag(status_flags::dynallocated)); } -const numeric & numeric::sub_dyn(const numeric & other) const + +/** Numerical multiplication method. Multiplies *this and argument and returns + * result as a numeric object on the heap. Use internally only for direct + * wrapping into an ex object, where the result would end up on the heap + * anyways. */ +const numeric &numeric::mul_dyn(const numeric &other) const { - return static_cast((new numeric((*value)-(*other.value)))-> - setflag(status_flags::dynallocated)); + // Efficiency shortcut: trap the neutral element by pointer. This hack + // is supposed to keep the number of distinct numeric objects low. + if (this==_num1_p) + return other; + else if (&other==_num1_p) + return *this; + + return static_cast((new numeric(value * other.value))-> + setflag(status_flags::dynallocated)); } -const numeric & numeric::mul_dyn(const numeric & other) const + +/** Numerical division method. Divides *this by argument and returns result as + * a numeric object on the heap. Use internally only for direct wrapping + * into an ex object, where the result would end up on the heap + * anyways. + * + * @exception overflow_error (division by zero) */ +const numeric &numeric::div_dyn(const numeric &other) const { - static const numeric * _num1p=&_num1(); - if (this==_num1p) { - return other; - } else if (&other==_num1p) { - return *this; - } - return static_cast((new numeric((*value)*(*other.value)))-> - setflag(status_flags::dynallocated)); + // Efficiency shortcut: trap the neutral element by pointer. This hack + // is supposed to keep the number of distinct numeric objects low. + if (&other==_num1_p) + return *this; + if (cln::zerop(cln::the(other.value))) + throw std::overflow_error("division by zero"); + return static_cast((new numeric(value / other.value))-> + setflag(status_flags::dynallocated)); } -const numeric & numeric::div_dyn(const numeric & other) const + +/** Numerical exponentiation. Raises *this to the power given as argument and + * returns result as a numeric object on the heap. Use internally only for + * direct wrapping into an ex object, where the result would end up on the + * heap anyways. */ +const numeric &numeric::power_dyn(const numeric &other) const { - if (::zerop(*other.value)) - throw (std::overflow_error("division by zero")); - return static_cast((new numeric((*value)/(*other.value)))-> - setflag(status_flags::dynallocated)); + // Efficiency shortcut: trap the neutral exponent (first try by pointer, then + // try harder, since calls to cln::expt() below may return amazing results for + // floating point exponent 1.0). + if (&other==_num1_p || cln::equal(other.value, _num1.value)) + return *this; + + if (cln::zerop(value)) { + if (cln::zerop(other.value)) + throw std::domain_error("numeric::eval(): pow(0,0) is undefined"); + else if (cln::zerop(cln::realpart(other.value))) + throw std::domain_error("numeric::eval(): pow(0,I) is undefined"); + else if (cln::minusp(cln::realpart(other.value))) + throw std::overflow_error("numeric::eval(): division by zero"); + else + return _num0; + } + return static_cast((new numeric(cln::expt(value, other.value)))-> + setflag(status_flags::dynallocated)); } -const numeric & numeric::power_dyn(const numeric & other) const + +const numeric &numeric::operator=(int i) { - static const numeric * _num1p=&_num1(); - if (&other==_num1p) - return *this; - if (::zerop(*value) && other.is_real() && ::minusp(realpart(*other.value))) - throw (std::overflow_error("division by zero")); - return static_cast((new numeric(::expt(*value,*other.value)))-> - setflag(status_flags::dynallocated)); + return operator=(numeric(i)); } -const numeric & numeric::operator=(int i) + +const numeric &numeric::operator=(unsigned int i) { - return operator=(numeric(i)); + return operator=(numeric(i)); } -const numeric & numeric::operator=(unsigned int i) + +const numeric &numeric::operator=(long i) { - return operator=(numeric(i)); + return operator=(numeric(i)); } -const numeric & numeric::operator=(long i) + +const numeric &numeric::operator=(unsigned long i) { - return operator=(numeric(i)); + return operator=(numeric(i)); } -const numeric & numeric::operator=(unsigned long i) + +const numeric &numeric::operator=(double d) { - return operator=(numeric(i)); + return operator=(numeric(d)); } -const numeric & numeric::operator=(double d) + +const numeric &numeric::operator=(const char * s) { - return operator=(numeric(d)); + return operator=(numeric(s)); } -const numeric & numeric::operator=(char const * s) + +/** Inverse of a number. */ +const numeric numeric::inverse() const { - return operator=(numeric(s)); + if (cln::zerop(value)) + throw std::overflow_error("numeric::inverse(): division by zero"); + return numeric(cln::recip(value)); } + /** Return the complex half-plane (left or right) in which the number lies. * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0, * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0. * - * @see numeric::compare(const numeric & other) */ -int numeric::csgn(void) const -{ - if (is_zero()) - return 0; - if (!::zerop(realpart(*value))) { - if (::plusp(realpart(*value))) - return 1; - else - return -1; - } else { - if (::plusp(imagpart(*value))) - return 1; - else - return -1; - } + * @see numeric::compare(const numeric &other) */ +int numeric::csgn() const +{ + if (cln::zerop(value)) + return 0; + cln::cl_R r = cln::realpart(value); + if (!cln::zerop(r)) { + if (cln::plusp(r)) + return 1; + else + return -1; + } else { + if (cln::plusp(cln::imagpart(value))) + return 1; + else + return -1; + } } + /** This method establishes a canonical order on all numbers. For complex * numbers this is not possible in a mathematically consistent way but we need * to establish some order and it ought to be fast. So we simply define it * to be compatible with our method csgn. * * @return csgn(*this-other) - * @see numeric::csgn(void) */ -int numeric::compare(const numeric & other) const + * @see numeric::csgn() */ +int numeric::compare(const numeric &other) const { - // Comparing two real numbers? - if (is_real() && other.is_real()) - // Yes, just compare them - return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value)); - else { - // No, first compare real parts - cl_signean real_cmp = ::cl_compare(realpart(*value), realpart(*other.value)); - if (real_cmp) - return real_cmp; - - return ::cl_compare(imagpart(*value), imagpart(*other.value)); - } + // Comparing two real numbers? + if (cln::instanceof(value, cln::cl_R_ring) && + cln::instanceof(other.value, cln::cl_R_ring)) + // Yes, so just cln::compare them + return cln::compare(cln::the(value), cln::the(other.value)); + else { + // No, first cln::compare real parts... + cl_signean real_cmp = cln::compare(cln::realpart(value), cln::realpart(other.value)); + if (real_cmp) + return real_cmp; + // ...and then the imaginary parts. + return cln::compare(cln::imagpart(value), cln::imagpart(other.value)); + } } -bool numeric::is_equal(const numeric & other) const + +bool numeric::is_equal(const numeric &other) const { - return (*value == *other.value); + return cln::equal(value, other.value); } + /** True if object is zero. */ -bool numeric::is_zero(void) const +bool numeric::is_zero() const { - return ::zerop(*value); // -> CLN + return cln::zerop(value); } + /** True if object is not complex and greater than zero. */ -bool numeric::is_positive(void) const +bool numeric::is_positive() const { - if (is_real()) - return ::plusp(The(cl_R)(*value)); // -> CLN - return false; + if (cln::instanceof(value, cln::cl_R_ring)) // real? + return cln::plusp(cln::the(value)); + return false; } + /** True if object is not complex and less than zero. */ -bool numeric::is_negative(void) const +bool numeric::is_negative() const { - if (is_real()) - return ::minusp(The(cl_R)(*value)); // -> CLN - return false; + if (cln::instanceof(value, cln::cl_R_ring)) // real? + return cln::minusp(cln::the(value)); + return false; } + /** True if object is a non-complex integer. */ -bool numeric::is_integer(void) const +bool numeric::is_integer() const { - return ::instanceof(*value, cl_I_ring); // -> CLN + return cln::instanceof(value, cln::cl_I_ring); } + /** True if object is an exact integer greater than zero. */ -bool numeric::is_pos_integer(void) const +bool numeric::is_pos_integer() const { - return (is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN + return (cln::instanceof(value, cln::cl_I_ring) && cln::plusp(cln::the(value))); } + /** True if object is an exact integer greater or equal zero. */ -bool numeric::is_nonneg_integer(void) const +bool numeric::is_nonneg_integer() const { - return (is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN + return (cln::instanceof(value, cln::cl_I_ring) && !cln::minusp(cln::the(value))); } + /** True if object is an exact even integer. */ -bool numeric::is_even(void) const +bool numeric::is_even() const { - return (is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN + return (cln::instanceof(value, cln::cl_I_ring) && cln::evenp(cln::the(value))); } + /** True if object is an exact odd integer. */ -bool numeric::is_odd(void) const +bool numeric::is_odd() const { - return (is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN + return (cln::instanceof(value, cln::cl_I_ring) && cln::oddp(cln::the(value))); } + /** Probabilistic primality test. * * @return true if object is exact integer and prime. */ -bool numeric::is_prime(void) const +bool numeric::is_prime() const { - return (is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN + return (cln::instanceof(value, cln::cl_I_ring) // integer? + && cln::plusp(cln::the(value)) // positive? + && cln::isprobprime(cln::the(value))); } + /** True if object is an exact rational number, may even be complex * (denominator may be unity). */ -bool numeric::is_rational(void) const +bool numeric::is_rational() const { - return ::instanceof(*value, cl_RA_ring); // -> CLN + return cln::instanceof(value, cln::cl_RA_ring); } + /** True if object is a real integer, rational or float (but not complex). */ -bool numeric::is_real(void) const +bool numeric::is_real() const { - return ::instanceof(*value, cl_R_ring); // -> CLN + return cln::instanceof(value, cln::cl_R_ring); } -bool numeric::operator==(const numeric & other) const + +bool numeric::operator==(const numeric &other) const { - return (*value == *other.value); // -> CLN + return cln::equal(value, other.value); } -bool numeric::operator!=(const numeric & other) const + +bool numeric::operator!=(const numeric &other) const { - return (*value != *other.value); // -> CLN + return !cln::equal(value, other.value); } + /** True if object is element of the domain of integers extended by I, i.e. is * of the form a+b*I, where a and b are integers. */ -bool numeric::is_cinteger(void) const +bool numeric::is_cinteger() const { - if (::instanceof(*value, cl_I_ring)) - return true; - else if (!is_real()) { // complex case, handle n+m*I - if (::instanceof(realpart(*value), cl_I_ring) && - ::instanceof(imagpart(*value), cl_I_ring)) - return true; - } - return false; + if (cln::instanceof(value, cln::cl_I_ring)) + return true; + else if (!this->is_real()) { // complex case, handle n+m*I + if (cln::instanceof(cln::realpart(value), cln::cl_I_ring) && + cln::instanceof(cln::imagpart(value), cln::cl_I_ring)) + return true; + } + return false; } + /** True if object is an exact rational number, may even be complex * (denominator may be unity). */ -bool numeric::is_crational(void) const +bool numeric::is_crational() const { - if (::instanceof(*value, cl_RA_ring)) - return true; - else if (!is_real()) { // complex case, handle Q(i): - if (::instanceof(realpart(*value), cl_RA_ring) && - ::instanceof(imagpart(*value), cl_RA_ring)) - return true; - } - return false; + if (cln::instanceof(value, cln::cl_RA_ring)) + return true; + else if (!this->is_real()) { // complex case, handle Q(i): + if (cln::instanceof(cln::realpart(value), cln::cl_RA_ring) && + cln::instanceof(cln::imagpart(value), cln::cl_RA_ring)) + return true; + } + return false; } + /** Numerical comparison: less. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator<(const numeric & other) const +bool numeric::operator<(const numeric &other) const { - if (is_real() && other.is_real()) - return (bool)(The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN - throw (std::invalid_argument("numeric::operator<(): complex inequality")); - return false; // make compiler shut up + if (this->is_real() && other.is_real()) + return (cln::the(value) < cln::the(other.value)); + throw std::invalid_argument("numeric::operator<(): complex inequality"); } + /** Numerical comparison: less or equal. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator<=(const numeric & other) const +bool numeric::operator<=(const numeric &other) const { - if (is_real() && other.is_real()) - return (bool)(The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN - throw (std::invalid_argument("numeric::operator<=(): complex inequality")); - return false; // make compiler shut up + if (this->is_real() && other.is_real()) + return (cln::the(value) <= cln::the(other.value)); + throw std::invalid_argument("numeric::operator<=(): complex inequality"); } + /** Numerical comparison: greater. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator>(const numeric & other) const +bool numeric::operator>(const numeric &other) const { - if (is_real() && other.is_real()) - return (bool)(The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN - throw (std::invalid_argument("numeric::operator>(): complex inequality")); - return false; // make compiler shut up + if (this->is_real() && other.is_real()) + return (cln::the(value) > cln::the(other.value)); + throw std::invalid_argument("numeric::operator>(): complex inequality"); } + /** Numerical comparison: greater or equal. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator>=(const numeric & other) const +bool numeric::operator>=(const numeric &other) const +{ + if (this->is_real() && other.is_real()) + return (cln::the(value) >= cln::the(other.value)); + throw std::invalid_argument("numeric::operator>=(): complex inequality"); +} + + +/** Converts numeric types to machine's int. You should check with + * is_integer() if the number is really an integer before calling this method. + * You may also consider checking the range first. */ +int numeric::to_int() const { - if (is_real() && other.is_real()) - return (bool)(The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN - throw (std::invalid_argument("numeric::operator>=(): complex inequality")); - return false; // make compiler shut up + GINAC_ASSERT(this->is_integer()); + return cln::cl_I_to_int(cln::the(value)); } -/** Converts numeric types to machine's int. You should check with is_integer() - * if the number is really an integer before calling this method. */ -int numeric::to_int(void) const + +/** Converts numeric types to machine's long. You should check with + * is_integer() if the number is really an integer before calling this method. + * You may also consider checking the range first. */ +long numeric::to_long() const { - GINAC_ASSERT(is_integer()); - return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN + GINAC_ASSERT(this->is_integer()); + return cln::cl_I_to_long(cln::the(value)); } + /** Converts numeric types to machine's double. You should check with is_real() * if the number is really not complex before calling this method. */ -double numeric::to_double(void) const +double numeric::to_double() const { - GINAC_ASSERT(is_real()); - return ::cl_double_approx(realpart(*value)); // -> CLN + GINAC_ASSERT(this->is_real()); + return cln::double_approx(cln::realpart(value)); } + +/** Returns a new CLN object of type cl_N, representing the value of *this. + * This method may be used when mixing GiNaC and CLN in one project. + */ +cln::cl_N numeric::to_cl_N() const +{ + return value; +} + + /** Real part of a number. */ -numeric numeric::real(void) const +const numeric numeric::real() const { - return numeric(::realpart(*value)); // -> CLN + return numeric(cln::realpart(value)); } + /** Imaginary part of a number. */ -numeric numeric::imag(void) const +const numeric numeric::imag() const { - return numeric(::imagpart(*value)); // -> CLN + return numeric(cln::imagpart(value)); } -#ifndef SANE_LINKER -// Unfortunately, CLN did not provide an official way to access the numerator -// or denominator of a rational number (cl_RA). Doing some excavations in CLN -// one finds how it works internally in src/rational/cl_RA.h: -struct cl_heap_ratio : cl_heap { - cl_I numerator; - cl_I denominator; -}; - -inline cl_heap_ratio* TheRatio (const cl_N& obj) -{ return (cl_heap_ratio*)(obj.pointer); } -#endif // ndef SANE_LINKER /** Numerator. Computes the numerator of rational numbers, rationalized * numerator of complex if real and imaginary part are both rational numbers * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other * cases. */ -numeric numeric::numer(void) const -{ - if (is_integer()) { - return numeric(*this); - } -#ifdef SANE_LINKER - else if (::instanceof(*value, cl_RA_ring)) { - return numeric(::numerator(The(cl_RA)(*value))); - } - else if (!is_real()) { // complex case, handle Q(i): - cl_R r = ::realpart(*value); - cl_R i = ::imagpart(*value); - if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring)) - return numeric(*this); - if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring)) - return numeric(complex(r*::denominator(The(cl_RA)(i)), ::numerator(The(cl_RA)(i)))); - if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring)) - return numeric(complex(::numerator(The(cl_RA)(r)), i*::denominator(The(cl_RA)(r)))); - if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring)) { - cl_I s = lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))); - return numeric(complex(::numerator(The(cl_RA)(r))*(exquo(s,::denominator(The(cl_RA)(r)))), - ::numerator(The(cl_RA)(i))*(exquo(s,::denominator(The(cl_RA)(i)))))); - } - } -#else - else if (instanceof(*value, cl_RA_ring)) { - return numeric(TheRatio(*value)->numerator); - } - else if (!is_real()) { // complex case, handle Q(i): - cl_R r = realpart(*value); - cl_R i = imagpart(*value); - if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring)) - return numeric(*this); - if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring)) - return numeric(complex(r*TheRatio(i)->denominator, TheRatio(i)->numerator)); - if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring)) - return numeric(complex(TheRatio(r)->numerator, i*TheRatio(r)->denominator)); - if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) { - cl_I s = lcm(TheRatio(r)->denominator, TheRatio(i)->denominator); - return numeric(complex(TheRatio(r)->numerator*(exquo(s,TheRatio(r)->denominator)), - TheRatio(i)->numerator*(exquo(s,TheRatio(i)->denominator)))); - } - } -#endif // def SANE_LINKER - // at least one float encountered - return numeric(*this); +const numeric numeric::numer() const +{ + if (cln::instanceof(value, cln::cl_I_ring)) + return numeric(*this); // integer case + + else if (cln::instanceof(value, cln::cl_RA_ring)) + return numeric(cln::numerator(cln::the(value))); + + else if (!this->is_real()) { // complex case, handle Q(i): + const cln::cl_RA r = cln::the(cln::realpart(value)); + const cln::cl_RA i = cln::the(cln::imagpart(value)); + if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring)) + return numeric(*this); + if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring)) + return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i))); + if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring)) + return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r))); + if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) { + const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i)); + return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))), + cln::numerator(i)*(cln::exquo(s,cln::denominator(i))))); + } + } + // at least one float encountered + return numeric(*this); } + /** Denominator. Computes the denominator of rational numbers, common integer * denominator of complex if real and imaginary part are both rational numbers * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */ -numeric numeric::denom(void) const -{ - if (is_integer()) { - return _num1(); - } -#ifdef SANE_LINKER - if (instanceof(*value, cl_RA_ring)) { - return numeric(::denominator(The(cl_RA)(*value))); - } - if (!is_real()) { // complex case, handle Q(i): - cl_R r = realpart(*value); - cl_R i = imagpart(*value); - if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_I_ring)) - return _num1(); - if (::instanceof(r, cl_I_ring) && ::instanceof(i, cl_RA_ring)) - return numeric(::denominator(The(cl_RA)(i))); - if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_I_ring)) - return numeric(::denominator(The(cl_RA)(r))); - if (::instanceof(r, cl_RA_ring) && ::instanceof(i, cl_RA_ring)) - return numeric(lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i)))); - } -#else - if (instanceof(*value, cl_RA_ring)) { - return numeric(TheRatio(*value)->denominator); - } - if (!is_real()) { // complex case, handle Q(i): - cl_R r = realpart(*value); - cl_R i = imagpart(*value); - if (instanceof(r, cl_I_ring) && instanceof(i, cl_I_ring)) - return _num1(); - if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_ring)) - return numeric(TheRatio(i)->denominator); - if (instanceof(r, cl_RA_ring) && instanceof(i, cl_I_ring)) - return numeric(TheRatio(r)->denominator); - if (instanceof(r, cl_RA_ring) && instanceof(i, cl_RA_ring)) - return numeric(lcm(TheRatio(r)->denominator, TheRatio(i)->denominator)); - } -#endif // def SANE_LINKER - // at least one float encountered - return _num1(); +const numeric numeric::denom() const +{ + if (cln::instanceof(value, cln::cl_I_ring)) + return _num1; // integer case + + if (cln::instanceof(value, cln::cl_RA_ring)) + return numeric(cln::denominator(cln::the(value))); + + if (!this->is_real()) { // complex case, handle Q(i): + const cln::cl_RA r = cln::the(cln::realpart(value)); + const cln::cl_RA i = cln::the(cln::imagpart(value)); + if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring)) + return _num1; + if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring)) + return numeric(cln::denominator(i)); + if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring)) + return numeric(cln::denominator(r)); + if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) + return numeric(cln::lcm(cln::denominator(r), cln::denominator(i))); + } + // at least one float encountered + return _num1; } + /** Size in binary notation. For integers, this is the smallest n >= 0 such * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that * 2^(n-1) <= x < 2^n. * * @return number of bits (excluding sign) needed to represent that number * in two's complement if it is an integer, 0 otherwise. */ -int numeric::int_length(void) const +int numeric::int_length() const { - if (is_integer()) - return ::integer_length(The(cl_I)(*value)); // -> CLN - else - return 0; + if (cln::instanceof(value, cln::cl_I_ring)) + return cln::integer_length(cln::the(value)); + else + return 0; } - -////////// -// static member variables -////////// - -// protected - -unsigned numeric::precedence = 30; - ////////// // global constants ////////// -const numeric some_numeric; -type_info const & typeid_numeric=typeid(some_numeric); /** Imaginary unit. This is not a constant but a numeric since we are - * natively handing complex numbers anyways. */ -const numeric I = numeric(complex(cl_I(0),cl_I(1))); + * natively handing complex numbers anyways, so in each expression containing + * an I it is automatically eval'ed away anyhow. */ +const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1))); + /** Exponential function. * * @return arbitrary precision numerical exp(x). */ -numeric exp(const numeric & x) +const numeric exp(const numeric &x) { - return ::exp(*x.value); // -> CLN + return cln::exp(x.to_cl_N()); } + /** Natural logarithm. * - * @param z complex number + * @param x complex number * @return arbitrary precision numerical log(x). - * @exception overflow_error (logarithmic singularity) */ -numeric log(const numeric & z) + * @exception pole_error("log(): logarithmic pole",0) */ +const numeric log(const numeric &x) { - if (z.is_zero()) - throw (std::overflow_error("log(): logarithmic singularity")); - return ::log(*z.value); // -> CLN + if (x.is_zero()) + throw pole_error("log(): logarithmic pole",0); + return cln::log(x.to_cl_N()); } + /** Numeric sine (trigonometric function). * * @return arbitrary precision numerical sin(x). */ -numeric sin(const numeric & x) +const numeric sin(const numeric &x) { - return ::sin(*x.value); // -> CLN + return cln::sin(x.to_cl_N()); } + /** Numeric cosine (trigonometric function). * * @return arbitrary precision numerical cos(x). */ -numeric cos(const numeric & x) +const numeric cos(const numeric &x) { - return ::cos(*x.value); // -> CLN + return cln::cos(x.to_cl_N()); } - + + /** Numeric tangent (trigonometric function). * * @return arbitrary precision numerical tan(x). */ -numeric tan(const numeric & x) +const numeric tan(const numeric &x) { - return ::tan(*x.value); // -> CLN + return cln::tan(x.to_cl_N()); } - + + /** Numeric inverse sine (trigonometric function). * * @return arbitrary precision numerical asin(x). */ -numeric asin(const numeric & x) +const numeric asin(const numeric &x) { - return ::asin(*x.value); // -> CLN + return cln::asin(x.to_cl_N()); } - + + /** Numeric inverse cosine (trigonometric function). * * @return arbitrary precision numerical acos(x). */ -numeric acos(const numeric & x) +const numeric acos(const numeric &x) { - return ::acos(*x.value); // -> CLN + return cln::acos(x.to_cl_N()); } - -/** Arcustangents. + + +/** Arcustangent. * - * @param z complex number - * @return atan(z) - * @exception overflow_error (logarithmic singularity) */ -numeric atan(const numeric & x) + * @param x complex number + * @return atan(x) + * @exception pole_error("atan(): logarithmic pole",0) */ +const numeric atan(const numeric &x) { - if (!x.is_real() && - x.real().is_zero() && - !abs(x.imag()).is_equal(_num1())) - throw (std::overflow_error("atan(): logarithmic singularity")); - return ::atan(*x.value); // -> CLN + if (!x.is_real() && + x.real().is_zero() && + abs(x.imag()).is_equal(_num1)) + throw pole_error("atan(): logarithmic pole",0); + return cln::atan(x.to_cl_N()); } -/** Arcustangents. + +/** Arcustangent. * * @param x real number * @param y real number * @return atan(y/x) */ -numeric atan(const numeric & y, const numeric & x) +const numeric atan(const numeric &y, const numeric &x) { - if (x.is_real() && y.is_real()) - return ::atan(realpart(*x.value), realpart(*y.value)); // -> CLN - else - throw (std::invalid_argument("numeric::atan(): complex argument")); + if (x.is_real() && y.is_real()) + return cln::atan(cln::the(x.to_cl_N()), + cln::the(y.to_cl_N())); + else + throw std::invalid_argument("atan(): complex argument"); } + /** Numeric hyperbolic sine (trigonometric function). * * @return arbitrary precision numerical sinh(x). */ -numeric sinh(const numeric & x) +const numeric sinh(const numeric &x) { - return ::sinh(*x.value); // -> CLN + return cln::sinh(x.to_cl_N()); } + /** Numeric hyperbolic cosine (trigonometric function). * * @return arbitrary precision numerical cosh(x). */ -numeric cosh(const numeric & x) +const numeric cosh(const numeric &x) { - return ::cosh(*x.value); // -> CLN + return cln::cosh(x.to_cl_N()); } - + + /** Numeric hyperbolic tangent (trigonometric function). * * @return arbitrary precision numerical tanh(x). */ -numeric tanh(const numeric & x) +const numeric tanh(const numeric &x) { - return ::tanh(*x.value); // -> CLN + return cln::tanh(x.to_cl_N()); } - + + /** Numeric inverse hyperbolic sine (trigonometric function). * * @return arbitrary precision numerical asinh(x). */ -numeric asinh(const numeric & x) +const numeric asinh(const numeric &x) { - return ::asinh(*x.value); // -> CLN + return cln::asinh(x.to_cl_N()); } + /** Numeric inverse hyperbolic cosine (trigonometric function). * * @return arbitrary precision numerical acosh(x). */ -numeric acosh(const numeric & x) +const numeric acosh(const numeric &x) { - return ::acosh(*x.value); // -> CLN + return cln::acosh(x.to_cl_N()); } + /** Numeric inverse hyperbolic tangent (trigonometric function). * * @return arbitrary precision numerical atanh(x). */ -numeric atanh(const numeric & x) -{ - return ::atanh(*x.value); // -> CLN +const numeric atanh(const numeric &x) +{ + return cln::atanh(x.to_cl_N()); +} + + +/*static cln::cl_N Li2_series(const ::cl_N &x, + const ::float_format_t &prec) +{ + // Note: argument must be in the unit circle + // This is very inefficient unless we have fast floating point Bernoulli + // numbers implemented! + cln::cl_N c1 = -cln::log(1-x); + cln::cl_N c2 = c1; + // hard-wire the first two Bernoulli numbers + cln::cl_N acc = c1 - cln::square(c1)/4; + cln::cl_N aug; + cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2 + cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i) + unsigned i = 1; + c1 = cln::square(c1); + do { + c2 = c1 * c2; + piac = piac * pisq; + aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1); + // 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)); + acc = acc + aug; + ++i; + } while (acc != acc+aug); + return acc; +}*/ + +/** Numeric evaluation of Dilogarithm within circle of convergence (unit + * circle) using a power series. */ +static cln::cl_N Li2_series(const cln::cl_N &x, + const cln::float_format_t &prec) +{ + // Note: argument must be in the unit circle + cln::cl_N aug, acc; + cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0); + cln::cl_I den = 0; + unsigned i = 1; + do { + num = num * x; + den = den + i; // 1, 4, 9, 16, ... + i += 2; + aug = num / den; + acc = acc + aug; + } while (acc != acc+aug); + return acc; +} + +/** Folds Li2's argument inside a small rectangle to enhance convergence. */ +static cln::cl_N Li2_projection(const cln::cl_N &x, + const cln::float_format_t &prec) +{ + const cln::cl_R re = cln::realpart(x); + const cln::cl_R im = cln::imagpart(x); + if (re > cln::cl_F(".5")) + // zeta(2) - Li2(1-x) - log(x)*log(1-x) + return(cln::zeta(2) + - Li2_series(1-x, prec) + - cln::log(x)*cln::log(1-x)); + if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5"))) + // -log(1-x)^2 / 2 - Li2(x/(x-1)) + return(- cln::square(cln::log(1-x))/2 + - Li2_series(x/(x-1), prec)); + if (re > 0 && cln::abs(im) > cln::cl_LF(".75")) + // Li2(x^2)/2 - Li2(-x) + return(Li2_projection(cln::square(x), prec)/2 + - Li2_projection(-x, prec)); + return Li2_series(x, prec); +} + +/** Numeric evaluation of Dilogarithm. The domain is the entire complex plane, + * the branch cut lies along the positive real axis, starting at 1 and + * continuous with quadrant IV. + * + * @return arbitrary precision numerical Li2(x). */ +const numeric Li2(const numeric &x) +{ + if (x.is_zero()) + return _num0; + + // what is the desired float format? + // first guess: default format + cln::float_format_t prec = cln::default_float_format; + const cln::cl_N value = x.to_cl_N(); + // second guess: the argument's format + if (!x.real().is_rational()) + prec = cln::float_format(cln::the(cln::realpart(value))); + else if (!x.imag().is_rational()) + prec = cln::float_format(cln::the(cln::imagpart(value))); + + if (value==1) // may cause trouble with log(1-x) + return cln::zeta(2, prec); + + if (cln::abs(value) > 1) + // -log(-x)^2 / 2 - zeta(2) - Li2(1/x) + return(- cln::square(cln::log(-value))/2 + - cln::zeta(2, prec) + - Li2_projection(cln::recip(value), prec)); + else + return Li2_projection(x.to_cl_N(), prec); } + /** Numeric evaluation of Riemann's Zeta function. Currently works only for * integer arguments. */ -numeric zeta(const numeric & x) -{ - // A dirty hack to allow for things like zeta(3.0), since CLN currently - // only knows about integer arguments and zeta(3).evalf() automatically - // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3 - // being an exact zero for CLN, which can be tested and then we can just - // pass the number casted to an int: - if (x.is_real()) { - int aux = (int)(::cl_double_approx(realpart(*x.value))); - if (zerop(*x.value-aux)) - return ::cl_zeta(aux); // -> CLN - } - clog << "zeta(" << x - << "): Does anybody know good way to calculate this numerically?" - << endl; - return numeric(0); -} - -/** The gamma function. +const numeric zeta(const numeric &x) +{ + // A dirty hack to allow for things like zeta(3.0), since CLN currently + // only knows about integer arguments and zeta(3).evalf() automatically + // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3 + // being an exact zero for CLN, which can be tested and then we can just + // pass the number casted to an int: + if (x.is_real()) { + const int aux = (int)(cln::double_approx(cln::the(x.to_cl_N()))); + if (cln::zerop(x.to_cl_N()-aux)) + return cln::zeta(aux); + } + throw dunno(); +} + + +/** The Gamma function. * This is only a stub! */ -numeric gamma(const numeric & x) +const numeric lgamma(const numeric &x) { - clog << "gamma(" << x - << "): Does anybody know good way to calculate this numerically?" - << endl; - return numeric(0); + throw dunno(); +} +const numeric tgamma(const numeric &x) +{ + throw dunno(); } + /** The psi function (aka polygamma function). * This is only a stub! */ -numeric psi(const numeric & x) +const numeric psi(const numeric &x) { - clog << "psi(" << x - << "): Does anybody know good way to calculate this numerically?" - << endl; - return numeric(0); + throw dunno(); } + /** The psi functions (aka polygamma functions). * This is only a stub! */ -numeric psi(const numeric & n, const numeric & x) +const numeric psi(const numeric &n, const numeric &x) { - clog << "psi(" << n << "," << x - << "): Does anybody know good way to calculate this numerically?" - << endl; - return numeric(0); + throw dunno(); } + /** Factorial combinatorial function. * + * @param n integer argument >= 0 * @exception range_error (argument must be integer >= 0) */ -numeric factorial(const numeric & nn) +const numeric factorial(const numeric &n) { - if (!nn.is_nonneg_integer()) - throw (std::range_error("numeric::factorial(): argument must be integer >= 0")); - return numeric(::factorial(nn.to_int())); // -> CLN + if (!n.is_nonneg_integer()) + throw std::range_error("numeric::factorial(): argument must be integer >= 0"); + return numeric(cln::factorial(n.to_int())); } + /** The double factorial combinatorial function. (Scarcely used, but still - * useful in cases, like for exact results of Gamma(n+1/2) for instance.) + * useful in cases, like for exact results of tgamma(n+1/2) for instance.) * * @param n integer argument >= -1 - * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == 1 == (-1)!! + * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1 * @exception range_error (argument must be integer >= -1) */ -numeric doublefactorial(const numeric & nn) -{ - // META-NOTE: The whole shit here will become obsolete and may be moved - // out once CLN learns about double factorial, which should be as soon as - // 1.0.3 rolls out! - - // We store the results separately for even and odd arguments. This has - // the advantage that we don't have to compute any even result at all if - // the function is always called with odd arguments and vice versa. There - // is no tradeoff involved in this, it is guaranteed to save time as well - // as memory. (If this is not enough justification consider the Gamma - // function of half integer arguments: it only needs odd doublefactorials.) - static vector evenresults; - static int highest_evenresult = -1; - static vector oddresults; - static int highest_oddresult = -1; - - if (nn == numeric(-1)) { - return _num1(); - } - if (!nn.is_nonneg_integer()) { - throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1")); - } - if (nn.is_even()) { - int n = nn.div(_num2()).to_int(); - if (n <= highest_evenresult) { - return evenresults[n]; - } - if (evenresults.capacity() < (unsigned)(n+1)) { - evenresults.reserve(n+1); - } - if (highest_evenresult < 0) { - evenresults.push_back(_num1()); - highest_evenresult=0; - } - for (int i=highest_evenresult+1; i<=n; i++) { - evenresults.push_back(numeric(evenresults[i-1].mul(numeric(i*2)))); - } - highest_evenresult=n; - return evenresults[n]; - } else { - int n = nn.sub(_num1()).div(_num2()).to_int(); - if (n <= highest_oddresult) { - return oddresults[n]; - } - if (oddresults.capacity() < (unsigned)n) { - oddresults.reserve(n+1); - } - if (highest_oddresult < 0) { - oddresults.push_back(_num1()); - highest_oddresult=0; - } - for (int i=highest_oddresult+1; i<=n; i++) { - oddresults.push_back(numeric(oddresults[i-1].mul(numeric(i*2+1)))); - } - highest_oddresult=n; - return oddresults[n]; - } +const numeric doublefactorial(const numeric &n) +{ + if (n.is_equal(_num_1)) + return _num1; + + if (!n.is_nonneg_integer()) + throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1"); + + return numeric(cln::doublefactorial(n.to_int())); } + /** The Binomial coefficients. It computes the binomial coefficients. For * integer n and k and positive n this is the number of ways of choosing k * objects from n distinct objects. If n is negative, the formula * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */ -numeric binomial(const numeric & n, const numeric & k) -{ - if (n.is_integer() && k.is_integer()) { - if (n.is_nonneg_integer()) { - if (k.compare(n)!=1 && k.compare(_num0())!=-1) - return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN - else - return _num0(); - } else { - return _num_1().power(k)*binomial(k-n-_num1(),k); - } - } - - // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit - throw (std::range_error("numeric::binomial(): don´t know how to evaluate that.")); +const numeric binomial(const numeric &n, const numeric &k) +{ + if (n.is_integer() && k.is_integer()) { + if (n.is_nonneg_integer()) { + if (k.compare(n)!=1 && k.compare(_num0)!=-1) + return numeric(cln::binomial(n.to_int(),k.to_int())); + else + return _num0; + } else { + return _num_1.power(k)*binomial(k-n-_num1,k); + } + } + + // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit + throw std::range_error("numeric::binomial(): don´t know how to evaluate that."); } + /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n! * in the expansion of the function x/(e^x-1). * * @return the nth Bernoulli number (a rational number). * @exception range_error (argument must be integer >= 0) */ -numeric bernoulli(const numeric & nn) -{ - if (!nn.is_integer() || nn.is_negative()) - throw (std::range_error("numeric::bernoulli(): argument must be integer >= 0")); - if (nn.is_zero()) - return _num1(); - if (!nn.compare(_num1())) - return numeric(-1,2); - if (nn.is_odd()) - return _num0(); - // Until somebody has the Blues and comes up with a much better idea and - // codes it (preferably in CLN) we make this a remembering function which - // computes its results using the formula - // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k)) - // whith B(0) == 1. - static vector results; - static int highest_result = -1; - int n = nn.sub(_num2()).div(_num2()).to_int(); - if (n <= highest_result) - return results[n]; - if (results.capacity() < (unsigned)(n+1)) - results.reserve(n+1); - - numeric tmp; // used to store the sum - for (int i=highest_result+1; i<=n; ++i) { - // the first two elements: - tmp = numeric(-2*i-1,2); - // accumulate the remaining elements: - for (int j=0; j= 0"); + + // Method: + // + // The Bernoulli numbers are rational numbers that may be computed using + // the relation + // + // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k) + // + // with B(0) = 1. Since the n'th Bernoulli number depends on all the + // previous ones, the computation is necessarily very expensive. There are + // several other ways of computing them, a particularly good one being + // cl_I s = 1; + // cl_I c = n+1; + // cl_RA Bern = 0; + // for (unsigned i=0; i results; + static unsigned next_r = 0; + + // algorithm not applicable to B(2), so just store it + if (!next_r) { + results.push_back(cln::recip(cln::cl_RA(6))); + next_r = 4; + } + if (n) + if (p < (1UL<= 1, n >= 0. + // + // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence + // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values + // agree.) + // Replace m by m+1: + // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0 + // Now put in m = n, to get + // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n)) + // F(2n+1) = F(n)^2 + F(n+1)^2 + // hence + // F(2n+2) = F(n+1)*(2*F(n) + F(n+1)) + if (n.is_zero()) + return _num0; + if (n.is_negative()) + if (n.is_even()) + return -fibonacci(-n); + else + return fibonacci(-n); + + cln::cl_I u(0); + cln::cl_I v(1); + cln::cl_I m = cln::the(n.to_cl_N()) >> 1L; // floor(n/2); + for (uintL bit=cln::integer_length(m); bit>0; --bit) { + // Since a squaring is cheaper than a multiplication, better use + // three squarings instead of one multiplication and two squarings. + cln::cl_I u2 = cln::square(u); + cln::cl_I v2 = cln::square(v); + if (cln::logbitp(bit-1, m)) { + v = cln::square(u + v) - u2; + u = u2 + v2; + } else { + u = v2 - cln::square(v - u); + v = u2 + v2; + } + } + if (n.is_even()) + // Here we don't use the squaring formula because one multiplication + // is cheaper than two squarings. + return u * ((v << 1) - u); + else + return cln::square(u) + cln::square(v); } + /** Absolute value. */ -numeric abs(const numeric & x) +const numeric abs(const numeric& x) { - return ::abs(*x.value); // -> CLN + return cln::abs(x.to_cl_N()); } + /** Modulus (in positive representation). * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the * sign of a or is zero. This is different from Maple's modp, where the sign @@ -1262,188 +1797,231 @@ numeric abs(const numeric & x) * * @return a mod b in the range [0,abs(b)-1] with sign of b if both are * integer, 0 otherwise. */ -numeric mod(const numeric & a, const numeric & b) +const numeric mod(const numeric &a, const numeric &b) { - if (a.is_integer() && b.is_integer()) - return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - else - return _num0(); // Throw? + if (a.is_integer() && b.is_integer()) + return cln::mod(cln::the(a.to_cl_N()), + cln::the(b.to_cl_N())); + else + return _num0; } + /** Modulus (in symmetric representation). * Equivalent to Maple's mods. * - * @return a mod b in the range [-iquo(abs(m)-1,2), iquo(abs(m),2)]. */ -numeric smod(const numeric & a, const numeric & b) + * @return a mod b in the range [-iquo(abs(b)-1,2), iquo(abs(b),2)]. */ +const numeric smod(const numeric &a, const numeric &b) { - // FIXME: Should this become a member function? - if (a.is_integer() && b.is_integer()) { - cl_I b2 = The(cl_I)(ceiling1(The(cl_I)(*b.value) / 2)) - 1; - return ::mod(The(cl_I)(*a.value) + b2, The(cl_I)(*b.value)) - b2; - } else - return _num0(); // Throw? + if (a.is_integer() && b.is_integer()) { + const cln::cl_I b2 = cln::ceiling1(cln::the(b.to_cl_N()) >> 1) - 1; + return cln::mod(cln::the(a.to_cl_N()) + b2, + cln::the(b.to_cl_N())) - b2; + } else + return _num0; } + /** Numeric integer remainder. * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned. * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the * sign of a or is zero. * - * @return remainder of a/b if both are integer, 0 otherwise. */ -numeric irem(const numeric & a, const numeric & b) + * @return remainder of a/b if both are integer, 0 otherwise. + * @exception overflow_error (division by zero) if b is zero. */ +const numeric irem(const numeric &a, const numeric &b) { - if (a.is_integer() && b.is_integer()) - return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - else - return _num0(); // Throw? + if (b.is_zero()) + throw std::overflow_error("numeric::irem(): division by zero"); + if (a.is_integer() && b.is_integer()) + return cln::rem(cln::the(a.to_cl_N()), + cln::the(b.to_cl_N())); + else + return _num0; } + /** Numeric integer remainder. * Equivalent to Maple's irem(a,b,'q') it obeyes the relation * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero, - * and irem(a,b) has the sign of a or is zero. + * and irem(a,b) has the sign of a or is zero. * * @return remainder of a/b and quotient stored in q if both are integer, - * 0 otherwise. */ -numeric irem(const numeric & a, const numeric & b, numeric & q) + * 0 otherwise. + * @exception overflow_error (division by zero) if b is zero. */ +const numeric irem(const numeric &a, const numeric &b, numeric &q) { - if (a.is_integer() && b.is_integer()) { // -> CLN - cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value)); - q = rem_quo.quotient; - return rem_quo.remainder; - } - else { - q = _num0(); - return _num0(); // Throw? - } + if (b.is_zero()) + throw std::overflow_error("numeric::irem(): division by zero"); + if (a.is_integer() && b.is_integer()) { + const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the(a.to_cl_N()), + cln::the(b.to_cl_N())); + q = rem_quo.quotient; + return rem_quo.remainder; + } else { + q = _num0; + return _num0; + } } + /** Numeric integer quotient. * Equivalent to Maple's iquo as far as sign conventions are concerned. * - * @return truncated quotient of a/b if both are integer, 0 otherwise. */ -numeric iquo(const numeric & a, const numeric & b) + * @return truncated quotient of a/b if both are integer, 0 otherwise. + * @exception overflow_error (division by zero) if b is zero. */ +const numeric iquo(const numeric &a, const numeric &b) { - if (a.is_integer() && b.is_integer()) - return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - else - return _num0(); // Throw? + if (b.is_zero()) + throw std::overflow_error("numeric::iquo(): division by zero"); + if (a.is_integer() && b.is_integer()) + return cln::truncate1(cln::the(a.to_cl_N()), + cln::the(b.to_cl_N())); + else + return _num0; } + /** Numeric integer quotient. * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation * r == a - iquo(a,b,r)*b. * * @return truncated quotient of a/b and remainder stored in r if both are - * integer, 0 otherwise. */ -numeric iquo(const numeric & a, const numeric & b, numeric & r) -{ - if (a.is_integer() && b.is_integer()) { // -> CLN - cl_I_div_t rem_quo = truncate2(The(cl_I)(*a.value), The(cl_I)(*b.value)); - r = rem_quo.remainder; - return rem_quo.quotient; - } else { - r = _num0(); - return _num0(); // Throw? - } -} - -/** Numeric square root. - * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4) - * should return integer 2. - * - * @param z numeric argument - * @return square root of z. Branch cut along negative real axis, the negative - * real axis itself where imag(z)==0 and real(z)<0 belongs to the upper part - * where imag(z)>0. */ -numeric sqrt(const numeric & z) + * integer, 0 otherwise. + * @exception overflow_error (division by zero) if b is zero. */ +const numeric iquo(const numeric &a, const numeric &b, numeric &r) { - return ::sqrt(*z.value); // -> CLN + if (b.is_zero()) + throw std::overflow_error("numeric::iquo(): division by zero"); + if (a.is_integer() && b.is_integer()) { + const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the(a.to_cl_N()), + cln::the(b.to_cl_N())); + r = rem_quo.remainder; + return rem_quo.quotient; + } else { + r = _num0; + return _num0; + } } -/** Integer numeric square root. */ -numeric isqrt(const numeric & x) -{ - if (x.is_integer()) { - cl_I root; - ::isqrt(The(cl_I)(*x.value), &root); // -> CLN - return root; - } else - return _num0(); // Throw? -} /** Greatest Common Divisor. * * @return The GCD of two numbers if both are integer, a numerical 1 * if they are not. */ -numeric gcd(const numeric & a, const numeric & b) +const numeric gcd(const numeric &a, const numeric &b) { - if (a.is_integer() && b.is_integer()) - return ::gcd(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - else - return _num1(); + if (a.is_integer() && b.is_integer()) + return cln::gcd(cln::the(a.to_cl_N()), + cln::the(b.to_cl_N())); + else + return _num1; } + /** Least Common Multiple. * * @return The LCM of two numbers if both are integer, the product of those * two numbers if they are not. */ -numeric lcm(const numeric & a, const numeric & b) +const numeric lcm(const numeric &a, const numeric &b) +{ + if (a.is_integer() && b.is_integer()) + return cln::lcm(cln::the(a.to_cl_N()), + cln::the(b.to_cl_N())); + else + return a.mul(b); +} + + +/** Numeric square root. + * If possible, sqrt(x) should respect squares of exact numbers, i.e. sqrt(4) + * should return integer 2. + * + * @param x numeric argument + * @return square root of x. Branch cut along negative real axis, the negative + * real axis itself where imag(x)==0 and real(x)<0 belongs to the upper part + * where imag(x)>0. */ +const numeric sqrt(const numeric &x) { - if (a.is_integer() && b.is_integer()) - return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - else - return *a.value * *b.value; + return cln::sqrt(x.to_cl_N()); } -ex PiEvalf(void) + +/** Integer numeric square root. */ +const numeric isqrt(const numeric &x) +{ + if (x.is_integer()) { + cln::cl_I root; + cln::isqrt(cln::the(x.to_cl_N()), &root); + return root; + } else + return _num0; +} + + +/** Floating point evaluation of Archimedes' constant Pi. */ +ex PiEvalf() { - return numeric(cl_pi(cl_default_float_format)); // -> CLN + return numeric(cln::pi(cln::default_float_format)); } -ex EulerGammaEvalf(void) + +/** Floating point evaluation of Euler's constant gamma. */ +ex EulerEvalf() { - return numeric(cl_eulerconst(cl_default_float_format)); // -> CLN + return numeric(cln::eulerconst(cln::default_float_format)); } -ex CatalanEvalf(void) + +/** Floating point evaluation of Catalan's constant. */ +ex CatalanEvalf() { - return numeric(cl_catalanconst(cl_default_float_format)); // -> CLN + return numeric(cln::catalanconst(cln::default_float_format)); } -// It initializes to 17 digits, because in CLN cl_float_format(17) turns out to -// be 61 (<64) while cl_float_format(18)=65. We want to have a cl_LF instead -// of cl_SF, cl_FF or cl_DF but everything else is basically arbitrary. + +/** _numeric_digits default ctor, checking for singleton invariance. */ _numeric_digits::_numeric_digits() - : digits(17) + : digits(17) { - assert(!too_late); - too_late = true; - cl_default_float_format = cl_float_format(17); + // It initializes to 17 digits, because in CLN float_format(17) turns out + // to be 61 (<64) while float_format(18)=65. The reason is we want to + // have a cl_LF instead of cl_SF, cl_FF or cl_DF. + if (too_late) + throw(std::runtime_error("I told you not to do instantiate me!")); + too_late = true; + cln::default_float_format = cln::float_format(17); } + +/** Assign a native long to global Digits object. */ _numeric_digits& _numeric_digits::operator=(long prec) { - digits=prec; - cl_default_float_format = cl_float_format(prec); - return *this; + digits = prec; + cln::default_float_format = cln::float_format(prec); + return *this; } + +/** Convert global Digits object to native type long. */ _numeric_digits::operator long() { - return (long)digits; + // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1 + return (long)digits; } -void _numeric_digits::print(ostream & os) const + +/** Append global Digits object to ostream. */ +void _numeric_digits::print(std::ostream &os) const { - debugmsg("_numeric_digits print", LOGLEVEL_PRINT); - os << digits; + os << digits; } -ostream& operator<<(ostream& os, _numeric_digits const & e) + +std::ostream& operator<<(std::ostream &os, const _numeric_digits &e) { - e.print(os); - return os; + e.print(os); + return os; } ////////// @@ -1454,10 +2032,9 @@ ostream& operator<<(ostream& os, _numeric_digits const & e) bool _numeric_digits::too_late = false; + /** Accuracy in decimal digits. Only object of this type! Can be set using * assignment from C++ unsigned ints and evaluated like any built-in type. */ _numeric_digits Digits; -#ifndef NO_GINAC_NAMESPACE } // namespace GiNaC -#endif // ndef NO_GINAC_NAMESPACE