chaospy.expansion.stieltjes

chaospy.expansion.stieltjes(order, dist, normed=False, graded=True, reverse=True, retall=False, cross_truncation=1.0)

Create orthogonal polynomial expansion from three terms recurrence formula.

Args:

order (int):

Order of polynomial expansion.

dist (Distribution):

Distribution space where polynomials are orthogonal If dist.ttr exists, it will be used. Must be stochastically independent.

normed (bool):

If True orthonormal polynomials will be used.

graded (bool):

Graded sorting, meaning the indices are always sorted by the index sum. E.g. q0**2*q1**2*q2**2 has an exponent sum of 6, and will therefore be consider larger than both q0**2*q1*q2, q0*q1**2*q2 and q0*q1*q2**2, which all have exponent sum of 5.

reverse (bool):

Reverse lexicographical sorting meaning that q0*q1**3 is considered bigger than q0**3*q1, instead of the opposite.

retall (bool):

If true return numerical stabilized norms as well. Roughly the same as cp.E(orth**2, dist).

cross_truncation (float):

Use hyperbolic cross truncation scheme to reduce the number of terms in expansion. only include terms where the exponents K satisfied the equation order >= sum(K**(1/cross_truncation))**cross_truncation.

Returns:

(numpoly.ndpoly, numpy.ndarray):

Orthogonal polynomial expansion. Norms of the orthogonal expansion on the form E(orth**2, dist). Calculated using recurrence coefficients for stability.

Examples:

>>> distribution = chaospy.J(chaospy.Normal(), chaospy.Normal())
>>> polynomials, norms = chaospy.expansion.stieltjes(2, distribution, retall=True)
>>> polynomials.round(10)
polynomial([1.0, q1, q0, q1**2-1.0, q0*q1, q0**2-1.0])
>>> norms.round(10)
array([1., 1., 1., 2., 1., 2.])
>>> polynomials = chaospy.expansion.stieltjes(2, distribution, normed=True)
>>> polynomials.round(3)
polynomial([1.0, q1, q0, 0.707*q1**2-0.707, q0*q1, 0.707*q0**2-0.707])