Method For Calculation Of Eigenvalues Research Articles

Novel method for measuring regional precipitation complexity characteristics based on multiscale permutation entropy combined with CMFO-PPTTE model

The influence of entropy values obtained under different multiscale permutation entropy scales on the complexity and unconstrained extreme value problem of traditional bionics algorithms was assessed. The multiscale permutation entropy (MSPE) eigenvalue calculation method based on the chaotic moth-flame optimization-based projection pursuit threat target evaluation (CMFO-PPTTE) model is proposed (CMFO-PPTTE-MSPE). The CMFO-PPTTE-MSPE not only solves the problems of rough calculation and information loss in traditional eigenvalue calculation but also improves the shortcomings of slow convergence and local optimality in the intelligent bionic algorithm. For this purpose, the monthly precipitation of 13 administrative regions in Heilongjiang Province under the Global Precipitation Climatology Centre dataset (GPCC) from 1967 to 2017 was evaluated to improve the precipitation complexity entropy accuracy. The main influencing factors were altitude (p < 0.05), water area (p < 0.05), urban construction area (p < 0.05) and forestland area (p < 0.01). Radial basis function (RBF) neural networks were used to forecast the precipitation in Heilongjiang Province over 36 months, and a better forecast was obtained. To verify the rationality of CMFO-PPTTE-MSPE under GPCC data, compared with the situ data, it is found that GPCC data can more accurately classify the complexity grade of Daxing’anling region. At the same time, the administrative discrimination of GPCC data (1.107) is significantly higher than the situ data (1.023). To verify the rationality of the CMFO-PPTTE-MSPE, the partial mean of MSPE (MSPE-PM), whale optimization algorithm PPTTE (WOA-PPTTE) and ω-particle swarm optimization PPTTE (ω-PSO-PPTTE) models were also used to calculate the MSPE eigenvalues under GPCC data. Comparisons and evaluations were performed after dividing the grade based on geographical discrimination (CMFO-PPTTE-MSPE (1.107) > WOA-PPTTE-MSPE (1.094) > W-PSO-PPTTE-MSPE (1.090) > MSPE-PM (1.063)) and administrative discrimination (CMFO-PPTTE-MSPE (1.146) > W-PSO-PPTTE-MSPE (1.002) > MSPE-PM (1.012) > WOA-PPTTE-MSPE (1.002)). The CMFO-PPTTE-MSPE had a significantly higher distinguishing capabilities than the other algorithms and thus could better distinguish the complexity of precipitation in different regions and geographic locations. In summary, the CMFO-PPTTE-MSPE is beneficial for analyzing precipitation complexity and represents a novel approach for mining the fine structural features of regional hydrological series. However, determining how to apply artificial intelligence algorithms to reasonably calibrate key parameters of the MSPE algorithm to further improve the accuracy of complexity diagnosis for hydrological series will be an important avenue of research.

Fluctuations of the membrane with piecewise smooth contour and mixed boundary conditions

The eigenvalue problem for the two-dimensional operator Laplace is classical in mathematics and physics. However, computing methods for calculation of eigenvalues have still many problems, especially in applications to acoustic and electromagnetic wave guides. The investigated below two-dimensional spectral for the Laplace operator have been previously considered by the author only in smooth areas. The solutions of these tasks (eigen functions) are infinitely differentiated or. even analytical and therefore in order to create effective algorithms it is necessary to consider this enormous a priori information. Traditional methods of finite differences and finite elements almost do not practically use the information on smoothness of the decision, i.e. these are methods with saturation. The term “saturation” was entered by K.I. Babenko. Using the method of computing experiment the author investigates the task about fluctuations of the membrane with the piecewise smooth contour for two-dimensional area, obtained by conformal representation of the square. It is shown that eigen functions are infinitely differentiated. Therefore, numerical algorithms without saturation are applicable. In article the calculation algorithm of eigenvalues in this two-dimensional area is developed, which allows determining up to 10 natural frequencies with the accuracy acceptable for practice on the grid 10×10.

Calculations of Electromechanical Eigenvalues Based on Generating Unit Instantaneous Power and Angular Speed Waveforms at a Step Disturbance

The paper presents the results of calculations of eigenvalues (associated with electromechanical phenomena) of a power system model state matrix made on the basis of analysis of disturbance waveforms of generating unit instantaneous power and angular speed. The waveforms that occur after introducing a disturbance in the form of a change in the voltage regulator reference voltage in one of the power system generating units were taken into account in the calculations. The power system model included the impact of a central frequency regulator. The eigenvalue calculation method used in the investigations consists in approximation of the analysed instantaneous power disturbance waveforms by the waveforms being a superposition of modal components associated with the searched eigenvalues and their participation factors. This approximation involves the minimisation of an objective function defined as the mean square error between the approximated and approximating waveforms. For minimisation of the objective function, a hybrid optimisation algorithm consisting of serially connected genetic and gradient algorithms was used. This combination allows one to eliminate fundamental drawbacks of both algorithms. DOI: 10.12736/issn.2300-3022.2015207

Coarse mesh finite difference formulation for accelerated Monte Carlo eigenvalue calculation

An efficient Monte Carlo (MC) eigenvalue calculation method for source convergence acceleration and stabilization is developed by employing the Coarse Mesh Finite Difference (CMFD) formulation. The detailed methods for constructing the CMFD system using proper MC tallies are devised such that the coarse mesh homogenization parameters are dynamically produced. These involve the schemes for tally accumulation and periodic reset of the CMFD system. The method for feedback which is to adjust the MC fission source distribution (FSD) using the CMFD global solution is then introduced through a weight adjustment scheme. The CMFD accelerated MC (CMFD-MC) calculation is examined first for a simple one-dimensional multigroup problem to investigate the effectiveness of the accelerated fission source convergence process and also to analyze the sensitivity of the CMFD-MC solutions on the size of coarse meshes and on the number of CMFD energy groups. The performance of CMFD acceleration is then assessed for a set of two-dimensional and three-dimensional multigroup (3D) pressurized water reactor core problems. It is demonstrated that very rapid convergence of the MC FSD is possible with the CMFD formulation in that a sufficiently converged MC FSD can be obtained within 20 cycles even for large three-dimensional problems which would require more than 600 inactive cycles with the standard MC fission source iteration scheme. It is also shown that the optional application of the CMFD formulation in the active cycles can stabilize FSDs such that the real-to-apparent variance ratio of the local tallies can be reduced. However, due to the reduced importance of the variance bias in fine local tallies of 3D MC eigenvalue problems, the effectiveness of CMFD in tally stabilization turns out to be not so great.

New Eigenvalue Method for Power System Stability Analysis

The most widely adopted methodology for the analysis of power system small signal stability relies on the approach in the frequency domain, i.e. to linearize the system equation to obtain a linear model as well as the system matrix of which the eigenvalues can be calculated to determine the system stability. However, we often have high order of system matrix and thus it will be undesirable to calculate and analyze all the system eigenvalues. This paper is to explore the problem of small signal stability for power system and the main purpose is to find out the worst-damped mode of system eigenvalues and thus to alleviate the effort for computing and analyzing all the system eigenvalues. The developed algorithm is to be tested on a sample power system to validate the feasibility of the proposed new eigenvalue calculation method.

The Eigenvalues and The Optimal Potential Functions of Sturm-Liouville Operators

In this article, we study the eigenvalues and potential functions of Sturm-Liouville operators with ordinary separated-type boundary condition. When the gap of the first two eigenvalues reaches minimum, we give the specific form of potential function. Meanwhile, for step potential function, we establish an one-to-one relationship between the eigenvalues and the nonnegative real roots of a class of algebraic equation, which provide an effective method for the approximate calculation of eigenvalues.

Non-parametric methods for [formula omitted]-gain estimation using iterative experiments

In this paper we develop non-parametric methods to estimate the L 2 -gain ( H ∞ -norm) of a linear dynamical system from iterative experiments. This work is mainly motivated by model error modeling, where the error dynamics are more complex than can be captured by a low order parametric model. The standard system identification approach to the gain estimation problem is to estimate a parametric model of the system, which is then used to calculate the gain. If it is possible to update the input signal during the experiment, an alternative way is to iteratively optimize the input signal in order to maximize the estimated input to output gain. A key observation is that the gradient of the gain with respect to the input signal can, without knowing a model, be found from two experiments. Iterative numerical methods for calculation of eigenvalues of matrices, e.g., the Power Method or the Lanczos Method, can then be applied to update the input signal sequence between experiments in order to find the maximum gain. The main difficulty compared to the corresponding eigenvalue problem in numerical analysis is the effects of additive measurement noise, which require modified schemes that avoid bias errors. Three such related methods are derived and evaluated by a numerical example. Partial results on convergence and statistical properties of the gain estimator are given. A constrained stochastic gradient method with local optimization of step-length gives the best numerical results in the case of noisy data.

Variational calculations of eigenstates for intramolecular nuclear motions using nontensor product bases

A method for the variational calculation of eigenvalues and eigenfunctions of a Hamiltonian for bounded nuclear motion in a molecule is presented. As a variational basis, a nontensor product function set is chosen in which the functions are eigenfunctions of the Hamiltonian in the adiabatic separable approximation. For the case where the potential along a favored straight coordinate ξ is strongly nonharmonic, and perpendicular to this coordinate only harmonic terms with ξ-dependent force constants appear, the secular matrix elements can be calculated from general selection rules with respect to vibrational quantum numbers and one-dimensional integrals. Numerical test calculations were carried out.

Determination of Safety Operating Regimes Based on the Analysis of Characteristic Equation of State-Space Model

Determining safe regions of operation is important at the design, control and optimization of process systems. Process alarms and safety systems should be designed based on such knowledge. In classical control systems these regions are represented as independent constraints on process variables. In case of a highly exothermic reaction takes place in any kind of reactor there is a high risk to develop reactor runaway if the produced heat can’t be removed. Reactor runaway phenomena is a serious problem in many chemical industrial processes, hence the most of reactions are exothermic. Reactor runaway means a sudden and considerable change in process variables such as reactor temperature. To detect and forecast the development of runaway a model based criteria is introduced in this manuscript. The criteria is based on indirect Ljapunov’s stability analysis of a process model. The aim of this work is to compare methods for calculation of eigenvalues. Eigenvalues of the Jacobian matrix were calculated with a numerical method and there were also computed analytically. Stability of two-phases fed-batch reactor with highly exothermic reactions are analyzed with the investigated stability analysis methods. Analytical solution is more sensitive and of course it is more accurate than the numerical one still the algorithm based on the analytical solution is a lot more faster.

Small-signal stability based on a synchronous generator model with field mutual-leakage reactance

Power system small-signal stability studies have been carried out using a synchronous machine model with a field mutual-leakage reactance called the Canay reactance in the d-axis equivalent circuit. The influence of the Canay reactance on the effectiveness of a power system stabilizer (PSS) was evaluated by the eigenvalue calculation method. The values of the Canay reactance reported in the literature were also investigated statistically, and an approximate expression applicable to power system stability studies was proposed. © 2005 Wiley Periodicals, Inc. Electr Eng Jpn, 154(2): 19–29, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20163

Correctness of Dorodnitsyn's Method for Approximate Calculation of Eigenvalues in a Class of Boundary Value Problems

Dorodnitsyn suggested [1] a method for calculating approximately eigenvalues of Sturm–Liouville boundary value problems with a spectral parameter λ : y′′ + (λr(x) + q(x))y = 0, 0 ≤ x ≤ `, h1y(0) = hy(0), H1y(`) +Hy(`) = 0. (1) Here the function q(x) is assumed to be continuous and real-valued over the closed interval [0, `]; h, h1, H, H1 ∈ R, and the coefficient r(x) has the form r(x) = r1(x)x, where α > −1, and r1(x) is a continuous positive function on [0, `]. In this situation, eigenvalues of problem (1) form a strictly increasing sequence {λn}n=1 with the asymptotic √ λn ∼ πn / ` ∫

Three-Dimensional First-Order Perturbation Calculation Method for Reactivity Changes Due to Core Deformations of Fast Breeder Reactors

A three-dimensional diffusion calculation method has been proposed to rapidly and accurately calculate reactivity changes of LMFBRs caused by assembly displacements in accidental events. The method requires shorter computation times and provides almost the same accuracy as a conventional direct eigenvalue calculation method. In this method, changes in macroscopic neutron cross-sections and diffusion coefficient are defined so that changes in both region volume and material composition can be treated in a mesh-centered finite-difference program under the same coarse mesh division as used for the normal, non-deformed core. Reactivity changes are calculated from the above-mentioned changes by the first-order perturbation method using normal and adjoint neutron fluxes calculated beforehand for the normal core. The method was applied to deformations of a 1,000-MWe LMFBR core. Reactivity changes calculated by the method agreed within 0.4% with those by a conventional direct eigenvalue calculation method, while ...

Numerical calculation of eigenvalues for the Schrödinger equation. III

AbstractA recently developed method for calculation of eigenvalues is applied to a four coupled oscillator system previously used to test more approximate methods. Analysis is presented to show how the present method scales for systems of two, three, and four coupled oscillator systems.

Numerical solution of eigenvalues for the schröndinger equation. II

A recently developed method for the quantum mechanical calculation of eigenvalues is applied to a Hamiltonian consisting of three coupled oscillators which are resonantly coupled. The agreement of the present calculation with a large variational calculation is excellent.

A New Method for Calculation of Eigenstates for a System of a Core and Two Valence Nucleons

A method for calculation of eigenvalues and eigenstates for a system of a core and two valence nucleons is suggested. It consists of approximating the potential by a sum of separable terms, for which the Lippman-Schwinger equation is solved exactly. The wave functions have the exact "three-particle" asymptotic form, and approximate the "two-particle" asymptotics.

Modification of Davidson's method for the calculation of eigenvalues and eigenvectors of large real-symmetric matrices: “root homing procedure”

Davidson's method for the iterative calculation of eigenvalues of large, real-symmetric matrices which makes possible the direct determination of higher roots without knowledge of the exact lower ones is modified to yield just such roots whose eigenvectors have a desired structure, that is, a desired set of dominant components. The algorithm is treated theoretically and applied to some configurational interaction (CI) problems. A study of the convergence behavior of Davidson's scheme is carried out.

Remarks on the Continued Fraction Calculation of Eigenvalues and Eigenvectors

For eigenvalue problems in which the secular determinant has tridiagonal form, e.g., the rigid asymmetric rotor; the secular equation may be written in the form f(λ′)=0, where f(λ′) is a continued fraction and λ′ an eigenvalue. Furthermore, if the secular problem is of nth order, then the continued fraction f(λ′) may be developed in n different ways. Since the eigenvalues are roots of a function f(λ), it is convenient to find the eigenvalues by means of the Newton-Raphson iterative procedure. This requires that the derivative of f(λ) with respect to λ(f′(λ)) be determined. An exact expression for f′(λ) is derived and it is shown that f′(λ′) is in fact the norm of the eigenvector belonging to the eigenvalue λ′. A simple recursion formula, in continued fraction form, for the eigenvector elements is also derived. The Newton-Raphson procedure is further shown to be equivalent to the variational method for iterative calculation of eigenvalues. The former procedure has, however, the advantage of bypassing the necessity of solving a set of simultaneous equations. Advantage is taken of the relation between f′(λ′) and the eigenvector of λ′ to formulate a reasonable criterion for choosing the best possible development of f(λ) is order to avoid convergence to an undesired root of f(λ).