* of special functions or implement the interface to the bignum package. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2002 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
{
// Not the whole int-range is available if we don't cast to long
// first. This is due to the behaviour of the cl_I-ctor, which
- // emphasizes efficiency. However, if the integer is small enough,
- // i.e. satisfies cl_immediate_p(), we save space and dereferences by
- // using an immediate type:
- if (cln::cl_immediate_p(i))
+ // emphasizes efficiency. However, if the integer is small enough
+ // we save space and dereferences by using an immediate type.
+ // (C.f. <cln/object.h>)
+ if (i < (1U<<cl_value_len-1))
value = cln::cl_I(i);
else
value = cln::cl_I((long) i);
{
// Not the whole uint-range is available if we don't cast to ulong
// first. This is due to the behaviour of the cl_I-ctor, which
- // emphasizes efficiency. However, if the integer is small enough,
- // i.e. satisfies cl_immediate_p(), we save space and dereferences by
- // using an immediate type:
- if (cln::cl_immediate_p(i))
+ // emphasizes efficiency. However, if the integer is small enough
+ // we save space and dereferences by using an immediate type.
+ // (C.f. <cln/object.h>)
+ if (i < (1U<<cl_value_len-1))
value = cln::cl_I(i);
else
value = cln::cl_I((unsigned long) i);
const std::string mul_sym = is_a<print_latex>(c) ? " " : "*";
const cln::cl_R r = cln::realpart(cln::the<cln::cl_N>(value));
const cln::cl_R i = cln::imagpart(cln::the<cln::cl_N>(value));
+ if (is_a<print_python_repr>(c))
+ c.s << class_name() << "('";
if (cln::zerop(i)) {
// case 1, real: x or -x
if ((precedence() <= level) && (!this->is_nonneg_integer())) {
c.s << par_close;
}
}
+ if (is_a<print_python_repr>(c))
+ c.s << "')";
}
}
return false;
}
+int numeric::degree(const ex & s) const
+{
+ return 0;
+}
+
+int numeric::ldegree(const ex & s) const
+{
+ return 0;
+}
+
+ex numeric::coeff(const ex & s, int n) const
+{
+ return n==0 ? *this : _ex0;
+}
+
/** Disassemble real part and imaginary part to scan for the occurrence of a
* single number. Also handles the imaginary unit. It ignores the sign on
* both this and the argument, which may lead to what might appear as funny
{
if (!nn.is_integer() || nn.is_negative())
throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
-
+
// Method:
//
// The Bernoulli numbers are rational numbers that may be computed using
// 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().
+ // up. The formula below accomplishes this. It is a modification of the
+ // defining formula above but the computation of the binomial coefficients
+ // is carried along in an inline fashion. It also honors the fact that
+ // B_n is zero when n is odd and greater than 1.
//
// (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
+ // in `Concrete Mathematics', which leads to a program a little faster 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.)
-
+
+ const unsigned n = nn.to_int();
+
// the special cases not covered by the algorithm below
- if (nn.is_equal(_num1))
- return _num_1_2;
- if (nn.is_odd())
- return _num0;
-
+ if (n & 1)
+ return (n==1) ? _num_1_2 : _num0;
+ if (!n)
+ return _num1;
+
// store nonvanishing Bernoulli numbers here
static std::vector< cln::cl_RA > results;
- static int highest_result = 0;
- // algorithm not applicable to B(0), so just store it
- if (results.empty())
- results.push_back(cln::cl_RA(1));
-
- int n = nn.to_long();
- for (int i=highest_result; i<n/2; ++i) {
- cln::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 = cln::cl_I(n*m) * (B+results[j]) / (d1*d2);
- n += 4;
- m += 2;
- d1 -= 1;
- d2 -= 2;
- }
- B = (1 - ((B+1)/(2*i+3))) / (cln::cl_I(1)<<(2*i+2));
- results.push_back(B);
- ++highest_result;
+ static unsigned next_r = 0;
+
+ // algorithm not applicable to B(2), so just store it
+ if (!next_r) {
+ results.push_back(); // results[0] is not used
+ results.push_back(cln::recip(cln::cl_RA(6)));
+ next_r = 4;
+ }
+ if (n<next_r)
+ return results[n/2];
+
+ results.reserve(n/2 + 1);
+ for (unsigned p=next_r; p<=n; p+=2) {
+ cln::cl_I c = 1; // seed for binonmial coefficients
+ cln::cl_RA b = cln::cl_RA(1-p)/2;
+ const unsigned p3 = p+3;
+ const unsigned pm = p-2;
+ unsigned i, k, p_2;
+ // test if intermediate unsigned int can be represented by immediate
+ // objects by CLN (i.e. < 2^29 for 32 Bit machines, see <cln/object.h>)
+ if (p < (1UL<<cl_value_len/2)) {
+ for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
+ c = cln::exquo(c * ((p3-i) * p_2), (i-1)*k);
+ b = b + c*results[k];
+ }
+ } else {
+ for (i=2, k=1, p_2=p/2; i<=pm; i+=2, ++k, --p_2) {
+ c = cln::exquo((c * (p3-i)) * p_2, cln::cl_I(i-1)*k);
+ b = b + c*results[k];
+ }
+ }
+ results.push_back(-b/(p+1));
}
+ next_r = n+2;
return results[n/2];
}
git://www.ginac.de / ginac.git / blobdiff