chaospy.expansion.chebyshev_2

chaospy.expansion.chebyshev_2(order, lower=- 1, upper=1, physicist=False, normed=False, retall=False)

Chebyshev polynomials of the second kind.

Args:

order (int):

The quadrature order.

lower (float):

Lower bound for the integration interval.

upper (float):

Upper bound for the integration interval.

physicist (bool):

Use physicist weights instead of probabilist.

Returns:

(numpoly.ndpoly, numpy.ndarray):

Chebyshev polynomial expansion. Norms of the orthogonal expansion on the form E(orth**2, dist).

Examples:

>>> polynomials, norms = chaospy.expansion.chebyshev_2(4, retall=True)
>>> polynomials
polynomial([1.0, q0, q0**2-0.25, q0**3-0.5*q0, q0**4-0.75*q0**2+0.0625])
>>> norms
array([1.        , 0.25      , 0.0625    , 0.015625  , 0.00390625])
>>> chaospy.expansion.chebyshev_2(3, physicist=True)
polynomial([1.0, 2.0*q0, 4.0*q0**2-0.5, 8.0*q0**3-2.0*q0])
>>> chaospy.expansion.chebyshev_2(3, lower=0.5, upper=1.5, normed=True).round(3)
polynomial([1.0, 4.0*q0-4.0, 16.0*q0**2-32.0*q0+15.0,
            64.0*q0**3-192.0*q0**2+184.0*q0-56.0])