相关论文: A Switching Approach to Avoid Breakdown in Lanczos…
Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal…
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…
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 polynomials. We give a backwards stability result which can be…
The Lanczos algorithm for matrix tridiagonalisation suffers from strong numerical instability in finite precision arithmetic when applied to evaluate matrix eigenvalues. The mechanism by which this instability arises is well documented in…
Lanczos methods for solving $\textit{A}\textbf{x}=\textbf{b}$ consist in constructing a sequence of vectors $(\textbf{x}_k), k=1,...$ such that $\textbf{r}_{k}=\textbf{b}-\textit{A}\textbf{x}_{k}=\textit{P}_{k}(\textit{A})\textbf{r}_{0}$,,…
We describe new methods for deciding the stability of switching systems. The methods build on two ideas previously appeared in the literature: the polytope norm iterative construction, and the lifting procedure. Moreover, the combination of…
We develop a block minimum residual (MINRES) algorithm for symmetric indefinite matrices. This version is built upon the band Lanczos method that generates one basis vector of the block Krylov subspace per iteration rather than a whole…
The Lanczos algorithm has proven itself to be a valuable matrix eigensolver for problems with large dimensions, up to hundreds of millions or even tens of billions. The computational cost of using any Lanczos algorithm is dominated by the…
This work introduces a method for determining the energy spectrum of lattice quantum chromodynamics (LQCD) by applying the Lanczos algorithm to the transfer matrix and using a bootstrap generalization of the Cullum-Willoughby method to…
The machine learning of lattice operators has three possible bottlenecks. From a statistical standpoint, it is necessary to design a constrained class of operators based on prior information with low bias, and low complexity relative to the…
Compared to the classical Lanczos algorithm, the $s$-step Lanczos variant has the potential to improve performance by asymptotically decreasing the synchronization cost per iteration. However, this comes at a cost. Despite being…
The method of quantum Lanczos recursion is extended to solve for multiple excitations on the quantum computer. While quantum Lanczos recursion is in principle capable of obtaining excitations, the extension to a block Lanczos routine can…
The Lanczos method is one of the most powerful and fundamental techniques for solving an extremal symmetric eigenvalue problem. Convergence-based error estimates depend heavily on the eigenvalue gap. In practice, this gap is often…
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…
Lanczos-based methods have become standard tools for tasks involving matrix functions. Progress on these algorithms has been driven by several largely disjoint communities, resulting many innovative and important advancements which would…
The Lanczos method with implicit restarting is one of the most popular methods for finding a few exterior eigenpairs of a large symmetric matrix $A$. Usually based on polynomial filtering, restarting is crucial to limit memory and the cost…
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…
Over-actuated systems often make it possible to achieve specific performances by switching between different subsets of actuators. However, when the system parameters are unknown, transferring authority to different subsets of actuators is…
If a linear switching system with frequent switches is stable, will it be stable under arbitrary switches? In general, the answer is negative. Nevertheless, this question can be answered in an explicit form for any concrete system. This is…
The Switch Point Algorithm is a new approach for solving optimal control problems whose solutions are either singular or bang-bang or both singular and bang-bang, and which possess a finite number of jump discontinuities in an optimal…