SzegoJacobiParameters - crowlogic/arb4j GitHub Wiki

The computation of Szegő-Jacobi parameters involves methods that translate moments of a measure into parameters defining orthogonal polynomials relative to that measure. Two notable algorithms for this computation are the Golub-Welsch algorithm and the Stieltjes procedure.

Golub-Welsch Algorithm

This algorithm efficiently computes the eigenvalues and eigenvectors of Jacobi matrices, which correspond to the zeros of orthogonal polynomials and their weights in the Gaussian quadrature formula, respectively. It involves constructing a tridiagonal matrix from the Szegő-Jacobi parameters, then using eigenvalue solvers to find its eigenvalues and eigenvectors.

  1. Construction: Construct a symmetric tridiagonal matrix $J$ using the recurrence coefficients (Szegő-Jacobi parameters) $a_n$ and $b_n$ as the entries, where $a_n$ are the off-diagonal elements and $b_n$ the diagonal elements.

  2. Eigenvalue Problem: Solve the eigenvalue problem $Jv = \lambda v$, where $v$ are the eigenvectors and $\lambda$ the eigenvalues.

  3. Orthogonal Polynomials Zeros and Weights: The eigenvalues $\lambda$ represent the zeros of the orthogonal polynomials. The first element of the normalized eigenvectors, squared and multiplied by the weight of the corresponding orthogonal polynomial, gives the weights for the Gaussian quadrature formula.

Stieltjes Procedure

The Stieltjes procedure is a method for generating orthogonal polynomials and their recurrence coefficients from a given measure. It relies on the Gram-Schmidt orthogonalization process applied to the sequence of monomials with respect to the inner product defined by the measure.

  1. Orthogonalization: Starting with a sequence of monomials $1, x, x^2, \ldots$, apply the Gram-Schmidt process using the inner product

$$\langle f, g \rangle = \int f(x)g(x) d\mu(x)$$

where $d\mu(x)$ is the measure with respect to which the polynomials are orthogonal.

  1. Recurrence Coefficients: This process inherently produces the recurrence coefficients $a_n$ and $b_n$, which are used to construct orthogonal polynomials through the three-term recurrence relation

$$P_{n+1}(x) = (x - a_n)P_n(x) - b_nP_{n-1}(x)$$

Both methods are fundamental in computational mathematics, specifically in numerical analysis for constructing orthogonal polynomials, which are crucial for various applications including quadrature (numerical integration) and solving differential equations. The choice between the Golub-Welsch algorithm and the Stieltjes procedure depends on the specific requirements of the problem at hand, such as the availability of moments of the measure or the need for numerical stability and efficiency.