Weights and Recurrence Helpers
PolyChaos.build_w_beta — Function
build_w_beta(alpha, beta)Return a beta-density closure on (0, 1).
Arguments
alpha,beta: beta shape parameters.
PolyChaos.build_w_gamma — Function
build_w_gamma(alpha)Return a unit-rate gamma-density closure.
Arguments
alpha: positive gamma shape parameter.
PolyChaos.build_w_genhermite — Function
build_w_genhermite(mu)Return the generalized-Hermite weight closure.
Arguments
mu: exponent parameter used byw_genhermite.
PolyChaos.build_w_genlaguerre — Function
build_w_genlaguerre(a)Return the generalized-Laguerre weight closure.
Arguments
a: exponent parameter used by the returned function.
PolyChaos.build_w_jacobi — Function
build_w_jacobi(a, b)Return the Jacobi weight closure t -> w_jacobi(t, a, b).
Arguments
a,b: exponents of(1 - t)and(1 + t).
PolyChaos.build_w_meixner_pollaczek — Function
build_w_meixner_pollaczek(lambda, phi)Return a Meixner-Pollaczek weight closure.
Arguments
lambda: positive family parameter.phi: angle parameter.
PolyChaos.rm_chebyshev1 — Function
rm_chebyshev1(N)Return the first N monic recurrence coefficients for first-kind Chebyshev polynomials.
Arguments
N: nonnegative recurrence coefficient count.
Returns
The pair (alpha, beta) of recurrence coefficient vectors.
PolyChaos.w_gaussian — Function
w_gaussian(t)Evaluate the standard normal probability density at t.
Arguments
t: real evaluation point.
Returns
The standard normal density evaluated at t.
PolyChaos.w_genhermite — Function
w_genhermite(t, mu)Evaluate abs(t)^(2mu) * exp(-t^2).
Arguments
t: real evaluation point.mu: generalized-Hermite parameter.
PolyChaos.w_hermite — Function
w_hermite(t)Evaluate the Hermite weight exp(-t^2).
Arguments
t: real evaluation point.
PolyChaos.w_jacobi — Function
w_jacobi(t, a, b)Evaluate the unnormalized Jacobi weight (1 - t)^a * (1 + t)^b on [-1, 1].
Arguments
t: evaluation point.a,b: Jacobi shape parameters.
PolyChaos.w_laguerre — Function
w_laguerre(t)Evaluate the Laguerre weight exp(-t) on [0, Inf).
Arguments
t: nonnegative evaluation point.
PolyChaos.w_legendre — Function
w_legendre(t)Evaluate the unit Legendre weight on [-1, 1].
Arguments
t: evaluation point in[-1, 1].
Throws DomainError outside the support.
PolyChaos.w_logistic — Function
w_logistic(t)Evaluate the standard logistic probability density at t.
Arguments
t: real evaluation point.
Returns
The standard logistic density evaluated at t.
PolyChaos.w_meixner_pollaczek — Function
w_meixner_pollaczek(t, lambda, phi)Evaluate the Meixner-Pollaczek weight.
Arguments
t: real evaluation point.lambda: positive family parameter.phi: angle parameter.
PolyChaos.w_uniform01 — Function
w_uniform01(t)Evaluate the unit uniform density on (0, 1).
Arguments
t: evaluation point in(0, 1).
Returns
The unit uniform density, or a DomainError outside the support.
PolyChaos.w_uniform_11 — Function
w_uniform_11(t)Evaluate the uniform density 1 / 2 on (-1, 1).
Arguments
t: evaluation point in(-1, 1).
Returns
The uniform density, or a DomainError outside the support.