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