Marcus Webb
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View article: Computing accurate eigenvalues using a mixed-precision Jacobi algorithm
Computing accurate eigenvalues using a mixed-precision Jacobi algorithm Open
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…
View article: Deflation Techniques for Finding Multiple Local Minima of a Nonlinear Least Squares Problem
Deflation Techniques for Finding Multiple Local Minima of a Nonlinear Least Squares Problem Open
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…
View article: A Sherman--Morrison--Woodbury approach to solving least squares problems with low-rank updates
A Sherman--Morrison--Woodbury approach to solving least squares problems with low-rank updates Open
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 f…
View article: A New Approach for Simulating Inhomogeneous Chemical Kinetics
A New Approach for Simulating Inhomogeneous Chemical Kinetics Open
In this paper, inhomogeneous chemical kinetics are simulated by describing the concentrations of interacting chemical species by a linear expansion of basis functions in such a manner that the coupled reaction and diffusion processes are p…
View article: Are sketch-and-precondition least squares solvers numerically stable?
Are sketch-and-precondition least squares solvers numerically stable? Open
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…
View article: Approximation of Wave Packets on the Real Line
Approximation of Wave Packets on the Real Line Open
In this paper we compare three different orthogonal systems in $$\textrm{L}_2({\mathbb {R}})$$ which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schrödinger equation …
View article: Sobolev‐orthogonal systems with tridiagonal skew‐Hermitian differentiation matrices
Sobolev‐orthogonal systems with tridiagonal skew‐Hermitian differentiation matrices Open
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 L 2 ‐orthogonal systems…
View article: Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation Matrices
Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation Matrices Open
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 i…
View article: Solving the linear semiclassical Schrödinger equation on the real line
Solving the linear semiclassical Schrödinger equation on the real line Open
The numerical solution of a linear Schrödinger 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…
View article: Approximation of wave packets on the real line
Approximation of wave packets on the real line Open
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 Schrödinger equation on the real li…
View article: A Differential Analogue of Favard's Theorem
A Differential Analogue of Favard's Theorem Open
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 convent…
View article: The AZ Algorithm for Least Squares Systems with a Known Incomplete Generalized Inverse
The AZ Algorithm for Least Squares Systems with a Known Incomplete Generalized Inverse Open
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 ra…
View article: The AZ algorithm for least squares systems with a known incomplete\n generalized inverse
The AZ algorithm for least squares systems with a known incomplete\n generalized inverse Open
We introduce an algorithm for the least squares solution of a rectangular\nlinear system $Ax=b$, in which $A$ may be arbitrarily ill-conditioned. We\nassume that a complementary matrix $Z$ is known such that $A - AZ^*A$ is\nnumerically low…
View article: A family of orthogonal rational functions and other orthogonal systems\n with a skew-Hermitian differentiation matrix
A family of orthogonal rational functions and other orthogonal systems\n with a skew-Hermitian differentiation matrix Open
In this paper we explore orthogonal systems in $\\mathrm{L}_2(\\mathbb{R})$\nwhich give rise to a skew-Hermitian, tridiagonal differentiation matrix.\nSurprisingly, allowing the differentiation matrix to be complex leads to a\nparticular f…
View article: Pointwise and Uniform Convergence of Fourier Extensions
Pointwise and Uniform Convergence of Fourier Extensions Open
Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent t…
View article: Fast Computation of Orthogonal Systems with a Skew-symmetric Differentiation Matrix
Fast Computation of Orthogonal Systems with a Skew-symmetric Differentiation Matrix Open
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…
View article: Fast Computation of Orthogonal Systems with a Skew-symmetric\n Differentiation Matrix
Fast Computation of Orthogonal Systems with a Skew-symmetric\n Differentiation Matrix Open
Orthogonal systems in $\\mathrm{L}_2(\\mathbb{R})$, once implemented in\nspectral methods, enjoy a number of important advantages if their\ndifferentiation matrix is skew-symmetric and highly structured. Such systems,\nwhere the differenti…
View article: Fast polynomial transforms based on Toeplitz and Hankel matrices
Fast polynomial transforms based on Toeplitz and Hankel matrices Open
Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive algorithms wi…
View article: Volume Preservation by Runge-Kutta Methods
Volume Preservation by Runge-Kutta Methods Open
It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume pres…
View article: The Acyclicity of the Frobenius Functor for Modules of Finite Flat\n Dimension
The Acyclicity of the Frobenius Functor for Modules of Finite Flat\n Dimension Open
Let $R$ be a commutative Noetherian local ring of prime characteristic $p$\nand $f:R\\to R$ the Frobenius ring homomorphism. For $e\\ge 1$ let $R^{(e)}$\ndenote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine,\nSzpiro, a…