Related papers: Shifted Lanczos method for quadratic forms with He…
A thick-restart Lanczos type algorithm is proposed for Hermitian $J$-symmetric matrices. Since Hermitian $J$-symmetric matrices possess doubly degenerate spectra or doubly multiple eigenvalues with a simple relation between the degenerate…
This work considers large-scale Lyapunov matrix equations of the form $AX + XA = \boldsymbol{c}\boldsymbol{c}^T$, where $A$ is a symmetric positive definite matrix and $\boldsymbol{c}$ is a vector. Motivated by the need to solve such…
A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple right-hand sides, is described. For the first right-hand side, eigenvectors…
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…
Quadratic minimization problems with orthogonality constraints (QMPO) play an important role in many applications of science and engineering. However, some existing methods may suffer from low accuracy or heavy workload for large-scale…
We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(\mathbf{A}) \mathbf{b}$ when $\mathbf{A}$ is a Hermitian matrix and $\mathbf{b}$ is a given vector. Assuming that $f :…
We consider a quadrature-based eigensolver to find eigenpairs of Hermitian matrices arising in lattice quantum chromodynamics. To reduce the computational cost for finding eigenpairs of such Hermitian matrices, we propose a new technique…
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…
For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has established new local general convergence results, independent of iterative solvers for inner linear systems. The theory shows that the method locally converges…
A common approach to approximating quadratic forms of matrix functions is to use a quadrature rule derived from the Lanczos process, known as a Lanczos quadrature. Although symmetric quadrature rules are computationally favorable, it has…
In this work we introduce a memory-efficient method for computing the action of a Hermitian matrix function on a vector. Our method consists of a rational Lanczos algorithm combined with a basis compression procedure based on rational…
The Arnoldi process provides an efficient framework for approximating functions of a matrix applied to a vector, i.e., of the form $f(M)\bm{b}$, by repeated matrix-vector multiplications. In this paper, we derive error estimates for…
We study the Lanczos algorithm where the initial vector is sampled uniformly from $\mathbb{S}^{n-1}$. Let $A$ be an $n \times n$ Hermitian matrix. We show that when run for few iterations, the output of Lanczos on $A$ is almost…
Objectives involving bilinear forms $u^\top f(A(\theta))v$ for Hermitian $A$ arise widely in scientific computing and probabilistic machine learning. For large matrices, Lanczos efficiently approximates these quantities, but differentiating…
This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos--type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the…
The spectral transformation Lanczos method for the sparse symmetric definite generalized eigenvalue problem for matrices $A$ and $B$ is an iterative method that addresses the case of semidefinite or ill conditioned $B$ using a shifted and…
Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for…
This paper revisits the error analysis of the Stochastic Lanczos Quadrature (SLQ) method for approximating the trace of matrix functions, with a specific focus on asymmetric Lanczos quadrature rules. We reexplain an existing theoretical…
The Lanczos process constructs a sequence of orthonormal vectors v_m spanning a nested sequence of Krylov subspaces generated by a hermitian matrix A and some starting vector b. In this paper we show how to cheaply recover a secondary…
Approximating the action of a matrix function $f(\mathbf{A})$ on a vector $\mathbf{b}$ is an increasingly important primitive in machine learning, data science, and statistics, with applications such as sampling high dimensional Gaussians,…