{
debugmsg("numeric constructor from long/long",LOGLEVEL_CONSTRUCT);
if (!denom)
- throw (std::overflow_error("division by zero"));
+ throw std::overflow_error("division by zero");
value = new ::cl_I(numer);
*value = *value / ::cl_I(denom);
calchash();
// 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) != '-')
+ if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
ss = '+' + ss;
std::string::size_type delim;
do {
* 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)
+static void print_real_number(std::ostream & os, const cl_R & num)
{
cl_print_flags ourflags;
if (::instanceof(num, ::cl_RA_ring)) {
* with the other routines and produces something compatible to ginsh input.
*
* @see print_real_number() */
-void numeric::print(ostream & os, unsigned upper_precedence) const
+void numeric::print(std::ostream & os, unsigned upper_precedence) const
{
debugmsg("numeric print", LOGLEVEL_PRINT);
if (this->is_real()) {
}
-void numeric::printraw(ostream & os) const
+void numeric::printraw(std::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
}
-void numeric::printtree(ostream & os, unsigned indent) const
+void numeric::printtree(std::ostream & os, unsigned indent) const
{
debugmsg("numeric printtree", LOGLEVEL_PRINT);
os << std::string(indent,' ') << *value
<< " (numeric): "
- << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
- << ", flags=" << flags << endl;
+ << "hash=" << hashvalue
+ << " (0x" << std::hex << hashvalue << std::dec << ")"
+ << ", flags=" << flags << std::endl;
}
-void numeric::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
+void numeric::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
{
debugmsg("numeric print csrc", LOGLEVEL_PRINT);
ios::fmtflags oldflags = os.flags();
/** Cast numeric into a floating-point object. For example exact numeric(1) is
* returned as a 1.0000000000000000000000 and so on according to how Digits is
- * currently set.
+ * currently set. In case the object already was a floating point number the
+ * precision is trimmed to match the currently set default.
*
- * @param level ignored, but needed for overriding basic::evalf.
+ * @param level ignored, only needed for overriding basic::evalf.
* @return an ex-handle to a numeric. */
ex numeric::evalf(int level) const
{
numeric numeric::div(const numeric & other) const
{
if (::zerop(*other.value))
- throw (std::overflow_error("division by zero"));
+ throw std::overflow_error("numeric::div(): division by zero");
return numeric((*value)/(*other.value));
}
return *this;
if (::zerop(*value)) {
if (::zerop(*other.value))
- throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
+ 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"));
+ 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"));
+ throw std::overflow_error("numeric::eval(): division by zero");
else
return _num0();
}
/** Inverse of a number. */
numeric numeric::inverse(void) const
{
+ if (::zerop(*value))
+ throw std::overflow_error("numeric::inverse(): division by zero");
return numeric(::recip(*value)); // -> CLN
}
const numeric & numeric::div_dyn(const numeric & other) const
{
if (::zerop(*other.value))
- throw (std::overflow_error("division by zero"));
+ throw std::overflow_error("division by zero");
return static_cast<const numeric &>((new numeric((*value)/(*other.value)))->
setflag(status_flags::dynallocated));
}
return *this;
if (::zerop(*value)) {
if (::zerop(*other.value))
- throw (std::domain_error("numeric::eval(): pow(0,0) is undefined"));
+ 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"));
+ 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"));
+ throw std::overflow_error("numeric::eval(): division by zero");
else
return _num0();
}
{
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"));
+ throw std::invalid_argument("numeric::operator<(): complex inequality");
return false; // make compiler shut up
}
{
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"));
+ throw std::invalid_argument("numeric::operator<=(): complex inequality");
return false; // make compiler shut up
}
{
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"));
+ throw std::invalid_argument("numeric::operator>(): complex inequality");
return false; // make compiler shut up
}
{
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"));
+ throw std::invalid_argument("numeric::operator>=(): complex inequality");
return false; // make compiler shut up
}
*
* @param z complex number
* @return arbitrary precision numerical log(x).
- * @exception overflow_error (logarithmic singularity) */
+ * @exception pole_error("log(): logarithmic pole",0) */
const numeric log(const numeric & z)
{
if (z.is_zero())
- throw (std::overflow_error("log(): logarithmic singularity"));
+ throw pole_error("log(): logarithmic pole",0);
return ::log(*z.value); // -> CLN
}
*
* @param z complex number
* @return atan(z)
- * @exception overflow_error (logarithmic singularity) */
+ * @exception pole_error("atan(): logarithmic pole",0) */
const numeric atan(const numeric & x)
{
if (!x.is_real() &&
x.real().is_zero() &&
- !abs(x.imag()).is_equal(_num1()))
- throw (std::overflow_error("atan(): logarithmic singularity"));
+ abs(x.imag()).is_equal(_num1()))
+ throw pole_error("atan(): logarithmic pole",0);
return ::atan(*x.value); // -> CLN
}
if (x.is_real() && y.is_real())
return ::atan(::realpart(*x.value), ::realpart(*y.value)); // -> CLN
else
- throw (std::invalid_argument("numeric::atan(): complex argument"));
+ throw std::invalid_argument("atan(): complex argument");
}
}
+/*static ::cl_N Li2_series(const ::cl_N & x,
+ const ::cl_float_format_t & prec)
+{
+ // Note: argument must be in the unit circle
+ // This is very inefficient unless we have fast floating point Bernoulli
+ // numbers implemented!
+ ::cl_N c1 = -::log(1-x);
+ ::cl_N c2 = c1;
+ // hard-wire the first two Bernoulli numbers
+ ::cl_N acc = c1 - ::square(c1)/4;
+ ::cl_N aug;
+ ::cl_F pisq = ::square(::cl_pi(prec)); // pi^2
+ ::cl_F piac = ::cl_float(1, prec); // accumulator: pi^(2*i)
+ unsigned i = 1;
+ c1 = ::square(c1);
+ do {
+ c2 = c1 * c2;
+ piac = piac * pisq;
+ aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / ::factorial(2*i+1);
+ // aug = c2 * ::cl_I(i%2 ? 1 : -1) / ::cl_I(2*i+1) * ::cl_zeta(2*i, prec) / piac / (::cl_I(1)<<(2*i-1));
+ acc = acc + aug;
+ ++i;
+ } while (acc != acc+aug);
+ return acc;
+}*/
+
+/** Numeric evaluation of Dilogarithm within circle of convergence (unit
+ * circle) using a power series. */
+static ::cl_N Li2_series(const ::cl_N & x,
+ const ::cl_float_format_t & prec)
+{
+ // Note: argument must be in the unit circle
+ ::cl_N aug, acc;
+ ::cl_N num = ::complex(::cl_float(1, prec), 0);
+ ::cl_I den = 0;
+ unsigned i = 1;
+ do {
+ num = num * x;
+ den = den + i; // 1, 4, 9, 16, ...
+ i += 2;
+ aug = num / den;
+ acc = acc + aug;
+ } while (acc != acc+aug);
+ return acc;
+}
+
+/** Folds Li2's argument inside a small rectangle to enhance convergence. */
+static ::cl_N Li2_projection(const ::cl_N & x,
+ const ::cl_float_format_t & prec)
+{
+ const ::cl_R re = ::realpart(x);
+ const ::cl_R im = ::imagpart(x);
+ if (re > ::cl_F(".5"))
+ // zeta(2) - Li2(1-x) - log(x)*log(1-x)
+ return(::cl_zeta(2)
+ - Li2_series(1-x, prec)
+ - ::log(x)*::log(1-x));
+ if ((re <= 0 && ::abs(im) > ::cl_F(".75")) || (re < ::cl_F("-.5")))
+ // -log(1-x)^2 / 2 - Li2(x/(x-1))
+ return(-::square(::log(1-x))/2
+ - Li2_series(x/(x-1), prec));
+ if (re > 0 && ::abs(im) > ::cl_LF(".75"))
+ // Li2(x^2)/2 - Li2(-x)
+ return(Li2_projection(::square(x), prec)/2
+ - Li2_projection(-x, prec));
+ return Li2_series(x, prec);
+}
+
+/** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
+ * the branch cut lies along the positive real axis, starting at 1 and
+ * continuous with quadrant IV.
+ *
+ * @return arbitrary precision numerical Li2(x). */
+const numeric Li2(const numeric & x)
+{
+ if (::zerop(*x.value))
+ return x;
+
+ // what is the desired float format?
+ // first guess: default format
+ ::cl_float_format_t prec = ::cl_default_float_format;
+ // second guess: the argument's format
+ if (!::instanceof(::realpart(*x.value),cl_RA_ring))
+ prec = ::cl_float_format(The(::cl_F)(::realpart(*x.value)));
+ else if (!::instanceof(::imagpart(*x.value),cl_RA_ring))
+ prec = ::cl_float_format(The(::cl_F)(::imagpart(*x.value)));
+
+ if (*x.value==1) // may cause trouble with log(1-x)
+ return ::cl_zeta(2, prec);
+
+ if (::abs(*x.value) > 1)
+ // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
+ return(-::square(::log(-*x.value))/2
+ - ::cl_zeta(2, prec)
+ - Li2_projection(::recip(*x.value), prec));
+ else
+ return Li2_projection(*x.value, prec);
+}
+
+
/** Numeric evaluation of Riemann's Zeta function. Currently works only for
* integer arguments. */
const numeric zeta(const numeric & x)
// 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))
+ if (::zerop(*x.value-aux))
return ::cl_zeta(aux); // -> CLN
}
- clog << "zeta(" << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
+ std::clog << "zeta(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
return numeric(0);
}
* This is only a stub! */
const numeric lgamma(const numeric & x)
{
- clog << "lgamma(" << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
+ std::clog << "lgamma(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
return numeric(0);
}
const numeric tgamma(const numeric & x)
{
- clog << "tgamma(" << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
+ std::clog << "tgamma(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
return numeric(0);
}
* This is only a stub! */
const numeric psi(const numeric & x)
{
- clog << "psi(" << x
- << "): Does anybody know good way to calculate this numerically?"
- << endl;
+ std::clog << "psi(" << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
return numeric(0);
}
* 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;
+ std::clog << "psi(" << n << "," << x
+ << "): Does anybody know good way to calculate this numerically?"
+ << std::endl;
return numeric(0);
}
const numeric factorial(const numeric & n)
{
if (!n.is_nonneg_integer())
- throw (std::range_error("numeric::factorial(): argument must be integer >= 0"));
+ throw std::range_error("numeric::factorial(): argument must be integer >= 0");
return numeric(::factorial(n.to_int())); // -> CLN
}
return _num1();
}
if (!n.is_nonneg_integer()) {
- throw (std::range_error("numeric::doublefactorial(): argument must be integer >= -1"));
+ throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
}
return numeric(::doublefactorial(n.to_int())); // -> CLN
}
}
}
- // 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."));
+ // should really be gamma(n+1)/gamma(r+1)/gamma(n-r+1) or a suitable limit
+ throw std::range_error("numeric::binomial(): don´t know how to evaluate that.");
}
const numeric 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);
+ throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
+
+ // Method:
+ //
+ // The Bernoulli numbers are rational numbers that may be computed using
+ // the relation
+ //
+ // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
+ //
+ // with B(0) = 1. Since the n'th Bernoulli number depends on all the
+ // previous ones, the computation is necessarily very expensive. There are
+ // several other ways of computing them, a particularly good one being
+ // cl_I s = 1;
+ // cl_I c = n+1;
+ // cl_RA Bern = 0;
+ // for (unsigned i=0; i<n; i++) {
+ // c = exquo(c*(i-n),(i+2));
+ // Bern = Bern + c*s/(i+2);
+ // s = s + expt_pos(cl_I(i+2),n);
+ // }
+ // return Bern;
+ //
+ // But if somebody works with the n'th Bernoulli number she is likely to
+ // also need all previous Bernoulli numbers. So we need a complete remember
+ // table and above divide and conquer algorithm is not suited to build one
+ // up. The code below is adapted from Pari's function bernvec().
+ //
+ // (There is an interesting relation with the tangent polynomials described
+ // in `Concrete Mathematics', which leads to a program twice as fast as our
+ // implementation below, but it requires storing one such polynomial in
+ // addition to the remember table. This doubles the memory footprint so
+ // we don't use it.)
+
+ // the special cases not covered by the algorithm below
+ if (nn.is_equal(_num1()))
+ return _num_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<numeric> 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<i; ++j)
- tmp += binomial(numeric(2*i+3),numeric(j*2+2))*results[j];
- // divide by -(nn+1) and store result:
- results.push_back(-tmp/numeric(2*i+3));
+ // store nonvanishing Bernoulli numbers here
+ static std::vector< ::cl_RA > 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; i<n/2; ++i) {
+ ::cl_RA B = 0;
+ long n = 8;
+ long m = 5;
+ long d1 = i;
+ long d2 = 2*i-1;
+ for (int j=i; j>0; --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;
}
- highest_result=n;
- return results[n];
+ return results[n/2];
}
const numeric fibonacci(const numeric & n)
{
if (!n.is_integer())
- throw (std::range_error("numeric::fibonacci(): argument must be 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.)
}
-void _numeric_digits::print(ostream & os) const
+void _numeric_digits::print(std::ostream & os) const
{
debugmsg("_numeric_digits print", LOGLEVEL_PRINT);
os << digits;
}
-ostream& operator<<(ostream& os, const _numeric_digits & e)
+std::ostream& operator<<(std::ostream& os, const _numeric_digits & e)
{
e.print(os);
return os;
git://www.ginac.de / ginac.git / blobdiff