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

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

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

eigenvalue patterned condition numbers toeplitz and hankel cases we continue the study started in noschese and pasquini eigenvalue condition numbers zero structured versus traditional j comput appl math 185 2006 174 189 concerning the sensitivity of simple eigenvalues of a matrix a to perturbations in a that belong to a chosen subspace of matrices in noschese and pasquini eigenvalue condition numbers zero structured versus traditional j comput appl math 185 2006 174 189 the zero structured perturbations have been considered here we focus on patterned perturbations and the cases of the toeplitz and of the hankel matrices are investigated in detail useful expressions of the absolute patterned condition number of the eigenvalue l and of the analogue of the matrix yx h which leads to the traditional condition number of l are given matlab codes are defined to compare traditional zero structured and patterned condition numbers a report on significant numerical tests is included

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

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

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

eigenvalues pseudospectrum and structured perturbations we investigate the behavior of eigenvalues under structured perturbations we show that for many common structures such as complex symmetric toeplitz symmetric toeplitz circulant and others the structured condition number is equal to the unstructured condition number for normwise perturbations and prove similar results for real perturbations an exception are complex skewsymmetric matrices we also investigate componentwise complex and real perturbations here hermitian and skew hermitian matrices are exceptional for real perturbations furthermore we characterize the structured complex and real pseudospectrum for a number of structures and show that often there is little or no significant difference to the usual unstructured pseudospectrum

on the use of stability regions in the numerical analysis of initial value problems this paper deals with the stability analysis of one step methods in the numerical solution of initial boundary value problems for linear ordinary and partial differential equations restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size these restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration the errors in the one step methods are measured in arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of the heat equation and some other differential equations where the error growth is measured in the maximum norm

computing unstructured and structured polynomial pseudospectrum approximations abstract in many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial the sensitivity commonly is described by condition numbers or pseudospectra however the determination of pseudospectra of matrix polynomials is very demanding computationally this paper describes a new approach to computing approximations of pseudospectra of matrix polynomials by using rank one or projected rank one perturbations these perturbations are inspired by wilkinson s analysis of eigenvalue sensitivity this approach allows the approximation of both structured and unstructured pseudospectra computed examples show the method to perform much better than a method based on random rank one perturbations both for the approximation of structured and unstructured i e standard polynomial pseudospectra

structured eigenvalue condition numbers and linearizations for matrix polynomials this work is concerned with eigenvalue problems for structured matrix polynomials including complex symmetric hermitian even odd palindromic and anti palindromic matrix polynomials most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size recently linearizations have been classified for which the pencil reflects the structure of the original polynomial a question of practical importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations for all structures under consideration we show that this cannot happen if the matrix polynomial is well scaled there is always a structured linearization for which the structured eigenvalue condition number does not differ much this implies for example that a structure preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial

eigenvalue condition numbers zero structured versus traditional we discuss questions of eigenvalue conditioning we study in some depth relationships between the classical theory of conditioning and the theory of the zero structured conditioning and we derive from the existing theory formulae for the mathematical objects involved then an algorithm to compare the zero structured individual condition numbers of a set of simple eigenvalues with the traditional ones is presented numerical tests are reported to highlight how the algorithm provides interesting information about eigenvalue sensitivity when the perturbations in the matrix have an arbitrarily assigned zero structure patterned matrices toeplitz and hankel will be investigated in a forthcoming paper eigenvalue patterned condition numbers toeplitz and hankel cases tech rep 3 mathematics department university of rome la sapienza 2005

communication by energy modulation the additive exponential noise channel communication across an additive exponential noise aen channel is studied the aen channel is derived from a continuous time gaussian channel by assuming decoherence between the signal and noise components decoherence renders impossible the use of the complex amplitude and information is transmitted by modulating the signal energy the aen channel is shown to perform closely in terms of channel information rate and error probability of binary codes including the use of bit interleaved coded modulation to an equivalent discrete time gaussian channel of identical signal to noise ratio constellations are designed for low signal to noise ratios where the minimum energy per bit is not universally attained by all modulations and for high signal to noise ratios where an equiprobable nonuniform constellation 0 76 db away from capacity is described the multiple access aen channel is also considered its capacity region coincides with that of the gaussian equivalent feedback does not however enlarge the capacity region

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

mismatched decoding revisited general alphabets channels with memory and the wide band limit the mismatch capacity of a channel is the highest rate at which reliable communication is possible over the channel with a given possibly suboptimal decoding rule this quantity has been studied extensively for single letter decoding rules over discrete memoryless channels dmcs here we extend the study to memoryless channels with general alphabets and to channels with memory with possibly non single letter decoding rules we also study the wide band limit and in particular the mismatch capacity per unit cost and the achievable rates on an additive noise spread spectrum system with single letter decoding and binary signaling

a note on structured pseudospectra of block matrices in this note we consider the question of equivalence of pseudospectra and structured pseudospectra of block matrices the structures we study are all so called double structures that is the blocks of the given matrix are of the same structure as the block matrix the approach is based on that of non block matrices which are also briefly studied and the use of distance to singularity we also list some open problems and conjectures

a note on structured pseudospectra in this note we study the notion of structured pseudospectra we prove that for toeplitz circulant hankel and symmetric structures the structured pseudospectrum equals the unstructured pseudospectrum we show that this is false for hermitian and skew hermitian structures we generalize the result to pseudospectra of matrix polynomials indeed we prove that the structured pseudospectrum equals the unstructured pseudospectrum for matrix polynomials with toeplitz circulant hankel and symmetric structures we conclude by giving a formula for structured pseudospectra of real matrix polynomials the particular type of perturbations used for these pseudospectra arise in control theory

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

backward error condition numbers and pseudospectra for the multiparameter eigenvalue problem we define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem mep the pseudospectrum for the mep is defined and characterized we show that the distance from a right definite mep to the closest non right definite mep is related to the smallest unbounded pseudospectrum some numerical results are given

llr compression for bicm systems using large constellations digital video broadcasting dvb c2 and other modern communication standards increase diversity by means of a symbol level interleaver that spans over several codewords de interleaving at the receiver requires a large memory which has a significant impact on the implementation cost in this paper we propose a technique that reduces the de interleaver memory size by quantizing log likelihood ratios with bit specific quantizers and compressing the quantized output we can significantly reduce the memory size with a negligible increase in computational complexity both the quantizer and compressor are designed via a gmi based maximization procedure for a typical dvb c2 scenario numerical results show that the proposed solution enables a memory saving up to 30

on the information loss of the max log approximation in bicm systems we present a comprehensive study of the information rate loss of the max log approximation for m ary pulse amplitude modulation pam in a bit interleaved coded modulation bicm system it is widely assumed that the calculation of l values using the max log approximation leads to an information loss we prove that this assumption is correct for all m pam constellations and labelings with the exception of a symmetric 4 pam constellation labeled with a gray code we also show that for max log l values the bicm generalized mutual information gmi which is an achievable rate for a standard bicm decoder is too pessimistic in particular it is proved that the so called harmonized gmi which can be seen as the sum of bit level gmis is achievable without any modifications to the decoder we then study how bit level channel symmetrization and mixing affect the mi and the gmi for max log l values our results show that these operations which are often used when analyzing bicm systems preserve the gmi however this is not necessarily the case when the mi is considered necessary and sufficient conditions under which these operations preserve the mi are provided

low rank differential equations for hamiltonian matrix nearness problems for a hamiltonian matrix with purely imaginary eigenvalues we aim to determine the nearest hamiltonian matrix such that some or all eigenvalues leave the imaginary axis conversely for a hamiltonian matrix with all eigenvalues lying off the imaginary axis we look for a nearest hamiltonian matrix that has a pair of imaginary eigenvalues the hamiltonian matrices can be allowed to be complex or restricted to be real such hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems they are closely related to the problem of determining extremal points of hamiltonian pseudospectra we obtain a characterization of optimal perturbations which turn out to be of low rank and are attractive stationary points of low rank differential equations that we derive we use a two level approach where in the inner level we determine extremal points of the hamiltonian varepsilon pseudospectrum for a given varepsilon by following the low rank differential equations into a stationary point and on the outer level we optimize for varepsilon this permits us to give fast algorithms exhibiting quadratic convergence for solving the considered hamiltonian matrix nearness problems

performance analysis for bicm transmission over gaussian mixture noise fading channels bit interleaved coded modulation bicm has been adopted in many systems and standards for spectrally efficient coded transmission the analytical evaluation of bicm performance parameters in particular bit error rate ber has received considerable attention in the recent past in this paper we derive ber approximations for bicm transmission over general fading channels impaired by gaussian mixture noise gmn to this end we build upon the saddlepoint approximation of the pairwise error probability pep and a recently established approximation for the probability density function pdf of bit wise reliability metrics for nonfading additive white gaussian noise awgn channels we extend this pdf approximation to the case of gmn and obtain closed form expressions for its laplace transform for fading gmn channels the latter allows us to express the pep and thus ber via the saddlepoint approximation for the special case of fading awgn channels the presented approximations are closed form since the saddlepoint is well approximated by 1 2 for bicm decoding furthermore we derive closed form pep expressions also for gmn channels in the high signal to noise ratio regime and establish the diversity and coding gain for bicm transmission over fading gmn channels selected numerical results for the ber of convolutional coded bicm highlight the usefulness of the proposed approximations and the differences between awgn and gmn channels

spectral value sets a graphical tool for robustness analysis in this note we present a method to determine and visualize the set of all complex numbers to which at least one eigenvalue of a matrix a can be shifted by real or complex perturbations of the form a a a d e where d and e are fixed matrices and the unknown disturbance matrix satisfies 0 both real and complex perturbations are studied the graphical method may be considered as an extension of the classical root locus method to multivariable systems and multiparameter perturbations the results are illustrated by various examples

structured pseudospectra and random eigenvalues problems in vibrating systems this paper introduces the concept of pseudospectra as a generalized tool for uncertainty quantification and propagation in structural dynamics different types of pseudospectra of matrices and matrix polynomials are explained particular emphasis is given to structured pseudospectra for matrix polynomials which offer a deterministic way of dealing with uncertainties for structural dynamic systems the pseudospectra analysis is compared with the results from monte carlo simulations of uncertain discrete systems two illustrative example problems one with probabilistic uncertainty with various types of statistical distributions and the other with interval type of uncertainty are studied in detail excellent agreement is found between the pseudospectra results and monte carlo simulation results

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

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

bit interleaved coded modulation revisited a mismatched decoding perspective we revisit the information theoretic analysis of bit interleaved coded modulation bicm by modeling the bicm decoder as a mismatched decoder the mismatched decoding model is well defined for finite yet arbitrary block lengths and naturally captures the channel memory among the bits belonging to the same symbol we give two independent proofs of the achievability of the bicm capacity calculated by caire et al where bicm was modeled as a set of independent parallel binary input channels whose output is the bitwise log likelihood ratio our first achievability proof uses typical sequences and shows that due to the random coding construction the interleaver is not required the second proof is based on the random coding error exponents with mismatched decoding where the largest achievable rate is the generalized mutual information moreover the generalized mutual information of the mismatched decoder coincides with the infinite interleaver bicm capacity we show that the error exponent and hence the cutoff rate of the bicm mismatched decoder is upper bounded by that of coded modulation and may thus be lower than in the infinite interleaved model for binary reflected gray mapping in gaussian channels the loss in error exponent is small we also consider the mutual information appearing in the analysis of iterative decoding of bicm with extrinsic information transfer exit charts if the symbol metric has knowledge of the transmitted symbol exit mutual information admits a representation as a pseudo generalized mutual information which is in general not achievable a different symbol decoding metric for which the extrinsic side information refers to the hypothesized symbol induces a generalized mutual information lower than the coded modulation capacity in this case perfect extrinsic side information turns the mismatched decoder error exponent into that of coded modulation

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

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

bit interleaved coded modulation with mismatched decoding metrics bit interleaved coded modulation bicm has become the de facto coding standard for communication systems recently bicm has been cast as a mismatched decoding scheme due to the assumption of independent bit metrics in addition to this inherent mismatch practical demodulators may produce mismatched decoding metrics because of implementation constraints such as clipping and metric approximation to reduce computational complexity in this paper we investigate bicm with such metrics in line with recent works on this topic we adopt the generalized mutual information gmi as the pertinent performance measure first we show that level dependent scaling of logarithmic bit metrics can improve the bicm gmi second we propose a uniform metric scaling which can lead to an improved performance of mismatched sum product symbol by symbol decoding even if the gmi is not changed third we investigate general metric mismatch correction methods and analyze their effects in terms of the gmi by means of three application examples we illustrate that metric mismatch correction including metric scaling can significantly increase bicm rates

signal shaping for bicm at low snr the generalized mutual information gmi of bit interleaved coded modulation bicm systems sometimes called the bicm capacity is investigated at low signal to noise ratio snr the combinations of input alphabet input distribution and binary labeling that achieve the shannon limit 1 59 db are completely characterized the main conclusion is that a bicm system with probabilistic shaping achieves the shannon limit at low snr if and only if it can be represented as a zero mean linear projection of a hypercube hence probabilistic shaping offers no extra degrees of freedom to optimize the low snr bicm gmi in addition to what is provided by geometrical shaping the analytical conclusions are confirmed by numerical results which also show that for a fixed input alphabet probabilistic shaping can improve the bicm gmi in the low and medium snr range

an algorithm for the multi input complex eigenvalue assignment problem in this paper a simple efficient and accurate method for complex eigenvalue assignment in the multi input linear system is presented the method considers both cases all eigenvalues need to assign are complex with their complex conjugates the other case is the arbitrary eigenvalues need to assign are mixed some are real and the others are complex which are closed under complex conjugations the algorithm proposed in this method is a modification to the one proposed by arnold and datta ieee trans automat contr 35 10 1986 1149 which considers the case where all the eigenvalues need to assign are only real also it is a kind of extension to the algorithms suggested by ramadan in press and ramadan and el shazly in press where arbitrary preassigned complex and mixed eigenvalues are assigned for single input case respectively the method first transforms the problem to block hessenberg form of needed form using orthogonal reduction then a simple linear recursion is carried out to find the feedback matrix so that the closed loop matrix has the prescribed eigenvalues the main advantages of the algorithm besides being simple and has recursive nature it avoids complex arithmetic and easy to program on computer numerical examples are given to illustrate the reduction process the algorithm performance and the accuracy of the method

on bicm receivers for tcm transmission recent results have shown that the performance of bit interleaved coded modulation bicm using convolutional codes in nonfading channels can be significantly improved when the interleaver takes a trivial form bicm t i e when it does not interleave the bits at all in this paper we give a formal explanation for these results and show that bicm t is in fact the combination of a tcm transmitter and a bicm receiver to predict the performance of bicm t a new type of distance spectrum for convolutional codes is introduced analytical bounds based on this spectrum are developed and asymptotic approximations are presented it is shown that the free hamming distance of the code is not the relevant optimization criterion for bicm t asymptotically optimal convolutional codes for different constraint lengths are tabulated and bicm t is shown to offer asymptotic gains of about 2 db over traditional bicm designs based on random interleavers the asymptotic gains over uncoded transmission are found to be the same as those obtained by ungerboeck s one dimensional trellis coded modulation 1d tcm and therefore in nonfading channels bicm t is shown to be as good as 1d tcm

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

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

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 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

an algorithm for pole assignment of time invariant linear systems an algorithm is suggested for the computation of a linear state feedback for a single input system such that the resultant closed loop system has specified eigenvalues the algorithm is more efficient than many comparable techniques and has some desirable numerical properties it is closely related to the qr algorithm for the eigenproblem and extends to descriptor systems

an algorithm for the multiinput eigenvalue assignment problem a very simple and inexpensive algorithm is presented for pole placement in the multiinput case the algorithm consists of orthogonal reduction to a block hessenberg form and a simple linear recursion it yields a matrix f such that a bf has any specified set of eigenvalues whenever the system is controllable it is extremely easy to program on a computer the algorithm is not a robust pole placement algorithm but appears to give comparable results in most well conditioned cases at a fraction of the cost it is a direct noniterative algorithm and no eigenvalues or singular values are computed the algorithm does not need any complex arithmetic even when complex conjugate eigenvalues need to be assigned

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

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

parallel and large scale matrix computations in control some ideas abstract the design and analysis of time invariant linear control systems give rise to a variety of interesting linear algebra problems numerically effective methods now exist for several of these problems however algorithms for large scale computations and efficient parallel algorithms for these problems are virtually nonexistent in this paper we propose several efficient general purpose parallel algorithms for single input and multiinput eigenvalue assignment problems a desirable feature of these algorithms is that they are composed of simple linear algebraic operations such as matrix vector multiplication solution of a linear system and computations of the eigensystem and singular values of a symmetric matrix for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms the proposed algorithms thus have potential for implementations on some existing and future parallel processors we also propose a numerical method for the sylvester matrix equation arising in the construction of a luenberger observer the method does not need reduction to condensed forms and is thus suitable for large and sparse matrices the method also exhibits a certain parallelism

accurate computation of the zeros of the generalized bessel polynomials a general method for approximating polynomial solutions of second order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized bessel polynomials and an algorithm is derived for simultaneously finding their zeros then a comparison with several alternative algorithms is carried out it shows that the computational problem of approximating the zeros of the generalized bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper

an algorithm for the multiinput pole assignment problem abstract we present an algorithm for the multiinput pole assignment problem this algorithm is based on basic linear operations such as matrix vector multiplications resolution of upper triangular systems of smaller sizes than the original problem and resolution of systems with multiple right hand sides for which efficient and parallel algorithms have been developed furthermore those independent systems can be computed in parallel the accuracy of the algorithm has been checked on some numerical examples

multilevel codes theoretical concepts and practical design rules this paper deals with 2 sup l ary transmission using multilevel coding mlc and multistage decoding msd the known result that mlc and msd suffice to approach capacity if the rates at each level are appropriately chosen is reviewed using multiuser information theory it is shown that there is a large space of rate combinations such that mlc and full maximum likelihood decoding mld can approach capacity it is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and therefore require the huge complexity of mld the capacity rule the balanced distances rules and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies moreover topics relevant in practical applications such as signal set labeling dimensionality of the constituent constellation and hard decision decoding are emphasized bit interleaved coded modulation proposed by caire et al see ibid vol 44 p 927 46 1998 is reviewed in the context of mlc finally the combination of signal shaping and coding is discussed significant shaping gains are achievable in practice only if these design rules are taken into account

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

componentwise pseudospectrum of a matrix abstract given a nonnegative weight matrix e defining componentwise structured perturbations e e we introduce a componentwise pseudospectrum of a square matrix a as contour sets of the function c a i 1 e where m is the spectral radius of a square matrix m and discuss how to compute it

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

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 asymptotic performance of bit wise decoders for coded modulation two decoder structures for coded modulation over the gaussian channel are studied 1 the maximum likelihood symbol wise decoder and 2 the suboptimal bit wise decoder based on the bit interleaved coded modulation paradigm we consider a 16 ary quadrature amplitude constellation labeled with a gray labeling it is shown that the asymptotic loss in terms of pairwise error probability for any two codewords caused by the bit wise decoder is bounded by 1 25 db the analysis also shows that the asymptotic loss is zero for a wide range of linear codes including all rate 1 2 convolutional codes

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

an algorithm for computing state feedback in multiinput linear systems a new algorithm is proposed in this note to compute the state feedback for arbitrary pole placement in multiinput linear systems the algorithm has good numerical behavior it also allows the user to adjust the eigenstructure of the resulting feedback system and therefore improve the condition of the problem and computation

a direct algorithm for pole assignment of time invariant multi input linear systems using state feedback abstract a direct algorithm is suggested for the computation of the linear state feedback for multi input systems such that the resultant closed loop system matrix has specified eigenvalues the extra freedom can be used in different ways for example to decrease some norm of the feedback matrix or to improve the condition of some of the eigenvalues of the closed loop matrix the algorithm uses unitary transformations for numerical reliability and is based on ideas from the qr algorithm for solving the eigenproblem the stability of the algorithm is proven by doing a backward rounding error analysis and numerical examples are given as well

bandwidth efficient and rate matched low density parity check coded modulation a new coded modulation scheme is proposed at the transmitter the concatenation of a distribution matcher and a systematic binary encoder performs probabilistic signal shaping and channel coding at the receiver the output of a bitwise demapper is fed to a binary decoder no iterative demapping is performed rate adaption is achieved by adjusting the input distribution and the transmission power the scheme is applied to bipolar amplitude shift keying ask constellations with equidistant signal points and it is directly applicable to two dimensional quadrature amplitude modulation qam the scheme is implemented by using the dvb s2 low density parity check ldpc codes at a frame error rate of 10 3 the new scheme operates within less than 1 1 db of the awgn capacity frac 1 2 log 2 1 mathsf snr at any spectral efficiency between 1 and 5 bits s hz by using only 5 modes i e 4 ask with code rate 2 3 8 ask with 3 4 16 ask and 32 ask with 5 6 and 64 ask with 9 10

correction of mismatched l values in bicm receivers in this work we analyze the problem of correction of the reliability metrics l values in bit interleaved coded modulation bicm receivers first we propose a method for finding the linear correction factors that minimize the probability of error of a maximum likelihood decoder that uses the corrected l values to this end we use the efficient approximation of the pairwise error probability in the domain of the cumulant generating functions cgf of the l values and conclude that the optimal correction factors are equal to the twice of value of the saddlepoint of the cgf next a simple extension of the proposed method to the non linear correction is presented we provide numerical examples demonstrating improvements attainable with the proposed method comparing it to the competitive solutions

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

bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications abstract we consider a class of symmetric tridiagonal matrices which may be viewed as perturbations of toeplitz matrices the toeplitz structure is destroyed since two elements on each off diagonal are perturbed based on a careful analysis we derive sharp bounds for the extremal eigenvalues of this class of matrices in terms of the original data of the given matrix in this way we also obtain a lower bound for the smallest singular value of certain matrices some numerical results indicate that our bounds are extremely good

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

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

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

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

efficient llr calculation for non binary modulations over fading channels log likelihood ratio llr computation for non binary modulations over fading channels is complicated a measure of llr accuracy on asymmetric binary channels is introduced to facilitate good llr approximations for non binary modulations considering piecewise linear llr approximations we prove convexity of optimizing the coefficients according to this measure for the optimized approximate llrs we report negligible performance loss compared to true llrs

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

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

pseudo spectra theory of tensors and tensor polynomial eigenvalue problems abstract this paper is devoted to the extension of the pseudo spectra theory from matrices to tensors based on the definition of an eigenpair of real symmetric tensors and results on the pseudo spectrum of square matrices we first introduce the pseudo spectrum of a complex tensor and investigate its fundamental properties such as its computational interpretations and the link with the stability of its related homogeneous dynamical system we then extend the pseudo spectrum to the setting of tensor polynomial eigenvalue problems we further derive basic structure of the pseudo spectrum for tensor polynomial eigenvalue problems including the symmetry boundedness and number of connected components under suitable mild assumptions finally we discuss the implication of the pseudo spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial

optimal alphabets and binary labelings for bicm at low snr optimal binary labelings input distributions and input alphabets are analyzed for the so called bit interleaved coded modulation bicm capacity paying special attention to the low signal to noise ratio snr regime for 8 ary pulse amplitude modulation pam and for 0 75 bit symbol the folded binary code results in a higher capacity than the binary reflected gray code brgc and the natural binary code nbc the 1 db gap between the additive white gaussian noise awgn capacity and the bicm capacity with the brgc can be almost completely removed if the input symbol distribution is properly selected first order asymptotics of the bicm capacity for arbitrary input alphabets and distributions dimensions mean variance and binary labeling are developed these asymptotics are used to define first order optimal foo constellations for bicm i e constellations that make bicm achieve the shannon limit 1 59 db it is shown that the eb n0 required for reliable transmission at asymptotically low rates in bicm can be as high as infinity that for uniform input distributions and 8 pam there are only 72 classes of binary labelings with a different first order asymptotic behavior and that this number is reduced to only 26 for 8 ary phase shift keying psk a general answer to the question of foo constellations for bicm is also given using the hadamard transform it is found that for uniform input distributions a constellation for bicm is foo if and only if it is a linear projection of a hypercube a constellation based on pam or quadrature amplitude modulation input alphabets is foo if and only if they are labeled by the nbc if the constellation is based on psk input alphabets instead it can never be foo if the input alphabet has more than four points regardless of the labeling

on the computation of structured singular values and pseudospectra structured singular values and pseudospectra play an important role in assessing the properties of a linear system under structured perturbations this paper discusses computational aspects of structured pseudospectra for structures that admit an eigenvalue minimization characterization including the classes of real skew symmetric hermitian and hamiltonian perturbations for all these structures we develop algorithms that require o n2 operations per grid point combining the schur decomposition with a lanczos method these algorithms form the basis of a graphical matlab interface for plotting structured pseudospectra 2009 elsevier b v all rights reserved

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

high snr asymptotics of mutual information for discrete constellations with applications to bicm asymptotic expressions of the mutual information between any discrete input and the corresponding output of the scalar additive white gaussian noise channel are presented in the limit as the signal to noise ratio snr tends to infinity asymptotic expressions of the symbol error probability sep and the minimum mean square error mmse achieved by estimating the channel input given the channel output are also developed it is shown that for any input distribution the conditional entropy of the channel input given the output mmse and sep have an asymptotic behavior proportional to the gaussian q function the argument of the q function depends only on the minimum euclidean distance med of the constellation and the snr and the proportionality constants are functions of the med and the probabilities of the pairs of constellation points at med the developed expressions are then generalized to study the high snr behavior of the generalized mutual information gmi for bit interleaved coded modulation bicm by means of these asymptotic expressions the long standing conjecture that gray codes are the binary labelings that maximize the bicm gmi at high snr is proven it is further shown that for any equally spaced constellation whose size is a power of two there always exists an anti gray code giving the lowest bicm gmi at high snr

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

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