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Summary: This paper deals with the filtering problem for a class of discrete‐time state‐saturated systems subject to randomly occurring nonlinearities and missing measurements. A set of mutually independent Bernoulli random variables is used to describe the random occurrence of the missing measurements. Due to the simultaneous consideration of the state saturation, the randomly occurring nonlinearities, and the missing measurements, it is extremely hard to calculate the actual filtering error covariance in a closed form. As such, the objective of this paper is to construct an upper bound for the filtering error covariance and then design the filter parameters to minimize such an upper bound. The performance of the proposed filters is analyzed in terms of boundedness and monotonicity. Specially, we have shown that the minimum upper bound is always bounded under a mild assumption. Moreover, the relationship between the estimator performance and the arrival probability of the measurements is discussed. A numerical simulation is used to demonstrate the effectiveness of the filtering method. [ABSTRACT FROM AUTHOR]

The recent paper [1] presents a formula for an interval containing all eigenvalues of a real symmetric matrix. The formula requires only the trace of the matrix and the trace of its square. This formula is here generalised to complex matrices. [ABSTRACT FROM AUTHOR]

This paper presents a new bounded real lemma for discrete-time linear descriptor systems. We first present a necessary and sufficient condition for the admissibility of discrete-time descriptor systems expressed in terms of strict linear matrix inequalities (LMIs). We then prove that a discrete-time descriptor system is l2-stable if and only if a symmetric matrix satisfying two strict LMIs can be found. Two illustrative examples are provided for verification. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we propose a fast algorithm for computing the spectral radii of symmetric nonnegative tensors. In particular, by this proposed algorithm, we are able to obtain the spectral radii of weakly reducible symmetric nonnegative tensors without requiring the partition of the tensors. As we know, it is very costly to determine the partition for large‐sized weakly reducible tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radii of large‐sized tensors. As an application, we present an algorithm for testing the positive definiteness of Z‐tensors. By this algorithm, it is guaranteed to determine the positive definiteness for any Z‐tensor. [ABSTRACT FROM AUTHOR]

Summary: The British Geological Survey's global geomagnetic model, Model of the Earth's Magnetic Environment (MEME), is an important tool for calculating the strength and direction of the Earth's magnetic field, which is continually in flux. While the ability to collect data from ground‐based observation sites and satellites has grown rapidly, the memory bound nature of the original code has proved a significant limitation on the size of the modelling problem required. In this paper, we describe work done replacing the bespoke, sequential, eigensolver with that of the PETSc/SLEPc package for solving the system of normal equations. Adopting PETSc/SLEPc also required fundamental changes in how we built and distributed the data structures, and as such, we describe an approach for building symmetric matrices that provides good load balance and avoids the need for close coordination between the processes or replication of work. We also study the memory bound nature of the code from an irregular memory accesses perspective and combine detailed profiling with software cache prefetching to significantly optimise this. Performance and scaling characteristics are explored on ARCHER, a Cray XC30, where we achieved a speed up for the solver of 294 times by replacing the model's bespoke approach with SLEPc. [ABSTRACT FROM AUTHOR]

We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incidence structure of the design. An ordinary 2‐(v,k,λ) $(v,k,\lambda)$ design has a single concurrence λ $\lambda $, and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design TDλ(k,u) ${\text{TD}}_{\lambda }(k,u)$ has two concurrences λ $\lambda $ and 0, and its concurrence matrix is circulant. In this paper, we survey the known PGDs by highlighting their concurrences and constructions. Then we investigate which symmetric circulant matrices are realized as the concurrence matrices of PGDs. In particular, we try to give a list of all PGDs of order up to 12 each of which has a circulant concurrence matrix. We then describe these designs along with their combinatorial properties and constructions. This work is part of the second author's Ph.D. dissertation [46]. [ABSTRACT FROM AUTHOR]

The delay-dependent stability problem for systems with time-delay varying in an interval is addressed in this study. Based on Lyapunov-Krasovskii theory the proposed methods formulated as linear matrix inequality problems are able to check the stability interval when the time-varying delay d(t) belongs to an interval [t1, t2]. The Lyapunov-Krasovskii functional (LKF) selected in the present paper is simpler than some ones considered in the literature. However, the criteria obtained, based on this simple LKF, outperformed the similar existing ones in all numerical tests accomplished in this paper. Notation: Through out this paper the superscript T stands for transpose. 0 refers to a null matrix with appropriate dimensions. For a real symmetric matrix M, the notation M > 0 ( < 0) means that M is the positive (negative) definite. The symmetric term in a matrix is denoted by [ABSTRACT FROM AUTHOR]

This paper presents a class of limited memory preconditioners (LMP) for solving linear systems of equations with symmetric indefinite matrices and multiple right-hand sides. These preconditioners based on limited memory quasi-Newton formulas require a small number k of linearly independent vectors and may be used to improve an existing first-level preconditioner. The contributions of the paper are threefold. First, we derive a formula to characterize the spectrum of the preconditioned operator. A spectral analysis of the preconditioned matrix shows that the eigenvalues are all real and that the LMP class is able to cluster at least k eigenvalues at 1. Secondly, we show that the eigenvalues of the preconditioned matrix enjoy interlacing properties with respect to the eigenvalues of the original matrix provided that the k linearly independent vectors have been prior projected onto the invariant subspaces associated with the eigenvalues of the original matrix in the open right and left half-plane, respectively. Third, we focus on theoretical properties of the Ritz-LMP variant, where Ritz information is used to determine the k vectors. Finally, we illustrate the numerical behaviour of the Ritz limited memory preconditioners on realistic applications in structural mechanics that require the solution of sequences of large-scale symmetric saddle-point systems. Numerical experiments show the relevance of the proposed preconditioner leading to a significant decrease in terms of computational operations when solving such sequences of linear systems. A saving of up to 43% in terms of computational effort is obtained on one of these applications. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The accurate modeling of triply and doubly degenerate seam together while computing the coupled multiple potential energy surfaces globally remains a challenging task due to the unavailability of a suitable model function taking care of this aspect. In this paper, we present such a function capable of predicting many‐fold electronic degeneracies simultaneously as well as the features found away from the vicinity of such degeneracies if present in the coupled surfaces disregarding the symmetry requirements. The basis of the proposed model comes from the analytical diagonalization of a 3×3 real valued diagonally symmetric diabatic matrix and thus, the resulting three symbolic Eigen valued functions form the basis of model to be used in this work. From the analytical model, an attempt was made to provide an analytical expression for the adiabatic‐diabatic transformation angle. Few applications were considered starting from the simple physical model systems to actual realistic molecule illustrating how the present model function are being applied to them to show the presence of degeneracies among the three electronic states. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a layer potential theory in L2‐based weighted Sobolev spaces on Lipschitz bounded and exterior domains of ℝn, n ≥ 3, for the anisotropic Stokes system with L∞ viscosity tensor coefficient satisfying an ellipticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well‐posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of ℝn, with the given data in L2‐based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well‐posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials. [ABSTRACT FROM AUTHOR]

Large size photovoltaic (PV) systems face a large number of issues based on malfunction and unfavourable climatic conditions such as partial shading conditions (PSCs). These PSCs are the major causes of PV systems' performance degradation. In this paper, a symmetric matrix (SM) game puzzle is used to reconfigure the electrical connections of the PV array system. Present shade dispersion methodology is based on the 'physical reallocation of PV module‐fixed electrical connections' principle. Modification in the electrical connections of conventional total cross‐tied (TCT) PV array configuration introduces a new 'SM‐TCT' configuration. An extensive comparative study of conventional TCT and novel‐TCT (NTCT) configurations with proposed SM‐TCT configuration prove the effectiveness to achieve higher performance. The MATLAB/Simulink results are obtained on the basis of the non‐linear nature of current‐voltage and power‐voltage characteristics. SM‐TCT, Shape‐do‐Ku, NTCT and TCT configurations are examined under three realistic PSCs in terms of power and voltage at global maximum power point, improved fill factor, reduced power losses, performance ratio and power enhancement. [ABSTRACT FROM AUTHOR]

Summary: This paper is concerned with the reachable set estimation (RSE) problem for singular systems with both time‐varying delays and bounded peak disturbances. The objective is to search a bounded set that contains all the system states under zero initial conditions. By utilizing the theory of {1}‐inverse and Wirtinger‐based integral inequality, an improved criterion is established in terms of the linear matrix inequalities (LMIs) to guarantee that the reachable set of time‐varying delay singular system is regular, impulse‐free and bounded by the intersection of ellipsoids. Here, a relaxed Lyapunov‐Krasovskii functional is employed to solve the addressed RSE problem which does not require all the involved symmetric matrices to be positive definite. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

Let Hc$$ {H}_c $$ be a (2n)×(2n)$$ (2n)\times (2n) $$ symmetric tridiagonal matrix with diagonal elements c∈ℝ$$ c\in \mathbb{R} $$ and off‐diagonal elements one, and S$$ S $$ be a (2n)×(2n)$$ (2n)\times (2n) $$ diagonal matrix with the first n$$ n $$ diagonal elements being plus ones and the last n$$ n $$ being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil Ac=Hc−λS$$ {\mathcal{A}}_c={H}_c-\lambda S $$ as 2n$$ 2n $$ approaches to infinity. It was conjectured by DL that for any n∈ℕ$$ n\in \mathbb{N} $$ the non‐real eigenvalues λ$$ \lambda $$ of Ac$$ {\mathcal{A}}_c $$ satisfy both |λ+c|<2$$ \mid \lambda +c\mid <2 $$ and |λ−c|<2$$ \mid \lambda -c\mid <2 $$. The conjecture has been verified numerically for a wide range of n$$ n $$ and c$$ c $$, but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non‐real eigenvalues of Ac$$ {\mathcal{A}}_c $$. We provide a proof of the conjecture for n≤3$$ n\le 3 $$, and also in the case where |λ+c|=|λ−c|$$ \mid \lambda +c\mid =\mid \lambda -c\mid $$. In addition, numerics indicate that some phenomena may occur for more general linear pencils. [ABSTRACT FROM AUTHOR]

In this paper, a successive supersymmetric rank-1 decomposition of a real higher-order supersymmetric tensor is considered. To obtain such a decomposition, we design a greedy method based on iteratively computing the best supersymmetric rank-1 approximation of the residual tensors. We further show that a supersymmetric canonical decomposition could be obtained when the method is applied to an orthogonally diagonalizable supersymmetric tensor, and in particular, when the order is 2, this method generates the eigenvalue decomposition for symmetric matrices. Details of the algorithm designed and the numerical results are reported in this paper. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper considers a new approach to a priori sparsification of the sparsity pattern of the factorized approximate inverses (FSAI) preconditioner using the so-called vector aggregation technique. The suggested approach consists in construction of the FSAI preconditioner to the aggregated matrix with a prescribed sparsity pattern. Then small entries of the computed ‘aggregated’ FSAI preconditioning matrix are dropped, and the resulting pointwise sparsity pattern is used to construct the low-density block sparsity pattern of the FSAI preconditioning matrix to the original matrix. This approach allows to minimize (sometimes significantly) the construction costs of low-density high-quality FSAI preconditioners. Numerical results with sample matrices from structural mechanics and thin shell problems are presented. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, systematic implementation of current-mode RMS-to-DC converters based upon MOS translinear (MTL) principle, utilizing symmetric cascoded MTL cell (SCMC) is proposed. Theory of operation and mathematical analysis of both explicit (direct) and implicit (indirect) techniques for realization of SCMC-based RMS-to-DC converters are discussed. The SCMC includes a folded MTL loop and realizes an MTL equation. MTL principle utilizes the square law characteristics of saturated MOS transistors to realize square-root domain (SRD) functions. The SCMC is constructed by two connected cascoded current mirrors and has a compact, symmetric, and multi-purpose structure, with capability of implementing the circuits into the programmable and configurable structures. The proposed RMS-to-DC converters utilize the SCMC along with a configurable current mirror array. The required squaring and square-rooting functions are realized using the SCMC, after proper configuration of the current mirror array. The proposed circuits have been implemented using a reconfigurable architecture fabricated in a 0.5 µm CMOS technology. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper proposes an improvement to the delay-dependent stability of discrete systems with time-varying delays. The approach is based on the observation that the positive definiteness of a chosen Lyapunov-Krasovskii functional does not necessarily require all the involved symmetric matrices to be positive definite, which has been overlooked in the literature. The derived delay-dependent stability conditions are in terms of linear matrix inequalities. It is theoretically proved that our results are less conservative than the corresponding ones obtained by requiring the positive definiteness of all the symmetric matrices in a chosen Lyapunov-Krasovskii functional. The importance of the present approach is that a great number of delay-dependent analysis and synthesis results obtained by the aforementioned requirement in the literature can be improved by the present approach without introducing any new decision variables. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The antibandwidth maximization problem aims to maximize the minimum distance of entries of a sparse symmetric matrix from the diagonal and as such may be regarded as the dual of the well-known bandwidth minimization problem. In this paper, we consider the feasibility of adapting heuristic algorithms for the bandwidth minimization problem to the antibandwidth maximization problem. In particular, using an inexpensive level-based heuristic, we obtain an initial ordering that we refine using a hill-climbing algorithm. This approach performs well on matrices coming from a range of practical problems with an underlying mesh. Comparisons with existing approaches show that, on this class of problems, our algorithm can be competitive with recently reported results in terms of quality while being significantly faster and applicable to much larger problems. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we consider efficient algorithms for solving the algebraic equation A α u = f, 0<α<1, where A is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in R d, d=1,2,3. This solution is then written as u = A β − α F with F = A − β f with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function tβ−α for 0

Let A be an n × n symmetric matrix of bandwidth 2m + 1. The matrix need not be positive definite. In this paper we will present an algorithm for factoring A which preserves symmetry and the band structure and limits element growth in the factorization. With this factorization one may solve a linear system with A as the coefficient matrix and determine the inertia of A, the number of positive, negative, and zero eigenvalues of A. The algorithm requires between 1/2nm2 and 5/4nm2 multiplications and at most (2m + 1)n locations compared to non-symmetric Gaussian elimination which requires between nm2 and 2nm2 multiplications and at most (3m + 1)n locations. Our algorithm reduces A to block diagonal form with 1 × 1 and 2 × 2 blocks on the diagonal. When pivoting for stability and subsequent transformations produce non-zero elements outside the original band, column/row transformations are used to retract the bandwidth. To decrease the operation count and the necessary storage, we use the fact that the correction outside the band is rank-1 and invert the process, applying the transformations that would restore the bandwidth first, followed by a modified correction. This paper contains an element growth analysis and a computational comparison with LAPACKs non-symmetric band routines and the Snap-back code of Irony and Toledo. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the regression setting, dimension reduction allows for complicated regression structures to be detected via visualisation in a low-dimensional framework. However, some popular dimension reduction methodologies fail to achieve this aim when faced with a problem often referred to as symmetric dependency. In this paper we show how vastly superior results can be achieved when carrying out response and predictor transformations for methods such as least squares and sliced inverse regression. These transformations are simple to implement and utilise estimates from other dimension reduction methods that are not faced with the symmetric dependency problem. We highlight the effectiveness of our approach via simulation and an example. Furthermore, we show that ordinary least squares can effectively detect multiple dimension reduction directions. Methods robust to extreme response values are also considered. [ABSTRACT FROM AUTHOR]

Random effects are particularly useful in insurance studies, to capture residual heterogeneity or to induce cross‐sectional and/or serial dependence, opening hence the door to many applications including experience rating and microreserving. However, their nonobservability often makes existing models computationally cumbersome in a multivariate context. In this paper, it is shown that the multivariate extension to the Gamma distribution based on Wishart distributions for random symmetric positive‐definite matrices (considering diagonal terms) is particularly tractable and convenient to model correlated random effects in multivariate frequency, severity and duration models. Three applications are discussed to demonstrate the versatility of the approach: (a) frequency‐based experience rating with several policies or guarantees per policyholder, (b) experience rating accounting for the correlation between claim frequency and severity components, and (c) joint modeling and forecasting of the time‐to‐payment and amount of payment in microlevel reserving, when both are subject to censoring. [ABSTRACT FROM AUTHOR]

In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability of success pn. Assuming that limn→∞npn=∞, we show that for the random sequence (ρn)n=1∞ given by ρn:=θn+npnθn,θn:=max(maxi≤n‖rowi(Wn)‖22−npn,npn), the ratio ‖Wn‖ρn converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime npn=Θ(logn). In particular, denoting by An the adjacency matrix of the Erdős–Renyi graph 𝒢(n,pn) and by λ|k|(An) its kth largest (by the absolute value) eigenvalue, under the assumptions limn→∞npn=∞ and limn→∞pn=0 we have (1) (No non‐trivial outliers): if liminfnpnlogn≥1log(4/e) then for any fixed k ≥ 2, |λ|k|(An)|2npn converges to 1 in probability; and (2) (Outliers): if limsupnpnlogn<1log(4/e) then there is ε > 0 such that for any k∈ℕ, we have limn→∞ℙ|λ|k|(An)|2npn>1+ε=1. On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices. [ABSTRACT FROM AUTHOR]

SUMMARY The method fast inverse using nested dissection (FIND) was proposed to calculate the diagonal entries of the inverse of a large sparse symmetric matrix. In this paper, we show how the FIND algorithm can be generalized to calculate off-diagonal entries of the inverse that correspond to 'short' geometric distances within the computational mesh of the original matrix. The idea is to extend the downward pass in FIND that eliminates all nodes outside of each node cluster. In our advanced downwards pass, it eliminates all nodes outside of each 'node cluster pair' from a subset of all node cluster pairs. The complexity depends on how far ( i, j) is from the main diagonal. In the extension of the algorithm, all entries of the inverse that correspond to vertex pairs that are geometrically closer than a predefined length limit l will be calculated. More precisely, let α be the total number of nodes in a two-dimensional square mesh. We will show that our algorithm can compute O( α3 ∕ 2 + 2 ε) entries of the inverse in O( α3 ∕ 2 + 2 ε) time where l = O( α1 ∕ 4 + ε) and 0 ≤ ε ≤1 ∕ 4. Numerical examples are given to illustrate the efficiency of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The aim of this paper is to apply the factorized sparse approximate inverse (FSAI) preconditioner to the iterative solution of linear systems with indefinite symmetric matrices. Until now the FSAI technique has been applied mainly to positive definite systems and with a limited success for the indefinite case. Here, it is demonstrated that the sparsity pattern for the preconditioner can be chosen in such a way that it guarantees the existence of the factorization. The proposed scheme shows excellent parallel scalability, performance and robustness. It is applicable with short recurrence iterative methods such as MinRes and SymmLQ. The properties are demonstrated on linear systems arising from mixed finite element discretizations in linear elasticity. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents an O(n2) method based on the twisted factorization for computing the Takagi vectors of an n-by-n complex symmetric tridiagonal matrix with known singular values. Since the singular values can be obtained in O(n2) flops, the total cost of symmetric singular value decomposition or the Takagi factorization is O(n2) flops. An analysis shows the accuracy and orthogonality of Takagi vectors. Also, techniques for a practical implementation of our method are proposed. Our preliminary numerical experiments have verified our analysis and demonstrated that the twisted factorization method is much more efficient than the implicit QR method, divide-and-conquer method and Matlab singular value decomposition subroutine with comparable accuracy. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R-1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The anti-reflective boundary condition for image restoration was recently introduced as a mathematically desirable alternative to other boundary conditions presently represented in the literature. It has been shown that, given a centrally symmetric point spread function (PSF), this boundary condition gives rise to a structured blurring matrix, a submatrix of which can be diagonalized by the discrete sine transform (DST), leading to an O(n2 log n) solution algorithm for an image of size n × n. In this paper, we obtain a Kronecker product approximation of the general structured blurring matrix that arises under this boundary condition, regardless of symmetry properties of the PSF. We then demonstrate the usefulness and efficiency of our approximation in an SVD-based restoration algorithm, the computational cost of which would otherwise be prohibitive. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The symmetric Sinc–Galerkin method developed by Lund (Math. Comput. 1986; 47:571–588), when applied to second-order self-adjoint boundary value problems on d dimensional rectangular domains, gives rise to an N × N positive definite coefficient matrix which can be viewed as the sum of d Kronecker products among d - 1 real diagonal matrices and one symmetric Toeplitz-plus-diagonal matrix. Thus, the resulting coefficient matrix has a strong structure so that it can be advantageously used in solving the discrete system. The main contribution of this paper is to present and analyse a multigrid method for these Sinc–Galerkin systems. In particular, we show by numerical examples that the solution of a discrete symmetric Sinc–Galerkin system can be obtained in an optimal way only using O(N log N) arithmetic operations. Numerical examples concerning one- and two-dimensional problems show that the multigrid method is practical and efficient for solving the above symmetric Sinc–Galerkin linear system. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

An n×n real matrix A is called a bisymmetric matrix if A=AT and A=SnASn, where Sn is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A0, find an n×n bisymmetric matrix A such that $$ AX = B,\qquad A_{0} = A([1\,{:}\,r])$$ where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A*, find an n×n matrix  in SE such that $$|A^{\ast} - {\hskip 2pt\hat{\hskip -2ptA}}\| = \min\limits_{A \in S_{E}}\,\|A^{\ast} - A\|$$ where ∥·∥ is Frobenius norm, and SE is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a growth result is obtained for normalized k-fold symmetric convex biholomorphic mappings on the unit ball of a complex Banach space. 2000 Mathematics Subject Classification 32A22, 32H02. [ABSTRACT FROM PUBLISHER]

Regular variation is an analytic condition on the tails of a probability distribution which is necessary for an extended central limit theorem to hold, when the tails are too heavy to allow attraction to a normal limit. The limiting distributions which can occur are called operator stable. In this paper we show that moving averages of random vectors with regularly varying tails are in the generalized domain of attraction of an operator stable law. We also prove that the sample autocovariance matrix of these moving averages is in the generalized domain of attraction of an operator stable law on the vector space of symmetric matrices. AMS 1990 subject classification. Primary 62M10, secondary 62E20, 62F12, 60F05. [ABSTRACT FROM AUTHOR]

We consider random symmetric matrices with independent entries distributed according to the Haar measure on ℤp for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over ℤp has a non-trivial zero. [ABSTRACT FROM AUTHOR]

Factor models have been widely used in recent years to model high‐dimensional spatio‐temporal data. However, the validity of employing factor models in a specific application has received less attention. This article proposes test statistics for testing the symmetry in cross‐correlation matrices of a high‐dimensional stochastic process implied by exact factor models. A rejection of symmetry indicates that the use of an exact factor model is questionable. Both simulations and real examples are used to demonstrate the applications and to study the finite‐sample performance of the proposed test statistics. Empirical results show that the proposed test statistics are effective in identifying cases where exact factor models are not appropriate, providing valuable guidance for choosing factor models in a high‐dimensional setting. [ABSTRACT FROM AUTHOR]

We show that the shortest non‐zero polynomials vanishing on bounded‐rank matrices and skew‐symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all t$t$‐minors or t$t$‐Pfaffians of a generic matrix or skew‐symmetric matrix, one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a very general t$t$‐dimensional subspace of an affine n$n$‐space does not contain polynomials with fewer than t+1$t+1$ terms. [ABSTRACT FROM AUTHOR]

The paper is concerned with the global convergence of the Jacobi method for symmetric matrices under a special class of cyclic pivot strategies. That class generalizes the well-known class of serial strategies. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The Rayleigh quotient iteration method finds an eigenvector and the corresponding eigenvalue of a symmetric matrix. This is a fundamental problem in science and engineering. ParteD and Kahan have shown, in 1968, that for almost any initial vector in the unit sphere, the Rayleigh quotient iteration method converges to some eigenvector. In this paper, the regions of the unit sphere which include all possible initial vectors converging to a specific eigenvector are studied. The generalized eigenvalue problem Ax = λBx is considered. It is shown that the regions do not change when the matrix is shifted or multiplied by a scalar. These regions are completely characterized in the three-dimensional case. It is shown that, in this case, the area of the region of convergence corresponding to the interior eigenvalue is larger than the area of those corresponding to any extreme one. This can be interpreted, with the appropriate choice of probability distribution, as: the probability of converging to an eigenvector corresponding to the interior eigenvalue is larger than the probability of converging to an eigenvector corresponding to any extreme eigenvalue. It is conjectured that the same is true for matrices of any order. Experiments in higher dimensions are presented which conform with the conjecture. [ABSTRACT FROM AUTHOR]

Unsupervised feature selection is a dimensionality reduction method and has been widely used as an important and indispensable preprocessing step in many tasks. However, real‐world data are not only high‐dimensional, but also have intrinsic correlations between data points, which are not fully utilized in feature selection. Furthermore, real‐world data usually inevitably contain noise or outliers. In order to select features from data more effectively, a sparse regression model based on latent low‐rank representation with the symmetric constraint for unsupervised feature selection is proposed. With the coefficient matrix of non‐negative symmetric low‐rank representation, an affinity matrix characterized by the correlation relationship between data points is adaptively obtained, which reveals the intrinsic geometric relationship, global structure, and discrimination of data points. A latent representation of all data points obtained from this affinity matrix is employed as a pseudo‐label, and feature selection is carried out by sparse linear regression. This method performs feature selection in the learned latent space instead of the original data space. An alternating iteration algorithm is designed to solve the proposed model, and its effectiveness and efficiency are verified on several benchmark data sets. [ABSTRACT FROM AUTHOR]

We 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. [ABSTRACT FROM AUTHOR]

This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (SPD) matrices, seen as a matrix-variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. SPD matrices: the log-Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of SPD matrices. The limiting distributions of these averages are used to provide large-sample confidence regions and two-sample tests for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability. [ABSTRACT FROM AUTHOR]

We give an explicit algorithm for calculating the Kauffman bracket of a link diagram from a Goeritz matrix for that link. Further, we show how the Jones polynomial can be recovered from a Goeritz matrix when the corresponding checkerboard surface is orientable, or when more information is known about its Gordon–Litherland form. In the process we develop a theory of Goeritz matrices for cographic matroids, which extends the bracket polynomial to any symmetric integer matrix. We place this work in the context of links in thickened surfaces. [ABSTRACT FROM AUTHOR]

We establish the existence of a solution to the following problem: Δ2u−α|∇u|r=μS2(D2u)+λfinΩ,u=0=∂u∂non∂Ω,$$\begin{equation*}\hskip7pc \def\eqcellsep{&}\begin{array}{ll}\Delta ^2u-\alpha |\nabla u|^r=\mu S_2(D^2 u)+\lambda f & \mbox{in } \Omega , \\[3pt] u=0=\displaystyle \frac{\partial u}{\partial n} &\mbox{on } \partial \Omega , \end{array} \end{equation*}$$where Ω⊂RN,N=2,3$\Omega \subset {\mathbb {R}}^N,\, N=2,3$, is a smooth and bounded domain and S2(D2u)(x)=∑1≤i0,λ>0$ f\in L^1(\Omega),\, \,\alpha >0,\,\lambda >0$ and 0≤μ≤1$0\le \mu \le 1$ are parameters. Moreover, we assume that r≥1$r\ge 1$ if N=2$N=2$ and 1≤r≤6$1\le r\le 6$ if N=3$N=3$. We use variational arguments and an iterative technique to prove our results. [ABSTRACT FROM AUTHOR]

In this short paper, practical estimates for the extreme eigenvalues of symmetric matrices and their inverses are obtained, and a criterion for Stieltjes matrices is presented. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we present a new type of restarted Krylov method for calculating the smallest eigenvalues of a symmetric positive definite matrix G. The new framework avoids the Lanczos tridiagonalization process and the use of polynomial filtering. This simplifies the restarting mechanism and allows the introduction of several modifications. Convergence is assured by a monotonicity property that pushes the eigenvalues toward their limits. Another innovation is the use of inexact inversions of G to generate the Krylov matrices. In this approach, the inverse of G is approximated by using an iterative method to solve the related linear system. Numerical experiments illustrate the usefulness of the proposed approach. [ABSTRACT FROM AUTHOR]

The study of topological indices of some special chemical graphs has become a hot topic in chemical theory. Let Ln be the dicyclobutadieno derivative of linear phenylene which consists of n hexagons and (n + 1) quadrangles and let Ln2 be the strong prism of the dicyclobutadieno derivative of linear phenylenes. Firstly, we obtain that the normalized Laplacian spectrum of Ln2 consists of the eigenvalues of symmetric matrices LA and LS according to the decomposition theorem of normalized Laplacian characteristic polynomial. Then, by taking full advantage of the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, we determine the explicit expressions of the multiplicative degree‐Kirchhoff index and the number of spanning trees of the graph Ln2. [ABSTRACT FROM AUTHOR]

The main result of this work is extension of the famous characterization of Bravais lattices according to their metrical, algebraic and geometric properties onto a wide class of primitive lattices (including Buerger-reduced, nearly Buerger-reduced and a substantial part of Delaunay-reduced) related to low-restricted semi-reduced descriptions ( s.r.d.'s). While the `geometric' operations in Bravais lattices map the basis vectors into themselves, the `arithmetic' operators in s.r.d. transform the basis vectors into cell vectors (basis vectors, face or space diagonals) and are represented by matrices from the set of all 960 matrices with the determinant ±1 and elements {0, ±1} of the matrix powers. A lattice is in s.r.d. if the moduli of off-diagonal elements in both the metric tensors M and M−1 are smaller than corresponding diagonal elements sharing the same column or row. Such lattices are split into 379 s.r.d. types relative to the arithmetic holohedries. Metrical criteria for each type do not need to be explicitly given but may be modelled as linear derivatives , where denotes the set of 39 highest-symmetry metric tensors, and describe changes of appropriate interplanar distances. A sole filtering of according to an experimental s.r.d. metric and subsequent geometric interpretation of the filtered matrices lead to mathematically stable and rich information on the Bravais-lattice symmetry and deviations from the exact symmetry. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from analysis of a lattice metric to analysis of symmetry matrices [Himes & Mighell (1987). Acta Cryst. A 43, 375-384], (ii) from the isometric approach and invariant subspaces to the orthogonality concept {some ideas in Le Page [ J. Appl. Cryst. (1982), 15, 255-259]} and splitting indices [Stróż (2011). Acta Cryst. A 67, 421-429] and (iii) from fixed cell transformations to transformations derivable via geometric information (Himes & Mighell, 1987; Le Page, 1982). It is illustrated that corresponding arithmetic and geometric holohedries share space distribution of symmetry elements. Moreover, completeness of the s.r.d. types reveals their combinatorial structure and simplifies the crystallographic description of structural phase transitions, especially those observed with the use of powder diffraction. The research proves that there are excellent theoretical and practical reasons for looking at crystal lattice symmetry from an entirely new and surprising point of view - the combinatorial set of matrices, their semi-reduced lattice context and their geometric properties. [ABSTRACT FROM AUTHOR]

SUMMARY For software to fully exploit the computing power of emerging heterogeneous computers, not only must the required computational kernels be optimized for the specific hardware architectures but also an effective scheduling scheme is needed to utilize the available heterogeneous computational units and to hide the communication between them. As a case study, we develop a static scheduling scheme for the tridiagonalization of a symmetric dense matrix on multicore CPUs with multiple graphics processing units (GPUs) on a single compute node. We then parallelize and optimize the Basic Linear Algebra Subroutines (BLAS)-2 symmetric matrix-vector multiplication, and the BLAS-3 low rank symmetric matrix updates on the GPUs. We demonstrate the good scalability of these multi-GPU BLAS kernels and the effectiveness of our scheduling scheme on twelve Intel Xeon processors and three NVIDIA GPUs. We then integrate our hybrid CPU-GPU kernel into computational kernels at higher-levels of software stacks, that is, a shared-memory dense eigensolver and a distributed-memory sparse eigensolver. Our experimental results show that our kernels greatly improve the performance of these higher-level kernels, not only reducing the solution time but also enabling the solution of larger-scale problems. Because such symmetric eigenvalue problems arise in many scientific and engineering simulations, our kernels could potentially lead to new scientific discoveries. Furthermore, these dense linear algebra algorithms present algorithmic characteristics that can be found in other algorithms. Hence, they are not only important computational kernels on their own but also useful testbeds to study the performance of the emerging computers and the effects of the various optimization techniques. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A new Bravais-lattice determination algorithm is introduced herein. For error-stable Bravais-lattice determination, Andrews & Bernstein [ Acta Cryst. (1988), A 44, 1009-1018] proposed the use of operations to search for nearly Buerger-reduced cells. Although these operations play an essential role in their method, they increase the computation time, in particular when lattice parameters obtained in (powder) auto-indexing are supposed to contain large errors. The new algorithm requires only several permutation matrices in addition to the operations that are necessary when the lattice parameters have exact values. As a result, the computational efficiency of error-stable Bravais-lattice determination is improved considerably. Furthermore, the new method is proved to be error stable under a very general assumption. The detailed algorithms and the set of matrices sufficient for error-stable determination are presented. [ABSTRACT FROM AUTHOR]