X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnumeric.cpp;h=f4dd738087de9fe0cc71b9f8a436694120b519e3;hp=99dcc8e4486fda82b1f0101daa38c2b56ec048b9;hb=c77a5c7fc1d9749628f856614dbaf85c7f086ce4;hpb=2565309dd7c38635c191eacf2a4af9b23fc0d310 diff --git a/ginac/numeric.cpp b/ginac/numeric.cpp index 99dcc8e4..f4dd7380 100644 --- a/ginac/numeric.cpp +++ b/ginac/numeric.cpp @@ -24,14 +24,22 @@ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ +#include "config.h" + #include #include #include -#include //!! + +#if defined(HAVE_SSTREAM) +#include +#elif defined(HAVE_STRSTREAM) +#include +#else +#error Need either sstream or strstream +#endif #include "numeric.h" #include "ex.h" -#include "config.h" #include "archive.h" #include "debugmsg.h" #include "utils.h" @@ -40,6 +48,7 @@ // instead of in some header file where it would propagate to other parts. // Also, we only need a subset of CLN, so we don't include the complete cln.h: #ifdef HAVE_CLN_CLN_H +#include #include #include #include @@ -51,7 +60,8 @@ #include #include #include -#else +#else // def HAVE_CLN_CLN_H +#include #include #include #include @@ -63,11 +73,11 @@ #include #include #include -#endif +#endif // def HAVE_CLN_CLN_H -#ifndef NO_GINAC_NAMESPACE +#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_GINAC_NAMESPACE +#endif // ndef NO_NAMESPACE_GINAC // linker has no problems finding text symbols for numerator or denominator //#define SANE_LINKER @@ -85,10 +95,11 @@ GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic) numeric::numeric() : basic(TINFO_numeric) { debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT); - value = new cl_N; - *value=cl_I(0); + value = new ::cl_N; + *value = ::cl_I(0); calchash(); - setflag(status_flags::evaluated| + setflag(status_flags::evaluated | + status_flags::expanded | status_flags::hash_calculated); } @@ -119,7 +130,7 @@ const numeric & numeric::operator=(const numeric & other) void numeric::copy(const numeric & other) { basic::copy(other); - value = new cl_N(*other.value); + value = new ::cl_N(*other.value); } void numeric::destroy(bool call_parent) @@ -138,39 +149,42 @@ numeric::numeric(int i) : basic(TINFO_numeric) { debugmsg("numeric constructor 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 + // first. This is due to the behaviour of the cl_I-ctor, which // emphasizes efficiency: - value = new cl_I((long) i); + value = new ::cl_I((long) i); calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); } + numeric::numeric(unsigned int i) : basic(TINFO_numeric) { debugmsg("numeric constructor 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 + // first. This is due to the behaviour of the cl_I-ctor, which // emphasizes efficiency: - value = new cl_I((unsigned long)i); + value = new ::cl_I((unsigned long)i); calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); } + numeric::numeric(long i) : basic(TINFO_numeric) { debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT); - value = new cl_I(i); + value = new ::cl_I(i); calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); } + numeric::numeric(unsigned long i) : basic(TINFO_numeric) { debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT); - value = new cl_I(i); + value = new ::cl_I(i); calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); @@ -184,13 +198,14 @@ numeric::numeric(long numer, long denom) : basic(TINFO_numeric) debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT); if (!denom) throw (std::overflow_error("division by zero")); - value = new cl_I(numer); - *value = *value / cl_I(denom); + value = new ::cl_I(numer); + *value = *value / ::cl_I(denom); calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); } + numeric::numeric(double d) : basic(TINFO_numeric) { debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT); @@ -204,13 +219,61 @@ numeric::numeric(double d) : basic(TINFO_numeric) status_flags::hash_calculated); } + +/** 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) -{ // MISSING: treatment of complex and ints and rationals. +{ debugmsg("numeric constructor from string",LOGLEVEL_CONSTRUCT); - if (strchr(s, '.')) - value = new cl_LF(s); - else - value = new cl_I(s); + value = new ::cl_N(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); + // make it safe by adding explicit sign + if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#') + ss = '+' + ss; + std::string::size_type delim; + 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 = term.replace(term.find("I"),1,""); + // erase '*': + if (term.find("*") != std::string::npos) + term = term.replace(term.find("*"),1,""); + // correct for trivial +/-I without explicit factor on I: + if (term.size() == 1) + term += "1"; + imaginary = true; + } + const char *cs = term.c_str(); + // CLN's short types are not useful within the GiNaC framework, hence + // we go straight to the construction of a long float. Simply using + // cl_N(s) would require us to use add a CLN exponent mark, otherwise + // we would not be save from over-/underflows. + if (strchr(cs, '.')) + if (imaginary) + *value = *value + ::complex(cl_I(0),::cl_LF(cs)); + else + *value = *value + ::cl_LF(cs); + else + if (imaginary) + *value = *value + ::complex(cl_I(0),::cl_R(cs)); + else + *value = *value + ::cl_R(cs); + } while(delim != std::string::npos); calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); @@ -218,10 +281,10 @@ numeric::numeric(const char *s) : basic(TINFO_numeric) /** 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 cl_N & z) : basic(TINFO_numeric) { debugmsg("numeric constructor from cl_N", LOGLEVEL_CONSTRUCT); - value = new cl_N(z); + value = new ::cl_N(z); calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); @@ -235,22 +298,36 @@ numeric::numeric(cl_N const & z) : basic(TINFO_numeric) numeric::numeric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst) { debugmsg("numeric constructor from archive_node", LOGLEVEL_CONSTRUCT); - value = new cl_N; -#if 0 //!! - // This is how it should be implemented but we have no istringstream here... - string str; + value = new ::cl_N; + + // Read number as string + std::string str; if (n.find_string("number", str)) { - istringstream s(str); - s >> *value; - } +#ifdef HAVE_SSTREAM + std::istringstream s(str); #else - // Workaround for the above: read from strstream - string str; - if (n.find_string("number", str)) { - istrstream f(str.c_str(), str.size() + 1); - f >> *value; - } + std::istrstream s(str.c_str(), str.size() + 1); #endif + ::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; + *value = re.sign * re.mantissa * ::expt(cl_float(2.0, cl_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; + *value = ::complex(re.sign * re.mantissa * ::expt(cl_float(2.0, cl_default_float_format), re.exponent), + im.sign * im.mantissa * ::expt(cl_float(2.0, cl_default_float_format), im.exponent)); + break; + default: // Ordinary number + s.putback(c); + s >> *value; + break; + } + } calchash(); setflag(status_flags::evaluated| status_flags::hash_calculated); @@ -266,17 +343,36 @@ ex numeric::unarchive(const archive_node &n, const lst &sym_lst) void numeric::archive(archive_node &n) const { inherited::archive(n); -#if 0 //!! - // This is how it should be implemented but we have no ostringstream here... - ostringstream s; - s << *value; - n.add_string("number", s.str()); + + // Write number as string +#ifdef HAVE_SSTREAM + std::ostringstream s; #else - // Workaround for the above: write to strstream char buf[1024]; - ostrstream f(buf, 1024); - f << *value << ends; - string str(buf); + std::ostrstream s(buf, 1024); +#endif + 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()) { + cl_idecoded_float re = integer_decode_float(The(::cl_F)(*value)); + s << "R"; + s << re.sign << " " << re.mantissa << " " << re.exponent; + } else { + cl_idecoded_float re = integer_decode_float(The(::cl_F)(::realpart(*value))); + cl_idecoded_float im = integer_decode_float(The(::cl_F)(::imagpart(*value))); + s << "C"; + s << re.sign << " " << re.mantissa << " " << re.exponent << " "; + s << im.sign << " " << im.mantissa << " " << im.exponent; + } + } +#ifdef HAVE_SSTREAM + n.add_string("number", s.str()); +#else + s << ends; + std::string str(buf); n.add_string("number", str); #endif } @@ -293,57 +389,91 @@ basic * numeric::duplicate() const return new numeric(*this); } + +/** 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. + * + * @see numeric::print() */ +static void print_real_number(ostream & os, const cl_R & num) +{ + cl_print_flags ourflags; + if (::instanceof(num, ::cl_RA_ring)) { + // case 1: integer or rational, nothing special to do: + ::print_real(os, ourflags, num); + } 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 = ::cl_float_format(The(::cl_F)(num)); + ::print_real(os, ourflags, num); + } + return; +} + +/** This method adds to the output so it blends more consistently together + * with the other routines and produces something compatible to ginsh input. + * + * @see print_real_number() */ 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()) { + if (this->is_real()) { // case 1, real: x or -x - if ((precedence<=upper_precedence) && (!is_pos_integer())) { - os << "(" << *value << ")"; + if ((precedence<=upper_precedence) && (!this->is_nonneg_integer())) { + os << "("; + print_real_number(os, The(::cl_R)(*value)); + os << ")"; } else { - os << *value; + print_real_number(os, The(::cl_R)(*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) { + if (::realpart(*value) == 0) { + if ((precedence<=upper_precedence) && (::imagpart(*value) < 0)) { + if (::imagpart(*value) == -1) { os << "(-I)"; } else { - os << "(" << imagpart(*value) << "*I)"; + os << "("; + print_real_number(os, The(::cl_R)(::imagpart(*value))); + os << "*I)"; } } else { - if (imagpart(*value) == 1) { + if (::imagpart(*value) == 1) { os << "I"; } else { - if (imagpart (*value) == -1) { + if (::imagpart (*value) == -1) { os << "-I"; } else { - os << imagpart(*value) << "*I"; + print_real_number(os, The(::cl_R)(::imagpart(*value))); + os << "*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) { + if (precedence <= upper_precedence) + os << "("; + print_real_number(os, The(::cl_R)(::realpart(*value))); + if (::imagpart(*value) < 0) { + if (::imagpart(*value) == -1) { os << "-I"; } else { - os << imagpart(*value) << "*I"; + print_real_number(os, The(::cl_R)(::imagpart(*value))); + os << "*I"; } } else { - if (imagpart(*value) == 1) { + if (::imagpart(*value) == 1) { os << "+I"; } else { - os << "+" << imagpart(*value) << "*I"; + os << "+"; + print_real_number(os, The(::cl_R)(::imagpart(*value))); + os << "*I"; } } - if (precedence <= upper_precedence) os << ")"; + if (precedence <= upper_precedence) + os << ")"; } } } @@ -356,21 +486,24 @@ void numeric::printraw(ostream & os) const 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 + os << std::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 (this->is_rational() && !this->is_integer()) { if (compare(_num0()) > 0) { os << "("; if (type == csrc_types::ctype_cl_N) @@ -399,49 +532,88 @@ void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) os.flags(oldflags); } + bool numeric::info(unsigned inf) const { switch (inf) { - case info_flags::numeric: - case info_flags::polynomial: - case info_flags::rational_function: + 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; +} + +/** 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_of_type(*other.bp, numeric)) + return false; + const numeric & o = static_cast(const_cast(*other.bp)); + if (this->is_equal(o) || this->is_equal(-o)) 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(); + 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. @@ -451,11 +623,20 @@ bool numeric::info(unsigned inf) const 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 + return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN } // protected +/** Implementation of ex::diff() for a numeric. It always returns 0. + * + * @see ex::diff */ +ex numeric::derivative(const symbol & s) const +{ + return _ex0(); +} + + int numeric::compare_same_type(const basic & other) const { GINAC_ASSERT(is_exactly_of_type(other, numeric)); @@ -468,26 +649,23 @@ int numeric::compare_same_type(const basic & other) const return compare(o); } + 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); + return this->is_equal(*o); } -/* + unsigned numeric::calchash(void) 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); + // Use CLN's hashcode. Warning: It 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. + return (hashvalue = cl_equal_hashcode(*value) | 0x80000000U); } -*/ ////////// @@ -542,11 +720,19 @@ numeric numeric::div(const numeric & other) const numeric numeric::power(const numeric & other) const { - static const numeric * _num1p=&_num1(); + 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")); + if (::zerop(*value)) { + if (::zerop(*other.value)) + throw (std::domain_error("numeric::eval(): pow(0,0) is undefined")); + else if (::zerop(::realpart(*other.value))) + throw (std::domain_error("numeric::eval(): pow(0,I) is undefined")); + else if (::minusp(::realpart(*other.value))) + throw (std::overflow_error("numeric::eval(): division by zero")); + else + return _num0(); + } return numeric(::expt(*value,*other.value)); } @@ -593,8 +779,16 @@ const numeric & numeric::power_dyn(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")); + if (::zerop(*value)) { + if (::zerop(*other.value)) + throw (std::domain_error("numeric::eval(): pow(0,0) is undefined")); + else if (::zerop(::realpart(*other.value))) + throw (std::domain_error("numeric::eval(): pow(0,I) is undefined")); + else if (::minusp(::realpart(*other.value))) + throw (std::overflow_error("numeric::eval(): division by zero")); + else + return _num0(); + } return static_cast((new numeric(::expt(*value,*other.value)))-> setflag(status_flags::dynallocated)); } @@ -636,15 +830,15 @@ const numeric & numeric::operator=(const char * s) * @see numeric::compare(const numeric & other) */ int numeric::csgn(void) const { - if (is_zero()) + if (this->is_zero()) return 0; - if (!::zerop(realpart(*value))) { - if (::plusp(realpart(*value))) + if (!::zerop(::realpart(*value))) { + if (::plusp(::realpart(*value))) return 1; else return -1; } else { - if (::plusp(imagpart(*value))) + if (::plusp(::imagpart(*value))) return 1; else return -1; @@ -661,16 +855,16 @@ int numeric::csgn(void) const int numeric::compare(const numeric & other) const { // Comparing two real numbers? - if (is_real() && other.is_real()) + if (this->is_real() && other.is_real()) // Yes, just compare them - return ::cl_compare(The(cl_R)(*value), The(cl_R)(*other.value)); + 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)); + 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)); + return ::cl_compare(::imagpart(*value), ::imagpart(*other.value)); } } @@ -688,47 +882,47 @@ bool numeric::is_zero(void) const /** True if object is not complex and greater than zero. */ bool numeric::is_positive(void) const { - if (is_real()) - return ::plusp(The(cl_R)(*value)); // -> CLN + if (this->is_real()) + return ::plusp(The(::cl_R)(*value)); // -> CLN return false; } /** True if object is not complex and less than zero. */ bool numeric::is_negative(void) const { - if (is_real()) - return ::minusp(The(cl_R)(*value)); // -> CLN + if (this->is_real()) + return ::minusp(The(::cl_R)(*value)); // -> CLN return false; } /** True if object is a non-complex integer. */ bool numeric::is_integer(void) const { - return ::instanceof(*value, cl_I_ring); // -> CLN + return ::instanceof(*value, ::cl_I_ring); // -> CLN } /** True if object is an exact integer greater than zero. */ bool numeric::is_pos_integer(void) const { - return (is_integer() && ::plusp(The(cl_I)(*value))); // -> CLN + return (this->is_integer() && ::plusp(The(::cl_I)(*value))); // -> CLN } /** True if object is an exact integer greater or equal zero. */ bool numeric::is_nonneg_integer(void) const { - return (is_integer() && !::minusp(The(cl_I)(*value))); // -> CLN + return (this->is_integer() && !::minusp(The(::cl_I)(*value))); // -> CLN } /** True if object is an exact even integer. */ bool numeric::is_even(void) const { - return (is_integer() && ::evenp(The(cl_I)(*value))); // -> CLN + return (this->is_integer() && ::evenp(The(::cl_I)(*value))); // -> CLN } /** True if object is an exact odd integer. */ bool numeric::is_odd(void) const { - return (is_integer() && ::oddp(The(cl_I)(*value))); // -> CLN + return (this->is_integer() && ::oddp(The(::cl_I)(*value))); // -> CLN } /** Probabilistic primality test. @@ -736,20 +930,20 @@ bool numeric::is_odd(void) const * @return true if object is exact integer and prime. */ bool numeric::is_prime(void) const { - return (is_integer() && ::isprobprime(The(cl_I)(*value))); // -> CLN + return (this->is_integer() && ::isprobprime(The(::cl_I)(*value))); // -> CLN } /** True if object is an exact rational number, may even be complex * (denominator may be unity). */ bool numeric::is_rational(void) const { - return ::instanceof(*value, cl_RA_ring); // -> CLN + return ::instanceof(*value, ::cl_RA_ring); // -> CLN } /** True if object is a real integer, rational or float (but not complex). */ bool numeric::is_real(void) const { - return ::instanceof(*value, cl_R_ring); // -> CLN + return ::instanceof(*value, ::cl_R_ring); // -> CLN } bool numeric::operator==(const numeric & other) const @@ -766,11 +960,11 @@ bool numeric::operator!=(const numeric & other) const * of the form a+b*I, where a and b are integers. */ bool numeric::is_cinteger(void) const { - if (::instanceof(*value, cl_I_ring)) + 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)) + else if (!this->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; @@ -780,11 +974,11 @@ bool numeric::is_cinteger(void) const * (denominator may be unity). */ bool numeric::is_crational(void) const { - if (::instanceof(*value, cl_RA_ring)) + 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)) + else if (!this->is_real()) { // complex case, handle Q(i): + if (::instanceof(::realpart(*value), ::cl_RA_ring) && + ::instanceof(::imagpart(*value), ::cl_RA_ring)) return true; } return false; @@ -795,8 +989,8 @@ bool numeric::is_crational(void) const * @exception invalid_argument (complex inequality) */ 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 + if (this->is_real() && other.is_real()) + return (The(::cl_R)(*value) < The(::cl_R)(*other.value)); // -> CLN throw (std::invalid_argument("numeric::operator<(): complex inequality")); return false; // make compiler shut up } @@ -806,8 +1000,8 @@ bool numeric::operator<(const numeric & other) const * @exception invalid_argument (complex inequality) */ 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 + if (this->is_real() && other.is_real()) + return (The(::cl_R)(*value) <= The(::cl_R)(*other.value)); // -> CLN throw (std::invalid_argument("numeric::operator<=(): complex inequality")); return false; // make compiler shut up } @@ -817,8 +1011,8 @@ bool numeric::operator<=(const numeric & other) const * @exception invalid_argument (complex inequality) */ 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 + if (this->is_real() && other.is_real()) + return (The(::cl_R)(*value) > The(::cl_R)(*other.value)); // -> CLN throw (std::invalid_argument("numeric::operator>(): complex inequality")); return false; // make compiler shut up } @@ -828,36 +1022,46 @@ bool numeric::operator>(const numeric & other) const * @exception invalid_argument (complex inequality) */ 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 + if (this->is_real() && other.is_real()) + return (The(::cl_R)(*value) >= The(::cl_R)(*other.value)); // -> CLN throw (std::invalid_argument("numeric::operator>=(): complex inequality")); return false; // make compiler shut up } -/** Converts numeric types to machine's int. You should check with is_integer() - * if the number is really an integer before calling this method. */ +/** 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(void) const { - GINAC_ASSERT(is_integer()); - return ::cl_I_to_int(The(cl_I)(*value)); // -> CLN + GINAC_ASSERT(this->is_integer()); + return ::cl_I_to_int(The(::cl_I)(*value)); // -> CLN +} + +/** 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(void) const +{ + GINAC_ASSERT(this->is_integer()); + return ::cl_I_to_long(The(::cl_I)(*value)); // -> CLN } /** 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 { - GINAC_ASSERT(is_real()); - return ::cl_double_approx(realpart(*value)); // -> CLN + GINAC_ASSERT(this->is_real()); + return ::cl_double_approx(::realpart(*value)); // -> CLN } /** Real part of a number. */ -numeric numeric::real(void) const +const numeric numeric::real(void) const { return numeric(::realpart(*value)); // -> CLN } /** Imaginary part of a number. */ -numeric numeric::imag(void) const +const numeric numeric::imag(void) const { return numeric(::imagpart(*value)); // -> CLN } @@ -879,46 +1083,46 @@ inline cl_heap_ratio* TheRatio (const cl_N& obj) * 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 +const numeric numeric::numer(void) const { - if (is_integer()) { + if (this->is_integer()) { return numeric(*this); } #ifdef SANE_LINKER - else if (::instanceof(*value, cl_RA_ring)) { - return numeric(::numerator(The(cl_RA)(*value))); + else if (::instanceof(*value, ::cl_RA_ring)) { + return numeric(::numerator(The(::cl_RA)(*value))); } - else if (!is_real()) { // complex case, handle Q(i): + else if (!this->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)) + 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)))))); + 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)) { + 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)) + else if (!this->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)), + 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)))); } } @@ -930,42 +1134,42 @@ numeric numeric::numer(void) const /** 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 +const numeric numeric::denom(void) const { - if (is_integer()) { + if (this->is_integer()) { return _num1(); } #ifdef SANE_LINKER - if (instanceof(*value, cl_RA_ring)) { - return numeric(::denominator(The(cl_RA)(*value))); + 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)) + if (!this->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)))); + 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)) { + 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)) + if (!this->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)) + 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)) + 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)); + 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 @@ -980,8 +1184,8 @@ numeric numeric::denom(void) const * in two's complement if it is an integer, 0 otherwise. */ int numeric::int_length(void) const { - if (is_integer()) - return ::integer_length(The(cl_I)(*value)); // -> CLN + if (this->is_integer()) + return ::integer_length(The(::cl_I)(*value)); // -> CLN else return 0; } @@ -1003,74 +1207,82 @@ const numeric some_numeric; const type_info & 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))); +const numeric I = numeric(::complex(cl_I(0),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 } + /** Natural logarithm. * * @param z complex number * @return arbitrary precision numerical log(x). * @exception overflow_error (logarithmic singularity) */ -numeric log(const numeric & z) +const numeric log(const numeric & z) { if (z.is_zero()) throw (std::overflow_error("log(): logarithmic singularity")); return ::log(*z.value); // -> CLN } + /** 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 } + /** 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 } - + + /** 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 } + /** 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 } - + + /** 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 } -/** Arcustangents. + +/** Arcustangent. * * @param z complex number * @return atan(z) * @exception overflow_error (logarithmic singularity) */ -numeric atan(const numeric & x) +const numeric atan(const numeric & x) { if (!x.is_real() && x.real().is_zero() && @@ -1079,70 +1291,78 @@ numeric atan(const numeric & x) return ::atan(*x.value); // -> CLN } -/** 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 + return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN else throw (std::invalid_argument("numeric::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 } + /** 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 } - + + /** 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 } + /** 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 } + /** 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 } + /** Numeric inverse hyperbolic tangent (trigonometric function). * * @return arbitrary precision numerical atanh(x). */ -numeric atanh(const numeric & x) +const numeric atanh(const numeric & x) { return ::atanh(*x.value); // -> CLN } + /** Numeric evaluation of Riemann's Zeta function. Currently works only for * integer arguments. */ -numeric zeta(const numeric & x) +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 @@ -1150,7 +1370,7 @@ numeric zeta(const numeric & x) // 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))); + int aux = (int)(::cl_double_approx(::realpart(*x.value))); if (zerop(*x.value-aux)) return ::cl_zeta(aux); // -> CLN } @@ -1160,19 +1380,28 @@ numeric zeta(const numeric & x) return numeric(0); } -/** The gamma function. + +/** The Gamma function. * This is only a stub! */ -numeric gamma(const numeric & x) +const numeric lgamma(const numeric & x) { - clog << "gamma(" << x + clog << "lgamma(" << x << "): Does anybody know good way to calculate this numerically?" << endl; return numeric(0); } +const numeric tgamma(const numeric & x) +{ + clog << "tgamma(" << x + << "): Does anybody know good way to calculate this numerically?" + << endl; + return numeric(0); +} + /** 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?" @@ -1180,9 +1409,10 @@ numeric psi(const numeric & x) return numeric(0); } + /** 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?" @@ -1190,38 +1420,42 @@ numeric psi(const numeric & n, const numeric & x) return numeric(0); } + /** 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()) + if (!n.is_nonneg_integer()) throw (std::range_error("numeric::factorial(): argument must be integer >= 0")); - return numeric(::factorial(nn.to_int())); // -> CLN + return numeric(::factorial(n.to_int())); // -> CLN } + /** 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 * @exception range_error (argument must be integer >= -1) */ -numeric doublefactorial(const numeric & nn) +const numeric doublefactorial(const numeric & n) { - if (nn == numeric(-1)) { + if (n == numeric(-1)) { return _num1(); } - if (!nn.is_nonneg_integer()) { + if (!n.is_nonneg_integer()) { throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1")); } - return numeric(::doublefactorial(nn.to_int())); // -> CLN + return numeric(::doublefactorial(n.to_int())); // -> CLN } + /** 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) +const numeric binomial(const numeric & n, const numeric & k) { if (n.is_integer() && k.is_integer()) { if (n.is_nonneg_integer()) { @@ -1234,58 +1468,157 @@ numeric binomial(const numeric & n, const numeric & k) } } - // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) or a suitable limit + // 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) +const 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(); + + // 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 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 results; + static int highest_result = 0; + // algorithm not applicable to B(0), so just store it + if (results.size()==0) + results.push_back(::cl_RA(1)); + + int n = nn.to_long(); + for (int i=highest_result; i0; --j) { + B = cl_I(n*m) * (B+results[j]) / (d1*d2); + n += 4; + m += 2; + d1 -= 1; + d2 -= 2; + } + B = (1 - ((B+1)/(2*i+3))) / (cl_I(1)<<(2*i+2)); + results.push_back(B); + ++highest_result; + } + return results[n/2]; +} + + +/** Fibonacci number. The nth Fibonacci number F(n) is defined by the + * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1. + * + * @param n an integer + * @return the nth Fibonacci number F(n) (an integer number) + * @exception range_error (argument must be an integer) */ +const numeric fibonacci(const numeric & n) +{ + if (!n.is_integer()) + throw (std::range_error("numeric::fibonacci(): argument must be integer")); + // Method: + // + // This is based on an implementation that can be found in CLN's example + // directory. There, it is done recursively, which may be more elegant + // than our non-recursive implementation that has to resort to some bit- + // fiddling. This is, however, a matter of taste. + // The following addition formula holds: + // + // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 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); + + ::cl_I u(0); + ::cl_I v(1); + ::cl_I m = The(::cl_I)(*n.value) >> 1L; // floor(n/2); + for (uintL bit=::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. + ::cl_I u2 = ::square(u); + ::cl_I v2 = ::square(v); + if (::logbitp(bit-1, m)) { + v = ::square(u + v) - u2; + u = u2 + v2; + } else { + u = v2 - ::square(v - u); + v = u2 + v2; + } } - highest_result=n; - return results[n]; + 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 ::square(u) + ::square(v); } + /** Absolute value. */ numeric abs(const numeric & x) { return ::abs(*x.value); // -> CLN } + /** 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 @@ -1296,25 +1629,26 @@ numeric abs(const numeric & x) 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 + return ::mod(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN else return _num0(); // Throw? } + /** 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) { - // 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; + 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? } + /** 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 @@ -1324,11 +1658,12 @@ numeric smod(const numeric & a, const numeric & b) 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 + return ::rem(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN else return _num0(); // Throw? } + /** 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, @@ -1339,7 +1674,7 @@ numeric irem(const numeric & a, const numeric & b) 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)); + 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; } @@ -1349,6 +1684,7 @@ numeric irem(const numeric & a, const numeric & b, numeric & q) } } + /** Numeric integer quotient. * Equivalent to Maple's iquo as far as sign conventions are concerned. * @@ -1356,11 +1692,12 @@ numeric irem(const numeric & a, const numeric & b, numeric & q) 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 + return truncate1(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN else return _num0(); // Throw? } + /** Numeric integer quotient. * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation * r == a - iquo(a,b,r)*b. @@ -1370,7 +1707,7 @@ numeric iquo(const numeric & a, const numeric & b) 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)); + 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 { @@ -1379,6 +1716,7 @@ numeric iquo(const numeric & a, const numeric & b, numeric & r) } } + /** Numeric square root. * If possible, sqrt(z) should respect squares of exact numbers, i.e. sqrt(4) * should return integer 2. @@ -1392,17 +1730,19 @@ numeric sqrt(const numeric & z) return ::sqrt(*z.value); // -> CLN } + /** Integer numeric square root. */ numeric isqrt(const numeric & x) { if (x.is_integer()) { cl_I root; - ::isqrt(The(cl_I)(*x.value), &root); // -> CLN + ::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 @@ -1410,11 +1750,12 @@ numeric isqrt(const numeric & x) 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 + return ::gcd(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN else return _num1(); } + /** Least Common Multiple. * * @return The LCM of two numbers if both are integer, the product of those @@ -1422,26 +1763,33 @@ numeric gcd(const numeric & a, const numeric & b) numeric lcm(const numeric & a, const numeric & b) { if (a.is_integer() && b.is_integer()) - return ::lcm(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN + return ::lcm(The(::cl_I)(*a.value), The(::cl_I)(*b.value)); // -> CLN else return *a.value * *b.value; } + +/** Floating point evaluation of Archimedes' constant Pi. */ ex PiEvalf(void) { - return numeric(cl_pi(cl_default_float_format)); // -> CLN + return numeric(::cl_pi(cl_default_float_format)); // -> CLN } -ex EulerGammaEvalf(void) + +/** Floating point evaluation of Euler's constant gamma. */ +ex EulerEvalf(void) { - return numeric(cl_eulerconst(cl_default_float_format)); // -> CLN + return numeric(::cl_eulerconst(cl_default_float_format)); // -> CLN } + +/** Floating point evaluation of Catalan's constant. */ ex CatalanEvalf(void) { - return numeric(cl_catalanconst(cl_default_float_format)); // -> CLN + return numeric(::cl_catalanconst(cl_default_float_format)); // -> CLN } + // 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. @@ -1450,27 +1798,31 @@ _numeric_digits::_numeric_digits() { assert(!too_late); too_late = true; - cl_default_float_format = cl_float_format(17); + cl_default_float_format = ::cl_float_format(17); } + _numeric_digits& _numeric_digits::operator=(long prec) { digits=prec; - cl_default_float_format = cl_float_format(prec); + cl_default_float_format = ::cl_float_format(prec); return *this; } + _numeric_digits::operator long() { return (long)digits; } + void _numeric_digits::print(ostream & os) const { debugmsg("_numeric_digits print", LOGLEVEL_PRINT); os << digits; } + ostream& operator<<(ostream& os, const _numeric_digits & e) { e.print(os); @@ -1485,10 +1837,11 @@ ostream& operator<<(ostream& os, const _numeric_digits & 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 +#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_GINAC_NAMESPACE +#endif // ndef NO_NAMESPACE_GINAC