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

spectral factorization via hermitian pencils abstract the spectral factorization problem is solved for state space systems via results on the canonical forms and inertia of hermitian matrix pencils these algebraic results then give a deflation method for spectral factorization

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

improved batch fuzzy learning vector quantization for image compression in this paper we develop a batch fuzzy learning vector quantization algorithm that attempts to solve certain problems related to the implementation of fuzzy clustering in image compression the algorithm s structure encompasses two basic components first a modified objective function of the fuzzy c means method is reformulated and then is minimized by means of an iterative gradient descent procedure second the overall training procedure is equipped with a systematic strategy for the transition from fuzzy mode where each training vector is assigned to more than one codebook vectors to crisp mode where each training vector is assigned to only one codebook vector the algorithm is fast and easy to implement finally the simulation results show that the method is efficient and appears to be insensitive to the selection of the fuzziness parameter

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

properties of a quadratic matrix equation and the solution of the continuous time algebraic riccati equation abstract we discuss some properties of a quadratic matrix equation with some restrictions then use these results on the algebraic riccati equation to get a new algorithm the algorithm sufficiently takes account of the structure of the associated matrix hence it is very effective

a complexity reduction technique for image vector quantization a technique for reducing the complexity of spatial domain image vector quantization vq is proposed the conventional spatial domain distortion measure is replaced by a transform domain subspace distortion measure due to the energy compaction properties of image transforms the dimensionality of the subspace distortion measure can be reduced drastically without significantly affecting the performance of the new quantizer a modified lbg algorithm incorporating the new distortion measure is proposed unlike conventional transform domain vq the codevector dimension is not reduced and a better image quality is guaranteed the performance and design considerations of a real time image encoder using the techniques are investigated compared with spatial domain a speed up in both codebook design time and search time is obtained for mean residual vq and the size of fast ram is reduced by a factor of four degradation of image quality is less than 0 4 db in psnr

orthosymmetric block reflectors we develop a general theory of reflectors and block reflectors in a class of non euclidean scalar product spaces generated by orthosymmetric scalar product matrices j these j reflectors are generalizations of ordinary householder transformations and we show that they can always be expressed in a householder like representation reflection and mapping properties of j reflectors are completely described block j reflectors can be used for block annihilation in qr like factorizations where q is j unitary and r is block upper triangular

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

convex analysis and global optimization of joint actuator location and control problems it is shown that the optimal value of the continuous time linear quadratic problem regarded as a function of the system model and index parameters exhibits properties convexity concavity and monoticity specially suitable for optimization purposes based on this fact a procedure for the global solution determination of eventually nonconvex problems involving the above mentioned function is proposed such problems embody some known designs such as filtering under noise uncertainty or precision constraints and optimal actuator sensor location the last problem is deeply analyzed and two practical applications namely satellite attitude control and large flexible system control are included

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

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

an efficient algorithm for the discrete time algebraic riccati equation the authors develop an algorithm to solve the standard discrete time algebraic riccati equation by using a skew hamiltonian transformation and the square root method the algorithm is structure preserving and efficient because the hamiltonian structure is fully exploited and only orthogonal transformations are used the efficiency and stability of the algorithm are analyzed numerical examples are included

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

tight closure plus closure and frobenius closure in cubical cones we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 74 3 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring k x y we show that frobenius closure is the same as tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c 1 we conclude that ir n r 1 for these ideals using injective modules over the ring r the union of all peth roots of elements of r we reduce the question of whether if i for 23 graded ideals to the case of 23 graded irreducible modules we classify the irreducible m primary z3 graded ideals we then show that jf 1 for most irreducible rn primary z3 graded ideals in k x y z x3 y3 z3 where k is a field of characteristic p and p 2 mod 3 hence 1 irmf n r for these ideals in this paper we discuss the conjecture that i ir n r where r denotes the integral closure of a domain r of characteristic p in an algebraic closure of its fraction field and i denotes the tight closure of i the ring r is characterized by the property that it is a domain integral over r and every monic polynomial with coefficients in r factors into monic linear factors this characterization can be used to prove the following property of r if w is a multiplicatively closed set of r then w lr w 1r aside from providing a much more concrete description of tight closure proving that i ir n r would solve the localization problem for tight closure it is known that i ir n r for parameter ideals sml and for rings in which every ideal of the normalization is tightly closed also for those ideals i of an excellent local domain r such that r i has finite phantom projective dimension it is known that i ir n r ab however the conjecture is open even for two dimensional normal gorenstein domains in particular the conjecture is open for the cubical cone k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 and more generally for rings of the form k x y z f x y z where f is a homogeneous cubic polynomial we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 in section 1 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular rings k x y in section 2 we show that the frobenius closure of an ideal i denoted if is the same as the tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c i received by the editors august 27 1997 1991 mathematics subject classification primary 13a35 13a02 13hi10

a vector distribution model and an effective nearest neighbor search method for image vector quantization in this correspondence a modified version of hunt s 1980 image model is used to interpret the distribution of image data vectors the model suggests that the diagonal line of the coordinates system is a good approximation of the principal axis of the image data vector set the validity of the model is supported by experiments following this suggestion an effective nearest neighbor search method for vector quantization of image data is developed the method is based on partitioning the vector space using hyperplanes which are perpendicular to the diagonal direction of the coordinate system the validity of the method is assessed by analyzing its complexity and comparing its performance to those of existing algorithms on a number of images

some numerical considerations and newton s method revisited for solving algebraic riccati equations analyzes some of the numerical aspects of solving the algebraic riccati equation are this analysis applies to both the symmetric and unsymmetric cases the author reconsiders the numerically relevant problems of balancing the are and the conditioning properties of the are and shows how these can be exploited by a solution algorithm he proposes an estimator for the condition number of the sylvester equation ax xb c based on iterative refinement also he interprets newton s method as a sequence of similarity transformations on the underlying system matrix this closes the gap between so called global and iterative methods for solving the are and also suggests an altogether revised implementation of newton s method one of the advantages of this revised implementation is that in the case where newton s method converges to a solution different from the desired solution enough information emerges to allow a switch to the desired solution the author examines the roundoff properties of the new algorithm and provides implementation considerations and numerical examples to highlight pros and cons

a fast method of designing better codebooks for image vector quantization a maximum descent md method is proposed as an alternative to the generalized lloyd lbg algorithm for vector quantization codebook generation compared with the lbg algorithm the md algorithm requires far less computation and generates codebooks that are superior in quality this paper investigates and compares the performances of various partition techniques in the md algorithm in image vector quantization the md algorithm can produce codebooks with about 1 db improvement in peak signal to noise ratio in less than 1 100 of the time required by the lbg algorithm

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

image coding using vector quantization a review a review of vector quantization techniques used for encoding digital images is presented first the concept of vector quantization is introduced then its application to digital images is explained spatial predictive transform hybrid binary and subband vector quantizers are reviewed the emphasis is on the usefulness of the vector quantization when it is combined with conventional image coding techniques or when it is used in different domains

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

an sdr algorithm for the solution of the generalized algebraic riccati equation an algorithm is presented for solving certain classes of the algebraic riccati equation by using the mdr transformation and extending the symplectic qr like or sr method of a bunse gerstner linear alg applic vol 83 p 49 78 1986 and v mehrmann ph d diss univ bielefeld w germany to the generalized eigenvalue problem the riccati equations arise from linear systems of the form edx dt ax bu where e is nonsingular but for numerical reasons e sup 1 a and e sup 1 b cannot be computed reliably

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

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

symplectic householder transformations for a qr like decomposition a geometric and algebraic approaches the aim of this paper is to show how geometric and algebraic approaches lead us to a new symplectic elementary transformations the 2 d symplectic householder transformations their features are studied in details their interesting properties allow us to construct a new algorithm for computing a sr factorization this algorithm is based only on these 2 d symplectic householder transformations its new features are highlighted the study shows that in the symplectic case the new algorithm is the corresponding one to the classical qr factorization algorithm via the householder transformations some numerical experiments are given

a new vector quantization clustering algorithm the pairwise nearest neighbor pnn algorithm is presented as an alternative to the linde buzo gray 1980 lbg generalized lloyd 1982 algorithm for vector quantization clustering the pnn algorithm derives a vector quantization codebook in a diminishingly small fraction of the time previously required without sacrificing performance in addition the time needed to generate a codebook grows only o n log n in training set size and is independent of the number of code words desired using this method one can either minimize the number of code words needed subject to a maximum rate the pnn algorithm can be used with squared error and weighted squared error distortion measure simulations on a variety of images encoded at 1 2 b pixel indicate that pnn codebooks can be developed in roughly 5 of the time required by the lbg algorithm

the pfaff lattice on symplectic matrices the pfaff lattice is an integrable system arising from the sr group factorization in an analogous way to how the toda lattice arises from the qr group factorization in our earlier paper kodama and pierce 2007 int math res not arxiv 0705 0510 we studied the pfaff lattice hierarchy for the case where the lax matrix is defined to be a lower hessenberg matrix in this paper we deal with the case of a symplectic lower hessenberg lax matrix this forces the lax matrix to take a 2 2 block tridiagonal shape we then show that the odd members of the pfaff lattice hierarchy are trivial while the even members are equivalent to the indefinite toda lattice hierarchy defined in kodama and ye 1996 physica d 91 321 39 this is analogous to the case of the toda lattice hierarchy in relation to the kac van moerbeke system in the case with the initial matrix having only real or imaginary eigenvalues the fixed points of the even flows are given by 2 2 block diagonal matrices with zero diagonals we also consider a family of skew orthogonal polynomials with a symplectic recursion relation related to the pfaff lattice and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite toda lattice

on the matrix sign function method for solving algebraic riccati equations this paper gives an improvement to the method of solving algebraic riccati equations via use of the matrix sign function we show that a 2n x n or n x 2n system can be truncated to an n x n system and the solution of the truncated system can be guaranteed to be hermitian we show also with standard numerical examples that although the truncated equation has a larger condition number its solution may actually make the algebraic riccati equation have smaller residual in our numerical experiment using matlab the solution of the truncated system needed only about 26 of the number of floating point operations that the original system did