arXiv subjects

Raf Vandebril

Publications and source records attributed to Raf Vandebril.

At least 19 recordsLinked to original sources

arXiv2026

Updating and Downdating the Recurrences of Discrete Multiple Orthogonal Polynomials on the Real Line

We study multiple orthogonal polynomials with orthogonality defined by multiple positive discrete measures on the real line. Focusing on type~I and type~II multiple orthogonal polynomials and their step-line recurrence relations, we consider the reconstruction of the associated banded upper Hessenberg recurrence matrix from given nodes and weights of the measures. We propose efficient updating and downdating procedures that modify an existing recurrence matrix when nodes are added to or removed from the discrete measures. The updating strategy is based on solving an inverse eigenvalue problem, while the downdating procedure relies on an eigenvalue deflation technique inspired by a QR-type of algorithm. Numerical experiments confirm the stability and efficiency of the proposed approaches.

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arXiv2026

Fast computation of eigenvalues of periodic CMV matrices

Periodic CMV matrices are unitary matrices that can be specified by $O(n)$ data. Their eigenvalues can be computed by standard methods, storing them as conventional matrices (using $O(n^{2})$ data) in $O(n^{3})$ time. Here a fast method that computes the eigenvalues in $O(n^{2})$ time (using $O(n)$ data) is presented.

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arXiv2026

Block Krylov subspaces and orthogonal matrix polynomials: a structural correspondence with applications to unitary matrices

We study the connection between block Krylov subspaces and matrix orthogonal functions. Under a no-deflation assumption, we show that polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations. The same correspondence holds for rational block Krylov subspaces and matrix-valued rational functions, and in the extended Krylov setting this leads naturally to Laurent matrix polynomials. When the matrix $A$ is normal, we prove that the induced inner product admits a representation in terms of a discrete spectral matrix measure, extending a classical result for Hermitian matrices. In the unitary case, where the measure is supported on the unit circle, this connection allows us to transfer the Szeg\H{o} recurrence for orthogonal matrix polynomials and the CMV framework for Laurent matrix polynomials to the block Krylov setting, yielding efficient procedures for the orthogonalization of polynomial and extended block Krylov subspaces.

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arXiv2025

ParaSLRF: A High Performance Rational Filter Method for Solving Large Scale Eigenvalue Problems

In \emph{Wang et al., A Shifted Laplace Rational Filter for Large-Scale Eigenvalue Problems}, the SLRF method was proposed to compute all eigenvalues of a symmetric definite generalized eigenvalue problem lying in an interval on the real positive axis. The current paper discusses a parallel implementation of this method, abbreviated as ParaSLRF. The parallelization consists of two levels: (1) on the highest level, the application of the rational filter to the various vectors is partitioned among groups of processors; (2) within each group, every linear system is solved in parallel. In ParaSLRF, the linear systems are solved by iterative methods instead of direct ones, in contrast to other rational filter methods, such as, PFEAST. Because of the specific selection of poles in ParaSLRF, the computational cost of solving the associated linear systems for each pole, is almost the same. This intrinsically leads to a better load balance between each group of resources, and reduces waiting times of processes. We show numerical experiments from finite element models of mechanical vibrations, and show a detailed parallel performance analysis. ParaSLRF shows the best parallel efficiency, compared to other rational filter methods based on quadrature rules for contour integration. To further improve performance, the converged eigenpairs are locked, and a good initial guess of iterative linear solver is proposed. These enhancements of ParaSLRF show good two-level strong scalability and excellent load balance in our experiments.

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arXiv2025

Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials

In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problemshave related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.

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arXiv2025

The RQR algorithm

Pole-swapping algorithms, generalizations of bulge-chasing algorithms, have been shown to be a viable alternative to the bulge-chasing QZ algorithm for solving the generalized eigenvalue problem for a matrix pencil A - λB. It is natural to try to devise a pole-swapping algorithm that solves the standard eigenvalue problem for a single matrix A. This paper introduces such an algorithm and shows that it is competitive with Francis's bulge-chasing QR algorithm.

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arXiv2025

Manifold-valued function approximation from multiple tangent spaces

Approximating a manifold-valued function from samples of input-output pairs consists of modeling the relationship between an input from a vector space and an output on a Riemannian manifold. We propose a function approximation method that leverages and unifies two prior techniques: (i) approximating a pullback to the tangent space, and (ii) the Riemannian moving least squares method. The core idea of the new scheme is to combine pullbacks to multiple tangent spaces with a weighted Fréchet mean. The effectiveness of this approach is illustrated with numerical experiments on model problems from parametric model order reduction.

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arXiv2025

A shifted Laplace rational filter for large-scale eigenvalue problems

We present a rational filter for computing all eigenvalues of a symmetric definite eigenvalue problem lying in an interval on the real axis. The linear systems arising from the filter embedded in the subspace iteration framework, are solved via a preconditioned Krylov method. The choice of the poles of the filter is based on two criteria. On the one hand, the filter should enhance the eigenvalues in the interval of interest, which suggests that the poles should be chosen close to or in the interval. On the other hand, the choice of poles has an important impact on the convergence speed of the iterative method. For the solution of problems arising from vibrations, the two criteria contradict each other, since fast convergence of the eigensolver requires poles to be in or close to the interval, whereas the iterative linear system solver becomes cheaper when the poles lie further away from the eigenvalues. In the paper, we propose a selection of poles inspired by the shifted Laplace preconditioner for the Helmholtz equation. We show numerical experiments from finite element models of vibrations. We compare the shifted Laplace rational filter with rational filters based on quadrature rules for contour integration.

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arXiv2025

On computing the zeros of a class of Sobolev orthogonal polynomials

A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre polynomials. The theoretical study of the asymptotic distribution of the spectrum of these polynomials is an active research topic. In this article we do not contribute to the theory, but provide a practical method to contribute to further and better understanding of the asymptotic behavior. The polynomials under consideration fit into the class of Sobolev orthogonal polynomials, satisfying a four--term recurrence relation. This allows computing the roots via a generalized eigenvalue problem. After condition enhancing similarity transformations, the problem is transformed into the computation of the eigenvalues of a comrade matrix, which is a symmetric tridiagonal modified by a rank--one matrix. The eigenvalues are then retrieved by relying on an existing structured rank based fast algorithm. Numerical examples are reported studying the accuracy, stability and conforming the efficiency for various parameter settings of the proposed approach.

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arXiv2025

Constructing Sobolev orthonormal rational functions via an updating procedure

In this paper, we generate the recursion coefficients for rational functions with prescribed poles that are orthonormal with respect to a continuous Sobolev inner product. Using a rational Gauss quadrature rule, the inner product can be discretized, thus allowing a linear algebraic approach. The presented approach involves reformulating the problem as an inverse eigenvalue problem involving a Hessenberg pencil, where the pencil will contain the recursion coefficients that generate the sequence of Sobolev orthogonal rational functions. This reformulation is based on the connection between Sobolev orthonormal rational functions and the orthonormal bases for rational Krylov subspaces generated by a Jordan-like matrix. An updating procedure, introducing the nodes of the inner product one after the other, is proposed and the performance is examined through some numerical examples.

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arXiv2025

Algorithms for Parallel Shared-Memory Sparse Matrix-Vector Multiplication on Unstructured Matrices

The sparse matrix-vector (SpMV) multiplication is an important computational kernel, but it is notoriously difficult to execute efficiently. This paper investigates algorithm performance for unstructured sparse matrices, which are more common than ever because of the trend towards large-scale data collection. The development of an SpMV multiplication algorithm for this type of data is hard due to two factors. First, parallel load balancing issues arise because of the unpredictable nonzero structure. Secondly, SpMV multiplication algorithms are inevitably memory-bound because the sparsity causes a low arithmetic intensity. Three state-of-the-art algorithms for parallel SpMV multiplication on shared-memory systems are discussed. Six new hybrid algorithms are developed which combine optimization techniques of the current algorithms. These techniques include parallelization strategies, storage formats, and nonzero orderings. A modern and high-performance implementation of all discussed algorithms is provided as open-source software. Using this implementation the algorithms are compared. Furthermore, SpMV multiplication algorithms require the matrix to be stored in a specific storage format. Therefore, the conversion time between these storage formats is also analyzed. Both tests are performed for multiple unstructured sparse matrices on different machines: two multi-CPU and two single-CPU architectures. We show that one of the newly developed algorithms outperforms the current state-of-the-art by 19% on one of the multi-CPU architectures. When taking conversion time into consideration, we show that 472 SpMV multiplications are needed to cover the cost of converting to a new storage format for one of the hybrid algorithms on a multi-CPU machine.

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arXiv2024

An Arnoldi-based approach to polynomial and rational least squares problems

In this research, we solve polynomial, Sobolev polynomial, rational, and Sobolev rational least squares problems. Although the increase in the approximation degree allows us to fit the data better in attacking least squares problems, the ill-conditioning of the coefficient matrix fuels the dramatic decrease in the accuracy of the approximation at higher degrees. To overcome this drawback, we first show that the column space of the coefficient matrix is equivalent to a Krylov subspace. Then the connection between orthogonal polynomials or rational functions and orthogonal bases for Krylov subspaces in order to exploit Krylov subspace methods like Arnoldi orthogonalization is established. Furthermore, some examples are provided to illustrate the theory and the performance of the proposed approach.

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arXiv2024

Approximating maps into manifolds with lower curvature bounds

Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for functions into linear spaces in such a way that we can upper bound the forward error in terms of a lower bound on the manifold's sectional curvature. Furthermore, when the sectional curvature is nonnegative, such as for compact Lie groups, the error is guaranteed to not be worse than in the linear case. We implement the algorithm in a Julia package ManiFactor.jl and apply it to two example problems.

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arXiv2023

A new deflation criterion for the QZ algorithm

The QZ algorithm computes the Schur form of a matrix pencil. It is an iterative algorithm and at some point, it must decide that an eigenvalue has converged and move on with another one. Choosing a criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might be inaccurate. Additionally, the criterion should not be computationally expensive to evaluate. This paper introduces a new criterion based on the size of and the gap between the eigenvalues. This is similar to the work of Ahues and Tissuer for the QR algorithm. Theoretical arguments and numerical experiments suggest that it outperforms the most popular criteria in terms of accuracy. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.

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arXiv2023

Algorithms for Modifying Recurrence Relations of Orthogonal Polynomial and Rational Functions when Changing the Discrete Inner Product

Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed. Moreover, also typically the inner product is changed to a discrete inner product, which is the finite sum of weighted functions evaluated in specific nodes. For particular applications it is beneficial to have an efficient procedure to update the recurrence relations when adding or removing nodes from the inner product. The construction of the recurrence relations is equivalent to computing a structured matrix (polynomial) or pencil (rational) having prescribed spectral properties. Hence the solution of this problem is often referred to as solving an Inverse Eigenvalue Problem. In Van Buggenhout et al. (2022) we proposed updating techniques to add nodes to the inner product while efficiently updating the recurrences. To complete this study we present in this article manners to efficiently downdate the recurrences when removing nodes from the inner product. The link between removing nodes and the QR algorithm to deflate eigenvalues is exploited to develop efficient algorithms. We will base ourselves on the perfect shift strategy and develop algorithms, both for the polynomial case and the rational function setting. Numerical experiments validate our approach.

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arXiv2023

Parallel two-stage reduction to Hessenberg-triangular form

We present a two-stage algorithm for the parallel reduction of a pencil to Hessenberg-triangular form. Traditionally, two-stage Hessenberg-triangular reduction algorithms achieve high performance in the first stage, but struggle to achieve high performance in the second stage. Our algorithm extends techniques described by Karlsson et al. to also achieve high performance in the second stage. Experiments in a shared memory environment demonstrate that the algorithm can outperform state-of-the-art implementations.

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arXiv2022

The seriation problem in the presence of a double Fiedler value

Seriation is a problem consisting of seeking the best enumeration order of a set of units whose interrelationship is described by a bipartite graph, that is, a graph whose nodes are partitioned in two sets and arcs only connect nodes in different groups. An algorithm for spectral seriation based on the use of the Fiedler vector of the Laplacian matrix associated to the problem was developed by Atkins et al., under the assumption that the Fiedler value is simple. In this paper, we analyze the case in which the Fiedler value of the Laplacian is not simple, discuss its effect on the set of the admissible solutions, and study possible approaches to actually perform the computation. Examples and numerical experiments illustrate the effectiveness of the proposed methods.

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arXiv2021

Generation of orthogonal rational functions by procedures for structured matrices

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

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