def exp(x, err=defaultError):
"""
Return exponential of x.
"""
if err <= defaultError:
reduced = rational.Rational(x)
if reduced < 0:
reverse = -1
reduced = -reduced
else:
reverse = 1
i = 0
while reduced >= 2:
reduced /= 2
i += 1
if reduced == 0:
retval = rational.Integer(1)
else:
series = ExponentialPowerSeries(
itertools.cycle((rational.Integer(1), )))
retval = series(reduced, err)
if i > 0:
retval **= 2**i
if reverse < 0:
retval = 1 / retval
else:
retval = rational.Rational(math.exp(x))
return retval
def cosh(z, err=defaultError):
if z == 0:
return rational.Integer(1)
if (defaultError >= err) or isinstance(err, AbsoluteError):
series = ExponentialPowerSeries(itertools.cycle((rational.Integer(1),0,)))
return series(z, err)
else:
return Complex(cmath.cosh(complex(z.real,z.imag)))
def _cosTaylor(x, err=defaultError):
"""
_cosTaylor(x [,err]) returns the cosine of x by Taylor series.
It is recomended to use only for 0 <= x <= pi / 4.
"""
cosSeries = ExponentialPowerSeries(
itertools.cycle((rational.Integer(1), 0, rational.Integer(-1), 0)))
rx = rational.Rational(x)
return cosSeries(rx, err)
def _sinTaylor(x, err=defaultError):
"""
_sinTaylor(x [,err]) returns the sine of x by Taylor expansion.
It is recommended to use only for 0 <= x <= pi / 4.
"""
rx = rational.Rational(x)
sinSeries = ExponentialPowerSeries(
itertools.cycle((0, rational.Integer(1), 0, rational.Integer(-1))))
return sinSeries(rx, err)
def __pow__(self, other):
if rational.isIntegerObject(other):
if other == 0:
return rational.Integer(1)
elif other == 1:
return +self
elif other < 0:
return (self**(-other)).inverse()
elif other == 2:
return self.__class__(self.real ** 2 - self.imag ** 2, 2 * self.real * self.imag)
else:
return rational.Integer(other).actMultiplicative(self)
return exp(other * log(self))
def testBuchberger(self):
"""
test Buchberger's algorithm.
http://www.geocities.com/famancin/buchberger.html
"""
F = (multiutil.polynomial(
{
(2, 0): rational.Integer(1),
(1, 2): rational.Integer(2)
}, rational.theRationalField, 2),
multiutil.polynomial(
{
(1, 1): rational.Integer(1),
(0, 3): rational.Integer(2),
(0, 0): rational.Integer(-1)
}, rational.theRationalField, 2))
G_from_F = groebner.buchberger(F, self.lex)
G_expected = list(F + (multiutil.polynomial(
{(1, 0): rational.Integer(1)}, rational.theRationalField, 2),
multiutil.polynomial(
{
(0, 3): rational.Integer(2),
(0, 0): rational.Integer(-1)
}, rational.theRationalField, 2)))
self.assertEqual(len(G_expected), len(G_from_F))
for p, q in zip(G_expected, G_from_F):
self.assertEqualUptoUnit(p, q)
def testNormalStrategy(self):
"""
test Buchberger's algorithm (normal strategy).
same example with testBuchberger.
"""
F = (multiutil.polynomial(
{
(2, 0): rational.Integer(1),
(1, 2): rational.Integer(2)
}, rational.theRationalField, 2),
multiutil.polynomial(
{
(1, 1): rational.Integer(1),
(0, 3): rational.Integer(2),
(0, 0): rational.Integer(-1)
}, rational.theRationalField, 2))
G_from_F = groebner.normal_strategy(F, self.lex)
G_expected = list(F + (multiutil.polynomial(
{(1, 0): rational.Integer(1)}, rational.theRationalField, 2),
multiutil.polynomial(
{
(0, 3): rational.Integer(2),
(0, 0): rational.Integer(-1)
}, rational.theRationalField, 2)))
self.assertEqual(len(G_expected), len(G_from_F))
for p, q in zip(G_expected, G_from_F):
self.assertEqualUptoUnit(p, q)
def terms(self, x):
"""
Generator of terms of series with assigned x value.
"""
if x == 0:
yield self.iterator.next()
else:
f = rational.Integer(1)
i = 0
y = rational.Integer(1)
for an in self.iterator:
yield an * y / f
y *= x
i += 1
f *= i
def cosh(x, err=defaultError):
"""
cosh(x [,err]) returns the hyperbolic cosine of x.
"""
if err <= defaultError:
series = ExponentialPowerSeries(
itertools.cycle((
rational.Integer(1),
0,
)))
rx = rational.Rational(x)
if rx == 0:
return rational.Integer(1)
return series(rx, err)
else:
return rational.Rational(math.cosh(x))
def asin(x, err=defaultError):
"""
asin(x [,err]) returns arc sine of x.
"""
if x > 1 or x < -1:
raise ValueError("%s is not in the range [-1, 1]." % str(x))
if x < 0:
return -asin(-x)
if err <= defaultError:
u = sqrt(rational.Rational(1, 2))
if x > u:
return pi(err) / 2 - asin(sqrt(1 - x**2))
if x == 0:
return rational.Integer(0)
y = rational.Rational(x)
y2 = y**2
i = 2
retval = y
term = rational.Rational(y)
oldvalue = 0
while not err.nearlyEqual(retval, oldvalue):
oldvalue = +retval
term *= y2 * (i - 1)**2 / (i * (i + 1))
i += 2
retval += term
else:
retval = rational.Rational(math.asin(x))
return retval
def atan(x, err=defaultError):
"""
atan(x [,err]) returns arc tangent of x.
"""
if not isinstance(err, defaultError.__class__) or err <= defaultError:
# atan(x) = -atan(-x)
if x < 0:
return -atan(-x, err)
# atan(x) = pi/2 - atan(1/x)
elif x > 1:
return pi(err) / 2 - atan(1 / x, err)
elif x == 1:
return pi(err) / 4
elif x == 0:
return rational.Integer(0)
y = rational.Rational(x)
y2 = y**2
retval = y
oldvalue = 0
term = rational.Rational(x)
i = 1
while not err.nearlyEqual(retval, oldvalue):
oldvalue = +retval
i += 2
term *= -y2 * (i - 2) / i
retval += term
else:
retval = rational.Rational(math.atan(x))
return retval
def testReduce(self):
F = (multiutil.polynomial(
{
(2, 0): rational.Integer(1),
(1, 2): rational.Integer(2)
}, rational.theRationalField, 2),
multiutil.polynomial(
{
(1, 1): rational.Integer(1),
(0, 3): rational.Integer(2),
(0, 0): rational.Integer(-1)
}, rational.theRationalField, 2))
rgb_expected = [
multiutil.polynomial({(1, 0): rational.Integer(1)},
rational.theRationalField, 2),
multiutil.polynomial(
{
(0, 3): rational.Integer(1),
(0, 0): rational.Rational(-1, 2)
}, rational.theRationalField, 2)
]
G_from_F = groebner.normal_strategy(F, self.lex)
rgb = groebner.reduce_groebner(G_from_F, self.lex)
self.assertEqual(2, len(rgb))
self.assertEqual(rgb_expected, rgb)
def ceil(x):
"""
ceil(x) returns the integer; if x is an integer then x itself,
otherwise the smallest integer greater than x.
"""
rx = rational.Rational(x)
if rx.denominator == 1:
return rational.Integer(rx.numerator)
return rx.numerator // rx.denominator + 1
def floor(x):
"""
floor(x) returns the integer; if x is an integer then x itself,
otherwise the biggest integer less than x.
"""
rx = rational.Rational(x)
if rx.denominator == 1:
return rational.Integer(rx.numerator)
return rx.numerator // rx.denominator
def minimumAbsolute(self):
"""
Return the minimum absolute representative integer of the
residue class.
"""
result = self.n % self.m
if result > self.m // 2:
result -= self.m
return rational.Integer(result)
def tranc(x):
"""
tranc(x) returns the integer; if x is an integer then x itself,
otherwise the nearest integer to x. If x has the fraction part
1/2, then bigger one will be chosen.
"""
rx = rational.Rational(x)
if rx.denominator == 1:
return rational.Integer(rx.numerator)
return floor(x + rational.Rational(1, 2))
def _li_terms(x):
"""
Generate terms of infinite part of Li(x):
x^i / (i * i!)
for each i.
"""
d = rational.Integer(1)
t = x
for i in bigrange.count(2):
yield t
t = t * x * (i - 1) / (i * i)
def expi(x, err=defaultError):
"""
expi(x [,err]) returns exp(i * x) where i is the imaginary unit
and x must be a real number.
"""
if x == 0:
return rational.Integer(1)
if isinstance(err, RelativeError):
_err = real.RelativeError(0, err.relativeerrorrange, 2)
elif isinstance(err, AbsoluteError):
_err = real.AbsoluteError(0, err.absoluteerrorrange, 2)
re = real.cos(x, _err)
im = real.sin(x, _err)
return Complex(re, im)
def _rational_mul(self, other):
"""
return other * self, assuming other is a rational element
"""
if rational.isIntegerObject(other):
other_numerator = rational.Integer(other)
other_denominator = rational.Integer(1)
else:
other_numerator = other.numerator
other_denominator = other.denominator
denom_gcd = gcd.gcd(self.denominator, other_numerator)
if denom_gcd != 1:
new_denominator = ring.exact_division(
self.denominator, denom_gcd) * other_denominator
multiply_num = other_numerator.exact_division(denom_gcd)
else:
new_denominator = self.denominator * other_denominator
multiply_num = other_numerator
new_module = self.__class__(
[self.mat_repr * multiply_num, new_denominator], self.number_field,
self.base, self.mat_repr.ishnf)
new_module._simplify()
return new_module
def sinh(x, err=defaultError):
"""
sinh(x [,err]) returns the hyperbolic sine of x.
"""
if not isinstance(err, defaultError.__class__) or err <= defaultError:
series = ExponentialPowerSeries(
itertools.cycle((
0,
rational.Integer(1),
)))
rx = rational.Rational(x)
if rx == 0:
return rational.Rational(0)
return series(rx, err)
else:
return rational.Rational(math.sinh(x))
def getRingInstance(obj):
"""
Return a RingElement instance which eqauls 'obj'.
Mainly for python built-in objects such as int or float.
"""
if isinstance(obj, RingElement):
return obj
elif isinstance(obj, int):
import nzmath.rational as rational
return rational.Integer(obj)
elif isinstance(obj, float):
import nzmath.real as real
return real.Real(obj)
elif isinstance(obj, complex):
import nzmath.imaginary as imaginary
return imaginary.Complex(obj)
return None
def testExtgcd(self):
u, v, d = gcd.extgcd(8, 11)
self.assertEqual(1, abs(d))
self.assertEqual(d, 8 * u + 11 * v)
#sf.bug 1924839
u, v, d = gcd.extgcd(-8, 11)
self.assertEqual(1, abs(d))
self.assertEqual(d, -8 * u + 11 * v)
u, v, d = gcd.extgcd(8, -11)
self.assertEqual(1, abs(d))
self.assertEqual(d, 8 * u - 11 * v)
u, v, d = gcd.extgcd(-8, -11)
self.assertEqual(1, abs(d))
self.assertEqual(d, -8 * u - 11 * v)
import nzmath.rational as rational
u, v, d = gcd.extgcd(rational.Integer(8), 11)
self.assertEqual(1, abs(d))
self.assertEqual(d, 8 * u + 11 * v)
def atan2(y, x, err=defaultError):
"""
atan2(y, x [,err]) returns the arc tangent of y/x.
Unlike atan(y/x), the signs of both x and y are considered.
It is unrecomended to obtain the value of pi with atan2(0,1).
"""
if x > 0 and y > 0:
return atan(y / x)
elif x > 0 and y < 0:
return pi(err) * 2 + atan(y / x)
elif x < 0:
return pi(err) + atan(y / x)
elif x == 0 and y > 0:
return pi(err) / 2
elif x == 0 and y < 0:
return -pi(err) / 2
return rational.Integer(0)
def sqrt(x, err=defaultError):
"""
sqrt(x [,err]) returns the positive square root of real number x.
"""
rx = rational.Rational(x)
if rx.numerator < 0:
raise ValueError("negative number is passed to sqrt")
if rx.numerator == 0:
return rational.Integer(0)
if err <= defaultError:
n = rx.denominator * rx.numerator
rt = rational.Rational(arith1.floorsqrt(n), rx.denominator)
newrt = (rt + rx / rt) / 2
while not err.nearlyEqual(rt, newrt):
rt = newrt
newrt = (rt + rx / rt) / 2
else:
newrt = rational.Rational(math.sqrt(x.toFloat()))
return newrt
def piGaussLegendre(err=defaultError):
"""
piGaussLegendre computes pi by Gauss-Legendre algorithm.
"""
if isinstance(err, RelativeError):
_err = err.absoluteerror(3.1415926535897932)
else:
_err = err
werr = AbsoluteError(0, _err.absoluteerrorrange**2)
maxdenom = int(1 / werr.absoluteerrorrange) * 2
a = rational.Integer(1)
b = (1 / sqrt(rational.Rational(2), werr)).trim(maxdenom)
t = rational.Rational(1, 4)
x = 1
while not _err.nearlyEqual(a, b):
a, b, c = (a + b) / 2, sqrt(a * b, werr).trim(maxdenom), (b - a)**2 / 4
t -= x * c
x *= 2
return (a + b)**2 / (t * 4)
def _convertToRational(x):
"""
Convert to rational from:
* int,
* long, or
* float.
A complex object cannot be converted and raise TypeError.
"""
if isinstance(x, float):
retval = +rational.Rational(long(math.frexp(x)[0] * 2**53), 2
**(53 - math.frexp(x)[1]))
elif isinstance(x, (int, long)):
retval = rational.Integer(x)
elif isinstance(x, complex):
raise TypeError, "The real module cannot handle %s. Please use imaginary module." % x
else:
# fall back
retval = rational.Rational(x)
return retval
def sin(x, err=defaultError):
"""
sin(x [,err]) returns the sine of x.
"""
if not isinstance(err, defaultError.__class__) or err <= defaultError:
rx = rational.Rational(x)
sign = rational.Rational(1)
# sin(-x) = -sin(x)
if rx < 0:
sign = -sign
rx = -rx
# sin(x + 2 * pi) = sin(x)
if rx >= 2 * pi:
rx -= floor(rx / (pi * 2)) * (pi * 2)
# sin(x + pi) = -sin(x)
if rx >= pi:
rx -= pi
sign = -sign
# sin(x) = sin(pi - x)
if rx > pi / 2:
rx = pi - rx
# sin(0) = 0 is a special case which must not be computed with series.
if rx == 0:
return rational.Rational(0)
# sin(x) = cos(pi/2 - x) (pi/2 >= x > 4/pi)
if rx > pi / 4:
if rx == pi / 3:
retval = sqrt(3) / 2
else:
retval = _cosTaylor(pi / 2 - rx, err)
elif rx == pi / 4:
retval = 1 / sqrt(2)
elif rx == pi / 6:
retval = rational.Rational(1, 2)
else:
retval = _sinTaylor(rx, err)
if retval > 1:
retval = rational.Integer(1)
retval *= sign
else:
retval = rational.Rational(math.sin(x))
return retval
def cos(x, err=defaultError):
"""
cos(x [,err]) returns the cosine of x.
"""
if err <= defaultError:
rx = rational.Rational(x)
sign = rational.Rational(1)
# cos(-x) = cos(x)
if rx < 0:
rx = -rx
# cos(x + 2 * pi) = cos(x)
if rx > 2 * pi:
rx -= floor(rx / (pi * 2)) * (pi * 2)
# cos(x + pi) = -cos(x)
if rx > pi:
rx -= pi
sign = -sign
# cos(x) = -cos(pi - x)
if rx > pi / 2:
rx = pi - rx
sign = -sign
# cos(x) = sin(pi/2 - x) (pi/2 >= x > 4/pi)
if rx > pi / 4:
if rx == pi / 3:
retval = rational.Rational(1, 2)
else:
retval = _sinTaylor(pi / 2 - rx, err)
elif rx == pi / 4:
retval = 1 / sqrt(2)
elif rx == pi / 6:
retval = sqrt(3) / 2
else:
retval = _cosTaylor(rx, err)
if retval > 1:
retval = rational.Integer(1)
retval *= sign
else:
retval = rational.Rational(math.cos(x))
return retval
def eContinuedFraction(err=defaultError):
"""
Compute the base of natural logarithm e by continued fraction expansion.
"""
if isinstance(err, RelativeError):
_err = err.absoluteerror(math.e)
else:
_err = err
ipart = rational.Integer(2)
fpart_old = rational.Rational(1, 1)
fpart = rational.Rational(2, 3)
i = 4
while not _err.nearlyEqual(fpart_old, fpart):
fpart, fpart_old = rational.Rational(
fpart.numerator + fpart_old.numerator,
fpart.denominator + fpart_old.denominator), fpart
fpart, fpart_old = rational.Rational(
fpart.numerator + fpart_old.numerator,
fpart.denominator + fpart_old.denominator), fpart
fpart, fpart_old = rational.Rational(
fpart.numerator * i + fpart_old.numerator,
fpart.denominator * i + fpart_old.denominator), fpart
i += 2
return ipart + fpart
def minimumNonNegative(self):
"""
Return the smallest non-negative representative element of the
residue class.
"""
return rational.Integer(self.n % self.m)