Complex Symmetric Matrices Research Articles

Modified Newton–CAPRESB method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matrices

In this paper, we contemplate addressing nonlinear problems involving complex symmetric Jacobian matrices. Firstly, we establish a parameter-free method called modified Newton–CAPRESB (MN–CAPRESB) method by harnessing the modified Newton method as the outer iteration and the CAPRESB (Chebyshev accelerated preconditioned square block) method as the inner iteration. Secondly, the local and semilocal convergence theorems of MN–CAPRESB method are proved under some conditions. Eventually, the numerical experiments of two kinds of complex nonlinear equations are presented to validate the feasibility of MN–CAPRESB method compared to other existing iteration methods.

Some properties of a complex symmetric operator

A square matrix with complex entries which is equal to its transpose is called a complex symmetric matrix. Such matrices are found in many areas of pure and applied mathematics. A conjugation on H is defined as a conjugate linear isometric involution C on a Hilbert Space H. A bounded linear operator T : H → H has a complex symmetric matrix with regard to an orthonormal basis if, for every conjugation C on H, T = CT *C. Such an operator is called a Complex symmetric operator. Several characteristics of Complex Symmetric matrices and associated operators are discussed in this article.

Universality of spectral fluctuations in open quantum chaotic systems

Quantum chaotic systems with one-dimensional spectra follow spectral correlations of Orthogonal (OE), Unitary (UE), or Symplectic Ensembles (SE) of random matrices depending on their invariance under time reversal and rotation. In this letter, we study the non-Hermitian and non-unitary ensembles based on the symmetry of matrix elements, viz. ensemble of complex symmetric, complex asymmetric (Ginibre), and self-dual matrices of complex quaternions. The eigenvalues for these ensembles lie in the two-dimensional plane. We show that the fluctuation statistics of these ensembles are universal and quantum chaotic systems belonging to OE, UE, and SE in the presence of a dissipative environment show similar spectral fluctuations. The short-range correlations are studied using spacing ratio and spacing distribution. For long-range correlations, unfolding at a non-local scale is crucial. We describe a generic method to unfold the two-dimensional spectra with non-uniform density and evaluate correlations using number variance. We find that both short-range and long-range correlations are universal. We verify our results with the quantum kicked top in a dissipative environment that can be tuned to exhibit symmetries of OE, UE, and SE in its conservative limit.

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

Ornstein-Uhlenbeck operators and holomorphic representations for a class of bounded symmetric domains

Let G be the group of holomorphic automorphisms of a bounded symmetric domain D. We discuss the correspondence between unitarizing measures for scalar-valued holomorphic discrete series representations of G and Ornstein-Uhlenbeck operators on D. We complement the results obtained for the domains in [1] and [6] by showing that for Cartan domain II of n×n symmetric complex matrices and domain III of skew-symmetric complex matrices Z such that ZZ⁎<I, the vector fields in the infinitesimal holomorphic representation are the entries of Hua matrices. Similar results stay valid for some Kähler submanifolds M of D, assuming that the entries of the matrices in M depend linearly on fixed independent entries and M is isomorphic to a product of Cartan domains.

Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces

In this paper, we answer an open conjecture concerning complex symmetric matrices and truncated Toeplitz operators. We study matrix representations of truncated Toeplitz operators with respect to orthonormal bases which are invariant under a canonical conjugation map. In particular, we determine necessary and sufficient conditions for when a symmetric matrix is the matrix representation of a truncated Toeplitz operator with respect to a given conjugation invariant orthonormal basis. We specialise our result to the case when the conjugation invariant orthonormal basis is a modified Clark basis. With this specialisation, we answer an open conjecture in the negative, and show not every unitary equivalence between a complex symmetric matrix and a truncated Toeplitz operator arises from modified Clark basis representations. We pose a new refined conjecture for how to realise a model theory for symmetric matrices through the use of truncated Toeplitz operators, and we show this conjecture is equivalent to a specified system of polynomial equations being satisfied.

Modified Newton-EHS method for solving nonlinear problems with complex symmetric Jacobian matrices

&lt;abstract&gt;&lt;p&gt;This manuscript is devoted to the study of numerical methods for a class of nonlinear problems. Instead of the standard Newton method, an efficient nonlinear solver is suggested to be used, and it is referred to as the Newton-EHS method, where "EHS" stands for Euler-extrapolated Hermitian-skew-Hermitian splitting. We construct this modified Newton-EHS method by utilizing a modified Newton method as the outer iteration and the EHS method as the inner iteration. Furthermore, we give the derivations of the local and semilocal convergence properties of the proposed method under the Hölder condition. Finally, in order to show the feasibility and validity of our new method, we compare it with some other iterative methods in two numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Fast optimization of viscosities for frequency‐weighted damping of second‐order systems

AbstractWe consider frequency‐weighted damping optimization for vibrating systems described by a second‐order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain the stability of the system. These frameworks also allow us to weight, which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems (QEPs) that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal‐plus‐rank‐one (DPR1) complex symmetric matrices, which we leverage by showing how each QEP can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new approach are verified and illustrated on several numerical examples.

Modified Newton-SSTS method for solving a class of nonlinear systems with complex symmetric Jacobian matrices

Modified Newton-SSTS method for solving a class of nonlinear systems with complex symmetric Jacobian matrices

Characterization of C-symmetric Toeplitz operators for a class of conjugations in Hardy spaces

ABSTRACT In this article, we introduce a new class of conjugations in the scalar-valued Hardy space and provide a characterization of a complex symmetric Toeplitz operator with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator on the vector-valued Hardy space with respect to certain conjugations introduced in [Câmara MC, Kliś-Garlicka K, Ptak M. Complex symmetric completions of partial operator matrices. Linear and Multilinear Algebra. 2019; DOI: 10.1080/03081087.2019.1631246], [Kang D, Ko E, Lee JE. Remarks on complex symmetric Toeplitz operators. Linear Multilinear Algebra. 2020; DOI: 10.1080/03081087.2020.1842847], [Ko E, Lee JE. Remark on complex symmetric operator matrices. Linear Multilinear Algebra. 2019;67(6):1198–1216].

Single step iterative method for linear system of equations with complex symmetric positive semi-definite coefficient matrices

In this study, we propose a new single-step iterative method for solving complex linear systems Az≡(W+iT)z=f, where z,f∈Rn, W∈Rn×n and T∈Rn×n are symmetric positive semi-definite matrices such that null(W)∩null(T)={0}. The convergence of the new method is analyzed in detail and discussion on the obtaining the optimal parameter is given. From Wang et al. (2017)[36] we can write W=PTDWP,T=PTDTP, where DW=Diag(μ1,…,μn),DT=Diag(λ1,…,λn), and P∈Rn×n is a nonsingular matrix and λk, μk satisfy μk+λk=1,0≤λk,μk≤1,k=1,…,n. Then we show that under some conditions on μmax=max{μk}k=1n, the new method has faster convergence rate in comparison with recently introduced methods. Finally, some numerical examples are given to demonstrate the efficiency of the new procedure in actual computation.

Isotropy groups of the action of orthogonal similarity on symmetric matrices

We find an algorithmic procedure that enables the computation and description of the structure of the isotropy subgroups of the group of complex orthogonal matrices with respect to the action of similarity on complex symmetric matrices. A key step in our proof is to solve a certain rectangular block upper triangular Toeplitz matrix equation.

The Impact of Damping in Second-Order Dynamical Systems with Applications to Power Grid Stability

We consider a broad class of second-order dynamical systems and study the impact of damping as a system parameter on the stability, hyperbolicity, and bifurcation in such systems. We prove a monotonic effect of damping on the hyperbolicity of the equilibrium points of the corresponding first-order system. This provides a rigorous formulation and theoretical justification for the intuitive notion that damping increases stability. To establish this result, we prove a matrix perturbation result for complex symmetric matrices with positive semidefinite perturbations to their imaginary parts, which may be of independent interest. Furthermore, we establish necessary and sufficient conditions for the breakdown of hyperbolicity of the first-order system under damping variations in terms of observability of a pair of matrices relating damping, inertia, and Jacobian matrices, and propose sufficient conditions for Hopf bifurcation resulting from such hyperbolicity breakdown. The developed theory has significant applications in the stability of electric power systems, which are one of the most complex and important engineering systems. In particular, we characterize the impact of damping on the hyperbolicity of the swing equation model which is the fundamental dynamical model of power systems, and demonstrate Hopf bifurcations resulting from damping variations.

Pipelined Preconditioned Conjugate Gradient Methods for real and complex linear systems for distributed memory architectures

Preconditioned Conjugate Gradient (PCG) is a popular method for solving large and sparse linear systems of equations. The performance of PCG at scale is affected due to the costly global synchronization steps that arise in dot-products on distributed memory systems. Pipelined PCG (PIPECG) removes the costly global synchronization steps from PCG by only executing a single non-blocking allreduce per iteration and overlapping it with independent computations.In our previous work, we have developed a novel pipelined PCG algorithm called PIPECG-OATI (One Allreduce per Two Iterations) for real linear systems which executes a single non-blocking allreduce per two iterations and provides a large overlap of global communication with independent computations at higher number of cores. Our method achieves this overlap by using iteration combination and by introducing new recurrence and non-recurrence computations. We implement optimizations in the PIPECG-OATI method to use cache memory efficiently.In this work, we present PIPECG-OATI-c method for linear systems with complex Hermitian positive definite and complex symmetric matrices. We compare our method with various pipelined CG methods on a variety of problems and demonstrate that our method always gives the least run times. We performed experiments with our method using 20M and 30M unknowns on up to 16K cores and obtained up to 2.48X performance improvement over PCG and 2.14X performance improvement over PIPECG methods. We also experimented with up to 1-billion unknowns on 16K cores, the largest problem size explored for the CG problem, to our knowledge, and obtained about 25% improvement over PCG.

Solving Sylvester equation with complex symmetric semi-definite positive coefficient matrices

Combination of real and imaginary parts (CRI) works well for solving complex symmetric linear systems. This paper develops a generalization of CRI method to determine the solution of Sylvester matrix equation. We show that this, regardless of condition, converges to solution of the Sylvester equation. At the end we test the new scheme by solving a numerical example.

階層型領域分割法による電磁界解析における四倍精度演算の検討

This paper deals with a finite element analysis (FEA) of an electromagnetic problem with a quadruple precision floating point number. A Conjugate Gradient (CG) method for real symmetric matrices and a Conjugate Orthogonal Conjugate Gradient (COCG) method for complex symmetric matrices are implemented in the quadruple precision. Then, an FEA solver with the quadruple precision arithmetic for the electromagnetic problem is implemented by the C language. As a result, the convergence of the iterative method is improved, and the minimum residual norm is reduced by approximately three orders of magnitude.

Eigenvalue problems for a class of infinite complex symmetric tridiagonal matrices with related three-term recurrence relation

We consider the eigenvalue problem of a class of infinite complex symmetric tridiagonal matrices whose diagonal and off-diagonal elements diverge in modulus, and which have a compact inverse. We regard the matrix as a linear operator mapping a maximal domain in Hilbert space ℓ2 into ℓ2. This paper aims to extend the work of Ikebe et al. on a class of eigenvalue problems and for which asymptotic error estimates have been obtained. In this paper we focus on the following points: (1) considering what class of zero-finding problems of three-term recurrence relations can be reformulated as eigenvalue problems of the class of infinite tridiagonal matrices stated above; (2) determining a class of matrices for which obtaining good approximate eigenvalues is guaranteed by using those of truncated principal sub-matrices; and (3) determining a class of matrices that permits us the asymptotic error estimates computed as in (2).

Structure of minimal 2-spheres of constant curvature in the complex hyperquadric

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both CPn and the hyperquadric of CPn. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices modulo an appropriate group action. Using this description, many examples, such as constantly curved holomorphic 2-spheres of higher degree, nonhomogenous minimal 2-spheres of constant curvature, etc., are constructed. Uniqueness is proven for the totally real constantly curved 2-sphere minimal in both the hyperquadric and CPn.

Complex skew-symmetric conference matrices

Complex conference matrices have received considerable attention in the last few years due to their application in quantum information theory and in geometry. Various results and constructions are known for symmetric and Hermitian conference matrices. In this article, we deal mainly with constructions of complex skew-symmetric conference matrices. We classify all complex conference matrices up to order 5. Furthermore, we classify all complex symmetric, skew-symmetric and Hermitian conference matrices of order 6.

A global variant of the COCR method for the complex symmetric Sylvester matrix equation AX + XB = C

Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric M-Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation AX+XB=C with complex symmetric coefficient matrices. To obtain the smooth and monotone convergence behavior, we also propose a smoothed Gl-COCR method, denoted by SGl-COCR. Finally, numerical examples are given to illustrate the performances of our methods.