def hermite_prob(n):
'''
Returns the *n-th* probabilistic Hermite polynomial, that is a polynomial of degree *n*.
:raises: :exc:`ValueError` if *n* is negative
:rtype: :class:`pypol.Polynomial`
**Examples**
::
>>> hermite_prob(0)
+ 1
>>> hermite_prob(1)
+ x
>>> hermite_prob(2)
+ x^2 - 1
>>> hermite_prob(4)
+ x^4 - 6x^2 + 3
>>> hermite_prob(45)
+ x^45 - 990x^43 + .. cut .. + 390756386568644372393927184375x^5 - 186074469794592558282822468750x^3 + 25373791335626257947657609375x
.. versionadded:: 0.3
'''
if n < 0:
raise ValueError('Hermite polynomials (probabilistic) only defined for n >= 0')
if n == 0:
return ONE
if n == 1:
return x
p = [x]
for _ in xrange(n - 1):
p.append(p[-1] * x - polyder(p[-1]))
return p[-1]
def hermite_phys(n):
'''
Returns the *n-th* Hermite polynomial (physicist).
:raises: :exc:`ValueError` if *n* is negative
:rtype: :class:`pypol.Polynomial`
**Examples**
::
>>> hermite_phys(0)
+ 1
>>> hermite_phys(1)
+ 2x
>>> hermite_phys(2)
+ 4x^2 - 2
>>> hermite_phys(3)
+ 8x^3 - 12x
>>> hermite_phys(4)
+ 16x^4 - 48x^2 + 12
>>> hermite_phys(9)
+ 512x^9 - 9216x^7 + 48384x^5 - 80640x^3 + 30240x
>>> hermite_phys(11)
+ 2048x^11 - 56320x^9 + 506880x^7 - 1774080x^5 + 2217600x^3 - 665280x
.. versionadded:: 0.3
'''
if n < 0:
raise ValueError('Hermite polynomials (physicist) only defined for n >= 0')
if n == 0:
return ONE
p = [ONE]
for _ in xrange(n):
p.append((p[-1] * x * 2) - polyder(p[-1]))
return p[-1]
def testPolyder(self):
p = pypol.poly1d([1]*4)
assert pypol.poly1d([3, 2, 1]) == funcs.polyder(p)
assert pypol.poly1d([6, 2]) == funcs.polyder(p, 2)
assert pypol.poly1d([6]) == funcs.polyder(p, 3)