def bernstein(v, n):
'''
Returns the Bernstein polynomial
:math:`B_{v,n}(x) = \\binom{n}{v}x^v(1 - x)^{n - v}`
:raises: :exc:`ValueError` if *v* or *n* are negative or *v* is greater than *n*
:rtype: :class:`pypol.Polynomial`
**Examples**
::
>>> bernstein(0, 0)
+ 1
>>> bernstein(0, 1)
- x + 1
>>> bernstein(0, 2)
+ x^2 - 2x + 1
>>> bernstein(1, 2)
- 2x^2 + 2x
>>> bernstein(-1, 2)
Traceback (most recent call last):
File "<pyshell#5>", line 1, in <module>
bernstein(-1, 2)
File "series.py", line 897, in bernstein
raise ValueError('Bernstein polynomials only defined for v >= 0 and n >= 0')
ValueError: Bernstein polynomials only defined for v >= 0 and n >= 0
>>> bernstein(3, 2)
Traceback (most recent call last):
File "<pyshell#6>", line 1, in <module>
bernstein(3, 2)
File "series.py", line 899, in bernstein
raise ValueError('v cannot be greater than n')
ValueError: v cannot be greater than n
>>> bernstein(3, 6)
- 20x^6 + 60x^5 - 60x^4 + 20x^3
>>> bernstein(13, 16)
- 560x^16 + 1680x^15 - 1680x^14 + 560x^13
>>> bernstein(18, 19)
- 19x^19 + 19x^18
**References**
`MathWorld <http://mathworld.wolfram.com/BernsteinPolynomial.html>`_
'''
if v < 0 or n < 0:
raise ValueError('Bernstein polynomials only defined for v >= 0 and n >= 0')
if v > n:
raise ValueError('v cannot be greater than n')
if not v and not n:
return ONE
if v == n:
return x ** v
if n == (v + 1):
return -n*x**n + n*x**v
return bin_coeff(n, v) * x**v * (ONE - x) ** (n - v)
def testBinCoeff(self):
py.test.raises(ValueError, lambda: funcs.bin_coeff(1, 2))
assert funcs.bin_coeff(2, 1) == 2
assert funcs.bin_coeff(4, 3) == 4
assert funcs.bin_coeff(12, 11) == 12
assert funcs.bin_coeff(10, 3) == 120
assert funcs.bin_coeff(20, 15) == 15504
assert funcs.bin_coeff(49, 8) == 450978066
def _sum(n):
return sum((- 1) ** k * bin_coeff(n, k) * (x + k) ** m for k in xrange(n + 1))
def b_c3():
return bin_coeff(m + 3, m)
def b_c(j):
return bin_coeff(m + 3, m - 6 * j)
def _sum(k):
return sum((-1) ** v * fractions.Fraction.from_float(bin_coeff(k, v)) * fractions.Fraction(v ** m, k + 1) for v in xrange(k + 1))
