Singular Eigenvalues Research Articles

Benchmarks of MATRIX Blockset for REX System

This paper presents performance benchmarks of several high-level LAPACK subroutines converted to C language in the author’s previous work. The tested algorithms cover singular value decomposition (DGESDD), eigenvalues (DGEES, DGEEV) and Sylvester equation (DTR-SYL modified for general matrices). Benchmarks compare: original LAPACK functions compiled by GFortran, C language versions compiled by GCC and by Visual Studio 2015 (VS2015) and C language versions embedded to REX function blocks compiled by VS2015 for Windows and by GCC for Raspbian. All tests are provided on the PC with Windows and on the Raspberry Pi 3 Model B+,The computation speed on Raspberry Pi is similar in all cases but on PC the speed for matrices of orders 250 and higher is up to 1.5 times faster compared to the speed of original code compiled by GFortran.

Detecting Non-Markovianity of Quantum Evolution via Spectra of Dynamical Maps.

We provide an analysis on non-Markovian quantum evolution based on the spectral properties of dynamical maps. We introduce the dynamical analog of entanglement witness to detect non-Markovianity and we illustrate its behavior with several instructive examples. It is shown that for several important classes of dynamical maps the corresponding evolution of singular values and/or eigenvalues of the map provides a simple non-Markovianity witness.

Bulk behaviour of some patterned block matrices

We investigate the bulk behaviour of singular values and/or eigenvalues of two types of block random matrices. In the first one, we allow unrestricted structure of order m × p with n × n blocks and in the second one we allow m × m Wigner structure with symmetric n × n blocks. Different rows of blocks are assumed to be independent while the blocks within any row satisfy a weak dependence assumption that allows for some repetition of random variables among nearby blocks. In general, n can be finite or can grow to infinity. Suppose the input random variables are i.i.d. with mean 0 and variance 1 with finite moments of all orders. We prove that under certain conditions, the Marcenko-Pastur result holds in the first model when m → ∞ and \(\tfrac{m} {p} \to c \in (0,\infty )\), and the semicircular result holds in the second model when m → ∞. These in particular generalize the bulk behaviour results of Loubaton [10].

MATRIX Blockset for REX Control System

Abstract: This paper deals with a function block library (blockset) for embedded linear algebra (matrix) computations. The blockset is based on a subset of BLAS and LAPACK subroutines converted to C language with respect to memory limitations of typical embedded platforms. The subset covers algorithms for real matrices with double precision elements in three areas: singular value decomposition, eigenvalues and Sylvester equation. The Implemented blockset contains besides suitably wrapped BLAS and LAPACK algorithms also function blocks for discretization of continuous linear systems and several helper blocks. The typical use of the blockset is demonstrated on examples in the REX control system.

Computing the Maslov index for large systems

We address the problem of computing the Maslov index for large linear symplectic systems on the real line. The Maslov index measures the signed intersections (with a given reference plane) of a path of Lagrangian planes. The natural chart parameterization for the Grassmannian of Lagrangian planes is the space of real symmetric matrices. Linear system evolution induces a Riccati evolution in the chart. For large order systems this is a practical approach as the computational complexity is quadratic in the order. The Riccati solutions, however, also exhibit singularites (which are traversed by changing charts). Our new results involve characterizing these Riccati singularities and two trace formulae for the Maslov index as follows. First, we show that the number of singular eigenvalues of the symmetric chart representation equals the dimension of intersection with the reference plane. Second, the Cayley map is a diffeomorphism from the space of real symmetric matrices to the manifold of unitary symmetric matrices. We show the logarithm of the Cayley map equals the arctan map (modulo 2 i 2\mathrm {i} ) and its trace measures the angle of the Langrangian plane to the reference plane. Third, the Riccati flow under the Cayley map induces a flow in the manifold of unitary symmetric matrices. Using the natural unitary action on this manifold, we pullback the flow to the unitary Lie algebra and monitor its trace. This avoids singularities, and is a natural robust procedure. We demonstrate the effectiveness of these approaches by applying them to a large eigenvalue problem. We also discuss the extension of the Maslov index to the infinite dimensional case.

Mixed finite elements for electromagnetic analysis

The occurrence of spurious solutions is a well-known limitation of the standard nodal finite element method when applied to electromagnetic problems. The two commonly used remedies that are used to address this problem are (i) The addition of a penalty term with the penalty factor based on the local dielectric constant, and which reduces to a Helmholtz form on homogeneous domains (regularized formulation); (ii) A formulation based on a vector and a scalar potential. Both these strategies have some shortcomings. The penalty method does not completely get rid of the spurious modes, and both methods are incapable of predicting singular eigenvalues in non-convex domains. Some non-zero spurious eigenvalues are also predicted by these methods on non-convex domains. In this work, we develop mixed finite element formulations which predict the eigenfrequencies (including their multiplicities) accurately, even for nonconvex domains. The main feature of the proposed mixed finite element formulation is that no ad-hoc terms are added to the formulation as in the penalty formulation, and the improvement is achieved purely by an appropriate choice of finite element spaces for the different variables. We show that the formulation works even for inhomogeneous domains where ‘double noding’ is used to enforce the appropriate continuity requirements at an interface. For two-dimensional problems, the shape of the domain can be arbitrary, while for the three-dimensional ones, with our current formulation, only regular domains (which can be nonconvex) can be modeled. Since eigenfrequencies are modeled accurately, these elements also yield accurate results for driven problems.

Singular-value (and eigenvalue) distribution and Krylov preconditioning of sequences of sampling matrices approximating integral operators

SUMMARYLet k(·,·) be a continuous kernel defined on Ω × Ω, Ω bounded subset of R d , d≥ 1, and let us consider theintegral operator K˜ from C(Ω) into C(Ω) (C(Ω) set of continuous functions on Ω) defined as the mapf(x) → l(x) =Z Ω k(x,y)f(y)dy, x∈ Ω.K˜ is a compact operator and therefore its spectrum forms a bounded sequence having zero as uniqueaccumulation point. Here we first consider in detail the appr oximation of K˜ by using rectangle formulain the case where Ω = [0,1] and the step is h= 1/n. The related linear application can be represented asa matrix A n of size n. In accordance with the compact character of the continuous operator, we prove that{A n } ∼ σ 0 and {A n } ∼ λ 0, i.e., the considered sequence has singular values and eigenvalues clustered atzero. Moreover the cluster is strong in perfect analogy with the compactness of K˜. Several generalizationsare sketched, with special attention to the general case of pure sampling sequences, and few examples andnumerical experiments are critically discussed, including the use of GMRES and preconditioned GMRESfor large linear systems coming from the numerical approximation of integral equations of the form((I− K˜)f(t))(x) = g(x), x∈ Ω, (1)with (Kf˜ (t))(x) =R

Singular values and eigenvalues of matrices in $\mathfrak{so}_n(\mathbb{C})$ and $\mathfrak{sp}_n(\mathbb{C})$

We give a complete relation between the singular values and eigenvalues of a complex skew symmetric matrix in terms of multiplicative majorization and double occurrences of singular values and eigenvalues. Similar studies are given for matrices in the algebras $\mathfrak{sp}_n(\mathbb{C})$ and $\mathfrak{sp}_n(\mathbb{R})$.

Study of Characteristic Equation of the Elastic Stress Field Near Bimaterial Notches

Fracture occurs at interface corners due to stress singularity which generates as a result of material discontinuity and geometrical configuration. In elastic stress field near a bimaterial notch tip, eigenvalues extracted from Airy’s stress function approach determine the order of singularity. In this paper, the characteristic equation of elastic stress field near bimaterial notches is investigated. The study is done on singular eigenvalues as well as the first non-singular eigenvalue which has not been well studied before. First, different combination of materials and geometrical configurations for two of the most applicable paths in the Bogy diagram (β = 0, β = α/4) were studied and the results were comprehensively discussed. It was shown that the geometrical and materials configurations near a bimaterial notch tip can significantly affect on the stress singularity near these corners. Finally, the areas between two lines β = 0 and β = α/4 in the Bogy diagram with high stress singularities were determined and discussed for both the first and second singular eigenvalue.

Eigenvalue decay of positive integral operators on the sphere

We obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in R m + 1 \mathbb {R}^{m+1} , m ≥ 2 m\geq 2 , under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined using standard differentiation in R m + 1 \mathbb {R}^{m+1} . In this paper, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by Rudin in the early fifties and genuinely spherical in nature. The rates we present depend on both, the differentiability order used to define the smoothness conditions and the dimension m m . They are shown to be optimal.

A. Horn's result on matrices with prescribed singular values and eigenvalues

We give a new proof of a classical result of A. Horn on the existence of a matrix with prescribed singular values and eigenvalues.

Wavepacket approach to the cumulative reaction probability within the flux operator formalism

Expressions for the singular flux operator eigenfunctions and eigenvalues are given in terms of the Dirac delta-function representable as a localized Gaussian wavepacket. This functional form enables computation of the cumulative reaction probability N(E) from the wavepacket time-correlation functions. The Gaussian based form of the flux eigenfunctions, which is not tied to a finite basis of a quantum-mechanical calculation, is particularly useful for approximate calculation of N(E) with the trajectory based wavepacket propagation techniques. Numerical illustration is given for the Eckart barrier using the conventional quantum-mechanical propagation and the quantum trajectory dynamics with the approximate quantum potential. N(E) converges with respect to the Gaussian width parameter, and the convergence is faster at low energy. The approximate trajectory calculation overestimates tunneling in the low energy regime, but gives a significant improvement over the parabolic estimate of the tunneling probability.

On finite basis set implementation of the exchange-only optimized effective potential method

In this paper, we analyze a structure of the basis set optimized effective potential (OEP) equations from the Fredholm alternative point of view and present one of possible numerical schemes to solve the OEP equation in a stable manner. The solution is constructed as a sum of a unique solution on the subspace of eigenfunctions of the response matrix with non-zero eigenvalues and a non-unique solution on a counterpart subspace with singular eigenvalues. Non-uniqueness of a solution is exploited to obtain a local effective potential that satisfies the condition for the highest occupied molecular orbital (HOMO) without restricting the variational freedom of the optimization procedure. Unlike the existing methods we implement the HOMO condition using the functions of the null-subspace. Numerical results for the total and orbital energies based on the proposed scheme are close to the corresponding literature data.

Legendre spectral methods for the − grad(div) operator

This paper describes two Legendre spectral methods for the − grad(div) eigenvalue problem in R 2 . The first method uses a single grid resulting from the P N ⊗ P N discretization in primal and dual variational formulations. As is well-known, this method is unstable and exhibits spectral ‘pollution’ effects: increased number of singular eigenvalues, and increased multiplicity of some eigenvalues belonging to the regular spectrum. Our study aims at the understanding of these effects. The second spectral method is based on a staggered grid of the P N ⊗ P N - 1 discretization. This discretization leads to a stable algorithm, free of spurious eigenmodes and with spectral convergence of the regular eigenvalues/eigenvectors towards their analytical values. In addition, divergence-free vector fields with sufficient regularity properties are spectrally projected onto the discrete kernel of − grad(div), a clear indication of the robustness of this algorithm.

Robust Wing Flutter Suppression Considering Aerodynamic Uncertainty

Ar obust aeroservoelastic stability analysis considering frequency-domain aerodynamic uncertainty is utilized fo rr obust control law design for flutter suppression of a flexible wing. The problem of stabilizing the wing in flutter using a minimum amount of control power is posed. For this purpose, numerical optimization is used to minimize the norm of a simple low-order controller subject to constraints on robust closed-loop stability. Robust stability is enforced in the optimization problem by posing constraints on the upper bounds on structured singular values and eigenvalues obtained from a linear stability analysis. The resulting controller is synthesized using gain scheduling, and robust wing flutter suppression is demonstrated in wind-tunnel testing.

Migration preconditioning with curvelets.

In this paper, the property of Curvelet transforms for preconditioning the migration and normal operators is investigated. These operators belong to the class of Fourier integral operators and pseudo-differential operators, respectively. The effect of this preconditioner is shown in term of improvement of sparsity, convergence rate, number of iteration for the Krylov-subspace solver and clustering of singular(eigen) values. The migration operator, which we employed in this work is the common-offset Kirchoff-Born migration.

Analysis of ill-conditioned power-flow problems using voltage stability methodology

Ill-conditioned power-flow problems have been widely investigated and reported in the literatun:. A typical approach develops enhanced solution algorithms when a power-flow case is found divergent with the conventional Newton method. It is known that a genuine ill-conditioned problem is caused by the presence of a large condition number in the power-flow Jacobian matrix. Since a large condition number is associated with small singular values or eigenvalues of a matrix and the voltage collapse is also related to small eigenvalues, it is therefore postulated that an ill- conditioned power-flow problem is actually a voltage collapse problem. The objective of ths paper is to investigate the relationship between power-flow ill-conditioning and voltage instability. The findings confirm that power-flow ill-conditioning only occurs at the voltage collapse point. As a result, developing improved algorithms to solve the problem is an unprofitable strategy. The well- known voltage stability assessment techniques such as the PV curve method are sufficient for the problem. This conclusion is supported with case studies on five widely known ill-conditioned power- flow problems and rigorous mathematical analysis.

Construction of Matrices with Prescribed Singular Values and Eigenvalues

Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m (≤ n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.

Singular regular differential equations and eigenvalues of products of matrices*

. We survey some recent results on linear representations of free groups motivated by problems on linear differential equations and we generalize a theorem of Simpson on necessary and sufficient conditions for the existence of linear representations of finitely generated free groups with prescribed similarity classes.

Some inequalities for singular values and eigenvalues of generalized Schur complements of products of matrices

In this paper, using transformation of Schur complements of matrices and some estimates of eigenvalues of positive semidefinite Hermitian matrices, we obtain some inequalities of singular values for Schur complements of products of matrices.