arXiv subjects

Elias Jarlebring

Publications and source records attributed to Elias Jarlebring.

At least 19 recordsLinked to original sources

arXiv2026

Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions. (Updated to reflect the status of the open problems as of August 20, 2026.)

math.NA↗

arXiv2026

Recursive expansion of the matrix step function using polynomials of degree eight

We consider the problem of efficiently computing the matrix step function of a large dense symmetric matrix. To this end, we introduce a recursive polynomial expansion method in which a composite polynomial of high degree is built recursively from component polynomials of degree eight. The component polynomial used in each iteration is designed to achieve strong amplification of the spectral gap across the step while favorably positioning the updated gap for subsequent iterations. A key ingredient is a novel evaluation scheme for arbitrary matrix polynomials of degree exactly eight requiring only three matrix-matrix multiplications and three matrices in memory. This scheme makes available a substantially larger class of component polynomials than previously possible within a three-multiplication budget, thereby expanding the class of composite polynomials that can be generated. Together with our polynomial selection strategy, this leads to a significant and consistent reduction in the number of matrix-matrix multiplications required to compute the matrix step function compared to existing recursive expansion methods.

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arXiv2026

Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.

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arXiv2025

From eigenvector nonlinearities with quadratic structure to eigenvalue nonlinearities with algebraic structure

Over the past decades, transformations between different classes of eigenvalue problems have played a central role in the development of numerical methods for eigenvalue computations. One of the most well-known and successful examples of this is the companion linearization for polynomial eigenvalue problems. In this paper, we construct a transformation that equivalently reframes a specific type of eigenvalue problem with eigenvector nonlinearities (NEPv) into an eigenvalue problem with eigenvalue nonlinearities (NEP). The NEPv class considered consists of nonlinearities expressed as sums of products of matrices and scalar functions, where the scalar functions depend nonlinearly on the eigenvector. Our transformation defines scalar eigenvalue nonlinearities through a polynomial system, resulting in NEP nonlinearities of algebraic type. We propose methods to solve the polynomial system, one of which involves a multiparameter eigenvalue problem (MEP). We adapt well-established NEP solvers to this setting, with the most effective strategy being a combination of deflation and a locally quadratically convergent iterative method. The efficiency and properties of the approach are illustrated by solving a problem related to a modification of a Gross-Pitaevskii equation (GPE). The simulations are reproducible and publicly available.

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arXiv2025

The Polynomial Set Associated with a Fixed Number of Matrix-Matrix Multiplications

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $Π_{2^{m}}^*$ represent the set of polynomials computable with $m$ matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of $Π_{2^{m}}^*$ and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by $m^2$, which is subsequently proven to be sharp, i.e., $\dim(Π_{2^m}^*)=m^2$. Consequently, we have identified a parameterization that, to the best of our knowledge, is the first minimal parameterization. We also conduct a study using computational tools from algebraic geometry to determine the largest degree $d$ such that all polynomials of that degree belong to $Π_{2^{m}}^*$, or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial.

math.NA↗

arXiv2025

Decoding complexity: how machine learning is redefining scientific discovery

As modern scientific instruments generate vast amounts of data and the volume of information in the scientific literature continues to grow, machine learning (ML) has become an essential tool for organising, analysing, and interpreting these complex datasets. This paper explores the transformative role of ML in accelerating breakthroughs across a range of scientific disciplines. By presenting key examples -- such as brain mapping and exoplanet detection -- we demonstrate how ML is reshaping scientific research. We also explore different scenarios where different levels of knowledge of the underlying phenomenon are available, identifying strategies to overcome limitations and unlock the full potential of ML. Despite its advances, the growing reliance on ML poses challenges for research applications and rigorous validation of discoveries. We argue that even with these challenges, ML is poised to disrupt traditional methodologies and advance the boundaries of knowledge by enabling researchers to tackle increasingly complex problems. Thus, the scientific community can move beyond the necessary traditional oversimplifications to embrace the full complexity of natural systems, ultimately paving the way for interdisciplinary breakthroughs and innovative solutions to humanity's most pressing challenges.

cs.LG↗

arXiv2025

Deep Learning on Graphs for Mobile Network Topology Generation

Mobile networks consist of interconnected radio nodes strategically positioned across various geographical regions to provide connectivity services. The set of relations between these radio nodes, referred to as the \emph{mobile network topology}, is vital in the construction of the networking infrastructure. Typically, the connections between radio nodes and their associated cells are defined by software features that establish mobility relations (referred to as \emph{edges} in this paper) within the mobile network graph through heuristic methods. Although these approaches are efficient, they encounter significant limitations, particularly since edges can only be established prior to the installation of physical hardware. In this work, we use graph-based deep learning methods to determine mobility relations (edges), trained on radio node configuration data and reliable mobility relations set by Automatic Neighbor Relations (ANR) in stable networks. This paper focuses on measuring the accuracy and precision of different graph-based deep learning approaches applied to real-world mobile networks. We evaluated two deep learning models. Our comprehensive experiments on Telecom datasets obtained from operational Telecom Networks demonstrate the effectiveness of the graph neural network (GNN) model and multilayer perceptron. Our evaluation showed that considering graph structure improves results, which motivates the use of GNNs. Additionally, we investigated the use of heuristics to reduce the training time based on the distance between radio nodes to eliminate irrelevant cases. Our investigation showed that the use of these heuristics improved precision and accuracy considerably.

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arXiv2024

Polynomial approximations for the matrix logarithm with computation graphs

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. The main computational effort lies in matrix-matrix multiplications and left matrix division. In this work we illustrate that the number of such operations can be substantially reduced, by using a graph based representation of an efficient polynomial evaluation scheme. A technique to analyze the rounding error is proposed, and backward error analysis is adapted. We provide substantial simulations illustrating competitiveness both in terms of computation time and rounding errors.

math.NA↗

arXiv2024

Computation of leaky waves in layered structures coupled to unbounded media by exploiting multiparameter eigenvalue problems

We present a semi-analytical approach to compute quasi-guided elastic wave modes in horizontally layered structures radiating into unbounded fluid or solid media. This problem is of relevance, e.g., for the simulation of guided ultrasound in embedded plate structures or seismic waves in soil layers over an elastic half-space. We employ a semi-analytical formulation to describe the layers, thus discretizing the thickness direction by means of finite elements. For a free layer, this technique leads to a well-known quadratic eigenvalue problem for the mode shapes and corresponding horizontal wavenumbers. Rigorously incorporating the coupling conditions to account for the adjacent half-spaces gives rise to additional terms that are nonlinear in the wavenumber. We show that the resulting nonlinear eigenvalue problem can be cast in the form of a multiparameter eigenvalue problem whose solutions represent the wave numbers in the plate and in the half-spaces. The multiparameter eigenvalue problem is solved numerically using recently developed algorithms.

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arXiv2023

Preconditioned Chebyshev BiCG for parameterized linear systems

We consider the problem of approximating the solution to $A(μ) x(μ) = b$ for many different values of the parameter $μ$. Here we assume $A(μ)$ is large, sparse, and nonsingular with a nonlinear dependence on $μ$. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of $A(μ)$ on the interval $[-a,a]$, $a \in \mathbb{R}$. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation to $x(μ)$ for many different values of the parameter $μ\in [-a,a]$ simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly. The competitiveness of the algorithms are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.

math.NA↗

arXiv2022

Preconditioned infinite GMRES for parameterized linear systems

We are interested in obtaining approximate solutions to parameterized linear systems of the form $A(μ) x(μ) = b$ for many values of the parameter $μ$. Here $A(μ)$ is large, sparse, and nonsingular, with a nonlinear analytic dependence on $μ$. Our approach is based on a companion linearization for parameterized linear systems. The companion matrix is similar to the operator in the infinite Arnoldi method, and we use this to adapt the flexible GMRES setting. In this way, our method returns a function $\tilde{x}(μ)$ which is cheap to evaluate for different $μ$, and the preconditioner is applied only approximately. This novel approach leads to increased freedom to carry out the action of the operation inexactly, which provides performance improvement over the method infinite GMRES, without a loss of accuracy in general. We show that the error of our method is estimated based on the magnitude of the parameter $μ$, the inexactness of the preconditioning, and the spectrum of the linear companion matrix. Numerical examples from a finite element discretization of a Helmholtz equation with a parameterized material coefficient illustrate the competitiveness of our approach. The simulations are reproducible and publicly available online.

math.NA↗

arXiv2021

The self-consistent field iteration for p-spectral clustering

The self-consistent field (SCF) iteration, combined with its variants, is one of the most widely used algorithms in quantum chemistry. We propose a procedure to adapt the SCF iteration for the p-Laplacian eigenproblem, which is an important problem in the field of unsupervised learning. We formulate the p-Laplacian eigenproblem as a type of nonlinear eigenproblem with one eigenvector nonlinearity , which then allows us to adapt the SCF iteration for its solution after the application of suitable regularization techniques. The results of our numerical experiments confirm the viablity of our approach.

math.NA↗

arXiv2021

Computational graphs for matrix functions

Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective approach to study existing techniques, improve them, and eventually derive new ones. The accuracy of these matrix techniques can be characterized by the accuracy of their scalar counterparts, thus designing algorithms for matrix functions can be regarded as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG, in a fashion analogous to backpropagation. This paper describes GraphMatFun.jl, a Julia package that offers the means to generate and manipulate computational graphs, optimize their coefficients, and generate Julia, MATLAB, and C code to evaluate them efficiently at a matrix argument. The software also provides tools to estimate the accuracy of a graph-based algorithm and thus obtain numerically reliable methods. For the exponential, for example, using a particular form (degree-optimal) of polynomials produces implementations that in many cases are cheaper, in terms of computational cost, than the Padé-based techniques typically used in mathematical software. The optimized graphs and the corresponding generated code are available online.

math.NA↗

arXiv2021

Linearizability of eigenvector nonlinearities

We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on an equivalent multiparameter problem (MEP) that contains the exact solutions of the NEPv. Due to the characterization of MEPs in terms of a generalized eigenvalue problem this provides a direct way to compute all NEPv solutions for small problems, and it opens up the possibility to develop locally convergent iterative methods for larger problems. Moreover, the linear formulation allows us to easily determine the number of solutions of the NEPv. We propose two numerical schemes that exploit the structure of the linearization: inverse iteration and residual inverse iteration. We show how symmetry in the MEP can be used to improve reliability and reduce computational cost of both methods. Two numerical examples verify the theoretical results and a third example shows the potential of a hybrid scheme that is based on a combination of the two proposed methods.

math.NA↗

arXiv2021

Infinite GMRES for parameterized linear systems

We consider linear parameter-dependent systems $A(μ) x(μ) = b$ for many different $μ$, where $A$ is large and sparse, and depends nonlinearly on $μ$. Solving such systems individually for each $μ$ would require great computational effort. In this work we propose to compute a partial parameterization $\tilde{x} \approx x(μ)$ where $\tilde{x}(μ)$ is cheap to compute for many different $μ$. Our methods are based on the observation that a companion linearization can be formed where the dependence on $μ$ is only linear. In particular, we develop methods which combine the well-established Krylov subspace method for linear systems, GMRES, with algorithms for nonlinear eigenvalue problems (NEPs) to generate a basis for the Krylov subspace. Within this new approach, the basis matrix is constructed in three different ways, using a tensor structure and exploiting that certain problems have low-rank properties. We show convergence factor bounds obtained similarly to those for the method GMRES for linear systems. More specifically, a bound is obtained based on the magnitude of the parameter $μ$ and the spectrum of the linear companion matrix, which corresponds to the reciprocal solutions to the corresponding NEP. Numerical experiments illustrate the competitiveness of our methods for large-scale problems. The simulations are reproducible and publicly available online.

math.NA↗

arXiv2020

Implicit algorithms for eigenvector nonlinearities

We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit viewpoint. More precisely, we change the Newton update equation in a way that the next iterate does not only appear linearly in the update equation. Although, the modifications of the update equation make the methods implicit we show how corresponding iterates can be computed explicitly. Therefore we can carry out steps of the implicit method using explicit procedures. In several cases, these procedures involve a solution of standard eigenvalue problems. We propose two modifications, one of the modifications leads directly to a well-established method (the self-consistent field iteration) whereas the other method is to our knowledge new and has several attractive properties. Convergence theory is provided along with several simulations which illustrate the properties of the algorithms.

math.NA↗

arXiv2019

Nonlinearizing two-parameter eigenvalue problems

We investigate a technique to transform a linear two-parameter eigenvalue problem, into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by considering it as a (standard) generalized eigenvalue problem. We characterize the equivalence between the original and the nonlinearized problem theoretically and show how to use the transformation computationally. Special cases of the transformation can be interpreted as a reversed companion linearization for polynomial eigenvalue problems, as well as a reversed (less known) linearization technique for certain algebraic eigenvalue problems with square-root terms. Moreover, by exploiting the structure of the NEP we present algorithm specializations for NEP methods, although the technique also allows general solution methods for NEPs to be directly applied. The nonlinearization is illustrated in examples and simulations, with focus on problems where the eliminated equation is of much smaller size than the other two-parameter eigenvalue equation. This situation arises naturally in domain decomposition techniques. A general error analysis is also carried out under the assumption that a backward stable eigenvalue solver method is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.

math.NA↗

arXiv2018

A density matrix approach to the convergence of the self-consistent field iteration

In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Sufficient and almost necessary conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.

math.NA↗