Related papers: On the correction equation of the Jacobi-Davidson …
Jacobi matrices are parametrized by their eigenvalues and norming constants (first coordinates of normalized eigenvectors): this coordinate system breaks down at reducible tridiagonal matrices. The set of real symmetric tridiagonal matrices…
Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance…
There are some types of ill-conditioned algebraic equations that have difficulty in obtaining accurate roots and coefficients that must be expressed with a multiple precision floating-point number. When all their roots are simple, the…
We describe two main classes of one-sided trigonometric and hyperbolic Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian matrices. These types of algorithms exhibit significant advantages over many other…
A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of $\bR_0$-functions gives rise to a linear pencil $H-\lambda J$, where $H$ and $J$ are Hermitian tridiagonal matrices. First, we show…
The solution of parameter-dependent linear systems, by classical methods, leads to an arithmetic effort that grows exponentially in the number of parameters. This renders the multigrid method, which has a well understood convergence theory,…
The Newton, Gauss--Newton and Levenberg--Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some…
The class of three-diagonal Jacobi matrix with exponentially increasing elements is considered. Under some assumptions the matrix corresponds to unbounded self-adjoint operator in the weighted space. The weight depends on elements of the…
A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair $(A, C)$ is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter…
The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or…
The Classic Howard's algorithm, a technique of resolution for discrete Hamilton-Jacobi equations, is of large use in applications for its high efficiency and good performances. A special beneficial characteristic of the method is the…
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…
Many problems in physics, chemistry and other fields are perturbative in nature, i.e. differ only slightly from related problems with known solutions. Prominent among these is the eigenvalue perturbation problem, wherein one seeks the…
In this paper, based on the Noda iteration, we present inexact Noda iterations (INI), to find the smallest eigenvalue and the associated positive eigenvector of a large irreducible nonsingular M-matrix. The positivity of approximations is…
Eigensolvers involving complex moments can determine all the eigenvalues in a given region in the complex plane and the corresponding eigenvectors of a regular linear matrix pencil. The complex moment acts as a filter for extracting…
A systematic investigation of the skew-symmetric solutions of the three-dimensional Jacobi equations is presented. As a result, three disjoint and complementary new families of solutions are characterized. Such families are very general,…
Preconditioning has long been a staple technique in optimization, often applied to reduce the condition number of a matrix and speed up the convergence of algorithms. Although there are many popular preconditioning techniques in practice,…
An effective numerical method is presented for optimizing model parameters that can be applied to any type of system of non-linear equations and any number of data-points, which does not require explicit formulation of the objective…
Three refined and refined harmonic extraction-based Jacobi--Davidson (JD) type methods are proposed, and their thick-restart algorithms with deflation and purgation are developed to compute several generalized singular value decomposition…
The Jacobi evolution method has been widely used in the QCD analysis of structure function data. However a recent paper claims that there are serious problems with its convergence and stability. Here we briefly review the evidence for the…