Related papers: Numerical Stability of Lanczos Methods
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric…
Polynomial Krylov subspace methods are among the most widely used methods for approximating $f(A)b$, the action of a matrix function on a vector, in particular when $A$ is large and sparse. When $A$ is Hermitian positive definite, the…
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is…
In this paper, a new rigorous numerical method to compute fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. Decomposing the fundamental matrix solutions $\Phi(t)$ by their…
Estimation of structure, such as in variable selection, graphical modelling or cluster analysis is notoriously difficult, especially for high-dimensional data. We introduce stability selection. It is based on subsampling in combination with…
Given a large square matrix $A$ and a sufficiently regular function $f$ so that $f(A)$ is well defined, we are interested in the approximation of the leading singular values and corresponding singular vectors of $f(A)$, and in particular of…
We present an efficient method for computing dominant eigenvalues of large, nonsymmetric, diagonalizable matrices based on an adaptive block Lanczos algorithm combined with Chebyshev polynomial filtering. The proposed approach improves…
A numerical integrator for $\dot{x}=f(x)$ is called \emph{stable} if, when applied to the 1D Dahlquist test equation $\dot{x}=\lambda x,\lambda\in\mathbb{C}$ with fixed timestep $h>0$, the numerical solution remains bounded as the number of…
Many modern datasets don't fit neatly into $n \times p$ matrices, but most techniques for measuring statistical stability expect rectangular data. We study methods for stability assessment on non-rectangular data, using statistical learning…
In Part I, we defined a LASSO condition number and developed an algorithm -- for computing support sets (feature selection) of the LASSO minimisation problem -- that runs in polynomial time in the number of variables and the logarithm of…
Various recurrence relations between formal orthogonal polynomials can be used to derive Lanczos-type algorithms. In this paper, we consider recurrence relation $A_{12}$ for the choice $U_i(x)=P_i(x)$, where $U_i$ is an auxiliary family of…
A backward stable numerical calculation of a function with condition number $\kappa$ will have a relative accuracy of $\kappa\epsilon_{\text{machine}}$. Standard formulations and software implementations of finite-strain elastic materials…
We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63--72]. As a…
We introduce a new algorithm for finding the eigenvalues and eigenvectors of Hermitian matrices within a specified region, based upon the LANSO algorithm of Parlett and Scott. It uses selective reorthogonalization to avoid the duplication…
In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit finite difference schemes for the one-dimensional advection equation with an inflow boundary condition. The strong…
The Lasso has been widely used as a method for variable selection, valued for its simplicity and empirical performance. However, Lasso's selection stability deteriorates in the presence of correlated predictors. Several approaches have been…
The block Lanczos algorithm proposed by Peter Montgomery is an efficient means to tackle the sparse linear algebra problem which arises in the context of the number field sieve factoring algorithm and its predecessors. We present here a…
Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of…
We solve the Hubbard model with the exact diagonalization method on a graphics processing unit (GPU). We benchmark our GPU program against a sequential CPU code by using the Lanczos algorithm to solve the ground state energy in two cases: a…
Recent work found that an analysis formalism based on the Lanczos algorithm allows energy levels to be extracted from Euclidean correlation functions with faster ground-state convergence than effective masses, convergent estimators for…