33 citations · 164 across the 51 of their papers we have counts for
55 papers
A Riemann-Hilbert approach to the computation of transform pairs
Kaitlynn Lilly, Thomas Trogdon
We develop a unified methodology that integrates spectral theory, Riemann-Hilbert problems, and inverse scattering theory for the construction and numerical evaluation of transform…
Banded Hermitian Matrices, Matrix Orthogonal Polynomials, and the Toda Lattice
Charbel Abi Younes, Thomas Trogdon
We study the direct and inverse spectral theory for a class of finite Hermitian banded matrices. Using the theory of matrix orthogonal polynomials, we provide an explicit procedure…
Efficient computation of soliton gas primitive potentials
Cade Ballew, Deniz Bilman, Thomas Trogdon
We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg--de Vries equation that arise from the accumulation of solitons on an infinite int…
A Lanczos-Based Algorithmic Approach for Spike Detection in Large Sample Covariance Matrices
Charbel Abi Younes, Xiucai Ding, Thomas Trogdon
We introduce a new approach for estimating the number of spikes in a general class of spiked covariance models without directly computing the eigenvalues of the sample covariance m…
The Akhiezer iteration and inverse-free solvers for Sylvester matrix equations
Cade Ballew, Thomas Trogdon, Heather Wilber
Two inverse-free iterative methods are developed for solving Sylvester matrix equations when the spectra of the coefficient matrices are on, or near, known disjoint subintervals of…
Numerical inverse scattering transform for the defocusing nonlinear Schrödinger equation with box-type initial conditions on a nonzero background
Aikaterini Gkogkou, Barbara Prinari, Thomas Trogdon
We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with a box-type initial condition (IC) having a nontrivial backg…
On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights
Thomas Trogdon
We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures suppo…
The Akhiezer iteration
Cade Ballew, Thomas Trogdon
We develop the Akhiezer iteration, a generalization of the classical Chebyshev iteration, for the inner product-free, iterative solution of indefinite linear systems using orthogon…
The ultraspherical rectangular collocation method and its convergence
Thomas Trogdon
We develop the ultraspherical rectangular collocation (URC) method, a collocation implementation of the sparse ultraspherical method of Olver \& Townsend for two-point boundary-val…
On the convergence of Fourier spectral methods involving non-compact operators
Thomas Trogdon
Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations . The framework posits the existence…
Computing the Tracy-Widom Distribution for Arbitrary
Thomas Trogdon, Yiting Zhang
We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloem…
GMRES, pseudospectra, and Crouzeix's conjecture for shifted and scaled Ginibre matrices
Tyler Chen, Anne Greenbaum, Thomas Trogdon
We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted matrix whose entries are independent complex Gaussians. When the righ…
Stability of the Lanczos algorithm on matrices with regular spectral distributions
Tyler Chen, Thomas Trogdon
We study the stability of the Lanczos algorithm run on problems whose eigenvector empirical spectral distribution is near to a reference measure with well-behaved orthogonal polyno…
A Riemann--Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals
Cade Ballew, Thomas Trogdon
We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our…
An artificially-damped Fourier method for dispersive evolution equations
Anne Liu, Thomas Trogdon
Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier…
Computation of Large-Genus Solutions of the Korteweg-de Vries Equation
Deniz Bilman, Patrik Nabelek, Thomas Trogdon
We consider the numerical computation of finite-genus solutions of the Korteweg-de Vries equation when the genus is large. Our method applies both to the initial-value problem when…
Randomized matrix-free quadrature: unified and uniform bounds for stochastic Lanczos quadrature and the kernel polynomial method
Tyler Chen, Thomas Trogdon, Shashanka Ubaru
We analyze randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied include the kernel polynomial method and stochastic Lanc…
Growth factors of random butterfly matrices and the stability of avoiding pivoting
John Peca-Medlin, Thomas Trogdon
Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have oft…
A Riemann--Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory
Xiucai Ding, Thomas Trogdon
We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory.…
The conjugate gradient algorithm on a general class of spiked covariance matrices
Xiucai Ding, Thomas Trogdon
We consider the conjugate gradient algorithm applied to a general class of spiked sample covariance matrices. The main result of the paper is that the norms of the error and residu…
The Numerical Unified Transform Method for the Nonlinear Schrödinger equation on the half-line
Xin Yang, Bernard Deconinck, Thomas Trogdon
We implement the Numerical Unified Transform Method to solve the Nonlinear Schrödinger equation on the half-line. For so-called linearizable boundary conditions, the method solves…
Analysis of stochastic Lanczos quadrature for spectrum approximation
Tyler Chen, Thomas Trogdon, Shashanka Ubaru
The cumulative empirical spectral measure (CESM) of a symmetric matrix is defined as the fraction of eigenvalues of…
Scattering and inverse scattering for the AKNS system: A rational function approach
Thomas Trogdon
We consider the use of rational basis functions to compute the scattering and inverse scattering transforms associated with the AKNS system. The proposed numerical forward scatteri…
On the open Toda chain with external forcing
Percy Deift, Luen-Chau Li, Herbert Spohn +2
We consider the open Toda chain with external forcing, and in the case when the forcing stretches the system, we derive the longtime behavior of solutions of the chain. Using an ob…