In numerical tests with the Kravchuk and Hahn families, scaled calculations produced errors several orders smaller than monic calculations and showed milder growth as the recurrence systems increased. At N = 100, approximately five digits of recurrence accuracy remained, and the bi-orthonormality error was below 10−1.
The study examines how to reconstruct, update and downdate banded upper-Hessenberg recurrence matrices for type I and type II multiple orthogonal polynomials from discrete-measure nodes and weights. Its proposed procedures are presented as memory-efficient because they reuse existing recurrence relations when nodes are added or deleted, avoiding a full restart in the stated use case.
A way to reuse an existing calculation
The implementation centers on r = 2 step-line recurrences. Their coefficients define a four-banded upper-Hessenberg matrix, so the reported methods are tested in that specific setting.
To balance the bases, the scaling strategy aims to make corresponding left and right basis columns approximately equal in norm. For downdating, the algorithm isolates a selected eigenvalue with two perfect-shift RL steps, first using a left eigenvector and then a right eigenvector.
Results were judged with recurrence-relation accuracy and biorthogonality error metrics. Here, biorthogonality refers to whether the paired left and right bases preserve their intended mutual relation.
Node placement mattered in the update tests
The updating tests used the Kravchuk and Hahn families and compared Chebyshev nodes with equidistant nodes. In the node-addition experiment, the process began from a 2 × 2 solution, added 98 nodes sequentially, and averaged results over 5 independent runs.
Errors grew more slowly for Chebyshev than for equidistant nodes. At N = 100, recurrence accuracy was approximately five digits for the Chebyshev case, compared with about three digits for the equidistant case.
One diagnostic also cautioned against treating the recurrence matrix alone as a complete measure of numerical quality. Bases generated from the exact recurrence matrix gradually lost biorthogonality as degree increased, while bases generated from the approximate matrix showed comparable biorthogonality.
Repeated removals remained controlled in the tested cases
Another test started at N = 40 and removed 20 nodes sequentially. During those removals, recurrence and biorthogonality errors stayed approximately between 10−9 and 10−11 across the tested matrix sizes, with no noticeable growth.
A separate alternating test began with a 40 × 40 solution, removed the last node to produce a 39 × 39 solution, added it back, and repeated the down/update cycle 20 times.
Evidence remains limited to numerical examples
The numerical record is narrower than the headline result. The recurrence and biorthogonality metrics summarize constructed numerical examples, and the reported comparisons are approximate summaries rather than raw numerical tables; no direct runtime or memory benchmark is provided.
In the Kravchuk and Hahn examples, the weights were not nested, so the algorithm was restarted from a basic 2 × 2 solution for each N rather than tested through fully incremental updating.
The implementation focuses on r = 2 step-line recurrences, while the reported update and downdate protocols cover only the tested matrix sizes and node sets. The monic-versus-scaled comparison also reflects both the scaling strategy and the nature of sequential updating, so the contribution of scaling alone is not isolated.
The updating discussion assumes that no breakdown occurs. For downdating, attention is restricted to unreduced Hessenberg matrices, and small eliminator denominators can cause instability. A thorough stability analysis remains future work.
The work is identified as arXiv:2608.25758v1, dated 26 August 2026. Its results describe a preprint’s numerical experiments, not a uniform stability guarantee beyond the reported setting.
Paper data and sources
Original title: Updating and Downdating the Recurrences of Discrete Multiple Orthogonal Polynomials on the Real Line
Authors: Amin Faghih, Marc Van Barel, Raf Vandebril
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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