solution of the large matrix equations which occur in response theory abstract the eigenvalue equation and the sets of linear equations that occur in linear and nonlinear response function calculations have a specific paired structure we have developed iterative algorithms which utilize this structure to efficiently solve the equations the algorithms have been designed so they do not require the matrices to be explicitly available which makes it possible to perform accurate calculations with a dimension of maybe 106 numerical tests show convergence behavior superior to previously suggested algorithms

a hamiltonian jacobi algorithm the nonsymmetric jacobi iteration is adapted to the special structure of hamiltonian matrices this hamiltonian jacobi algorithm uses symplectic unitary similarity transformations to solve algebraic riccati equations through the hamiltonian schur form it preserves hamiltonian structure without using a condensed form although it converges too slowly for use on conventional serial computers it may be attractive for some highly parallel architectures

a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

on the bilinear structure associated to bezoutians abstract this paper is partly a survey of known results on quadratic forms that are hard to find in the literature our main focus is a twisted form of a construction due to bezout this skew bezoutian is a symplectic resp quadratic space associated to a pair of reciprocal or skew reciprocal coprime polynomials of same degree the isometry group of this space turns out to contain a certain associated hypergeometric group using the skew bezoutian we construct explicit isometries of bilinear spaces with given invariants such as the characteristic polynomial or jordan form and in the quadratic case the spinor norm

quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations we present a practical and efficient means to compute the singular value decomposition svd of a real or complex quaternion matrix a based on bidiagonalization of a to a real or complex bidiagonal matrix b using quaternionic householder transformations computation of the svd of b using an existing subroutine library such as lapack provides the singular values of a the singular vectors of a are obtained trivially from the product of the householder transformations and the real or complex singular vectors of b we show in the paper that left and right quaternionic householder transformations are different because of the non commutative multiplication of quaternions and we present formulae for computing the householder vector and matrix in each case

algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory by means of complex representation and real representation of a quaternion matrix this paper studies the problem of diagonalization of a quaternion matrix gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory

structured maximal perturbations for hamiltonian eigenvalue problems we discuss the effect of structure preserving perturbations on complex or real hamiltonian eigenproblems and characterize the structured worst case effect perturbations we derive explicit expressions for the maximal hamiltonian perturbations

algebraic techniques for diagonalization of a split quaternion matrix in split quaternionic mechanics in the study of the relation between complexified classical and non hermitian quantum mechanics physicists found that there are links to quaternionic and split quaternionic mechanics and this leads to the possibility of employing algebraic techniques of split quaternions to tackle some problems in complexified classical and quantum mechanics this paper by means of real representation of a split quaternion matrix studies the problem of diagonalization of a split quaternion matrix and gives algebraic techniques for diagonalization of split quaternion matrices in split quaternionic mechanics

quaternion involutions and anti involutions an involution or anti involution is a self inverse linear mapping in this paper we study quaternion involutions and anti involutions we review formal axioms for such involutions and anti involutions we present two mappings one a quaternion involution and one an anti involution and a geometric interpretation of each as reflections we present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti involutions finally we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti involutions

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

some remarks on the complex j symmetric eigenproblem abstract the eigenproblem for complex j symmetric matrices h c a c d a t a c c t d d t c n n is considered a proof of the existence of a transformation to the complex j symmetric schur form proposed in 21 is given the complex symplectic unitary qr decomposition and the complex symplectic sr decomposition are discussed it is shown that a qr like method based on the complex symplectic unitary qr decomposition is not feasible here a complex symplectic sr algorithm is presented which can be implemented such that one step of the sr algorithm can be carried out in o n arithmetic operations based on this a complex symplectic lanczos method can be derived moreover it is discussed how the 2 n 2 n complex j symmetric matrix h c can be embedded in a 4 n 4 n real hamiltonian matrix

the smoothed spectral abscissa for robust stability optimization this paper concerns the stability optimization of parameterized matrices a x a problem typically arising in the design of fixed order or fixed structured feedback controllers it is well known that the minimization of the spectral abscissa function alpha a gives rise to very difficult optimization problems since alpha a is not everywhere differentiable and even not everywhere lipschitz we therefore propose a new stability measure namely the smoothed spectral abscissa tilde alpha epsilon a which is based on the inversion of a relaxed h 2 type cost function the regularization parameter epsilon allows tuning the degree of smoothness for epsilon approaching zero the smoothed spectral abscissa converges towards the nonsmooth spectral abscissa from above so that tilde alpha epsilon a leq0 guarantees asymptotic stability evaluation of the smoothed spectral abscissa and its derivatives w r t matrix parameters x can be performed at the cost of solving a primal dual lyapunov equation pair allowing for an efficient integration into a derivative based optimization framework two optimization problems are considered on the one hand the minimization of the smoothed spectral abscissa tilde alpha epsilon a x as a function of the matrix parameters for a fixed value of epsilon and on the other hand the maximization of epsilon such that the stability requirement tilde alpha epsilon a x leq0 is still satisfied the latter problem can be interpreted as an h 2 norm minimization problem and its solution additionally implies an upper bound on the corresponding h infty norm or a lower bound on the distance to instability in both cases additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem

on the quadratic type of some simple self dual modules over fields of characteristic two let g be a finite group and let k be an algebraically closed field of characteristic 2 let v be a non trivial simple self dual kg module we say that v is self dual if it is isomorphic to its dual v it is a theorem of w x fong 4 lemma 1 that in this case there is a non degenerate g invariant alternating bilinear form f say defined on v v we say that v is a kg module of quadratic type if f is the polarization of a non degenerate w x g invariant quadratic form defined on v in a previous paper 6 the present authors described some methods to decide if such a module v is of w x quadratic type one of the main results of 6 is the following suppose that g is a group with a split b n pair of odd characteristic p let v be a simple self dual kg module that is not of quadratic type and let w be the brauer character of v then there exists a complex irreducible character of g that occurs as a constituent of the induced character 1 and contains b w as a modular constituent with non zero multiplicity this result suggests that we should investigate the decomposition modulo 2 of the irreducible characters in 1 when g is a group of lie type of odd characteristic and b see which real valued irreducible brauer characters occur as constituents the decomposition modulo 2 of these characters is not known in complete

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

arbitrary rotation invariant random matrix ensembles and supersymmetry orthogonal and unitary symplectic case recently the supersymmetry method was extended from gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the hubbard stratonovich transformation here we complete this extension by including arbitrary orthogonally and unitary symplectically invariant matrix ensembles the results are equivalent to but the approach is different from the superbosonization formula we express our results in a unifying way we also give explicit expressions for all one point functions and discuss features of the higher order correlations

a sort jacobi algorithm for semisimple lie algebras the classical jacobi algorithm is extended to an unified lie algebraic approach the conventional jacobi algorithm minimize the distance to diagonality they reduce the off norm i e the sum of squares of off diagonal entries sorting the diagonal elements after each step would accelerate the convergence but there are difficulties to apply this sorting to the off norm that has to minimize using the gradient flow of a trace function it r w brockett dynamical systems that sort lists diagonalize matrices and solve linear programming problems linear algebra appl 146 79 91 1991 zbl 0719 90045 as a more appropriate distance measure matrices can be diagonalized and the eigenvalues can be simultaneously sorted in this paper the sort jacobi algorithm is extended to a large class of structured matrices besides the semisimple lie algebra cases such as e g the symmetric hermitian and skew symmetric eigenvalue problem and the real and complex singular value decomposition par the local quadratic convergence of the sort jacobi method is proved for the regular case but not for the case in which the eigenvalues or singular values occur in clusters for these more complicated irregular problem a future publication is announced par the results are illustrated considering the lie algebra of derivations of the complex octonions

an updating algorithm for subspace tracking in certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components the usual tool for doing this is the singular value decomposition however the singular value decomposition has the drawback that it requires o p sup 3 operations to recompute when a new sample arrives it is shown that a different decomposition called the urv decomposition is equally effective in exhibiting the null space and can be updated in o p sup 2 time the updating technique can be run on a linear array of p processors in o p time

basic classes of matrices with respect to quaternionic indefinite inner product spaces basic classes of matrices or linear transformations in finite dimensional quaternionic vector spaces with a regular indefinite inner product are introduced and studied the classes include plus matrices selfadjoint skew adjoint and unitary matrices in particular results are proved concerning extensions of invariant semidefinite subspaces canonical form for unitary matrices is developed and subsequently applied to stability of periodic hamiltonian systems

diagonalization of bounded linear operators on separable quaternionic hilbert space by using the representation of its complex conjugate pairs we have investigated the diagonalization of a bounded linear operator on separable infinite dimensional right quaternionic hilbert space the sufficient condition for diagonalizability of quaternionic operators is derived the result is applied to anti hermitian operators which is essential for solving schrodinger equation in quaternionic quantum mechanics

a robust numerical method for the γ iteration in h control abstract we present a numerical method for the solution of the optimal h control problem based on the iteration and a novel extended matrix pencil formulation of the state space solution to the sub optimal h control problem in particular instead of algebraic riccati equations or unstructured matrix pencils our approach is solely based on solving even generalized eigenproblems the enhanced numerical robustness of the method is derived from the fact that using the structure of the problem spectral symmetries are preserved moreover these methods are also applicable even if the pencil has eigenvalues on the imaginary axis we compare the new method with conventional methods and present several examples

hamiltonian square roots of skew hamiltonian matrices abstract we present a constructive existence proof that every real skew hamiltonian matrix w has a real hamiltonian square root the key step in this construction shows how one may bring any such w into a real quasi jordan canonical form via symplectic similarity we show further that every w has infinitely many real hamiltonian square roots and give a lower bound on the dimension of the set of all such square roots some extensions to complex matrices are also presented

the weak and strong stability of algorithms in numerical linear algebra the stability of algorithms in numerical linear algebra is discussed the concept of stability is extended to notions of weak stability and strong stability justifications are given for these extensions and the implications of error analyses in terms of these definitions are discussed the concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for toeplitz systems

balancing sparse hamiltonian eigenproblems balancing a matrix by a simple and accurate similarity transformation can improve the speed and accuracy of numerical methods for computing eigenvalues we describe balancing strategies for a large and sparse hamiltonian matrix h it is first shown how to permute h to irreducible form while retaining its structure this form can be used to decompose the hamiltonian eigenproblem into smaller sized problems next we discuss the computation of a symplectic scaling matrix d so that the norm of d 1 h d is reduced the considered scaling algorithm is solely based on matrix vector products and thus particularly suitable if the elements of h are not explicitly given the merits of balancing for eigenvalue computations are illustrated by several practically relevant examples c 2004 elsevier inc all rights reserved

on the exponential of matrices in su 4 this paper provides explicit techniques to compute the exponentials of a variety of anti hermitian matrices in dimension 4 many of these formulae can be written down directly from the entries of the matrix whenever any spectral calculations are required these can be done in closed form in many instances only 2 2 spectral calculations are required these formulae cover a wide variety of applications conditions on the matrix which render it to admit one of three minimal polynomials are also given matrices with these minimal polynomials admit simple and tractable representations for their exponentials one of these is the euler rodrigues formula the key technique is the relation between real 4 4 matrices and the quaternions

a step toward a unified treatment of continuous and discrete time control problems in this paper we introduce a new approach for a unified theory for continuous and discrete time optimal control problems based on the generalized cayley transformation we also relate the associated discrete and continuous generalized algebraic riccati equations we demonstrate the potential of this new approach by proving a new result for discrete algebraic riccati equations but we also discuss where this new approach as well as all other approaches still is nonsatisfactory we explain a discrepancy observed between the discrete and continuous case and show that this discrepancy is partly due to the consideration of the wrong analogues we also present an idea for an implication scheme that relates general theorems for discrete and continuous control problems

a new real structure preserving quaternion qr algorithm abstract new real structure preserving decompositions are introduced to develop fast and robust algorithms for the right eigenproblem of general quaternion matrices under the orthogonally j r s symplectic transformations the francis j r s qr step and the j r s qr algorithm are firstly proposed for j r s symmetric matrices and then applied to calculate the schur form of quaternion matrices a novel quaternion givens matrix is defined and utilized to compute the qr factorization of quaternion hessenberg matrices an implicit double shift quaternion qr algorithm is presented with a technique for automatically choosing shifts and within real operations numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms

computing the density of states for optical spectra of molecules by low rank and qtt tensor approximation abstract in this paper we introduce a new interpolation scheme to approximate the density of states dos for a class of rank structured matrices with application to the tamm dancoff approximation tda of the bethe salpeter equation bse the presented approach for approximating the dos is based on two main techniques first we propose an economical method for calculating the traces of parametric matrix resolvents at interpolation points by taking advantage of the block diagonal plus low rank matrix structure described in 6 3 for the bse tda problem this allows us to overcome the computational difficulties of the traditional schemes since we avoid the construction of the matrix inverse and hence the need of stochastic sampling second we show that a regularized or smoothed dos discretized on a fine grid of size n can be accurately represented in a low rank quantized tensor train qtt format that can be determined through a least squares fitting procedure the qtt tensor provides good approximation properties for strictly oscillating dos functions with multiple gaps in contrast to interpolation by problem independent functions like polynomials trigonometric functions etc moreover the qtt approximant requires asymptotically much fewer e g o log n functional calls compared with the full grid size n numerical tests indicate that the qtt approach yields accurate recovery of dos associated with problems that contain relatively large spectral gaps the qtt tensor rank only weakly depends on the size of a molecular system that paves the way for treatment of large scale spectral problems

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

the eigenvalue problem for hermitian matrices with time reversal symmetry this paper describes a computational method for dealing with a class of matrices which arise in quantum mechanics involving time reversal and inversion symmetry the algorithms presented here have greatly reduced the computational effort required to solve this problem and also produce a stable more accurate solution

a numerical method for quadratic eigenvalue problems of gyroscopic systems abstract we consider the quadratic eigenvalues problem qep of gyroscopic systems 2 m g k x 0 where m m g g and k k r n n with m being positive definite guo numerical solution of a quadratic eigenvalue problem linear algebra and its applications 385 2004 391 406 showed that all eigenvalues of the qep can be found by solving the maximal solution of a nonlinear matrix equation z a z 1 a q with quadratic convergence when the qep has no eigenvalues on the imaginary axis the convergence becomes linear or more slower guo 2004 when the qep allows purely imaginary eigenvalues having even partial multiplicities in this paper we consider the general case when the qep has eigenvalues on the imaginary axis we propose an eigenvalue shifting technique to transform the original gyroscopic system to a new gyroscopic system which shifts purely imaginary eigenvalues to eigenvalues with nonzero real parts while keeps other eigenpairs unchanged this transformation ensures that the new method for computing the maximal solution of the nonlinear matrix equation converges quadratically numerical examples illustrate the efficiency of our method

quaternionic quantum field theory we show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions as well as the bosons obey a second order wave equation the theory takes the form of either a functional integral with quaternion imaginary lagrangian or a schrodinger equation and transformation theory for quaternion valued wave functions with a quaternion imaginary hamiltonian the connection between the two formulations is developed in detail and many related issues including the breakdown of the correspondence principle and the hilbert space structure are discussed

an implicitly restarted krylov subspace method for real symmetric skew symmetric eigenproblems abstract a new implicitly restarted krylov subspace method for real symmetric skew symmetric generalized eigenvalue problems is presented the new method improves and generalizes the shira method of mehrmann and watkins 2001 37 to the case where the skew symmetric matrix is singular it computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point several applications from control theory are presented and the properties of the new method are illustrated by benchmark examples

on structured componentwise condition numbers for hamiltonian eigenvalue problems abstract in this paper we introduce the structured componentwise perturbation analysis to hamiltonian eigenvalue problems the explicit expressions for the relative structured componentwise condition number for hamiltonian eigenvalue problems are derived also we will compare the relative structured componentwise condition number with the relative unstructured componentwise condition number and the relative structured normwise condition number respectively we show the superiority of the relative structured componentwise condition number through numerical examples

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

on two questions about quaternion matrices abstract this paper answers two questions proposed in fuzhen zhang linear algebra appl 251 1997 21 57 showing that there exists a 2 2 quaternion matrix with only two non similar left eigenvalues that is not diagonalizable and giving some necessary and sufficient conditions for a b b a 0

a structure preserving jacobi algorithm for quaternion hermitian eigenvalue problems abstract a new real structure preserving jacobi algorithm is proposed for solving the eigenvalue problem of quaternion hermitian matrix by employing the generalized jrs symplectic jacobi rotations the new quaternion jacobi algorithm can preserve the symmetry and jrs symmetry of the real counterpart of quaternion hermitian matrix moreover the proposed algorithm only includes real operations without dimension expanding and is generally superior to the state of the art algorithm numerical experiments are reported to indicate its efficiency and accuracy

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

proper quaternion gaussian graphical models in this paper we extend gaussian graphical models to proper quaternion gaussian distributions the properness assumption reduces the number of unknowns by a factor of four while graphical models reduce the number of degrees of freedom via sparsity each of the methods allows accurate estimation using a small number of samples to enjoy both gains we show that the proper quaternion gaussian inverse covariance estimation problem is convex and has a closed form solution we proceed to demonstrate that the additional sparsity constraints on the inverse covariance matrix also lead to a convex problem and the optimizations can be efficiently solved by standard numerical methods in the special but practical case of a chordal graph we provide a closed form solution we demonstrate the improved performance of our suggested estimators on both synthetic and real data

constrained two sided coupled sylvester type quaternion matrix equations abstract in this paper we consider the solvability of a system of constrained two sided coupled generalized sylvester quaternion matrix equations some necessary and sufficient conditions for the existence of solutions to this system are derived in terms of the ranks and the generalized inverses of the coefficient matrices moreover we find an expression of the general solution to this system when the solvability conditions are satisfied finally a numerical example is presented to illustrate our results

on a class of generalized eigenvalue problems and equivalent eigenvalue problems that arise in systems and control theory systems and control theory has long been a rich source of problems for the numerical linear algebra community in many problems conditions on analytic functions of a complex variable are usually evaluated by solving a special generalized eigenvalue problem in this paper we develop a general framework for studying such problems we show that for these problems solutions can be obtained by either solving a generalized eigenvalue problem or by solving an equivalent eigenvalue problem a consequence of this observation is that these problems can always be solved by finding the eigenvalues of a hamiltonian or discrete time counterpart matrix even in cases where an associated hamiltonian matrix cannot normally be defined we also derive a number of new compact tests for determining whether or not a transfer function matrix is strictly positive real these tests which are of independent interest due to the fact that many problems can be recast as spr problems are defined even in the case when the matrix d d is singular and can be formulated without requiring inversion of the system matrix a

a fast structure preserving method for computing the singular value decomposition of quaternion matrices abstract in this paper we propose a fast structure preserving algorithm for computing the singular value decomposition of quaternion matrices the algorithm is based on the structure preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal jrs symplectic matrices the algorithm is efficient and numerically stable

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

quaternionic eigenvalue problem we discuss the right eigenvalue equation for h c and r linear quaternionic operators the possibility to introduce an isomorphism between these operators and real complex matrices allows us to translate the quaternionic problem into an equivalent real or complex counterpart interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics

hamilton and jacobi come full circle jacobi algorithms for structured hamiltonian eigenproblems we develop jacobi algorithms for solving the complete eigenproblem for hamiltonian and skew hamiltonian matrices that are also symmetric or skew symmetric based on the direct solution of 4 4 and in one case 8 8 subproblems these structure preserving algorithms produce symplectic orthogonal bases for the invariant subspaces associated with a matrix in any one of the four classes under consideration the key step in the construction of the algorithms is a quaternion characterization of the 4 4 symplectic orthogonal group and the subspaces of 4 4 hamiltonian skew hamiltonian symmetric and skew symmetric matrices in addition to preserving structure these algorithms are inherently parallelizable numerically stable and show asymptotic quadratic convergence

on the sign of the solutions to some semilinear dirichlet problems we study the sign of solutions to a class of semilinear dirichlet problems when the nonlinearity is for instance a concave convex function which interacts with the spectrum of the linear part we are able to prove in some cases exact multiplicity results for positive and negative solutions for the proofs we employ a device which splits the given problem into two others satisfying a suitable version of the ambrosetti prodi result

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

a structure preserving algorithm for the quaternion cholesky decomposition in this paper we discuss the cholesky decomposition of the hermitian positive definite quaternion matrix for the first time the structure preserving gauss transformation is defined and then a novel structure preserving algorithm which is applied to its real representation matrix is proposed our algorithm needs only real number operations does not depend on the quaternion toolbox for matlab qtfm and has more portability although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or qtfm numerical experiments show that our algorithm runs faster

on computing the eigenvalues of a symplectic pencil this paper presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete time algebraic riccati equation the algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil it requires about one fourth the number of floating point operations that the qz algorithm uses to compute the eigenvalues of the pencil directly the proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by van loan for computing the eigenvalues of hamiltonian matrices

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

on the exponentials of some structured matrices this paper provides explicit techniques to compute the exponentials of a variety of structured 4 4 matrices the procedures are fully algorithmic and can be used to find the desired exponentials in closed form with one exception they require no spectral information about the matrix being exponentiated they rely on a mixture of lie theory and one particular clifford algebra isomorphism these can be extended in some cases to higher dimensions when combined with techniques such as givens rotations

a jacobi like method for solving algebraic riccati equations on parallel computers an algorithm to solve continuous time algebraic riccati equations through the hamiltonian schur form is developed it is an adaption for hamiltonian matrices of an asymmetric jacobi method of eberlein 1987 it uses unitary symplectic similarity transformations and preserves the hamiltonian structure of the matrix each iteration step needs only local information about the current matrix thus admitting efficient parallel implementations on certain parallel architectures convergence performance of the algorithm is compared with the hamiltonian jacobi algorithm of byers 1990 the numerical experiments suggest that the method presented here converges considerably faster for non hermitian hamiltonian matrices than byers hamiltonian jacobi algorithm besides that numerical experiments suggest that for the method presented here the number of iterations needed for convergence can be predicted by a simple function of the matrix size

matrix lsqr algorithm for structured solutions to quaternionic least squares problem abstract in this paper we employ matrix lsqr algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures and propose a strategy to accelerate convergence rate of the algorithm via right left preconditioning of the coefficient matrices we mainly focus on analyzing the minimum norm hermitian solution and the minimum norm bihermitian solution to the quaternionic least squares problem i j k other structured solutions also can be obtained using the proposed technique a number of numerical experiments are performed to show the efficiency of the preconditioned matrix lsqr algorithm

an algorithm for eigenvalues and eigenvectors of quaternion matrices in quaternionic quantum mechanics by means of complex representation and companion vector this paper studies the problems of eigenvalues and eigenvectors of quaternion matrices and gives a technique of computing the eigenvalues and eigenvectors of the quaternion matrices in quaternionic quantum mechanics

a new block method for computing the hamiltonian schur form abstract a generalization of the method of chu et al d chu x liu v mehrmann a numerical method for computing the hamiltonian schur form numer math 105 2007 375 412 for the computation of the hamiltonian real schur form is presented the new method avoids some of the difficulties that may arise when a hamiltonian matrix has tightly clustered groups of eigenvalues a detailed analysis of the method is presented and several numerical examples demonstrate the superior behavior of the method

gersgorin type theorems for quaternionic matrices this paper aims to set an account of the left eigenvalue problems for real quaternionic finite matrices in particular we will present the gersgorin type theorems for the left and right eigenvalues of square quaternionic matrices we shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

real structure preserving algorithms of householder based transformations for quaternion matrices in this paper we survey three different forms of householder based transformations for quaternion matrices in the literature and propose a new form of quaternion householder based transformation we propose real structure preserving algorithms of these householder based transformations which make the procedure computationally more flexible and efficient we compare the computation counts and assignment numbers of these algorithms we also compare the effectiveness of these real structure preserving algorithms applying to the quaternion qrd and the quaternion svd all these four real structure preserving algorithms are more efficient comparing to the algorithms which apply quaternion toolbox using quaternion arithmetics or algorithms which directly performs real householder transformations on the real representation of a quaternion matrix among these four real structure preserving algorithms the most efficient ones are based on quaternion householder reflection and new proposed householder based transformation

on left eigenvalues of a quaternionic matrix abstract in this paper we introduce the concept of left and right eigenvalues for a quaternionic matrix and investigate their properties quantities and relationship

fast iterative solution of the bethe salpeter eigenvalue problem using low rank and qtt tensor approximation abstract in this paper we propose and study two approaches to approximate the solution of the bethe salpeter equation bse by using structured iterative eigenvalue solvers both approaches are based on the reduced basis method and low rank factorizations of the generating matrices we also propose to represent the static screen interaction part in the bse matrix by a small active sub block with a size balancing the storage for rank structured representations of other matrix blocks we demonstrate by various numerical tests that the combination of the diagonal plus low rank plus reduced block approximation exhibits higher precision with low numerical cost providing as well a distinct two sided error estimate for the smallest eigenvalues of the bethe salpeter operator the complexity is reduced to o n b 2 in the size of the atomic orbitals basis set n b instead of the practically intractable o n b 6 scaling for the direct diagonalization in the second approach we apply the quantized tt qtt tensor representation to both the long eigenvectors and the column vectors in the rank structured bse matrix blocks and combine this with the als type iteration in block qtt format the qtt rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems n o n b hence the overall asymptotic complexity for solving the bse problem by the qtt approximation is estimated by o log n o n o 2 we confirm numerically a considerable decrease in computational time for the presented iterative approaches applied to various compact and chain type molecules while supporting sufficient accuracy

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

first order structured perturbation theory for multiple zero eigenvalues of skew adjoint matrices abstract given a matrix belonging to some class of structured matrices we consider the question of comparing the sensitivity of its eigenvalues under two different kinds of perturbations either unstructured i e arbitrary or structured i e those belonging to the same class of matrices as the unperturbed one in a previous paper kressner et al 2009 13 the authors compared the structured and unstructured condition numbers of possibly multiple eigenvalues for several different matrix and pencil structures only one case was left out of the analysis namely the one where the asymptotic order of perturbed eigenvalues under structured perturbations is different from the asymptotic order under unstructured ones this is precisely the case we consider in the present paper given a matrix which is skew adjoint with respect to a symmetric scalar product and has a zero eigenvalue with a certain jordan structure first order expansions are obtained for the perturbed eigenvalues under structured perturbation as well as bounds on the structured condition number similar results are obtained for structured perturbations of symmetric skew symmetric and palindromic matrix pencils

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

the noncompact portion of sp 4 r via quaternions this work provides a quaternionic representation for real symplectic matrices in dimension four analogous to that for the orthogonal group this is achieved by characterizing positive definite symplectic matrices via quaternions it also provides a technique to compute the polar decomposition for sp 4 r which requires no diagonalization but relies only on the solution of a 2 2 linear system this constructive technique to compute the non compact portion of sp 4 r is then used to compute the smallest eigenvalue of the noise covariance matrix of the so called gaussian two mode systems other applications where this non compact portion is relevant are also discussed

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

an svd like matrix decomposition and its applications this author is supported by nsf under grant no eps 9874732 and matching support from the state of kansas

quaternion var modelling and estimation quaternion vector autoregression var modeling is a natural extension of real and complex var we demonstrate how a quaternion var can be treated as a special case of structured real var we show that generalized least squares and under gaussianity maximum likelihood estimation of the parameters reduces to simple least squares estimation if the innovations are quaternion proper

multiple solutions for some semilinear elliptic equations on donne des resultats sur le nombre des solutions du probleme de dirichlet pour des equations elliptiques semilineaires

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

discriminants of polynomials and of quadratic forms abstract in this paper we describe the quadratic forms over any field k which admit a similarity with a given separable characteristic polynomial f x as the transfer of some binary quadratic form associated to the polynomial f x using this description we show that the discriminant of the quadratic form is the discriminant of this polynomial

on the power method for quaternion right eigenvalue problem abstract in this paper we study the power method of the right eigenvalue problem of a quaternion matrix a if a is hermitian we propose the power method that is a direct generalization of that of complex hermitian matrix when a is non hermitian by applying the properties of quaternion right eigenvalues we propose the power method for computing the standard right eigenvalue with the maximum norm and the associated eigenvector we also briefly discuss the inverse power method and shift inverse power method for the both cases the real structure preserving algorithm of the power method in the two cases are also proposed and numerical examples are provided to illustrate the efficiency of the proposed power method and inverse power method