X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnumeric.cpp;h=49e65be47ed77b2cc8eaf285f25d3d4ae11e28b1;hp=8609740e70d34a65af7e74311004a820aed78663;hb=ad19befada3733da04b376bae520cef491ae64c8;hpb=9eab44408b9213d8909b7a9e525f404ad06064dd diff --git a/ginac/numeric.cpp b/ginac/numeric.cpp index 8609740e..49e65be4 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) 1999-2000 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 @@ -24,27 +24,66 @@ * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ +#include "config.h" + #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" // 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: +// 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 -#else -#include -#endif - +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#else // def HAVE_CLN_CLN_H +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#endif // def HAVE_CLN_CLN_H + +#ifndef NO_NAMESPACE_GINAC namespace GiNaC { +#endif // ndef NO_NAMESPACE_GINAC // linker has no problems finding text symbols for numerator or denominator //#define SANE_LINKER +GINAC_IMPLEMENT_REGISTERED_CLASS(numeric, basic) + ////////// // default constructor, destructor, copy constructor assignment // operator and helpers @@ -57,9 +96,10 @@ numeric::numeric() : basic(TINFO_numeric) { debugmsg("numeric default constructor", LOGLEVEL_CONSTRUCT); value = new cl_N; - *value=cl_I(0); + *value = cl_I(0); calchash(); - setflag(status_flags::evaluated| + setflag(status_flags::evaluated | + status_flags::expanded | status_flags::hash_calculated); } @@ -69,13 +109,13 @@ numeric::~numeric() destroy(0); } -numeric::numeric(numeric const & other) +numeric::numeric(const numeric & other) { debugmsg("numeric copy constructor", LOGLEVEL_CONSTRUCT); copy(other); } -numeric const & numeric::operator=(numeric const & other) +const numeric & numeric::operator=(const numeric & other) { debugmsg("numeric operator=", LOGLEVEL_ASSIGNMENT); if (this != &other) { @@ -87,7 +127,7 @@ numeric const & numeric::operator=(numeric const & other) // protected -void numeric::copy(numeric const & other) +void numeric::copy(const numeric & other) { basic::copy(other); value = new cl_N(*other.value); @@ -109,7 +149,7 @@ 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); calchash(); @@ -117,11 +157,12 @@ numeric::numeric(int i) : basic(TINFO_numeric) 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); calchash(); @@ -129,6 +170,7 @@ numeric::numeric(unsigned int i) : basic(TINFO_numeric) status_flags::hash_calculated); } + numeric::numeric(long i) : basic(TINFO_numeric) { debugmsg("numeric constructor from long",LOGLEVEL_CONSTRUCT); @@ -138,6 +180,7 @@ numeric::numeric(long i) : basic(TINFO_numeric) status_flags::hash_calculated); } + numeric::numeric(unsigned long i) : basic(TINFO_numeric) { debugmsg("numeric constructor from ulong",LOGLEVEL_CONSTRUCT); @@ -162,6 +205,7 @@ numeric::numeric(long numer, long denom) : basic(TINFO_numeric) status_flags::hash_calculated); } + numeric::numeric(double d) : basic(TINFO_numeric) { debugmsg("numeric constructor from double",LOGLEVEL_CONSTRUCT); @@ -175,7 +219,8 @@ numeric::numeric(double d) : basic(TINFO_numeric) status_flags::hash_calculated); } -numeric::numeric(char const *s) : basic(TINFO_numeric) + +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, '.')) @@ -189,7 +234,7 @@ numeric::numeric(char const *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); @@ -198,6 +243,93 @@ numeric::numeric(cl_N const & z) : basic(TINFO_numeric) status_flags::hash_calculated); } +////////// +// archiving +////////// + +/** Construct object from archive_node. */ +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; + + // Read number as string + string str; + if (n.find_string("number", str)) { +#ifdef HAVE_SSTREAM + istringstream s(str); +#else + 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); +} + +/** Unarchive the object. */ +ex numeric::unarchive(const archive_node &n, const lst &sym_lst) +{ + return (new numeric(n, sym_lst))->setflag(status_flags::dynallocated); +} + +/** Archive the object. */ +void numeric::archive(archive_node &n) const +{ + inherited::archive(n); + + // Write number as string +#ifdef HAVE_SSTREAM + ostringstream s; +#else + char buf[1024]; + 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; + string str(buf); + n.add_string("number", str); +#endif +} + ////////// // functions overriding virtual functions from bases classes ////////// @@ -210,69 +342,150 @@ basic * numeric::duplicate() const return new numeric(*this); } -// The method printraw doesn't do much, it simply uses CLN's operator<<() for -// output, which is ugly but reliable. Examples: -// 2+2i -void numeric::printraw(ostream & os) const + +/** 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() */ +void print_real_number(ostream & os, const cl_R & num) { - debugmsg("numeric printraw", LOGLEVEL_PRINT); - os << "numeric(" << *value << ")"; + 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; } -// The method print adds to the output so it blends more consistently together -// with the other routines and produces something compatible to Maple input. +/** 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 { 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 << ")"; + } + } +} + + +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 (this->is_rational() && !this->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); } + bool numeric::info(unsigned inf) const { switch (inf) { @@ -285,19 +498,25 @@ bool numeric::info(unsigned inf) const 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(numZERO())>=0; + return !is_negative(); case info_flags::posint: return is_pos_integer(); case info_flags::negint: - return is_integer() && (compare(numZERO())<0); + return is_integer() && is_negative(); case info_flags::nonnegint: return is_nonneg_integer(); case info_flags::even: @@ -310,24 +529,69 @@ bool numeric::info(unsigned inf) const 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; + 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. * * @param level ignored, but needed for overriding basic::evalf. - * @return an ex-handle to a numeric. */ + * @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 + return numeric(::cl_float(1.0, ::cl_default_float_format) * (*value)); // -> CLN } // protected -int numeric::compare_same_type(basic const & other) const +/** Implementation of ex::diff() for a numeric. It always returns 0. + * + * @see ex::diff */ +ex numeric::derivative(const symbol & s) const { - ASSERT(is_exactly_of_type(other, numeric)); - numeric const & o = static_cast(const_cast(other)); + return _ex0(); +} + + +int numeric::compare_same_type(const basic & other) const +{ + GINAC_ASSERT(is_exactly_of_type(other, numeric)); + const numeric & o = static_cast(const_cast(other)); if (*value == *o.value) { return 0; @@ -336,12 +600,23 @@ int numeric::compare_same_type(basic const & other) const return compare(o); } -bool numeric::is_equal_same_type(basic const & other) const + +bool numeric::is_equal_same_type(const basic & other) const { - ASSERT(is_exactly_of_type(other,numeric)); - numeric const *o = static_cast(&other); + 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 +{ + return (hashvalue=cl_equal_hashcode(*value) | 0x80000000U); + /* + cout << *value << "->" << hashvalue << endl; + hashvalue=HASHVALUE_NUMERIC+1000U; + return HASHVALUE_NUMERIC+1000U; + */ } /* @@ -372,26 +647,26 @@ unsigned numeric::calchash(void) const /** Numerical addition method. Adds argument to *this and returns result as * a new numeric object. */ -numeric numeric::add(numeric const & other) const +numeric numeric::add(const numeric & other) const { return numeric((*value)+(*other.value)); } /** Numerical subtraction method. Subtracts argument from *this and returns * result as a new numeric object. */ -numeric numeric::sub(numeric const & other) const +numeric numeric::sub(const numeric & other) const { return numeric((*value)-(*other.value)); } /** Numerical multiplication method. Multiplies *this and argument and returns * result as a new numeric object. */ -numeric numeric::mul(numeric const & other) const +numeric numeric::mul(const numeric & other) const { - static const numeric * numONEp=&numONE(); - if (this==numONEp) { + static const numeric * _num1p=&_num1(); + if (this==_num1p) { return other; - } else if (&other==numONEp) { + } else if (&other==_num1p) { return *this; } return numeric((*value)*(*other.value)); @@ -401,141 +676,164 @@ numeric numeric::mul(numeric const & other) const * a new numeric object. * * @exception overflow_error (division by zero) */ -numeric numeric::div(numeric const & other) const +numeric numeric::div(const numeric & other) const { - if (zerop(*other.value)) + if (::zerop(*other.value)) throw (std::overflow_error("division by zero")); return numeric((*value)/(*other.value)); } -numeric numeric::power(numeric const & other) const +numeric numeric::power(const numeric & other) const { - static const numeric * numONEp=&numONE(); - if (&other==numONEp) { + static const numeric * _num1p = &_num1(); + if (&other==_num1p) return *this; + 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(); } - if (zerop(*value) && other.is_real() && minusp(realpart(*other.value))) - throw (std::overflow_error("division by zero")); - return numeric(expt(*value,*other.value)); + return numeric(::expt(*value,*other.value)); } /** Inverse of a number. */ numeric numeric::inverse(void) const { - return numeric(recip(*value)); // -> CLN + return numeric(::recip(*value)); // -> CLN } -numeric const & numeric::add_dyn(numeric const & other) const +const numeric & numeric::add_dyn(const numeric & other) const { - return static_cast((new numeric((*value)+(*other.value)))-> + return static_cast((new numeric((*value)+(*other.value)))-> setflag(status_flags::dynallocated)); } -numeric const & numeric::sub_dyn(numeric const & other) const +const numeric & numeric::sub_dyn(const numeric & other) const { - return static_cast((new numeric((*value)-(*other.value)))-> + return static_cast((new numeric((*value)-(*other.value)))-> setflag(status_flags::dynallocated)); } -numeric const & numeric::mul_dyn(numeric const & other) const +const numeric & numeric::mul_dyn(const numeric & other) const { - static const numeric * numONEp=&numONE(); - if (this==numONEp) { + static const numeric * _num1p=&_num1(); + if (this==_num1p) { return other; - } else if (&other==numONEp) { + } else if (&other==_num1p) { return *this; } - return static_cast((new numeric((*value)*(*other.value)))-> + return static_cast((new numeric((*value)*(*other.value)))-> setflag(status_flags::dynallocated)); } -numeric const & numeric::div_dyn(numeric const & other) const +const numeric & numeric::div_dyn(const numeric & other) const { - if (zerop(*other.value)) + if (::zerop(*other.value)) throw (std::overflow_error("division by zero")); - return static_cast((new numeric((*value)/(*other.value)))-> + return static_cast((new numeric((*value)/(*other.value)))-> setflag(status_flags::dynallocated)); } -numeric const & numeric::power_dyn(numeric const & other) const +const numeric & numeric::power_dyn(const numeric & other) const { - static const numeric * numONEp=&numONE(); - if (&other==numONEp) { + static const numeric * _num1p=&_num1(); + if (&other==_num1p) return *this; + 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(); } - // The ifs are only a workaround for a bug in CLN. It gets stuck otherwise: - if ( !other.is_integer() && - other.is_rational() && - (*this).is_nonneg_integer() ) { - if ( !zerop(*value) ) { - return static_cast((new numeric(exp(*other.value * log(*value))))-> - setflag(status_flags::dynallocated)); - } else { - if ( !zerop(*other.value) ) { // 0^(n/m) - return static_cast((new numeric(0))-> - setflag(status_flags::dynallocated)); - } else { // raise FPE (0^0 requested) - return static_cast((new numeric(1/(*other.value)))-> - setflag(status_flags::dynallocated)); - } - } - } else { // default -> CLN - return static_cast((new numeric(expt(*value,*other.value)))-> - setflag(status_flags::dynallocated)); - } + return static_cast((new numeric(::expt(*value,*other.value)))-> + setflag(status_flags::dynallocated)); } -numeric const & numeric::operator=(int i) +const numeric & numeric::operator=(int i) { return operator=(numeric(i)); } -numeric const & numeric::operator=(unsigned int i) +const numeric & numeric::operator=(unsigned int i) { return operator=(numeric(i)); } -numeric const & numeric::operator=(long i) +const numeric & numeric::operator=(long i) { return operator=(numeric(i)); } -numeric const & numeric::operator=(unsigned long i) +const numeric & numeric::operator=(unsigned long i) { return operator=(numeric(i)); } -numeric const & numeric::operator=(double d) +const numeric & numeric::operator=(double d) { return operator=(numeric(d)); } -numeric const & numeric::operator=(char const * s) +const numeric & numeric::operator=(const char * s) { return operator=(numeric(s)); } +/** 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 (this->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; + } +} + /** 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 - * similar to Maple's csgn. */ -int numeric::compare(numeric const & other) const + * to be compatible with our method csgn. + * + * @return csgn(*this-other) + * @see numeric::csgn(void) */ +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)); } } -bool numeric::is_equal(numeric const & other) const +bool numeric::is_equal(const numeric & other) const { return (*value == *other.value); } @@ -543,59 +841,53 @@ bool numeric::is_equal(numeric const & other) const /** True if object is zero. */ bool numeric::is_zero(void) const { - return zerop(*value); // -> CLN + return ::zerop(*value); // -> CLN } /** 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 (bool)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. @@ -603,48 +895,67 @@ 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 { - 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; + 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 (bool)instanceof(*value, cl_R_ring); // -> CLN + return ::instanceof(*value, ::cl_R_ring); // -> CLN } -bool numeric::operator==(numeric const & other) const +bool numeric::operator==(const numeric & other) const { return (*value == *other.value); // -> CLN } -bool numeric::operator!=(numeric const & other) const +bool numeric::operator!=(const numeric & other) const { return (*value != *other.value); // -> CLN } +/** 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 +{ + if (::instanceof(*value, ::cl_I_ring)) + return true; + 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; +} + +/** True if object is an exact rational number, may even be complex + * (denominator may be unity). */ +bool numeric::is_crational(void) const +{ + if (::instanceof(*value, ::cl_RA_ring)) + return true; + 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; +} + /** Numerical comparison: less. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator<(numeric const & 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 - } + 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 } @@ -652,11 +963,10 @@ bool numeric::operator<(numeric const & other) const /** Numerical comparison: less or equal. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator<=(numeric const & 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 - } + 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 } @@ -664,11 +974,10 @@ bool numeric::operator<=(numeric const & other) const /** Numerical comparison: greater. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator>(numeric const & 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 - } + 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 } @@ -676,41 +985,50 @@ bool numeric::operator>(numeric const & other) const /** Numerical comparison: greater or equal. * * @exception invalid_argument (complex inequality) */ -bool numeric::operator>=(numeric const & 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 - } + 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 { - ASSERT(is_integer()); - return cl_I_to_int(The(cl_I)(*value)); + 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 { - ASSERT(is_real()); - return cl_double_approx(realpart(*value)); + 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 + 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 + return numeric(::imagpart(*value)); // -> CLN } #ifndef SANE_LINKER @@ -728,47 +1046,48 @@ inline cl_heap_ratio* TheRatio (const cl_N& obj) /** 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 itself in all other cases. */ -numeric numeric::numer(void) const + * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other + * cases. */ +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): - 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*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)))); } } @@ -780,46 +1099,46 @@ 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()) { - return numONE(); + 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)) - return numONE(); - 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 (!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)))); } #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)) - return numONE(); - if (instanceof(r, cl_I_ring) && instanceof(i, cl_RA_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(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 - return numONE(); + return _num1(); } /** Size in binary notation. For integers, this is the smallest n >= 0 such @@ -830,11 +1149,10 @@ 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 - } else { + if (this->is_integer()) + return ::integer_length(The(cl_I)(*value)); // -> CLN + else return 0; - } } @@ -851,295 +1169,382 @@ unsigned numeric::precedence = 30; ////////// const numeric some_numeric; -type_info const & typeid_numeric=typeid(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))); - -////////// -// global functions -////////// - -numeric const & numZERO(void) -{ - const static ex eZERO = ex((new numeric(0))->setflag(status_flags::dynallocated)); - const static numeric * nZERO = static_cast(eZERO.bp); - return *nZERO; -} +const numeric I = numeric(::complex(cl_I(0),cl_I(1))); -numeric const & numONE(void) -{ - const static ex eONE = ex((new numeric(1))->setflag(status_flags::dynallocated)); - const static numeric * nONE = static_cast(eONE.bp); - return *nONE; -} - -numeric const & numTWO(void) -{ - const static ex eTWO = ex((new numeric(2))->setflag(status_flags::dynallocated)); - const static numeric * nTWO = static_cast(eTWO.bp); - return *nTWO; -} - -numeric const & numTHREE(void) -{ - const static ex eTHREE = ex((new numeric(3))->setflag(status_flags::dynallocated)); - const static numeric * nTHREE = static_cast(eTHREE.bp); - return *nTHREE; -} - -numeric const & numMINUSONE(void) -{ - const static ex eMINUSONE = ex((new numeric(-1))->setflag(status_flags::dynallocated)); - const static numeric * nMINUSONE = static_cast(eMINUSONE.bp); - return *nMINUSONE; -} - -numeric const & numHALF(void) -{ - const static ex eHALF = ex((new numeric(1, 2))->setflag(status_flags::dynallocated)); - const static numeric * nHALF = static_cast(eHALF.bp); - return *nHALF; -} /** Exponential function. * * @return arbitrary precision numerical exp(x). */ -numeric exp(numeric const & 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(numeric const & 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(numeric const & x) +const numeric sin(const numeric & x) { return ::sin(*x.value); // -> CLN } + /** Numeric cosine (trigonometric function). * * @return arbitrary precision numerical cos(x). */ -numeric cos(numeric const & x) +const numeric cos(const numeric & x) { return ::cos(*x.value); // -> CLN } - + + /** Numeric tangent (trigonometric function). * * @return arbitrary precision numerical tan(x). */ -numeric tan(numeric const & 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(numeric const & 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(numeric const & 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(numeric const & x) +const numeric atan(const numeric & x) { if (!x.is_real() && x.real().is_zero() && - !abs(x.imag()).is_equal(numONE())) + !abs(x.imag()).is_equal(_num1())) throw (std::overflow_error("atan(): logarithmic singularity")); return ::atan(*x.value); // -> CLN } -/** Arcustangents. + +/** Arcustangent. * * @param x real number * @param y real number * @return atan(y/x) */ -numeric atan(numeric const & y, numeric const & 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(numeric const & 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(numeric const & 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(numeric const & 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(numeric const & 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(numeric const & 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(numeric const & x) +const numeric atanh(const numeric & x) { return ::atanh(*x.value); // -> CLN } -/** The gamma function. - * stub stub stub stub stub stub! */ -numeric gamma(numeric const & x) + +/** Numeric evaluation of Riemann's Zeta function. Currently works only for + * integer arguments. */ +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()) { + 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. + * This is only a stub! */ +const numeric lgamma(const numeric & 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! */ +const numeric psi(const numeric & x) { - clog << "gamma(): Nobody expects the Spanish inquisition" << endl; + clog << "psi(" << x + << "): Does anybody know good way to calculate this numerically?" + << endl; return numeric(0); } + +/** The psi functions (aka polygamma functions). + * This is only a stub! */ +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); +} + + /** Factorial combinatorial function. * + * @param n integer argument >= 0 * @exception range_error (argument must be integer >= 0) */ -numeric factorial(numeric const & 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)!! + * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1 * @exception range_error (argument must be integer >= -1) */ -numeric doublefactorial(numeric const & nn) -{ - // 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 numONE(); +const numeric doublefactorial(const numeric & n) +{ + 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")); } - if ( nn.is_even() ) { - int n = nn.div(numTWO()).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(numONE()); - 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(numONE()).div(numTWO()).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(numONE()); - highest_oddresult=0; - } - for (int i=highest_oddresult+1; i<=n; i++) { - oddresults.push_back(numeric(oddresults[i-1].mul(numeric(i*2+1)))); + 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. */ +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(::binomial(n.to_int(),k.to_int())); // -> CLN + else + return _num0(); + } else { + return _num_1().power(k)*binomial(k-n-_num1(),k); } - highest_oddresult=n; - return oddresults[n]; } + + // 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.")); } -/** The Binomial function. It computes the binomial coefficients. If the - * arguments are both nonnegative integers and 0 <= k <= n, then - * binomial(n, k) = n!/k!/(n-k)! which is the number of ways of choosing k - * objects from n distinct objects. If k > n, then binomial(n,k) returns 0. */ -numeric binomial(numeric const & n, numeric const & k) -{ - if (n.is_nonneg_integer() && k.is_nonneg_integer()) { - return numeric(::binomial(n.to_int(),k.to_int())); // -> CLN - } else { - // should really be gamma(n+1)/(gamma(r+1)/gamma(n-r+1) - return numeric(0); + +/** 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) */ +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(); + 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 defining formula + // B(nn) == - 1/(nn+1) * sum_{k=0}^{nn-1}(binomial(nn+1,k)*B(k)) + // whith B(0) == 1. + // Be warned, though: the Bernoulli numbers are computationally very + // expensive anyhow and you shouldn't expect miracles to happen. + 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= 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; + } + } + 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(numeric const & x) +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 @@ -1147,46 +1552,44 @@ numeric abs(numeric const & x) * * @return a mod b in the range [0,abs(b)-1] with sign of b if both are * integer, 0 otherwise. */ -numeric mod(numeric const & a, numeric const & b) +numeric mod(const numeric & a, const numeric & b) { - if (a.is_integer() && b.is_integer()) { + if (a.is_integer() && b.is_integer()) return ::mod(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - } - else { - return numZERO(); // Throw? - } + 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(numeric const & a, numeric const & b) +numeric smod(const numeric & a, const numeric & b) { 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 numZERO(); // Throw? - } + } 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 * sign of a or is zero. * * @return remainder of a/b if both are integer, 0 otherwise. */ -numeric irem(numeric const & a, numeric const & b) +numeric irem(const numeric & a, const numeric & b) { - if (a.is_integer() && b.is_integer()) { + if (a.is_integer() && b.is_integer()) return ::rem(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - } - else { - return numZERO(); // Throw? - } + 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, @@ -1194,7 +1597,7 @@ numeric irem(numeric const & a, numeric const & b) * * @return remainder of a/b and quotient stored in q if both are integer, * 0 otherwise. */ -numeric irem(numeric const & a, numeric const & b, numeric & q) +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)); @@ -1202,42 +1605,44 @@ numeric irem(numeric const & a, numeric const & b, numeric & q) return rem_quo.remainder; } else { - q = numZERO(); - return numZERO(); // Throw? + q = _num0(); + return _num0(); // Throw? } } + /** 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(numeric const & a, numeric const & b) +numeric iquo(const numeric & a, const numeric & b) { - if (a.is_integer() && b.is_integer()) { + if (a.is_integer() && b.is_integer()) return truncate1(The(cl_I)(*a.value), The(cl_I)(*b.value)); // -> CLN - } else { - return numZERO(); // Throw? - } + 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. * * @return truncated quotient of a/b and remainder stored in r if both are * integer, 0 otherwise. */ -numeric iquo(numeric const & a, numeric const & b, numeric & r) +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 = numZERO(); - return numZERO(); // Throw? + 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. @@ -1246,61 +1651,71 @@ numeric iquo(numeric const & a, numeric const & b, numeric & r) * @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(numeric const & z) +numeric sqrt(const numeric & z) { return ::sqrt(*z.value); // -> CLN } + /** Integer numeric square root. */ -numeric isqrt(numeric const & x) +numeric isqrt(const numeric & x) { - if (x.is_integer()) { - cl_I root; - ::isqrt(The(cl_I)(*x.value), &root); // -> CLN - return root; - } else - return numZERO(); // Throw? + 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(numeric const & a, numeric const & b) +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 numONE(); + 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(numeric const & a, numeric const & 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. @@ -1309,28 +1724,32 @@ _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, _numeric_digits const & e) + +ostream& operator<<(ostream& os, const _numeric_digits & e) { e.print(os); return os; @@ -1344,8 +1763,11 @@ 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_NAMESPACE_GINAC } // namespace GiNaC +#endif // ndef NO_NAMESPACE_GINAC