arXiv subjects

Marcus Webb

Publications and source records attributed to Marcus Webb.

At least 19 recordsLinked to original sources

arXiv2026

Computing accurate singular vectors and eigenvectors using mixed-precision Jacobi algorithms

Mixed-precision variants of the Jacobi algorithm for symmetric positive definite eigenproblems and the one-sided Jacobi algorithm for singular value decompositions have recently been shown to compute eigenvalues and singular values to high relative accuracy. However, these analyses do not address the accuracy of the computed eigenvectors and singular vectors. In this paper, we prove error bounds for the computed eigenvectors and singular vectors, where the error is measured by the sine of the angle between the vector and its computed counterpart. The obtained bounds preserve the relative gap structure of the bounds for Jacobi algorithms proved by Demmel and Veseli\'{c}, but involve the scaled condition number of the preconditioned matrix rather than that of the original matrix (the former of which is typically much smaller). Numerical experiments support our theoretical bounds and demonstrate that the mixed-precision preconditioned Jacobi algorithms are especially effective for ill-conditioned matrices with small absolute gaps and moderate relative gaps between eigenvalues or singular values.

math.NA↗

arXiv2026

Stable Hermite transforms via the Golub-Welsch algorithm

We introduce an efficient stable algorithm for transforms associated with expansions in Hermite functions interpolated at Hermite polynomial roots. The Hermite transform matrix can be factorised into a diagonal component and an orthogonal matrix, leading to a form which allows both the forward and inverse Hermite transforms to be computed stably. Our novel algorithm computes this factorisation based on the eigendecomposition of the Jacobi matrix associated with Hermite functions. Through numerical experiments, we demonstrate the stability and efficiency gains of this novel method over prior work. Numerical experiments show that the new approach matches or improves on the accuracy of existing stabilized methods, is substantially faster in practice, and enables reliable use of large Hermite expansions in downstream PDE computations. We also provide an open-source implementation, together with reference implementations of previous methods, to facilitate adoption by the community.

math.NA↗

arXiv2026

T-systems: a theory of orthonormal functions with a tridiagonal differentiation matrix

The starting point of this paper is that a spectral method is essentially a combination of an orthonormal basis of the underlying Hilbert space with Galerkin conditions. The choice of an orthonormal basis depends on a number of desirable features which we explore in the context of spectral methods for time-dependent partial differential equations in a single space dimension. A central role in ensuring many of the above features is played by the differentiation matrix of the underlying orthonormal system. In particular, it is beneficial if this matrix is skew-symmetric and tridiagonal. While orthonormal systems with this feature have been characterised in A. Iserles & M. Webb, ``Orthogonal systems with a skew-symmetric differentiation matrix'', Found. Comput. Maths, 19 (2019), 1191--1221, employing Fourier transforms, in this paper we provide an alternative characterisation using the differential Lanczos algorithm, which can be implemented constructively. It is valid for inner products that obey an `integration-by-parts condition', inclusive of $L_2$ and Sobolev norms on the real line. Motivated by quest for integration methods that conserve Hamiltonian energy, we conclude the paper replacing inner products by more general sesquilinear forms and presenting preliminary results. Here the Fourier transform characterisation generalises to spectral theory of Schr\"odinger operators and the differential Lanczos algorithm generalises to the differential Arnoldi algorithm.

math.NA↗

arXiv2026

Computing accurate singular values using a mixed-precision one-sided Jacobi algorithm

We present a relative forward error analysis of a mixed-precision preconditioned one-sided Jacobi algorithm, analogous to a two-sided version introduced in [N. J. Higham, F. Tisseur, M. Webb and Z. Zhou, SIAM J. Matrix Anal. Appl. 46 (2025), pp. 2423-2448], which uses low precision to compute the preconditioner, applies it in high precision, and computes the singular value decomposition using the one-sided Jacobi algorithm at working precision. Our analysis yields relative forward error bounds involving the scaled condition number of the preconditioned matrix rather than that of the input matrix, potentially improving accuracy when the former is smaller. We present and analyze two approaches for constructing effective preconditioners. Our numerical experiments demonstrate that our algorithm achieves smaller relative forward errors than the LAPACK routines $\texttt{DGESVJ}$ and $\texttt{DGEJSV}$, as well as the MATLAB function $\texttt{svd}$, particularly for ill-conditioned matrices. Timing tests show that our approach accelerates the convergence of the Jacobi iterations and that the dominant cost arises from a single high-precision matrix-matrix multiplication. With improved software or hardware support for this bottleneck, our algorithm would be faster than the LAPACK one-sided Jacobi algorithm $\texttt{DGESVJ}$ and comparable in speed to the state-of-the-art preconditioned one-sided Jacobi algorithm $\texttt{DGEJSV}$, but much more accurate.

math.NA↗

arXiv2026

Simulation of Radiation Chemistry by a One-Shot Hybrid Continuum / Monte Carlo Method

Understanding the spatio-temporal evolution of radiolytic species created by high-energy electrons in water underpins key applications from radiotherapy and nuclear safety to environmental processing and electron microscopy. Here, using the Manchester Inhomogeneous Radiation Chemistry by Linear Expansions (MIRaCLE) toolkit, we introduce and benchmark a novel approach to simulating these processes. Although the initial conditions are determined stochastically, the subsequent time evolution is calculated deterministically using a continuum representation, derived from those initial conditions. This hybrid approach essentially averages over many chemistry ``trajectories'' simultaneously, often converging to the 1% level in one shot, not requiring multiple runs. We demonstrate this new approach through the calculation of time-dependent G-values for e_{aq}^-$, \dot{\mathrm{OH}} and other radiolytic products, including at unprecedented dose rates where calculations which would take years with a conventional Monte Carlo approach can be performed in mere hours on a commercial laptop. We demonstrate that the main artifact of continuum modelling can be mitigated by a correction term. These results establish MIRaCLE as a flexible and efficient platform for modelling long-timescale radiolysis, providing a bridge between Monte Carlo approaches and macroscopic reaction--diffusion schemes, with broad implications for radiation chemistry in medicine, energy, and materials science.

physics.chem-ph↗

arXiv2025

Low-rank approximation of analytic kernels

Many algorithms in scientific computing and data science take advantage of low-rank approximation of matrices and kernels, and understanding why nearly-low-rank structure occurs is essential for their analysis and further development. This paper provides a framework for bounding the best low-rank approximation error of matrices arising from samples of a kernel that is analytically continuable in one of its variables to an open region of the complex plane. Elegantly, the low-rank approximations used in the proof are computable by rational interpolation using the roots and poles of Zolotarev rational functions, leading to a fast algorithm for their construction.

math.NA↗

arXiv2025

A Riemannian Gradient Descent Method for the Least Squares Inverse Eigenvalue Problem

We address an algorithm for the least squares fitting of a subset of the eigenvalues of an unknown Hermitian matrix lying an an affine subspace, called the Lift and Projection (LP) method, due to Chen and Chu (SIAM Journal on Numerical Analysis, 33 (1996), pp.2417-2430). The LP method iteratively `lifts' the current iterate onto the spectral constraint manifold then 'projects' onto the solution's affine subspace. We prove that this is equivalent to a Riemannian Gradient Descent with respect to a natural Riemannian metric. This insight allows us to derive a more efficient implementation, analyse more precisely its global convergence properties, and naturally append additional constraints to the problem. We provide several numerical experiments to demonstrate the improvement in computation time, which can be more than an order of magnitude if the eigenvalue constraints are on the smallest eigenvalues, the largest eigenvalues, or the eigenvalues closest to a given number. These experiments include an inverse eigenvalue problem arising in Inelastic Neutron Scattering of Manganese-6, which requires the least squares fitting of 16 experimentally observed eigenvalues of a $32400\times32400$ sparse matrix from a 5-dimensional subspace of spin Hamiltonian matrices.

math.NA↗

arXiv2025

Computing accurate eigenvalues using a mixed-precision Jacobi algorithm

We provide a rounding error analysis of a mixed-precision preconditioned Jacobi algorithm, which uses low precision to compute the preconditioner, applies it at high precision (amounting to two matrix-matrix multiplications) and solves the eigenproblem using the Jacobi algorithm at working precision. Our analysis yields meaningfully smaller relative forward error bounds for the computed eigenvalues compared with those of the Jacobi algorithm. We further prove that, after preconditioning, if the off-diagonal entries of the preconditioned matrix are sufficiently small relative to its smallest diagonal entry, the relative forward error bound is independent of the condition number of the original matrix. We present two constructions for the preconditioner that exploit low precision, along with their error analyses. Our numerical experiments confirm our theoretical results and compare the relative forward error of the proposed algorithm with the Jacobi algorithm, a preconditioned Jacobi algorithm, and MATLAB's $\texttt{eig}$ function. Timings using Julia suggest that the dominant cost of obtaining this level of accuracy comes from the high precision matrix-matrix multiplies; if support in software or hardware for this were improved, then this would become a negligible cost.

math.NA↗

arXiv2024

Deflation Techniques for Finding Multiple Local Minima of a Nonlinear Least Squares Problem

In this paper we generalize the technique of deflation to define two new methods to systematically find many local minima of a nonlinear least squares problem. The methods are based on the Gauss-Newton algorithm, and as such do not require the calculation of a Hessian matrix. They also require fewer deflations than for applying the deflated Newton method on the first order optimality conditions, as the latter finds all stationary points, not just local minima. One application of interest covered in this paper is the inverse eigenvalue problem (IEP) associated with the modelling of spectroscopic data of relevance to the physical and chemical sciences. Open source MATLAB code is provided at https://github.com/AlbanBloorRiley/DeflatedGaussNewton.

math.NA↗

arXiv2024

A Sherman--Morrison--Woodbury approach to solving least squares problems with low-rank updates

We present a simple formula to update the pseudoinverse of a full-rank rectangular matrix that undergoes a low-rank modification, and demonstrate its utility for solving least squares problems. The resulting algorithm can be dramatically faster than solving the modified least squares problem from scratch, just like the speedup enabled by Sherman--Morrison--Woodbury for solving linear systems with low-rank modifications.

math.NA↗

arXiv2023

Are sketch-and-precondition least squares solvers numerically stable?

Sketch-and-precondition techniques are efficient and popular for solving large least squares (LS) problems of the form $Ax=b$ with $A\in\mathbb{R}^{m\times n}$ and $m\gg n$. This is where $A$ is ``sketched" to a smaller matrix $SA$ with $S\in\mathbb{R}^{\lceil cn\rceil\times m}$ for some constant $c>1$ before an iterative LS solver computes the solution to $Ax=b$ with a right preconditioner $P$, where $P$ is constructed from $SA$. Prominent sketch-and-precondition LS solvers are Blendenpik and LSRN. We show that the sketch-and-precondition technique in its most commonly used form is not numerically stable for ill-conditioned LS problems. For provable and practical backward stability and optimal residuals, we suggest using an unpreconditioned iterative LS solver on $(AP)z=b$ with $x=Pz$. Provided the condition number of $A$ is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a backward error comparable to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to argue that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems. Additionally, we provide experimental evidence that using the sketch-and-solve solution as a starting vector in sketch-and-precondition algorithms (as suggested by Rokhlin and Tygert in 2008) should be highly preferred over the zero vector. The initialization often results in much more accurate solutions -- albeit not always backward stable ones.

math.NA↗

arXiv2022

Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation Matrices

We introduce and develop a theory of orthogonality with respect to Sobolev inner products on the real line for sequences of functions with a tridiagonal, skew-Hermitian differentiation matrix. While a theory of such L2-orthogonal systems is well established, Sobolev orthogonality requires new concepts and their analysis. We characterise such systems completely as appropriately weighed Fourier transforms of orthogonal polynomials and present a number of illustrative examples, inclusive of a Sobolev-orthogonal system whose leading N coefficients can be computed in $\mathcal{O}(N \log N)$ operations.

math.CA↗

arXiv2021

Solving the linear semiclassical Schr\"odinger equation on the real line

The numerical solution of a linear Schr\"odinger equation in the semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $\mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schr\"odinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schr\"odinger operator.

math.NA↗

arXiv2021

Approximation of wave packets on the real line

In this paper we compare three different orthogonal systems in $\mathrm{L}_2(\mathbb{R})$ which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schr\"odinger equation on the real line, specifically, stretched Fourier functions, Hermite functions and Malmquist--Takenaka functions. All three have banded skew-Hermitian differentiation matrices, which greatly simplifies their implementation in a spectral method, while ensuring that the numerical solution is unitary -- this is essential in order to respect the Born interpretation in quantum mechanics and, as a byproduct, ensures numerical stability with respect to the $\mathrm{L}_2(\mathbb{R})$ norm. We derive asymptotic approximations of the coefficients for a wave packet in each of these bases, which are extremely accurate in the high frequency regime. We show that the Malmquist--Takenaka basis is superior, in a practical sense, to the more commonly used Hermite functions and stretched Fourier expansions for approximating wave packets

math.NA↗

arXiv2020

A Differential Analogue of Favard's Theorem

Favard's theorem characterizes bases of functions $\{p_n\}_{n\in\mathbb{Z}_+}$ for which $x p_n(x)$ is a linear combination of $p_{n-1}(x)$, $p_n(x)$, and $p_{n+1}(x)$ for all $n \geq 0$ with $p_{0}\equiv1$ (and $p_{-1}\equiv 0$ by convention). In this paper we explore the differential analogue of this theorem, that is, bases of functions $\{\varphi_n\}_{n\in\mathbb{Z}_+}$ for which $\varphi_n'(x)$ is a linear combination of $\varphi_{n-1}(x)$, $\varphi_n(x)$, and $\varphi_{n+1}(x)$ for all $n \geq 0$ with $\varphi_{0}(x)$ given (and $\varphi_{-1}\equiv 0$ by convention). We answer questions about orthogonality and completeness of such functions, provide characterisation results, and also, of course, give plenty of examples and list challenges for further research. Motivation for this work originated in the numerical solution of differential equations, in particular spectral methods which give rise to highly structured matrices and stable-by-design methods for partial differential equations of evolution. However, we believe this theory to be of interest in its own right, due to the interesting links between orthogonal polynomials, Fourier analysis and Paley--Wiener spaces, and the resulting identities between different families of special functions.

math.CA↗

arXiv2019

The AZ algorithm for least squares systems with a known incomplete generalized inverse

We introduce an algorithm for the least squares solution of a rectangular linear system $Ax=b$, in which $A$ may be arbitrarily ill-conditioned. We assume that a complementary matrix $Z$ is known such that $A - AZ^*A$ is numerically low rank. Loosely speaking, $Z^*$ acts like a generalized inverse of $A$ up to a numerically low rank error. We give several examples of $(A,Z)$ combinations in function approximation, where we can achieve high-order approximations in a number of non-standard settings: the approximation of functions on domains with irregular shapes, weighted least squares problems with highly skewed weights, and the spectral approximation of functions with localized singularities. The algorithm is most efficient when $A$ and $Z^*$ have fast matrix-vector multiplication and when the numerical rank of $A - AZ^*A$ is small.

math.NA↗

arXiv2019

A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix

In this paper we explore orthogonal systems in $\mathrm{L}_2(\mathbb{R})$ which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for $\mathrm{L}_2(\mathbb{R})$, have a simple explicit formulae as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this family of functions is essentially the only orthonormal basis possessing a differentiation matrix of the above form and whose coefficients are equal to classical Fourier coefficients of a modified function though a monotone, differentiable change of variables. Examples of other orthogonal bases with skew-Hermitian, tridiagonal differentiation matrices are discussed as well.

math.NA↗

arXiv2019

Fast Computation of Orthogonal Systems with a Skew-symmetric Differentiation Matrix

Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation matrix is skew-symmetric, tridiagonal and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first $N$ coefficients {of the expansion} can be computed to high accuracy in $\mathcal{O}(N\log_2N)$ operations. We consider two settings, one approximating a function $f$ directly in $(-\infty,\infty)$ and the other approximating $[f(x)+f(-x)]/2$ and $[f(x)-f(-x)]/2$ separately in $[0,\infty)$. In each setting we prove that there is a single family, parametrised by $\alpha,\beta > -1$, of orthogonal systems with a skew-symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where $\alpha, \beta= \pm 1/2$ are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz-plus-Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials.

math.NA↗