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Two decades ago, Martin andWoodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type Ã1. In this paper, we take that observation far beyond its original scope. We conjecture that for Ãn there is an equivalence of categories between the characteristic p diagrammatic Hecke category and a 'blob category' that we introduce (using certain quotients of KLR algebras called generalized blob algebras). Using alcove geometry we prove the 'graded degree' part of this equivalence for all n and all prime numbers p. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic p give the p-Kazhdan-Lusztig polynomials in type Ãn. We prove this for Ã1, the only case where the p-Kazhdan-Lusztig polynomials are known. This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures. [ABSTRACT FROM AUTHOR]

The exterior algebra E$E$ on a finite‐rank free module V$V$ carries a Z/2$\mathbb {Z}/2$‐grading and an increasing filtration, and the Z/2$\mathbb {Z}/2$‐graded filtered deformations of E$E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on V$V$. We extend this result to A∞$A_\infty$‐algebra deformations A$\mathcal {A}$, showing that they are classified by formal functions on V$V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho–Hong–Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such A$\mathcal {A}$. By applying these ideas to a related construction of Cho–Hong–Lau, we prove a local form of homological mirror symmetry: the Floer A∞$A_\infty$‐algebra of a monotone Lagrangian torus is quasi‐isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential. [ABSTRACT FROM AUTHOR]

This paper deals with the fault detection and isolation (FDI) problem for uncertain closed-loop systems with external disturbances and nonlinear perturbations. To address the system uncertainties and the nonlinear perturbations in different faulty models, adaptive and switching techniques are introduced to construct a bank of FDI observers, such that one of them can match the current system, and the corresponding observer estimate errors can converge asymptotically to zero. An effective FDI scheme is then presented by introducing some model-matching indexes. Moreover, the introduced switching laws liberate the equality constraints often used in the existing FDI approaches, which are hard to satisfy if the system matrices include uncertainties. Finally, a simulation example of F/A-18A automatic carrier landing system is used to illustrate the effectiveness of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper we introduce the Open Cezeri Library (OCL) framework as a domain specific language (DSL) for researchers, scientists, and engineering students to enable them to develop basic linear algebra operations via simple matrix calculations, image processing, computer vision, and machine learning applications in JAVA programming language. OCL provides a strong intuition of coding for the developer while implementing by means of a fluent interface. The significant aspect of the OCL is to combine the methods of well-known platforms; MATLAB and JAVA, accordingly. Moreover, OCL supports a fluent interface so that users can extend a single line of codes by putting a dot between the methods because all the methods implemented actually return the host class. It was observed that the learning curve of the OCL is lower than the MATLAB and the native JAVA languages, and makes coding more readable, understandable, traceable, and enjoyable. In addition to this, the experiments revealed that the running performance of the OCL is quite comparable and can be used in a variety of diverse applications. © 2016 Wiley Periodicals, Inc. Comput Appl Eng Educ 24:736-743, 2016; View this article online at ; DOI [ABSTRACT FROM AUTHOR]

Consistency and transitivity are important and leading research topics in the study of decision-making in terms of pairwise comparison matrices. In this paper, we search for conditions that, in case of inconsistency, guarantee ordinal compatibility between ordinal ranking (actual ranking) derived from a transitive matrix and cardinal rankings provided by the most usual priority vectors proposed in the scientific literature. We provide the notion of weak consistency; it is a condition weaker than consistency and stronger than transitivity and ensures that vectors associated with a matrix, by means of a strictly increasing synthesis functional, provide a preference order, on the related set of decision elements, equal to the actual ranking. This notion extends, to the case in which the decision-maker can be indifferent between two or more alternatives/criteria, weak consistency introduced in previous papers under constraint of no indifference. Finally, we introduce an order relation on the rows of the matrix, that is, a simple order if and only if weak consistency is satisfied; this simple order allows us to easily determine the actual ranking on the set of decision elements. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of linear second-order matrix descriptor differential equations of Apostol-Kolodner type with constant coefficients. In the new approach, we propose a different transformation from what Kalogeropoulos et al. (2009) have used in their recent paper. However, similarly with them, the Weierstrass canonical form has been considered, and the analytical formula for the solution of this general class is derived naturally for consistent initial conditions. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper provides a delay-dependent criterion for a class of descriptor systems with a delay varying in an interval. This criterion, which can ensure the descriptor system is regular, impulse free and stable, involves fewer matrix variables but has less conservatism. Numerical examples illustrate the advantage of the criterion. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [ABSTRACT FROM AUTHOR]

Külshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the ℓ-Cartan matrix for Sn (equivalently, the invariant factors of the Cartan matrix for the Iwahori–Hecke algebra ℋn(q), where q is a primitive ℓth root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when ℓ=pr, p is prime and r≤p, and went on to conjecture that the formulae should hold for all r. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition , the Cartan matrix of an ℓ-block of Sn is a product of Cartan matrices associated to -blocks of Sn. In particular, the invariant factors of the Cartan matrix associated to an ℓ-block of Sn can be recovered from the Cartan matrices associated to the -blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of Sn, not only for the full Cartan matrix, but also for an individual block. We collect evidence for this conjecture by showing that the formulae predict the correct determinant of the ℓ-Cartan matrix. We then go on to show that Hill's conjecture implies the conjecture of Külshammer, Olsson and Robinson. [ABSTRACT FROM PUBLISHER]

In practical applications, when testing parametric restrictions for hidden Markov models (HMMs), one frequently encounters non-standard situations such as testing for zero entries in the transition matrix, one-sided tests for the parameters of the transition matrix or for the components of the stationary distribution of the underlying Markov chain, or testing boundary restrictions on the parameters of the state-dependent distributions. In this paper, we briefly discuss how the relevant asymptotic distribution theory for the likelihood ratio test (LRT) when the true parameter is on the boundary extends from the independent and identically distributed situation to HMMs. Then we concentrate on discussing a number of relevant examples. The finite-sample performance of the LRT in such situations is investigated in a simulation study. An application to series of epileptic seizure counts concludes the paper. [ABSTRACT FROM AUTHOR]

In 1996 Rosenthal and York proposed (time-invariant) BCH convolutional codes [4] in which the parity check matrix of a BCH code is used in the construction of the convolutional code. The lower bound on the minimum free distance of a BCH convolutional code is guaranteed by the BCH limit. In this paper we propose a periodically time-variant convolutional code that can be constructed not only using the BCH parity check matrix but using the check matrix of any binary linear block code and show that the lower bound on the minimum free distance is guaranteed by the minimum free distance of the binary linear block code. In addition, taking 12 binary linear block codes as examples, we perform comparisons of the proposed codes with BCH convolutional codes using three evaluation criteria (minimum free distance, number of delay elements, coding rate) and show that there exist proposed codes that are superior to existing ones. © 2007 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 90(9): 31– 40, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20271 [ABSTRACT FROM AUTHOR]

A matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row averages is called a B-matrix by J. M. Peña in References (SIAM J. Matrix Anal. Appl. 2001; 22:1027–1037) and (Numer. Math. 2003; 95:337–345). In this paper, it is generalized to more extended matrices—MB-matrices, which is proved to be a subclass of the class of P-matrices. Subsequently, we establish relationships between defined and some already known subclasses of P-matrices (see, References SIAM J. Matrix Anal. Appl. 2000; 21:67–78; Linear Algebra Appl. 2004; 393:353–364; Linear Algebra Appl. 1995; 225:117–123). As an application, some subclasses of P-matrices are used to localize the real eigenvalues of a real matrix. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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]

The rank of a vector space $\mathcal{A}$ of $n\times n$-matrices is by definition the maximal rank of an element of $\mathcal{A}$. The space $\mathcal{A}$ is called rank-critical if any matrix space that properly contains $\mathcal{A}$ has a strictly higher rank. This paper exhibits a sufficient condition for rank-criticality, which is then used to prove that the images of certain Lie algebra representations are rank-critical. A rather counter-intuitive consequence, and the main novelty in this paper, is that for infinitely many $n$, there exists an eight-dimensional rank-critical space of $n \times n$-matrices having generic rank $n-1$, or, in other words: an eight-dimensional maximal space of non-invertible matrices. This settles the question, posed by Fillmore, Laurie, and Radjavi in 1985, of whether such a maximal space can have dimension smaller than $n$. Another consequence is that the image of the adjoint representation of any semisimple Lie algebra is rank-critical; in both results, the ground field is assumed to have characteristic zero. [ABSTRACT FROM AUTHOR]

In this paper, the fast-multipole method (FMM) is applied over a 3D formulation of the integral-equation asymptotic-phase (IE-AP) method. The rapid phase variations of the induced currents are included into the IE formulation by using the linearly-phased Rao–Wilton–Glisson (RWG) basis functions, which allows a drastic reduction of the required number of unknowns. The resulting matrix equation is then solved using the FMM, thus combining the efficient representation of the current using the IE-AP formulation with the capability of FMM to reduce the computational cost of the matrix equation resolution. The joint effect is a reduction of the overall computational cost by some orders of magnitude. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1594–1599, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21698 [ABSTRACT FROM AUTHOR]

In this paper the definition of semiseparable matrices is investigated. Properties of the frequently used definition and the corresponding representation by generators are deduced. Corresponding to the class of tridiagonal matrices another definition of semiseparable matrices is introduced preserving the nice properties dual to the class of tridiagonal matrices. Several theorems and properties are included showing the viability of this alternative definition. Because of the alternative definition, the standard representation of semiseparable matrices is not satisfying anymore. The concept of a representation is explicitly formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The past 13 years have seen the development of many algorithms for approximating matrix functions in O(N) time, where N is the basis size. These O(N) algorithms rely on assumptions about the spatial locality of the matrix function; therefore their validity depends very much on the argument of the matrix function. In this article I carefully examine the validity of certain O(N) algorithms when applied to Hamiltonians of disordered systems. I focus on the prototypical disordered system, the Anderson model. I find that O(N) algorithms for the density matrix function can be used well below the Anderson transition (i.e. in the metallic phase;) they fail only when the coherence length becomes large. This paper also includes some experimental results about the Anderson model's behaviour across a range of disorders. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

How to obtain 6-DOF parallel manipulators with optimum global isotropy is investigated in this paper. A systematic method is first presented to get isotropic parallel designs. A measure for spatial isotropy is then proposed to evaluate and compare the global isotropy of obtained manipulators. Efficient methods to find the minimum and maximum singular values of matrices are developed to facilitate the evaluation process. © 2005 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper we consider solving Hermitian Toeplitz systems Tnx=b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non-negative continuous 2π-periodic function ƒ, i.e. Tn=𝒯n[ƒ]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if ƒ is positive then the spectrum of 𝒯n[1/ƒ]𝒯n[ƒ] is clustered around 1. We prove that the trigonometric polynomial qn(s) (s⩾2, cf. (2) and (3)) converges to 1/ƒ uniformly as n→∞ under the condition that 1/ƒ is in Wiener class. It follows that the computational cost of the PCG method can be reduced by replacing 1/ƒ with qN(2), where Nn when ƒ has finite zeros of even orders. We prove that with our preconditioners, the preconditioned matrix has spectrum clustered around 1. It follows that the PCG methods converge very fast when applied to solve the preconditioned systems. Numerical results are given to demonstrate the efficiency of our preconditioners. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Many popular solution methods for large discrete ill-posed problems are based on Tikhonov regularization and compute a partial Lanczos bidiagonalization of the matrix. The computational effort required by these methods is not reduced significantly when the matrix of the discrete ill-posed problem, rather than being a general nonsymmetric matrix, is symmetric and possibly indefinite. This paper describes new methods, based on partial Lanczos tridiagonalization of the matrix, that exploit symmetry. Computed examples illustrate that one of these methods can require significantly less computational work than available structure-ignoring schemes. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

To make possible the application of the three-dimensional finite element method to electrically large problems, it is combined with the analytical solutions of arbitrary large cavity with aperture. In this paper, detailed analysis for this hybrid method is presented. The element matrices necessary for coupling the finite element method to analytical solutions are given. The proposed hybrid method significantly reduces the number of unknown since only the inhomogeneity needs to be discretized. Thus computer memory and storage demands are reduced. In addition, this hybrid method employs the edge element which is not expected to produce spurious solutions. Also, the formulation presented in this paper preserves the sparsity of the finite element matrix, and does not require any matrix inversion. This new hybrid method is used to compute the field distributions in various partially filled rectangular enclosures. The results match well with the pure edge- based finite element method and analytical solutions. [ABSTRACT FROM AUTHOR]

In this study, we introduce the concept of elliptic biquaternion matrices. Firstly, we obtain elliptic matrix representations of elliptic biquaternion matrices and establish a universal similarity factorization equality for elliptic biquaternion matrices. Afterwards, with the aid of these representations and this equality, we obtain various results on some basic topics such as generalized inverses, eigenvalues and eigenvectors, determinants, and similarity of elliptic biquaternion matrices. These valuable results may be useful for developing a perfect theory on matrix analysis over elliptic biquaternion algebra in the future. [ABSTRACT FROM AUTHOR]

This paper analyses a novel method for constructing preconditioners for diagonally dominant symmetric positive-definite matrices. The method discussed here is based on a simple idea: we construct M by simply dropping offdiagonal non-zeros from A and modifying the diagonal elements to maintain a certain row-sum property. The preconditioners are extensions of Vaidya's augmented maximum-spanning-tree preconditioners. The preconditioners presented here were also mentioned by Vaidya in an unpublished manuscript, but without a complete analysis. The preconditioners that we present have only O(n+t2) nonzeros, where n is the dimension of the matrix and 1⩽t⩽n is a parameter that one can choose. Their construction is efficient and guarantees that the condition number of the preconditioned system is O(n2/t2) if the number of nonzeros per row in the matrix is bounded by a constant. We have developed an efficient algorithm to construct these preconditioners and we have implemented it. We used our implementation to solve a simple model problem; we show the combinatorial structure of the preconditioners and we present encouraging convergence results. [ABSTRACT FROM AUTHOR]

In this paper, we construct new ω-circulant preconditioners for non-Hermitian Toeplitz systems, where we allow the generating function of the sequence of Toeplitz matrices to have zeros on the unit circle. We prove that the eigenvalues of the preconditioned normal equation are clustered at 1 and that for (N, N)-Toeplitz matrices with spectral condition number 𝒪(Nα) the corresponding PCG method requires at most 𝒪(N log2N) arithmetical operations. If the generating function of the Toeplitz sequence is a rational function then we show that our preconditioned original equation has only a fixed number of eigenvalues which are not equal to 1 such that preconditioned GMRES needs only a constant number of iteration steps independent of the dimension of the problem. Numerical tests are presented with PCG applied to the normal equation, GMRES, CGS and BICGSTAB. In particular, we apply our preconditioners to compute the stationary probability distribution vector of Markovian queuing models with batch arrival. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We consider the GMRES(s), i.e. the restarted GMRES with restart s for the solution of linear systems Ax = b with complex coefficient matrices. It is well known that the GMRES(s) applied on a real system is convergent if the symmetric part of the matrix A is positive definite. This paper introduces sufficient conditions implying the convergence of a restarted GMRES for a more general class of non-Hermitian matrices. For real systems these conditions generalize the known result initiated as above. The discussion after the main theorem concentrates on the question of how to find an integer j such that the GMRES(s) converges for all s ≥ j. Additional properties of GMRES obtained by derivation of the main theorem are presented in the last section. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present an algorithm for matrix multiplication based on a formula discovered by Pan [7]. For matrices of order up to 10 000, the nearly optimum tuning of the algorithm results in a rather clear non-recursive one- or two-level structure with the operation count comparable to that of the Strassen algorithm [9]. The algorithm takes less workspace and has better numerical stability as compared to the Strassen algorithm, especially in Winograd's modification [2]. Moreover, its clearer and more flexible structure is potentially more suitable for efficient implementation on modern supercomputers. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Karle-Hauptman matrices may be used in an algebraic approach to the phase problem. When eigenvalues and eigenvectors are used, it is possible to obtain structural information from Karle-Hauptman matrices of orders greater than N, despite the fact that the determinants are zero. In this paper, the properties of large Karle Hauptman matrices are examined in the infinite and non-infinite cases. The characteristics of electron densities corresponding to separate eigenvectors are examined. [ABSTRACT FROM AUTHOR]

This paper describes a method to avoid the singular problem at the "S-control" algorithm. The "S-control" is proposed to determine the correct operating state in any bia polar transistor network (BTN). Each diode of the Ebers-Moll model is replaced with a directional switch called an S-branch which has "on" and "off" states. The switch status is changed by the S-control algorithm. How- ever, the matrix singular problem occurs in the process of the S-control. The singular matrix is formed by a cutset of only current sources or a loop of only voltage sources. The singular problem is solved by the sign- discrimination method by which the sign of each node voltage is judged by the introduction of small immittances in cutset (loop) of only current (voltage) sources formed in process of the S-control. The S-control algorithm will be continued to obtain the next switch state unless the limit cycle is not generated. [ABSTRACT FROM AUTHOR]

This paper addresses various questions about pairs of similarity classes of matrices that contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question about nilpotent classes; this reduction makes use of class types in the sense of Steinberg and Green. We investigate the set of scalars that arise as determinants of elements of the centralizer algebra of a matrix, providing a complete description of this set in terms of the class type of the matrix. Several results are established concerning the commuting of nilpotent classes. Classes that are represented in the centralizer of every nilpotent matrix are classified; this result holds over any field. Nilpotent classes are parametrized by partitions; we find pairs of partitions whose corresponding nilpotent classes commute over some finite fields, but not over others. We conclude by classifying all pairs of classes, parametrized by two-part partitions, that commute. Our results on nilpotent classes complement work of Oblak and of Košir and Oblak. [ABSTRACT FROM AUTHOR]

We prove the title by constructing 2‐colourable completely positive approximations for the Toeplitz algebra. Besides results about nuclear dimension and completely positive contractive order zero maps, our argument involves projectivity of the cone over a finite‐dimensional C∗‐algebra and Lin's theorem on almost normal matrices. [ABSTRACT FROM AUTHOR]

This paper presents a new algebraic extension of the Rayleigh quotient multigrid (RQMG) minimization algorithm to compute the smallest eigenpairs of a symmetric positive definite pencil (A, M). Earlier versions of RQMG minimize the Rayleigh quotient over a hierarchy of geometric grids. We replace the geometric mesh information with the algebraic information defined by an algebraic multigrid preconditioner. At each level, we minimize the Rayleigh quotient with a block preconditioned algorithm. Numerical experiments illustrate the efficiency of this new algorithm to compute several eigenpairs. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper investigates the relations between the structured backward error (SBE) and the partial structured backward error (P-SBE), and the partial backward error (P-BE) and the P-SBE for a computed solution to general Toeplitz system. In particular, we show in some cases the P-SBE for general Toeplitz system is usually a good estimate of the SBE. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper deals with the application of the restricted-denominator rational Krylov method, recently discussed in (BIT 2004; 44(3):595–615; SIAM J. Sci. Comput. 2005; 27:1438–1457), to the computation of the action of the so-called ϕ-functions, which play a fundamental role in several modern exponential integrators. The analysis here presented is devoted in particular to the construction of error estimates of easy practical use. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, some optimal inclusion intervals of matrix singular values are discussed in the set ΩA of matrices equimodular with matrix A. These intervals can be obtained by extensions of the Gerschgorin-type theorem for singular values, based only on the use of positive scale vectors and their intersections. Theoretic analysis and numerical examples show that upper bounds of these intervals are optimal in some cases and lower bounds may be non-trivial (i.e. positive) when PA is a H-matrix, where P is a permutation matrix, which improves the conjecture in Reference (Linear Algebra Appl 1984; 56:105-119). Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A*X-αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, two direct algorithms for solving the two-sided obstacle problem with an M-matrix are presented. The algorithms are well defined and have polynomial computational complexity. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Structured matrix algebras ℒ and a generalized BFGS-type iterative scheme have been recently investigated to introduce low-complexity quasi-Newton methods, named ℒQN, for solving general (non-structured) minimization problems. In this paper we introduce the ℒkQN methods, which exploit ad hoc algebras at each step. Since the structure of the updated matrices can be modified at each iteration, the new methods can better fit the Hessian matrix, thereby improving the rate of convergence of the algorithm. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kernel polynomials are considered. Theoretical results as well as numerical experiments illustrate the effectiveness of our approach. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we present some new absolute and relative perturbation bounds of eigenvalues of normal matrices. The bounds depend upon the closeness of perturbed matrices to normal matrices and improve those previous results (Duke Math. J. 1953; 20:37–39, Linear Algebra Appl. 1996; 246:215–223). Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The pivoted QLP decomposition, introduced by Stewart, represents the first two steps in an algorithm which approximates the SVD. If A is an m-by-n matrix, the matrix A∏0 is first factored as A∏0 = QR, and then the matrix RT∏1 is factored as RT∏1 = PLT, resulting in A = Q∏1LPT∏0T, with Q and P orthogonal, L lower-triangular, and ∏0 and ∏1 permutation matrices. The Q and P matrices provide approximations of the left and right singular subspaces, and the diagonal elements of L are excellent approximations of the singular values of A. Stewart observed that pivoting is not necessary in the second step, allowing one to efficiently truncate the decomposition, computing only the first few columns of R and L and choosing the stopping point dynamically. In this paper, we demonstrate that this truncating actually works by extending our theory for the complete pivoted QLP decomposition (UCLA CAM Report # 02-29, 2002). In particular, say there is a gap between σk and σk+1, and partition the matrix L into diagonal blocks L11 and L22 and off-diagonal block L21, where L11 is k-by-k. If we compute only the block L11, the convergence of (σj(L11)-1 - σj-1)/σj-1 for j = 1,...,k are all quadratic in the gap ratio σk+1/σk. Hence, if the gap ratio is small, as it usually is when A has numerical rank k (independent of m and n), then all of the singular values are likely to be well approximated. This truncated pivoted QLP decomposition can be computed in 𝒪(mnk) time, making it ideal for accurate SVD approximations of low-rank matrices. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The computation of the minimal positive solution of a non-symmetric algebraic Riccati equation arising in transport theory is considered. It was shown in (SIAM J Matrix Anal Appl, submitted) that this can be done via only computing the minimal positive solution of a vector equation, which is derived from special form of the solutions of the Riccati equation and by exploitation of the special structure of the coefficient matrices of the Riccati equation. In this paper, the Newton method is developed for the vector equation. The Newton method is more simple and efficient than the corresponding Newton method directly for original Riccati equation and can preserve the form that any solution of the Riccati equation must satisfy. Combination of the simple iteration and the Newton iteration is also considered. Numerical examples are given. Copyright © 2004 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]

For decades considerable efforts have been exerted to resolve the inverse eigenvalue problem for non-negative matrices. Yet fundamental issues such as the theory of existence and the practice of computation remain open. Recently, it has been proved that, given an arbitrary (n–1)-tuple ℒ = (λ2,...,λn) ∈ ℂn–1 whose components are closed under complex conjugation, there exists a unique positive real number ℛ(ℒ), called the minimal realizable spectral radius of ℒ, such that the set {λ1,...,λn} is precisely the spectrum of a certain n × n non-negative matrix with λ1 as its spectral radius if and only if λ1 ⩾ ℛ(ℒ). Employing any existing necessary conditions as a mode of checking criteria, this paper proposes a simple bisection procedure to approximate the location of ℛ(ℒ). As an immediate application, it offers a quick numerical way to check whether a given n-tuple could be the spectrum of a certain non-negative matrix. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The authors propose a multi-distance co-occurrence matrix and a method of generating seamless textures from natural textures by using this multi-distance co-occurrence matrix. The authors had proposed a seamless texture generation method for solving the boundary discontinuity problem that had conventionally occurred when texture mapping had been repeatedly performed. However, the proposed seamless texture generation method had the problem that seamless textures could not be generated from existing textures. In this paper, the authors propose a multi-distance co-occurrence matrix that extends the co-occurrence matrix used in texture analysis for solving this kind of problem and then propose a texture generation method that combines this multi-distance co-occurrence matrix and a pixel value determination procedure called the fractal method, which is one of the proposed seamless texture generation methods. As a result, seamless textures can be generated from existing textures. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 2, 87(12): 28–37, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjb.20133 [ABSTRACT FROM AUTHOR]

Many very large Markov chains can be modelled efficiently as stochastic automata networks (SANs). A SAN is composed of individual automata which, for the most part, act independently, requiring only infrequent interaction. SANs represent the generator matrix Q of the underlying Markov chain compactly as the sum of Kronecker products of smaller matrices. Thus, storage savings are immediate. The benefit of a SAN's compact representation, known as the descriptor, is often outweighed by its tendency to make analysis of the underlying Markov chain tough. While iterative or projections methods have been used to solve the system πQ=0, the time until these methods converge to the stationary solution π is still unsatisfactory. SAN's compact representation has made the next logical research step of preconditioning thorny. Several preconditioners for SANs have been proposed and tested, yet each has enjoyed little or no success. Encouraged by the recent success of approximate inverses as preconditioners, we have explored their potential as SAN preconditioners. One particularly relevant finding on approximate inverse preconditioning is the nearest Kronecker product approximation discovered by Pitsianis and Van Loan. In this paper, we extend the nearest Kronecker product technique to approximate the Q matrix for an SAN with a Kronecker product, A1 ⊗A2. … ⊗AN. Then, we take M = A1-1 ⊗A2-1⊗ … ⊗AN-1 as our SAN NKP preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, an eigenvalue analysis of the SIMPLE preconditioning for incompressible flow is presented. Some formulations have been set up to characterize the spectrum of the preconditioned matrix. This leads to a generalized eigenvalue problem. The generalized eigenvalue problem is investigated. Some eigenvalue bounds and the estimation for the spectral condition number in the symmetric case are given. Numerical tests are reported to illustrate the theoretical discussions. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The partial pole assignment (PPA) problem is the one of reassigning a few unwanted eigenvalues of a control system by feedback to suitably chosen ones, while keeping the remaining large number of eigenvalues unchanged. The problem naturally arises in modifying dynamical behaviour of the system. The PPA has been considered by several authors in the past for standard state–space systems and for quadratic matrix polynomials associated with second-order systems. In this paper, we consider the PPA for a cubic matrix polynomial arising from modelling of a vibrating system with aerodynamics effects and derive explicit formulas for feedback matrices in terms of the coefficient matrices of the polynomial. Our results generalize those of a quadratic matrix polynomial by Datta et al. (Linear Algebra Appl. 1997;257: 29) and is based on some new orthogonality relations for eigenvectors of the cubic matrix polynomial, which also generalize the similar ones reported in Datta et al. (Linear Algebra Appl. 1997;257: 29) for the symmetric definite quadratic pencil. Besides playing an important role in our solution for the PPA, these orthogonality relations are of independent interests, and believed to be an important contribution to linear algebra in its own right. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The class $P_n(I)$ of matrix monotone functions of order $n$, defined on an open interval $I$, is considered in this paper. Explicit examples are given to prove that if $I$ is different from the whole real line, then $P_{n+1}(I)$ is a proper subset of $P_n(I)$ for each natural number $n$. The subhomogeneous C*-algebras are then characterized in terms of matrix monotone functions. [ABSTRACT FROM AUTHOR]

Akemann and Anderson made a conjecture about ‘paving’ projections in finite-dimensional matrix algebras which, if true, would settle the well-known Kadison–Singer problem. Their conjecture is falsified in this paper by an explicit sequence of counterexamples. [ABSTRACT FROM PUBLISHER]

In a recent work, the author introduced a robust multilevel incomplete factorization algorithm using spanning trees of matrix graphs (Proceedings of the 1999 International Conference on Preconditioning Techniques for Large Sparse Matrix Problems in Industrial Applications, Hubert H. Humphrey Center, University of Minnesota, 1999, 251–257). Based on this idea linear and non-linear algebraic multilevel iteration (AMLI) methods are investigated in the present paper. In both cases, the preconditioner is constructed recursively from the coarsest to finer and finer levels. The considered W-cycles only need diagonal solvers on all levels and additionally evaluate a second-degree matrix polynomial (linear case), or, perform ν inner GCG-type iterations (non-linear case) on every other level. This involves the same type of preconditioner for the corresponding Schur complement. The non-linear variant has the additional benefit of being free from any method parameters to be estimated. Based on the same type of approximation property similar convergence rates are obtained for linear and non-linear AMLI, even for a very small number ν of inner iterations, e.g. ν =2,3. The presented methods are robust with respect to anisotropy and discontinuities in the coefficients of the PDEs and can also be applied to unstructured-grid problems. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]