orthax.recurrence.Jacobi
- class orthax.recurrence.Jacobi(alpha: Array | ndarray | bool | number | bool | int | float | complex, beta: Array | ndarray | bool | number | bool | int | float | complex, scale: str = 'standard')Source
Recurrence relation for Jacobi polynomials \(P^{(\alpha, \beta)}_n(x)\)
Jacobi polynomials are orthogonal on the interval (-1, 1) with the weight function \(w(x) = (1-x)^\alpha (1+x)^\beta\)
- Parameters:
alpha (float > -1) – Hyperparameters α, β.
beta (float > -1) – Hyperparameters α, β.
scale ({"standard", "monic", "normalized"}) – “standard” corresponds to the common scaling found in textbooks such as Abramowitz & Stegun. “monic” scales them such that the leading coefficient is 1. “normalized” scales them to have a weighted norm of 1.
- __init__(alpha: Array | ndarray | bool | number | bool | int | float | complex, beta: Array | ndarray | bool | number | bool | int | float | complex, scale: str = 'standard')Source
Methods
__init__(alpha, beta[, scale])a(k)a coefficients of the monic three term recurrence relation.
b(k)b coefficients of the monic three term recurrence relation.
g(k)Weighted norm of the kth monic orthogonal polynomial.
m(k)Coefficient of x**k in the kth polynomial in the desired normalization.
weight(x)Weight function defining inner product.
Attributes
domainLower and upper bounds for inner product defining orthogonality.
alphabeta