{
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);
}
{
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();
{
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();
case 'N': // Ordinary number
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);
+ *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));
+ *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);
switch (c) {
case 'R': // Integer-decoded real number
f >> re.sign >> re.mantissa >> re.exponent;
- *value = re.sign * re.mantissa * expt(cl_float(2.0, cl_default_float_format), 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
f >> re.sign >> re.mantissa >> re.exponent;
f >> 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));
+ *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
f.putback(c);
case info_flags::negative:
return is_negative();
case info_flags::nonnegative:
- return compare(_num0())>=0;
+ return !is_negative();
case info_flags::posint:
return is_pos_integer();
case info_flags::negint:
- return is_integer() && (compare(_num0())<0);
+ return is_integer() && is_negative();
case info_flags::nonnegint:
return is_nonneg_integer();
case info_flags::even:
}
/** Disassemble real part and imaginary part to scan for the occurrence of a
- * single number. Also handles the imaginary unit. */
+ * 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<numeric &>(const_cast<basic &>(*other.bp));
- if (this->is_equal(o))
+ 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) ||
}
-/** Evaluation of numbers doesn't do anything. */
+/** 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
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;
+ */
+}
+
/*
unsigned numeric::calchash(void) const
{
bool numeric::operator<(const numeric & other) const
{
if (this->is_real() && other.is_real())
- return (bool)(The(cl_R)(*value) < The(cl_R)(*other.value)); // -> CLN
+ 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
}
bool numeric::operator<=(const numeric & other) const
{
if (this->is_real() && other.is_real())
- return (bool)(The(cl_R)(*value) <= The(cl_R)(*other.value)); // -> CLN
+ 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
}
bool numeric::operator>(const numeric & other) const
{
if (this->is_real() && other.is_real())
- return (bool)(The(cl_R)(*value) > The(cl_R)(*other.value)); // -> CLN
+ 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
}
bool numeric::operator>=(const numeric & other) const
{
if (this->is_real() && other.is_real())
- return (bool)(The(cl_R)(*value) >= The(cl_R)(*other.value)); // -> CLN
+ 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
}
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))));
+ 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))));
+ 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)))),
+ 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_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));
+ 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));
+ 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)),
+ 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))));
}
}
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))));
+ return numeric(::lcm(::denominator(The(cl_RA)(r)), ::denominator(The(cl_RA)(i))));
}
#else
if (instanceof(*value, cl_RA_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));
+ return numeric(::lcm(TheRatio(r)->denominator, TheRatio(i)->denominator));
}
#endif // def SANE_LINKER
// at least one float encountered
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 _num0();
// Until somebody has the Blues and comes up with a much better idea and
// codes it (preferably in CLN) we make this a remembering function which
- // computes its results using the formula
+ // 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 probably 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();
* @exception range_error (argument must be an integer) */
const numeric fibonacci(const numeric & n)
{
- if (!n.is_integer()) {
+ if (!n.is_integer())
throw (std::range_error("numeric::fibonacci(): argument must be integer"));
- }
- // For positive arguments compute the nearest integer to
- // ((1+sqrt(5))/2)^n/sqrt(5). For negative arguments, apply an additional
- // sign. Note that we are falling back to longs, but this should suffice
- // for all times.
- int sig = 1;
- const long nn = ::abs(n.to_double());
- if (n.is_negative() && n.is_even())
- sig =-1;
+ // 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);
- // Need a precision of ((1+sqrt(5))/2)^-n.
- cl_float_format_t prec = ::cl_float_format((int)(0.208987641*nn+5));
- cl_R sqrt5 = ::sqrt(::cl_float(5,prec));
- cl_R phi = (1+sqrt5)/2;
- return numeric(::round1(::expt(phi,nn)/sqrt5)*sig);
+ 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);
}
git://www.ginac.de / ginac.git / blobdiff