Fast measure modification of orthogonal polynomials via matrices with displacement structure

TL;DR

This paper introduces fast algorithms for modifying orthogonal polynomials using matrices with displacement structure, significantly reducing computational complexity in measure modification problems.

Contribution

It reveals the displacement structure of Gram matrices in measure modification, enabling efficient algorithms for connection coefficients and Cholesky factorizations.

Findings

Reduced complexity from O(n^3) to O(n^2) for general cases.

Further reduced complexity to O(b n) for banded matrices.

Hierarchical low-rank structure observed in modified Chebyshev Gram matrices.

Abstract

It is well known that matrices with low Hessenberg-structured displacement rank enjoy fast algorithms for certain matrix factorizations. We show how $n\times n$ principal finite sections of the Gram matrix for the orthogonal polynomial measure modification problem has such a displacement structure, unlocking a collection of fast algorithms for computing connection coefficients (as the upper-triangular Cholesky factor) between a known orthogonal polynomial family and the modified family. In general, the ${\cal O}(n^3)$ complexity is reduced to ${\cal O}(n^2)$, and if the symmetric Gram matrix has upper and lower bandwidth b, then the ${\cal O}(b^2n)$ complexity for a banded Cholesky factorization is reduced to ${\cal O}(b n)$. In the case of modified Chebyshev polynomials, we show that the Gram matrix is a symmetric Toeplitz-plus-Hankel matrix, and if the modified Chebyshev moments decay algebraically, then a hierarchical off-diagonal low-rank structure is observed in the Gram matrix, enabling a further reduction in the complexity of an approximate Cholesky factorization powered by randomized numerical linear algebra.