Partial Eigenvalue Problem Research Articles

A robust regional eigenvalue assignment problem using rank-one control for undamped gyroscopic systems

<abstract><p>Considering the advantages of economic benefit and cost reduction by using rank-one control, we investigated the problem of robust regional eigenvalue assignment using rank-one control for undamped gyroscopic systems. Based on the orthogonality relation, we presented a method for solving partial eigenvalue assignment problems to reassign partial undesired eigenvalues accurately. Since it is difficult to achieve robust control by assigning desired eigenvalues to precise positions with rank-one control, we assigned eigenvalues within specified regions to provide the necessary freedom. According to the sensitivity analysis theories, we derived the sensitivity of closed-loop eigenvalues to parameter perturbations to measure robustness and proposed a numerical algorithm for solving robust regional eigenvalue assignment problems so that the closed-loop eigenvalues were insensitive to parameter perturbations. Numerical experiments demonstrated the effectiveness of our method.</p></abstract>

OPERATOR SPECTRUM TRANSFORMATION IN HARTREE<i>–</i>FOCK AND KOHN–SHAM EQUATIONS

The paper proposes a method for preliminary transformation of the spectrum of the equation operator both in the Hartree–Fock method and in density functional theory. This method makes it possible to solve a partial eigenvalue problem instead of the complete one, and the eigenfunctions turn out to be ordered in a way convenient for calculation. The transformation makes an old idea of grid approximation of a solution competitive in terms of computational speed as compared to widely used approaches based on basis sets methods.

A mathematical model of an inductively coupled radio-frequency discharge at low pressure as nonlinear partial eigenproblem

A new approach for modeling steady state inductively coupled radio frequency discharges at low pressure is described. A simple one-dimensional model is considered, which includes Maxwell’s equations and the electron balance equation with boundary conditions of the third kind. It is shown that the system of boundary value problems is a two-parameter partial eigenvalue problem. The smallest eigenvalue of the problem is the boundary value of the magnetic field strength. The second parameter of the problem is the concentration of electrons at the center of the plasma bunch. The developed approach makes it possible to calculate the inductor current required to maintain a steady state of the discharge. The results of calculations of the dependence of the inductor current, electron density, electric and magnetic fields on pressure are presented.

Solution of the symmetric band partial inverse eigenvalue problem for the damped mass spring system

ABSTRACT The structured partial quadratic inverse eigenvalue problem (SPQIEP) is to construct the structured quadratic matrix polynomial using the partial eigendata. The structures arising in physical applications include symmetry, band (tridiagonal, diagonal, pentagonal) etc. The construction of the structured matrix polynomial is the most difficult aspect of this problem and the research on structured inverse eigenvalue problem is rare. In this paper, the symmetric band partial quadratic inverse eigenvalue problem (SBPQIEP) for the damped mass spring system is considered. This problem concerns in finding the symmetric band matrices , and C with bandwidth p from m ( ) prescribed eigenpairs so that the corresponding quadratic matrix polynomial has the given eigenpairs as its eigenvalues and eigenvectors. In general, SBPQIEP is very hard to solve due to the additional band structure constraint. We propose a novel method, based on the matrix-vectorization and Kronecker product of matrices for solving this problem. Furthermore, explicit expressions for general solutions are presented. Numerical experiments on a spring mass problem are presented to illustrate the applicability and the practical usefulness of the proposed method.

Block subspace projection preconditioned conjugate gradient method in modal structural analysis

This paper considers the preconditioned block conjugate gradient method for solving the algebraic, generalized partial eigenvalue problem, which arises when the finite element method is applied to dynamic analysis in structural and solid mechanics. The focus is on the development of an algorithm that keeps constant the iterable subspace dimension, independently of the number of eigenpairs required. Novelties in this approach include a preconditioning strategy based on block incomplete factorization of the shifted stiffness matrix, and analysis of unremovable errors, which accumulate with increasing numbers of eigenpairs.

Hybrid Algorithms for Solving the Algebraic Eigenvalue Problem with Sparse Matrices

Hybrid algorithms for solving the partial generalized eigenvalue problem for symmetric positive definite sparse matrices of different structures by hybrid computers with graphic processors are proposed, coefficients for the efficiency of the algorithms are obtained, and approbation of the developed algorithms for test and practical problems is carried out.

English

We extend the analysis of the recently proposed nonlinear EIS scheme applied to the partial eigenvalue problem. We address the case where the Rayleigh quotient iteration is used as the smoother on the fine-level. Unlike in our previous theoretical results, where the smoother given by the linear inverse power method is assumed, we prove nonlinear speed-up when the approximation becomes close to the exact solution. The speed-up is cubic. Unlike existent convergence estimates for the Rayleigh quotient iteration, our estimates take advantage of the powerful effect of the coarse-space.

Parallel Rayleigh Quotient Optimization with FSAI‐Based Preconditioning

The present paper describes a parallel preconditioned algorithm for the solution of partial eigenvalue problems for large sparse symmetric matrices, on parallel computers. Namely, we consider the Deflation‐Accelerated Conjugate Gradient (DACG) algorithm accelerated by factorized‐sparse‐approximate‐inverse‐ (FSAI‐) type preconditioners. We present an enhanced parallel implementation of the FSAI preconditioner and make use of the recently developed Block FSAI‐IC preconditioner, which combines the FSAI and the Block Jacobi‐IC preconditioners. Results onto matrices of large size arising from finite element discretization of geomechanical models reveal that DACG accelerated by these type of preconditioners is competitive with respect to the available public parallel hypre package, especially in the computation of a few of the leftmost eigenpairs. The parallel DACG code accelerated by FSAI is written in MPI‐Fortran 90 language and exhibits good scalability up to one thousand processors.

Internal and external eigenvalue problems of Hermitian operators and their use in electronic structure theory

Within the fragment resolution of molecular systems the conceptual and interpretative advantages of using the separate eigenvalue problems for the internal and external part of the Hermitian matrix representing a physical quantity in quantum mechanics are examined. By definition, these two parts accordingly combine only the diagonal and off-diagonal subsystem-resolved blocks of matrix elements. These two partial eigenvalue problems bring about the matrix internal or external decouplings, respectively, which have recently been used in several interpretations of the molecular electronic structure. A character and structure of the external eigensolutions is examined in some detail and their recent applications in the Charge Sensitivity Analysis—to extract the most important electron-transfer effects between constituent atoms of model chemisorption systems, and in the Molecular-Orbital theory—to precisely identify the inter-orbital flows of electrons, are summarized and commented upon. The grouping relation, for combining the external/internal eigensolutions into those for the whole matrix, is derived in the context of the complementary “rotations” of the basis set vectors.

The implicit restarted Arnoldi method, an efficient alternative to solve the neutron diffusion equation

To calculate the neutronic steady state of a nuclear power reactor core and its subcritical modes, it is necessary to solve a partial eigenvalue problem. In this paper, an implicit restarted Arnoldi method is presented as an advantageous alternative to classical methods as the Power Iteration method and the Subspace Iteration method. The efficiency of these methods, has been compared calculating the dominant Lambda modes of several configurations of the Three Mile Island reactor core.

Application of differential equations to synthesize a class of algorithms for numerical solution of a partial eigenvalue problem

We synthesize a class of recursive algorithms in the formation of which essential use is made of the properties of a certain type of differential equation and the qualitative analysis of it. These algorithms admit generalization to the case of nonsymmetric matrices.

Eigenvalues calculation algorithms for λ-modes determination. Parallelization approach

In this paper, we review two methods to obtain the λ-modes of a nuclear reactor, Subspace Iteration method and Arnoldi's method, which are popular methods to solve the partial eigenvalue problem for a given matrix. In the developed application for the neutron diffusion equation we include improved acceleration techniques for both methods. Also, we propose two parallelization approaches for these methods, a coarse grain parallelization and a fine grain one. We have tested the developed algorithms with two realistic problems, focusing on the efficiency of the methods according to the CPU times.

Algorithm to solve the partial eigenvalue problem for large profile matrices

An algorithm is proposed for computing some of the eigenvalues and the corresponding eigenvectors of the generalized eigenvalue problem for symmetric positive-definite matrices stored in profile form on disk.

Solution of the partial eigenvalue problem for a regular matrix pencil

Algorithms are proposed for the solution of the partial eigenvalue problem for regular polynomial matrix pencils.

Numerical solution of ordinary and partial functional-differential eigenvalue problems with the Tau Method

We apply a recent new formulation of the Tau Method to reduce the numerical treatment of eigenvalue problems for ordinary and partialfunctional-differential equations to that of generalized algebraic eigenvalue problems. We find accurate numerical results through the use of a simple algorithm which we discuss in applications to several concrete examples. Extrapolation is used to refine the results already obtained.

Certain modifications of the AB-algorithm

One considers various modifications of the AB-algorithm for the solution of the complete (partial) eigenvalue problem of a regular pencil A-λB of square matrices. A modification of the AB-algorithm is suggested which allows to eliminate in a finite number of steps the zero and the infinite eigenvalues of the pencil A-λB and to lower its dimensions. For regular pencils with real eigenvalues a modification ot the AB-algorithm with a shift is presented. For a well-defined choice of the shifts one proves the quadratic convergence of the algorithm, successively to each eigenvalue of the pencil, starting with the smallest one. For a pencil whose eigenvalues can be divided into the groups of “large” and “small” eigenvalues, one considers a modification of the AB-algorithm, allowing to obtain approximations to the indicated groups of eigenvalues as solutions of a problem for pencils of lower dimensions.

A numerical study of the solution of the partial eigenvalue problem

The partial eigenvalue problem that arises from the application of the finite element method is considered. A number of iterative methods are examined which consist in seeking the stationary points of the Rayleigh quotient and thus avoiding the physical assembling of the matrices involved. The computational efficiencies of the steepest descent, the conjugate gradient and the co-ordinate overrelaxation methods are compared. Several other modifications to the original conjugate gradient algorithm as well as an orthogonalization process are studied. The dependence of the convergence of the methods on the initial estimate of the eigenvectors and on different values of the relaxation parameter, in the case of the co-ordinate overrelaxation, are also examined.

Solving the eigenvalue problem for sparse matrices

A modification of the Danilewski method is presented, permitting the solution of the eigenvalue problem for a constant sparse matrix of large order to be reduced to the solution of the same problem for a polynomial matrix of lower order. Certain solution algorithms are proposed for a partial eigenvalue problem for the polynomial matrix. Questions of the realization of the algorithms on a model PRORAB computer are examined.

A method of solving the partial eigenvalue problem

A method of solving the partial eigenvalue problem