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Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With his note I hope to clarify the history of this bound and some of its generalizations.

Francis turbines operating in full load conditions feature an axisymmetric vortex rotating in the opposite direction of the turbine runner. In low-pressure conditions, the resulting cavitation vortex may enter in an unstable self-exciting process, leading to large pressure pulsations and oscillations in the generating unit power output. In this research work, prototype on-site and reduced scale model test results are presented where the turbine changes from a stable to an unstable full load condition due to an increase in discharge. Measurements are compared in the frequency and time domain, where similarities are evidenced between model and prototype. Using the measurements on the reduced scale model and 1-D numerical models of both the reduced scale model and the turbine prototype, eigenvalue calculations are performed to predict the discharge value of transition from stable to unstable conditions. The transition point on the prototype is then predicted with a small deviation. Transient simulations in the time domain are performed replicating the self-exciting behavior of the unstable full load condition.

International audience; This paper deals with the non-linear vibration of sandwich viscoelastic shell structures. Coupling a harmonic balance method with the Galerkin's procedure, one obtains an amplitude equation depending on two complex coefficients. The latter are determined by solving a classical eigenvalue problem and two linear ones. This permits to get the non-linear frequency and the non-linear loss factor as functions of the displacement amplitude. To validate our approach, these relationships are illustrated in the case of a circular sandwich ring.

In this paper, we study the order of a maximal clique in an amply regular graph with a fixed smallest eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that if a strongly regular graph contains a Delsarte clique, then the parameter μ is either small or large. Furthermore, we obtain a cubic polynomial that assures that a maximal clique in an amply regular graph is either small or large (under certain assumptions). Combining this cubic polynomial with the claw-bound, we rule out an infinite family of feasible parameters (v, k, λ, μ) for strongly regular graphs. Lastly, we provide tables of parameters (v, k, λ, μ) for nonexistent strongly regular graphs with smallest eigenvalue −4, −5, −6 or −7. Ministry of Education (MOE) Gary Greaves is partially supported by the Singapore Ministry of Education Academic Research Fund (Tier 1); grant numbers: RG29/18 and RG21/20. Jack H. Koolen is partially supported by the National Natural Science Foundation of China (No. 12071454) and Anhui Initiative in Quantum Information Technologies (No. AHY150000). And the research was partially supported by the project ‘‘Analysis and Geometry on Bundles’’ of Ministry of Science and Technology of the People’s Republic of China. Jongyook Park is partially supported by Basic Science Research Program through the National Research Foundation of Korea funded by Ministry of Education (NRF-2017R1D1A1B03032016) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2020R1A2C1A01101838).

In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

The discretization of the shallow water system on unstructured grids can lead to spurious modes which usually can affect accuracy and/or cause stability problems. This paper introduces a new approach for stability analysis of unstructured linear finite volume schemes for linear shallow water equations with the Coriolis Effect using spectra, pseudospectra, and singular value decomposition. The discrete operator of the scheme is the principal parameter used in the analysis. It is shown that unstructured grids have a large influence on operator normality. In some cases the eigenvectors of the operator can be far from orthogonal, which leads to amplification of solutions and/or stability problems. Large amplifications of the solution can be observed, even for discrete operators which respect the condition of asymptotic stability, and in some cases even for Lax-Richtmyer stable methods. The pseudospectra are shown to be efficient for the verification of stability of finite volume methods for linear shallow water equations. In some cases, the singular value decomposition is employed for further analysis in order to provide more information about the existence of unstable modes. The results of the analysis can be helpful in choosing the type of mesh, the appropriate placements of the variables of the system on the grid, and the suitable discretization method which is stable for a wide range of modes. The authors thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. This publication was made possible by NPRP Grant 4-935-2-354 from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are solely the responsibility of the authors. Appendix A Scopus

The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs., 20 pages, 3 figures v2. updated acknowledgements

Let G be a connected triangle-free graph of order n > 5 with μ ∉ { − 1 , 0 } as an eigenvalue of multiplicity k > 1 . We show that if d is the maximum degree in G then k ≤ n − d − 1 ; moreover, if k = n − d − 1 then either (a) G is non-bipartite and k ≤ ( μ 2 + 3 μ + 1 ) ( μ 2 + 2 μ − 1 ) , with equality only if G is strongly regular, or (b) G is bipartite and k ≤ d − 1 , with equality only if G is a bipolar cone. In each case we discuss the extremal graphs that arise.

We consider the Karpelevic region Θ n ⊂ C consisting of all eigenvalues of all stochastic matrices of order n. We provide an alternative characterisation of Θ n that sharpens the original description given by Karpelevic. In particular, for each θ ∈ [ 0 , 2 π ) , we identify the point on the boundary of Θ n with argument θ. We further prove that if n ∈ N with n ≥ 2 , and t ∈ Θ n , then t is a subdominant eigenvalue of some stochastic matrix of order n.

Free vibrations of fluid-solids structures are governed by a nonsymmetric eigenvalue problem. This problem can be transformed into a definite quadratic eigenvalue problem. The present paper considers properties of definite problems and variational characterization for definite quadratic eigenvalue problem. We propose improvement of existing methods which determine whether a quadratic pencil Q ( λ ) is definite. The improved method is reflected in a better localization of start parameters μ and ξ where Q ( μ ) > 0 > Q ( ξ ) and μ > ξ .

We analyze a possibility of turning off post-smoothing (relaxation) in geometric multigrid when used as a preconditioner in preconditioned conjugate gradient (PCG) linear and eigenvalue solvers for the 3D Laplacian. The geometric Semicoarsening Multigrid (SMG) method is provided by the hypre parallel software package. We solve linear systems using two variants (standard and flexible) of PCG and preconditioned steepest descent (PSD) methods. The eigen-value problems are solved using the locally optimal block preconditioned conjugate gradient (LOBPCG) method available in hypre through BLOPEX software. We observe that turning off the post-smoothing in SMG dramatically slows down the standard PCG-SMG. For flexible PCG and LOBPCG, our numerical tests show that removing the post-smoothing results in overall 40–50 percent acceleration, due to the high costs of smoothing and relatively insignificant decrease in convergence speed. We demonstrate that PSD-SMG and flexible PCG-SMG converge similarly if SMG post-smoothing is off. A theoretical justification is provided.

Motivated by results of Aujla and Silva [3] , we give several more precise weak majorization and eigenvalue inequalities for some matrix versions of the famous Jensen inequality with regard to a convexity. Our main results are then applied to log convex functions. As an application, we obtain refinements of some well-known matrix inequalities.

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let $I_k:=(-\pi,\pi)^k$, let $\mathcal M_s$ be the linear space of complex $s\times s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components $f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$. Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices of size $n_1\cdots n_ks$ generated by $f,g$. Then $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f$, i.e. the family of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic) spectral distribution described by the function $g^{-1}f$, provided $g$ possesses certain properties (which ensure in particular the invertibility of $T_n^{-1}(g)$ for all $n$) and the following topological conditions are met: the essential range of $g^{-1}f$, defined as the union of the essential ranges of the eigenvalue functions $\lambda_j(g^{-1}f),\ j=1,\ldots,s$, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case $g=1$. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems., Comment: 18 pages, 26 figures

Let G be a finite graph with μ as an eigenvalue of multiplicity k . A star set for μ is a set X of k vertices in G such that μ is not an eigenvalue of G − X . We investigate independent star sets of largest possible size in a variety of situations. We note connections with symmetric designs, codes, strongly regular graphs, and graphs with least eigenvalue −2.

We study the endomorphisms ϕ of abelian groups G having a “small” algebraic entropy h (where “small” usually means h(ϕ)

In this paper, a new lower bound on a positive stable block triangular preconditioner for saddle point problems is derived; it is superior to the corresponding result obtained by Cao [Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math. 57 (2007) 899–910]. A numerical example is reported to confirm the presented result.

In this paper, we consider the existence problem to some type of the Lidstone boundary value problem at resonance with discontinuity of the Caratheodory type. We prove that a solution exists if the conditions similar to the Landesman–Lazer ones are satisfied.

We settle in the affirmative a question of Bhatia and Kittaneh. For P and Q positive semidefinite n × n matrices, the inequality σ r ( PQ ) ⩽ 1 2 λ r ( P + Q ) holds for r = 1 , 2 , … , n .

We prove that there exists a unique irreducible Brauer character of height zero in any 2-blocks of the symmetric group. This generalizes the theorem of P. Fong and G.D. James that the dimension of every non-trivial 2-modular simple module of the symmetric group is even.

In 1989, Bai and Demmel proposed the multishift QR algorithm for eigenvalue problems. Although the global convergence property of the algorithm (i.e., the convergence from any initial matrix) still remains an open question for general nonsymmetric matrices, in 1992 Jiang focused on symmetric tridiagonal case and gave a global convergence proof for the generalized Rayleigh quotient shifts. In this paper, we propose Wilkinson-like shifts, which reduce to the standard Wilkinson shift in the single shift case, and show a global convergence theorem.

It is known that the Schur complements of doubly diagonally dominant matrices are doubly diagonally dominant. In this paper, we obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices. Then, as an application we obtain that the eigenvalues of the Schur complements are located in the Brauer Ovals of Cassini of the original matrices under certain conditions. As another application, we obtain an upper bound for the infinity norm on the inverse on the Schur complement of strictly doubly diagonally dominant matrices. Further, based on the derived results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster.

This paper presents a novel technique for the computation of eigenvalues and eigenvectors of partially enclosed basins such as harbours and bays. The procedure makes use of the finite element approximation of the linear shallow water equations, and converts the time-depending problem into an eigenvalues one. The main point of novelty of this research is the mathematical condition used at the boundary that separates the computational domain from the open sea. While classical techniques impose a zero surface elevation (i.e. a nodal line), here an approximate radiation condition is applied. The use of a radiation condition at the open boundary gives a quadratic eigenvalue problem that admits as solution complex eigenvalues and eigenvectors, thus describing the flow in terms of both standing and progressive waves. The new method is applied to an idealized long and narrow harbour, for which an analytical solution of long wave resonance exists, and to the harbour of Marina di Carrara (Italy), for which measurements and previous numerical computation results are available. In both cases the results show good agreement with the available data.

We investigate the family of graphs with many large eigenvalues. It is not hard to see that every graph with many vertices of large degree that are pairwise at distance at least four from each other, has many large eigenvalues. We show that this does not hold if the vertices of large degree are at mutual distance three from each other. We explore this class of graphs further and provide some bounds on their eigenvalues.

In this paper, motivated by a problem posed by Wilkinson, we study the coefficient perturbations of a (square) matrix polynomial to a matrix polynomial that has a prescribed eigenvalue of specified algebraic multiplicity and index of annihilation. For an n × n matrix polynomial P ( λ ) and a given scalar μ ∈ C , we introduce two weighted spectral norm distances, E r ( μ ) and E r , k ( μ ) , from P ( λ ) to the n × n matrix polynomials that have μ as an eigenvalue of algebraic multiplicity at least r and to those that have μ as an eigenvalue of algebraic multiplicity at least r and maximum Jordan chain length (exactly) k, respectively. Then we obtain a lower bound for E r , k ( μ ) , and derive an upper bound for E r ( μ ) by constructing an associated perturbation of P ( λ ) .

Yazar adının diğer kullanılan biçimi: Mahir Hasanov This paper is devoted to multi-parameter eigenvalue problems for one-dimensional perturbed p-Laplacians, modelling travelling waves for a class of nonlinear evolution PDE. Dispersion relations between the eigen-parameters, the existence of eigenfunctions and positive eigenfunctions, variational principles for eigenvalues and constructing solutions in the analytical and implicit forms are the main subject of this paper. We use both variational and analytical methods.

We apply several matrix inequalities to the derivative companion matrices of complex polynomials to establish new bounds and majorization relations for the critical points of these polynomials in terms of their zeros.

This short note presents a simple algebraic result about matrices and associated polynomials which clarifies a central point in the Homotopic Deviation theory presented in chapter 7 of [3] . The polynomial π ˆ is defined by Chatelin in [3] . The polynomial π ˜ is used by Lidskii in [11] , [13] , and by Chatelin in [3] . In this note we prove that these two polynomials are equal up to a nonzero constant.

We prove that, aside from the complete multipartite graphs and graphs of Steiner type, there are only finitely many connected strongly regular graphs with a regular star complement of prescribed degree s ∈ N . We investigate the possible parameters when s ⩽ 5 .

By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in R3 and Brunn–Minkowski and isoperimetric inequalities for related functionals.

In this paper, by using linear programming under weak conditions, we improve the Greub–Rheinboldt inequality and obtain the best known bound. An application is also given.

In this paper, we will establish several Lyapunov inequalities for linear Hamiltonian systems, which unite and generalize the most known ones. For planar linear Hamiltonian systems, the connection between Lyapunov inequalities and estimates of eigenvalues of stationary Dirac operators will be given, and some optimal stability criterion will be proved.

Bounds for the size of the support of a compactly supported momentum density of the Camassa–Holm equation are derived. This is achieved by estimating the first Dirichlet eigenvalue of the support. This elaborates the result on the preservation of its compactness, and gives more information on the velocity by estimating the size of the region where it is not that well understood.

We summarize seventeen equivalent conditions for the equality of algebraic and geometric multiplicities of an eigenvalue for a complex square matrix. As applications, we give new proofs of some important results related to mean ergodic and positive matrices.

We determine the shape which minimizes, among domains with given measure, the first eigenvalue of a nonlocal operator consisting of a perturbation of the standard Dirichlet Laplacian by an integral of the unknown function. We show that this problem displays a saturation behaviour in that the corresponding value of the minimal eigenvalue increases with the weight affecting the average up to a (finite) critical value of this weight, and then remains constant. This critical point corresponds to a transition between optimal shapes, from one ball as in the Faber–Krahn inequality to two equal balls.

[EN] When linearity can be assumed (linear response of heads to stresses), stream-aquifer flow exchange can be simulated as the drainage of a number of independent linear reservoirs. This conceptual model, which can be mathematically deduced in a univocal way from an eigenvalue solution of the linear groundwater flow problem, facilitates the understanding of the physical phenomenon and the analysis of influencing factors. The number of reservoirs required to simulate stream depletion in some ideal homogeneous cases of stream-aquifer connection was analyzed in detail in a previous investigation using analytical eigenvalue solutions [16]. However, most aquifers are heterogeneous in nature and numerical solutions must be employed to analyze whether they could also be simulated using few reservoirs. This paper presents a stochastic analysis of the influence of heterogeneity on the simulation of natural groundwater discharges in aquifers connected to rivers, as a series of linear reservoirs. A Monte-Carlo approach was employed to perform this study. The results show that, on a monthly time scale, many cases (even heterogeneous aquifers) can be simulated using just a few reservoirs with sufficient accuracy and at minimum computational cost. Therefore, this modeling technique can be useful to efficiently simulate the integrated management of complex water resources systems at the basin scale (with many aquifers, reservoirs, demands, etc.) that need to simultaneously consider surface and groundwater flow and stream-aquifer interaction. © 2011 Elsevier Ltd., The authors thank the reviewers for their comments, which have helped to improve significantly the clarity of the original manuscript. This research has been supported by the SAWARES project (GESHYDRO CGL2009-13238-C02-01 & HYDROECOCLIMATE CGL2009-13238-C02-02) of the Spanish Ministry of Science and Innovation (Plan Nacional I+D+I 2008-2011) and the European Community 7th Framework Project GENESIS (226536) on groundwater systems.

Suppose G is a graph and λ 1 , λ 2 , … , λ n are the eigenvalues of G . The Estrada index EE ( G ) of G is defined as the sum of e λ i , 1 ≤ i ≤ n . In this paper some new upper bounds for the Estrada index of bipartite graphs are presented. We apply our result on a ( 4 , 6 ) -fullerene to improve our bound given in an earlier paper.

We prove that the potential q ( x ) of an indefinite Sturm–Liouville problem on the closed interval [ a , b ] with the indefinite weight function w ( x ) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [ a , b ] and two right-definite problems defined on [ a , 0 ] and [ 0 , b ] , where point 0 lies in ( a , b ) and is the turning point of the weight function w ( x ) .

We consider the following problem: Given a set of m×n real (or complex) matrices A1,…,AN, find an m×m orthogonal (or unitary) matrix P and an n×n orthogonal (or unitary) matrix Q such that P*A1Q,…,P*ANQ are in a common block-diagonal form with possibly rectangular diagonal blocks. We call this the simultaneous singular value decomposition (simultaneous SVD). The name is motivated by the fact that the special case with N=1, where a single matrix is given, reduces to the ordinary SVD. With the aid of the theory of *-algebra and bimodule it is shown that a finest simultaneous SVD is uniquely determined. An algorithm is proposed for finding the finest simultaneous SVD on the basis of recent algorithms of Murota–Kanno–Kojima–Kojima and Maehara–Murota for simultaneous block-diagonalization of square matrices under orthogonal (or unitary) similarity.

In this paper, the notion of Birkhoff–James approximate orthogonality sets is introduced for rectangular matrices and matrix polynomials. The proposed definition yields a natural generalization of standard numerical range and q -numerical range (and also of recent extensions), sharing with them several geometric properties.

A tricyclic graph of order n is a connected graph with n vertices and n+2 edges. In this paper, all tricyclic graphs whose second largest eigenvalue does not exceed 1 are identified.

In this paper, inverse spectra problems for a differential pencil are studied. By using the approach similar to those in Hochstadt and Lieberman (1978) [14] and Ramm (2000) [26] , we prove that (1) if p ( x ) (or q ( x ) ) is full given on the interval [ 0 , π ] , then a set of values of eigenfunctions at the mid-point of the interval [ 0 , π ] and one spectrum suffice to determine q ( x ) (or p ( x ) ) on the interval [ 0 , π ] and all parameters in the boundary conditions; (2) if p ( x ) (or q ( x ) ) is full given on the interval [ 0 , π ] , then some information on eigenfunctions at some internal point b ∈ ( π 2 , π ) and parts of two spectra suffice to determine q ( x ) (or p ( x ) ) on the interval [ 0 , π ] and all parameters in the boundary conditions.

In the paper, we shall consider the boundary value problem {u^(^n^)+@la(t)f(t,u,u^',u^'',u^(^3^),...,u^(^n^-^2^))=0,n>=2,t@?(0,1),u^(^i^)(0)=0,0@?i@?n-3,@au^(^n^-^2^)(0)-@bu^(^n^-^1^)(0)=0,@cu^(^n^-^2^)(1)+@du^(^n^-^1^)(1)=0 where @l>0, @a,@b,@c and @d are constants satisfying @a, @c>0 and @b,@d>=0. Intervals of @l are determined to ensure the existence of a positive solution of the boundary value problem according to the signs of a(t) and f.

This paper deals with the instability problem of flexible bridge decks under an approaching crosswind flow within framework of bimodal flutter involving fundamental vertical and torsional modes of bridge vibration. Start from simplifying coefficients from terms of the cubic and quartic polynomials derived from singularity conditions of an integral wind-structure impedance matrix and by using the quasi-steady approach, the approximated formula for calculating onset flutter of the aeroelastic bridge system are proposed. A good agreement is obtained comparing predictions on the onset flutter by the proposed and available formulas for several study cases from the existing bridges with difference of structural and aerodynamic characteristics of given bridge deck.

In this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor. We prove this method converges for any irreducible nonnegative tensor. We also apply this method to study the positive definiteness of a multivariate form.

Let A be the principal 3-block of a finite group G with an abelian Sylow 3-subgroup P. Let C A be the Cartan matrix of A, and we denote by ρ ( C A ) the unique largest eigenvalue of C A . The value ρ ( C A ) is called the Frobenius–Perron eigenvalue of C A . We shall prove that ρ ( C A ) is a rational number if and only if A and the principal 3-block of N G ( P ) are Morita equivalent. This generalizes earlier Wada's theorem in 2007, where he proves it only for the case that the order of P is nine, while we prove it for the case that P is an arbitrary finite abelian 3-group. The result presented here uses the classification of finite simple groups.

The shift-and-invert method is very efficient in eigenvalue computations, in particular when interior eigenvalues are sought. This method involves solving linear systems of the form ([email protected])z=b. The shift @s is variable, hence when a direct method is used to solve the linear system, the LU factorization of ([email protected]) needs to be computed for every shift change. We present two strategies that reduce the number of floating point operations performed in the LU factorization when the shift changes. Both methods perform first a preprocessing step that aims at eliminating parts of the matrix that are not affected by the diagonal change. This leads to about 43% and 50% flops savings respectively for the dense matrices.

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in R 3 , as an application we get the Brunn–Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in R 3 .

This paper introduces a novel neural network based approach for extracting the eigenvalues with the largest or smallest modulus of real skew-symmetric matrices, as well as the corresponding eigenvectors. To this end, unlike the previous neural network based methods that can be summarized by some 2n-dimensional ordinary differential equations (ODEs), where n is the order of the given skew-symmetric matrix, our proposed approach corresponds to an ODE of order n, instead of 2n. Hence, the scale of networks can be reduced a lot. Simulations verify the computational capability of such approach.

We present the extension of the successful Constant Perturbation Method (CPM) for Schrodinger problems to the more general class of Sturm-Liouville eigenvalue problems. Whereas the orginal CPM can only be applied to Sturm-Liouville prob- lems after a Liouville transformation, the more general CPM presented here solves the Sturm-Liouville problem directly. This enlarges the range of applicability of the CPM to a wider variety of problems and allows a more efficient solution of many problems. The CPMs are closely related to the second-order coefficient approxima- tion method underlying the SLEDGE software package, but provide for higher order approximations. These higher order approximations can also be obtained by apply- ing a modified Neumann method. The CPM approach, however, leads to simpler formulae in a more convenient form. In this paper we are concerned with differential equation eigenvalue problems that can be put in the general form of the classical Sturm-Liouville (SL) problem −(p(x)y ' (x)) ' + q(x)y(x) = Ew(x)y(x)