Complete Eigenvalue Problem Research Articles

Эквивалентирование динамического межотраслевого баланса при решении задач устойчивости

Since 2017, the Federal State Statistics Service (Rosstat) has published “cost-output” tables on its website. Together with some parts of its statistical database of studies, this helps to create dynamic inter-branch models of a sufficiently high dimension. These models are designed to tackle issues of stability, analysis and improvement of the own dynamic properties of economic systems. Models can be used either independently or integrated into complexes with computable equilibrium models abbreviated as CGE (Computable General Equilibrium), as well as with other agent-focused models. Improvement here is driven by the active use of the theory and methods of linear algebra, differential equations, and operations research. The need to repeatedly solve the complete eigenvalue problem on this path makes the improvement algorithm computationally challenging. Still, among the defined components of the movement, determined by the structure of the economy, there are those uncontrolled and poorly controlled by the parameters of regulation. This makes relevant a formalized reduction of the dimension of the original model, adequate to the problem being solved. The paper introduces a technique of equivalenting high-dimensional dynamic inter-industry models for effective work as part of improving algorithms for the intrinsic/internal dynamic properties of economic systems.

Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

Analysis of a fast-acting waveguide ferrite phase shifter with longitudinal magnetization phasing structure

The electrodynamic analysis and calculation of the phasing structure of a waveguide ferrite Faraday phase shifter with longitudinal magnetization of a ferrite rod with a square cross-section are performed. In the strict electrodynamic formulation, the Galerkin method solves the key problem for the design of the phase shifter: the problem of eigenwaves finding with a second-order differential magnetic operator in the projection procedure for a rectangular waveguide with a transversely inhomogeneous ferrite-dielectric filling under longitudinal magnetization. The problem is reduced to solving the complete eigenvalue problem of a complex matrix. The computational algorithm, implemented as a computer program in DELPHI, gives in one procedure the numerical values of the coefficients of the decomposition of fields in the form of eigenvectors of the matrix, and the coefficients of wave propagation in the form of its eigenvalues. The results of calculations of the phasing structure activity for two particular cases of its implementation are presented: on the basis of a ferrite rod both with a conductive coating and without a coating, it is shown that in the second case the activity of the phase shifter is 1.5 times higher. For the real parameters of the ferrite medium, the length of the phasing structure is calculated depending on the transverse dimensions of the ferrite rod and the square waveguide, which ensures the creation of a 360° controlled phase shift.

Computationally efficient finite-difference modal method for the solution of Maxwell’s equations

In this work, a new implementation of the finite-difference (FD) modal method (FDMM) based on an iterative approach to calculate the eigenvalues and corresponding eigenfunctions of the Helmholtz equation is presented. Two relevant enhancements that significantly increase the speed and accuracy of the method are introduced. First of all, the solution of the complete eigenvalue problem is avoided in favor of finding only the meaningful part of eigenmodes by using iterative methods. Second, a multigrid algorithm and Richardson extrapolation are implemented. Simultaneous use of these techniques leads to an enhancement in terms of accuracy, which allows a simple method such as the FDMM with a typical three-point difference scheme to be significantly competitive with an analytical modal method.

Instantaneous Stokes Flow in a Conical Apex Aligned with Gravity and Bounded by a Stress-Free Surface

The instantaneous viscous motion near the apex of a fluid cone of included angle $2\theta_0$ with stress-free boundary conditions is studied in the limit of zero Reynolds number. Gravity acts parallel to the conical axis, and surface tension is neglected. When $2\theta_0 > 134.6^{\circ}$ the dominant term of the solution close to the apex is independent of gravity and depends only on the far-field boundary conditions; the leading behavior is thus a self-similar solution of the second kind. In this case the instantaneous flow is akin to the steady flow in a rigid cone reported by Liu and Joseph [SIAM J. Appl. Math., 34 (1978), pp. 286--296] who showed that toroidal eddies appear below a critical apex angle. For $2\theta_0 < 134.6^{\circ}$ the flow close to the apex is dominated by gravity and represents a self-similar solution of the first kind. The complete eigenvalue problem is solved and example streamline patterns are presented. We estimate the short-time behavior of flow in the neighborhood of the ape...

On multiple solutions of the Fock-space coupled-cluster method

Intermediate Hamiltonian techniques are used to investigate the problem of multiple solutions of the Fock-space coupled-cluster methods. Two intermediate Hamiltonian schemes are presented, based upon similarity transformations that unlike the wave operator formalism allow for following the complete eigenvalue problem. Solutions of the standard effective Hamiltonian formulation that requires iterative approaches to solve the equations are equivalently given by diagonalization of the intermediate Hamiltonian matrix representations. This procedure provides a systematic way of obtaining all possible solutions of the effective Hamiltonian scheme in all sectors higher than the lowest one.

Solution of an eigenvalue problem for pencils of band matrices

An iterative algorithm is proposed for solving the complete eigenvalue problem of a regular, linear pencil A-λB of matrices A and B of band structure which under certain conditions preserves the band structure of matrices of the pencil. It is modification of the algorithm AB-1 based on applying nonorthogonal transformations. A detailed description of the algorithm is presented in application to the pencils with tridiagonal and pentadiagonal matrices.

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.

Solving the eigenvalue problem for matrices

One presents some algorithms related among themselves for solving the partial and the complete eigenvalue problem for an arbitrary matrix. Algorithm 1 allows us to construct the invariant subspaces and to obtain with their aid a matrix whose eigenvalues coincide with the eigenvalues of the initial matrix and belong to a given semiplane. Algorithm 2 solves the same problem for a given strip. The algorithms 3 and 4 reduces the complete eigenvalue problem of an arbitrary matrix to some problem for a quasitriangular matrix whose diagonal blocks have eigenvalues with identical real parts. Algorithm 4 finds also the unitary matrix which realizes this transformation. One gives Algol programs which realize the algorithms 1–3 for real matrices and testing examples.

On the efficient and accurate solution of the skew-symmetric eigenvalue problem: An arrangement of new and already known algorithmic formulations

Algorithms are presented to solve the special eigenvalue problem AZ = Zλ , where A is skew-symmetric. The effective use of Householder's method, the bisection method and inverse iteration for solving the complete eigen-value problem are described in some detail. Simultaneous vector iteration is formulated for skew-symmetric matrices. The amount of work for the skew-symmetric Jacobi algorithm and the simultaneous vector iteration may be reduced by using the solution of a simplified eigenvalue problem. For Hermitian matrices also quadratic eigenvalue bounds for groups of eigenvalues and linear bounds for groups of eigenvectors are derived. The case where the set of calculated eigenvectors is not orthonormal is considered in some detail. In principle, the skew-symmetric eigenvalue problem may be easily transformed into a symmetric eigenvalue problem; but such a procedure has the following disadvantages: first, the results are in general less accurate, and, second, the eigenvectors which belong to well separated eigenvalues are not uniquely determined.

Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form

The solution of the complete eigenvalue problem for a non-normal matrix A presents severe practical difficulties when A is defective or close to a defective matrix. However in the presence of rounding errors one cannot even determine whether or not a matrix is defective. Several of the more stable methods for computing the Jordan canonical form are discussed together with the alternative approach of computing well-defined bases (usually orthogonal) of the relevant invariant subspaces.

An algorithm for numerical determination of the structure of a general matrix

An algorithm, proposed by V. N. Kublanovskaya, for solving the complete eigenvalue problem of a degenerate (that is defective and/or derogatory) matrix, is studied theoretically and numerically. It uses successiveQR-factorizations to determine annihilated subspaces.

On a method of solving the complete eigenvalue problem for a degenerate matrix

On a method of solving the complete eigenvalue problem for a degenerate matrix

A method for the solution of the complete eigenvalue problem

IN THIS paper we shall give some iterative methods for the solution of the complete eigenvalue problem of a matrix. These can be easily carried out on electronic computers and the main portion of the work can be done with fixed-point arithmetic without materially reducing the accuracy. The method for solving the complete eigenvalue problem for normal matrices is given in Q 1. It resembles Jacobi’s process in its structure (see [I], 5 Sl). The method of solution in the case of an arbitrary matrix with a simple structure is given in 5 2. It resembles the triangular power method (see [l], 9 78). For simplicity we shall consider real matrices only, although all the methods given can be applied to complex matrices also without particular difficulty.

Triangular orthogonal methods for the solution of the complete eigenvalue problem

Triangular orthogonal methods for the solution of the complete eigenvalue problem

On some algorithms for the solution of the complete eigenvalue problem

In this article we give the basis and details of algorithms used in the solution of the complete problem of the eigenvalues of a matrix, already briefly explained in a note of the author's [1], and their extension to provide the solution of the partial problem. In § 1 we consider algorithms I and II, which solve the problem for a real non-symmetric non-singular matrix with real eigenvalues, different in modulus. In § 2 we consider algorithms III and IV, which enable us to find the complete spectrum and eigenvectors of the matrices AA′ and A′ A, starting only from the matrices A, and without having to construct A′ A and AA′. In § 3 we consider a modification of these four algorithms which enables us to solve the partial problem. All these algorithms are iterative. They are similar to Jacobian processes in their realization [2] (and also can be carried out using plane rotations), but differ from them as far as their region of convergence is concerned, approaching the triangular step method [3] and the LR algorithm [4]. They can easily be put into effect on a computer, and make it possible to determine both the eigenvalues and the eigenvectors in one operation, thus simplifying the programming. They also allow the greater part of the computations to be made in fixed point without any consequent loss of accuracy. We should expect the algorithms to be stable with respect to rounding errors, since each elementary step in the process reduces to the formation of a linear combination of two columns (rows) of a matrix with coefficients which are less than unity in absolute magnitude. In all the algorithms the sequence of matrices A k and the sequence of left triangular matrices Λ k = ( l ij ( k) ) are constructed recurrently, where Λ k is obtained from the corresponding matrix A k by multiplying it on its right by the orthogonal matrix τ k = ( t ij ( k) ). The orthogonal matrices are constructed using elementary rotation or reflection matrices. In our examples below we shall dwell briefly on the construction of the latter. The reader can find a detailed account in the book [5].