Related papers: Acceleration of multiple precision solver for ill-…

This report shows on real data that the direct methods such as LDL decomposition and Gaussian elimination for solving linear systems with ill-conditioned matrices provide inaccurate results due to divisions by very small numbers, which in…

This study investigates the iterative refinement method applied to the solution of linear discrete inverse problems by considering its application to the Tikhonov problem in mixed precision. Previous works on mixed precision iterative…

We propose a new type of multilevel method for solving eigenvalue problems based on Newton iteration. With the proposed iteration method, solving eigenvalue problem on the finest finite element space is replaced by solving a small scale…

The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we first give the error analysis for a single step or sweep of Jacobi's method in floating point arithmetic. Then we propose a mixed precision…

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to…

Efficiently solving sparse linear algebraic equations is an important research topic of numerical simulation. Commonly used approaches include direct methods and iterative methods. Compared with the direct methods, the iterative methods…

We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with…

We consider linear ill-conditioned operator equations in a Hilbert space setting. Motivated by the aggregation method, we consider approximate solutions constructed from linear combinations of Tikhonov regularization, which amounts to…

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…

Singular equations with rank-deficient Jacobians arise frequently in algebraic computing applications. As shown in case studies in this paper, direct and intuitive modeling of algebraic problems often results in nonisolated singular…

We derive a formula that simplifies the original asymptotic iteration method formulation to find the energy eigenvalues for the analytically solvable cases. We then show that there is a connection between the asymptotic iteration and the…

This paper investigates using the conjugate gradient iterative solver for ill-posed problems. We show that preconditioner and Tikhonov-regularization work in conjunction. In particular when they employ the same symmetric positive…

In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution…

Different variants of approximate inverse iteration like the locally optimal block preconditioned conjugate gradient method became in recent years increasingly popular for the solution of the large matrix eigenvalue problems arising from…

For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit…

The Riemann problem for first-order hyperbolic systems of partial differential equations is of fundamental importance for both theoretical and numerical purposes. Many approximate solvers have been developed for such systems; exact solution…

We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…

We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schr\"odinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either…

In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda, u)$ as…

Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using…