introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

continuous phase modulation part i full response signaling the continuous phase modulation cpm signaling scheme has gained interest in recent years because of its attractive spectral properties data symbol pulse shaping has previously been studied with regard to spectra for binary data and modulation index 0 5 in this paper these results have been extended to the m ary case where the pulse shaping is over a one symbol interval the so called full response systems results are given for modulation indexes of practical interest concerning both performance and spectrum comparisons are made with minimum shift keying msk and systems have been found which are significantly better in e b n 0 for a large signal to noise ratio snr without expanded bandwidth schemes with the same bit error probability as msk but with considerably smaller bandwidth have also been found significant improvement in both power and bandwidth are obtained by increasing the number of levels m from 2 to 4

an accelerated technique for solving one type of discrete time algebraic riccati equations abstract algebraic riccati equations are encountered in many applications of control and engineering problems e g lqg problems and h control theory in this work we study the properties of one type of discrete time algebraic riccati equations our contribution is twofold first we present sufficient conditions for the existence of a unique positive definite solution second we propose an accelerated algorithm to obtain the positive definite solution with the rate of convergence of any desired order numerical experiments strongly support that our approach performs extremely well even in the almost critical case as a byproduct we show that this method is capable of computing the unique negative definite solution once it exists

positive definite solutions of the matrix equations x a x qa q abstract in this paper we investigate nonlinear matrix equations x a x q a q and x a x q a q where q 0 1 we derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations we provide a sufficient condition for the equation x a x q a q to have two different positive definite solutions and a sufficient condition for the equation x a x q a q to have a unique positive definite solution we also propose iterative methods for obtaining positive definite solutions for these equations

delayed decision feedback sequence estimation the authors delayed decision feedback sequence estimation ddfse for uncoded pam signals is considered estimates on the performance of the algorithm are given and simulation results are provided for several examples a more general form of ddfse applicable to coded modulation systems is also presented as an example detection of trellis coded qpsk quadrature phase shift keyed signals over intersymbol interference channels is discussed

the inverse mean problem of geometric mean and contraharmonic means abstract in this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices proposed in w n anderson m e mays t d morley g e trapp the contraharmonic mean of hsd matrices siam j algebra disc meth 8 1987 674 682 a x y 2 x 1 y 1 1 b x y by proving its equivalence to the well known nonlinear matrix equation x t bx 1b where t 1 2 a a a 8 ba 1 b is the unique positive definite solution of x a 2bx 1b the inverse mean problem is solvable if and only if b a

the iterative method for solving nonlinear matrix equation xs a x ta q abstract nonlinear matrix equation x s a x t a q where q is an hermitian positive definite matrix a is the conjugate transpose of the matrix a and both of s and t are positive integers has widely applied background in this paper necessary conditions and sufficient conditions for the existence of hermitian positive definite solutions of this equation are derived an iterative method for obtaining hermitian positive definite solutions is proposed moreover a perturbation bound for the hermitian positive definite solutions is given

newton s method for solving a class of nonlinear matrix equations in this paper a class of nonlinear matrix equations x q a h i x c 1a is considered an equivalent form of this equation is derived and the properties of the solutions are studied then newton s iterative method is applied to solve this equivalent one finally two numerical examples are given to illustrate the efficiency of newton s method

the structure preserving doubling algorithm and convergence analysis for a nonlinear matrix equation abstract in this paper applying special properties of doubling transformation a structure preserving doubling algorithm is developed for computing the positive definite solutions for a nonlinear matrix equation further by mathematical induction we establish the convergence theory of the structure preserving doubling algorithm finally we offer corresponding numerical examples to illustrate the effectiveness of the derived algorithm

a list type reduced constraint generalization of the viterbi algorithm the viterbi algorithm va an optimum decoding rule for a q ary trellis code of constraint length k operates by taking the best survivor from each of q k 1 lists of candidates at each decoding step a generalized va gva is proposed that makes comparisons on the basis of a label of length l l leq k it selects incorporating the notion of list decoding the s best survivors from each of q l 1 lists of candidates at each decoding step coding theorems for a discrete memoryless channel are proved for gva decoding and shown to be natural generalizations of those for va decoding an example of intersymbol interference removal is given to illustrate the practical benefits that the gva can provide

on the matrix equation x a tx 12ma i in this paper we study iterative methods for finding the extremal positive definite solutions of the matrix equation x a tx 1ma i first a condition on the existence of a positive definite solution of this matrix equation is given then the existence as well as the rate of convergence of some proposed algorithms to obtain the extremal positive definite solutions of this equation is presented moreover a generalization of computationally simple and efficient known algorithm is applied for obtaining the extremal positive definite solutions in addition both the necessary and sufficient conditions for this matrix equation to have positive definite solution are presented numerical examples are given to illustrate the performance and the effectiveness of the algorithms

algorithms for hyperbolic quadratic eigenvalue problems we consider the quadratic eigenvalue problem or the qep 2 a b c x 0 where a b and c are hermitian with a positive definite the qep is called hyperbolic if x bx 2 4 x ax x cx for all nonzero x c r we show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the qep a hyperbolic qep is overdamped if b is positive definite and c is positive semidefinite we show that a hyperbolic qep whose eigenvalues are necessarily real is overdamped if and only if its largest eigenvalue is nonpositive for overdamped qeps we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction we also present a new measure for the degree of hyperbolicity of a hyperbolic qep

some interlacing properties of the schur complement of a hermitian matrix abstract for a hermitian matrix h with nonsingular principal submatrix a it is shown that the eigenvalues of the moore penrose inverse of the schur complement h a of a in h interlace the eigenvalues of the moore penrose inverse of h moreover if h is semidefinite it is shown that the eigenvalues of h a interlace the eigenvalues of h

new convergence results on functional iteration techniques for the numerical solution of m g 1 type markov chains by performing an accurate analysis of the convergence we give a complete theoretical explanation of the experimental behaviour of functional iteration techniques for the computation of the minimal nonnegative solution g of the matrix equation x sum i 0 infty x ia i arising in the numerical solution of m g 1 type markov chains here the a i s are nonnegative k times k matrices such that the matrix sum i 0 infty a i is column stochastic moreover we introduce a general class of functional iteration methods which includes the standard methods and we give an optimality convergence result in this class

scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method a simple scaling procedure for discrete time riccati equations is introduced this procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized schur method without changing the condition of the underlying problem a computable bound for the relative error of the solution of the riccati equation is also derived

positive definite solutions of the nonlinear matrix equation x a h x 1 a i this paper is concerned with the positive definite solutions to the nonlinear matrix equation x a email protected 1a i where x is the unknown and a is a given complex matrix by introducing and studying a matrix operator on complex matrices it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation w b tw 1b i which has been extensively studied in the literature where b is a real matrix and is uniquely determined by a it is also shown that if the considered nonlinear matrix equation has a positive definite solution then it has the maximal and minimal solutions bounds of the positive definite solutions are also established in terms of matrix a finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

an accelerated cyclic reduction based solvent method for solving quadratic eigenvalue problem of gyroscopic systems abstract the quadratic eigenvalue problem qep 2 m g k x 0 with m t m being positive definite k t k being negative definite and g t g is associated with gyroscopic systems in guo 2004 a cyclic reduction based solvent crs method was proposed to compute all eigenvalues of the above mentioned qep firstly the problem is converted to find a suitable solvent of the quadratic matrix equation qme m x 2 g x k 0 then using a cayley transformation and a proper substitution the qme is transformed into the nonlinear matrix equation nme z a t z 1 a q with a m k g and q 2 m k the problem finally can be solved by applying the cr method to obtain the maximal symmetric positive definite solution of the nme as long as the qep has no eigenvalues on the imaginary axis or for some cases where the qep has eigenvalues on the imaginary axis however when all eigenvalues of the qep are far away from or near the origin the cayley transformation seems not to be the best one and the convergence rate of the crs method proposed in guo 2004 might be further improved in this paper inspired by using a doubling algorithm to solve the qme we use a mobius transformation instead of the cayley transformation to present an accelerated crs acrs method for solving the qep of gyroscopic systems in addition we discuss the selection strategies of optimal parameter for the acrs method numerical results demonstrate the efficiency of our method

fast verified computation for stabilizing solutions of discrete time algebraic riccati equations fast iterative algorithms for computing interval matrices containing solutions of discrete time algebraic riccati equations are proposed these algorithms involve only cubic complexity per iteration the stabilizability and uniqueness of the contained solution can moreover be verified by these algorithms numerical results show the properties of the algorithms

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

efficient computation of the extreme solutions of x a x 1 a q and x a x 1 a q we propose a new quadratically convergent algorithm having a low computational cost per step and good numerical stability properties which allows the simultaneous approximation of the extreme solutions of the matrix equations x a x 1 a q and x a x 1 a q the algorithm is based on the cyclic reduction method

the extremal solution of the matrix equation x s a x q a i in this paper the distribution of the positive definite solution of the nonlinear equation x s a x qa i is deduced the existence of the maximal solution and the minimal solution is discussed and two new iterative algorithms for obtaining these solutions are proposed these algorithms avoid matrix inversion

a new inversion free method for a rational matrix equation abstract motivated by the classical newton schulz method for finding the inverse of a nonsingular matrix we develop a new inversion free method for obtaining the minimal hermitian positive definite solution of the matrix rational equation x a x 1 a i where i is the identity matrix and a is a given nonsingular matrix we present convergence results and discuss stability properties when the method starts from the available matrix aa we also present numerical results to compare our proposal with some previously developed inversion free techniques for solving the same rational matrix equation

on the solution of the nonlinear matrix equation xn f x abstract we consider a class of nonlinear matrix equations x n f x 0 where f is a self map on the convex cone p k of k k positive definite real matrices it is shown that for n 2 the matrix equation has a unique positive definite solution depending continuously on the function f if f belongs to the semigroup of nonexpansive mappings with respect to the gl k r invariant riemannian metric distance on p k which contains congruence transformations translations the matrix inversion and in particular symplectic hamiltonians appearing in kalman filtering we show that the sequence of positive definite solutions varying over n 2 converges always to the identity matrix

signal codes convolutional lattice codes the coded modulation scheme proposed in this paper has a simple construction an integer sequence representing the information is convolved with a fixed continuous valued finite impulse response fir filter to generate the codeword a lattice point due to power constraints the code construction includes a shaping mechanism inspired by precoding techniques such as the tomlinson harashima filter we naturally term these codes convolutional lattice codes or alternatively signal codes due to the signal processing interpretation of the code construction surprisingly properly chosen short fir filters can generate good codes with large minimal distance decoding can be done efficiently by sequential decoding or for better performance by bidirectional sequential decoding error analysis and simulation results indicate that for the additive white gaussian noise awgn channel convolutional lattice codes with computationally reasonable decoders can achieve low error rate close to the channel capacity

computations with infinite toeplitz matrices and polynomials we relate polynomial computations with operations involving infinite band toeplitz matrices and show applications to the numerical solution of markov chains of nonlinear matrix equations to spectral factorizations and to the solution of finite toeplitz systems in particular two matrix versions of graeffe s iteration are introduced and their convergence properties are analyzed correlations between graeffe s iteration for matrix polynomials and cyclic reduction for block toeplitz matrices are pointed out the paper contains a systematic treatment of known topics and presentation of new results improvements and extensions

some efficient boundary integral strategies for time harmonic wave problems in an elastic halfspace abstract when boundary integral equation boundary element methods bie bems are used for halfspace problems such as occur frequently in elastodynamics the fullspace stokes fundamental solution or halfspace lamb fundamental solution can be used to formulate the bie both stokes and lamb approaches have advantages and disadvantages this paper presents systematic strategies based on the bem for 3d halfspace elastodynamic problems wherein the best features of formulations based on the stokes and the lamb solutions are exploited strategies are illustrated and numerical results are given for point sources and radiation from a spherical void in a halfspace

on the semigroup property for some structured iterations abstract nonlinear matrix equations play a crucial role in science and engineering problems however solutions of nonlinear matrix equations cannot in general be given analytically one standard way of solving nonlinear matrix equations is to apply the fixed point iteration with usually only the linear convergence rate to advance the existing methods we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order we realize our way by starting with examples of solving the scalar equations and also connect this method with some well known equations including but not limited to the stein matrix equation the generalized eigenvalue problem the generalized nonlinear matrix equation the discrete time algebraic riccati equations to express the capacity of this method

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

toward solution of matrix equation x af x b c abstract this paper studies the solvability existence of unique solution closed form solution and numerical solution of matrix equation x af x b c with f x x t f x x and f x x h where x is the unknown it is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard stein equations in the form of w a w b c where the dimensions of the coefficient matrices a b and c are the same as those of the original equation closed form solutions of equation x af x b c can then be obtained by utilizing standard results on the standard stein equation on the other hand some generalized stein iterations and accelerated stein iterations are proposed to obtain numerical solutions of equation x af x b c necessary and sufficient conditions are established to guarantee the convergence of the iterations

new upper solution bounds of the discrete algebraic riccati matrix equation in this note we present upper matrix bounds for the solution of the discrete algebraic riccati equation dare using the matrix bound of theorem 2 2 we then give several eigenvalue upper bounds for the solution of the dare and make comparisons with existing results the advantage of our results over existing upper bounds is that the new upper bounds of theorem 2 2 and corollary 2 1 are always calculated if the stabilizing solution of the dare exists whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions finally we give numerical examples to demonstrate the effectiveness of the derived results

breadth first maximum likelihood sequence detection geometry sa b c is an algorithm penetrating a tree or trellis breadth first it performs maximum likelihood sequence detection mlsd under that structural constraint and also under the complexity constraints determined by the parameters b and c first c is the number of partitions into which the states are distributed and b denotes the number of paths in each partition recursively selecting paths which are closest to the received signal in the euclidean distance hamming distance sense guarantees constrained mlsd for the additive white gaussian binary symmetric channel the previously presented vector euclidean distance ved is an important tool for analyzing sa b c over the additive white gaussian noise awgn channel a geometric interpretation of the signals involved clarifies the basic properties of this ved and other relevant general results invariance monotonicity these results also form a basis for the construction of an algorithm for the efficient and fast calculation of minimum veds of interest for large signal to noise ratio snr detection performance this in turn reveals the necessary complexity requirements to meet specified performance requirements for concrete trellis coded systems here the simple example of convolutionally coded rate 1 2 with gray coded quaternary phase shift keying over the awgn channel is considered when c 1 and choosing b spl sim spl radic s s being the number of states in the code trellis gives the same asymptotic detection performance large snr as unconstrained mlsd e g implemented using the viterbi algorithm

partitioning boundary integral equations and exact green s functions discretization of boundary integral equations leads to large full systems of algebraic equations in practice partitioning is a method for solving such systems by breaking them down into smaller systems it may be viewed merely as a technique from linear algebra however it is profitable to view it as arising directly from partitions of the boundary these partitions could be natural such as two separate boundaries but they need not be we investigate partitioning in the context of multiple scattering of acoustic waves by two sound hard obstacles the ideas extend to other physical situations specifically we make a connection between partitioning and the use of the exact green s function for a single obstacle in isolation this suggests computing the action of this green s function once and for all storing it perhaps on a compact disc and then using it to solve other problems in which the second obstacle is altered one example of this approach is computing the stress distribution around a cavity of a standard but complicated shape inside a structure whose shape is varied the theoretical foundation for these ideas is given as well as a connection with the use of generalized born series for multiple scattering problems important distinctions between the partitioning green s function idea in this paper and seemingly similar ideas such as substructuring multi zoning and domain decomposition are made

generalized app detection of continuous phase modulation over unknown isi channels techniques for computing soft information in the presence of unknown intersymbol interference are presented with a particular focus on iterative detection of serially concatenated continuous phase modulation the techniques are centered around the recursive least squares algorithm thus enabling unsupervised detection in particular we employ bidirectional estimation

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

limited search trellis decoding of convolutional codes the least storage and node computation required by a breadth first tree or trellis decoder that corrects t errors over the binary symmetric channels is calculated breadth first decoders work with code paths of the same length without backtracking the viterbi algorithm is an exhaustive trellis decoder of this type other schemes look at a subset of the tree or trellis paths for random tree codes theorems about the asymptotic number of paths required and their depth are proved for concrete convolutional codes the worst case storage for t error sequences is measured in both cases the optimal decoder storage has the same simple dependence on t the m algorithm and algorithms proposed by g j foschini ibid vol it 23 p 605 9 sept 1977 and by s j simmons phd diss queens univ kingston ont canada are optimal or nearly so they are all far more efficient than the viterbi algorithm

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

a note on the fixed point iteration for the matrix equations x a x 1a i the fixed point iteration is a simple method for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a i the plus minus equations the convergence of this method may be very slow if the initial matrix is not chosen carefully a strategy for choosing better initial matrices has been recently proposed by ivanov hasanov and uhlig they proved that this strategy can improve the convergence in general and observed from numerical experiments that dramatic improvement happens for the plus equation with some matrices a it turns out that the matrices a are normal for those examples in this note we prove a result that explains the dramatic improvement in convergence for normal and thus nearly normal matrices for the plus equation a similar result is also proved for the minus equation ams classification 15a24 65f30 65h10

solutions and perturbation analysis for the nonlinear matrix equation x i 1mai x 1ai i abstract in this paper we study the nonlinear matrix equation x i 1 m a i x 1 a i i arising in some physical calculations some new necessary and sufficient conditions for the existence of positive definite solutions are given based on the matrix differentiation a precise perturbation bound for the maximal positive definite solution is derived and the new result is illustrated by a numerical example in the end

the matrix equation x a tx 1 a q and its application in nano research the matrix equation x a tx 1 a q has been studied extensively when a and q are real square matrices and q is symmetric positive definite the equation has positive definite solutions under suitable conditions and in that case the solution of interest is the maximal positive definite solution the same matrix equation plays an important role in green s function calculations in nano research but the matrix q there is usually indefinite so the matrix equation has no positive definite solutions and one is interested in the case where the matrix equation has no positive definite solutions even when q is positive definite the solution of interest in this nano application is a special weakly stabilizing complex symmetric solution in this paper we show how a doubling algorithm can be used to find good approximations to the desired solution efficiently and reliably

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

towards positive definite solutions of a class of nonlinear matrix equations abstract this paper is concerned with the positive definite solutions of the nonlinear matrix equation nme x a h x 1 a i n where a is a given complex matrix we show that such an nme has a positive definite solution if and only if an auxiliary standard nme in the form of y a a h y 1 a a q has furthermore the relationship regarding the maximal positive definite solution and the minimal positive definite solution between these two nmes are established three iterative algorithms are proposed to solve the considered nme the first method is directly related to the considered original equation but involves matrix inversion the second method avoids the computation of matrix inversion while the third method is an accelerated version of the first one convergence performances of these three kinds of algorithms are analyzed in detail numerical examples are worked out to validate the effectiveness of the proposed algorithms

numerical solution of a quadratic eigenvalue problem we consider the quadratic eigenvalue problem qep 2 m g k x 0 where m m t is positive denite k k t is negative denite and g g t the eigenvalues of the qep occur in quadruplets or in real or purely imaginary pairs we show that all eigenvalues of the qep can be found eciently and with the correct symmetry by nding a proper solvent x of the matrix equation mx 2 gx k 0 as long as the qep has no eigenvalues on the imaginary axis this solvent approach works well also for some cases where the qep has eigenvalues on the imaginary axis ams classication 15a18 15a24 65f30

list decoding of turbo codes list decoding of turbo codes is analyzed under the assumption of a maximum likelihood ml list decoder it is shown that large asymptotic gains can be achieved on both the additive white gaussian noise awgn and fully interleaved flat rayleigh fading channels it is also shown that the relative asymptotic gains for turbo codes are larger than those for convolutional codes finally a practical list decoding algorithm based on the list output viterbi algorithm lova is proposed as an approximation to the ml list decoder simulation results show that the proposed algorithm provides significant gains corroborating the analytical results the asymptotic gain manifests itself as a reduction in the bit error rate ber and frame error rate fer floor of turbo codes

on ando li mathias geometric mean equations abstract in this paper we consider a family of nonlinear matrix equations based on the higher order geometric means of positive definite matrices that proposed by ando li mathias we prove that the geometric mean equation x b g a 1 a 2 a m x x x n has a unique positive definite solution depending continuously on the parameters of positive definite ai and positive semidefinite b it is shown that the unique positive definite solutions gn a1 a2 am for b 0 satisfy the minimum properties of geometric means yielding a sequence of higher order geometric means of positive definite matrices

on equations that are equivalent to the nonlinear matrix equation x a x α a q the nonlinear matrix equation x 1 a x aa q 0 email protected 1 is equivalent to the nonlinear matrix equation x a x aa q 0 email protected 1 the nonlinear matrix equation x 1 axa 1 a q 1 a is equivalent to the nonlinear matrix equation x 1 a x aa q 1 a the necessary and sufficient conditions for the existence of a positive definite solution of x 1 a x aa q 0 email protected 1 and x 1 axa 1 a q 1 a are given in the process two iterative algorithms are obtained estimations of the errors of the iterative algorithms are derived two numerical examples are given that demonstrate that the iterative algorithms are applicable

on the tripling algorithm for large scale nonlinear matrix equations with low rank structure for the large scale nonlinear matrix equations with low rank structure the well developed doubling algorithm in low rank form da lr is known as an efficient method to compute the stabilizing solution by further analyzing the global efficiency index constructed in this paper we propose a tripling algorithm in low rank form ta lr from two points of view the cyclic reduction and the symplectic structure preservation the new presented algorithm shares the same pre processing complexity with that of da lr but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset under the solvability condition the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order numerical experiments including some from nano research show that the ta lr is highly efficient to compute the stabilizing solution of large scale nonlinear matrix equations with low rank structure

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

a soft output mimo detector with achievable information rate based partial marginalization in this paper we propose a soft output detector for multiple input multiple output mimo channels that utilizes achievable information rate air based partial marginalization pm the proposed air based pm air pm detector has superior performance compared to previously proposed pm designs and other soft output detectors such as k best while at the same time yielding lower computational complexity a detection latency that is independent of the number of transmit layers and straightforward inclusion of soft input information using a tree representation of the mimo signal the key property of the airpm is that the connections among all child layers are broken therefore least square ls estimates used for marginalization are obtained independently and in parallel which have better quality than the zero forcing decision feedback zf df estimates used in previous pm designs such a property of the airpm detector is designed via a mismatched detection model that maximizes the air furthermore we show that the chain rule holds for the air calculation which facilitates an information theoretic characterization of the air pm detector less

comment on improved analysis of list decoding and its application to convolutional codes and turbo codes in a recent paper bai ieee trans inf theory 2007 an improved analysis concerning the analysis of list decoding was presented the event that the correct codeword is excluded from the list is central for the additive white gaussian noise awgn channel an important quantity is the in bai ieee trans inf theory 2007 called effective euclidean distance this was earlier considered in aulin ieee trans commun 1999 under the name vector euclidean distance ved where also a simple mathematical expression for this quantity was easily derived for any list size in bai ieee trans inf theory 2007 a geometrical analysis gives this when the list size is 1 2 or 3

on reduced complexity equalization based on ungerboeck and forney observation models this correspondence investigates the asymptotic behavior of reduced state intersymbol interference isi equalizers based on the ungerboeck and forney observation models it is shown that ungerboeck based equalizers can suffer from correct path loss cpl even in the noiseless regime isi conditions that lead to cpl are analytically derived and illustrated by examples

low complexity decoders for constant envelope digital modulations digital angle modulations having input symbol memory can be demodulated using maximum likelihood sequence estimation mlse or viterbi decoding the demodulation of the more bandwidth efficient of these can require a large number of computations in this paper lower complexity decoding approaches are presented these decoders use a predetermined processing order and a reduced number of survivor signals s at every time nt processing is performed on the signal sequences using metrics likelihoods obtained by a matched filter bank similar to that needed for mlse the decoders can achieve asymptotic optimality of error rate while being computationally faster and simpler than mlse for many modulations in addition error rate performance can be traded for complexity reduction expected performance has been verified for representative modulations

a structure preserving doubling algorithm for nonsymmetric algebraic riccati equation in this paper we propose a structure preserving doubling algorithm sda for the computation of the minimal nonnegative solution to the nonsymmetric algebraic riccati equation nare based on the techniques developed for the symmetric cases this method allows the simultaneous approximation to the minimal nonnegative solutions of the nare and its dual equation requiring only the solutions to two linear systems and several matrix multiplications per iteration similar to newton s method and the fixed point iteration methods for solving nares we also establish global convergence for sda under suitable conditions using only elementary matrix theory we show that sequences of matrices generated by sda are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the nare and its dual equation numerical experiments show that the sda algorithm is feasible and effective and outperforms newton s iteration and the fixed point iteration methods

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

on time harmonic elastic wave analysis by the boundary element method for moderate to high frequencies abstract certain inherent difficulties in obtaining accurate boundary element solutions to problems of elastodynamics at moderate to high frequencies are examined some procedures for overcoming or compensating for these difficulties are described in addition results demonstrating the effectiveness of these procedures are presented finally the use of these procedures in solving transient problems via fourier transform inversion is illustrated

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

a boundary integral equation method for radiation and scattering of elastic waves in three dimensions etude du rayonnement et de la diffusion d ondes elastiques par des obstacles de forme arbitraire

two structure preserving doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation in this paper we present two structure preserving doubling like algorithms for obtaining the positive definite solution of the nonlinear matrix equation x a h x 1 a q where x c n i n is an unknown matrix and q c n i n is a hermitian positive definite matrix we prove that the sequences generated by the algorithms converge to the positive definite solution of the considered matrix equation r quadratically in addition we also present some numerical results to illustrate the behavior of the considered algorithm

a new fast multipole formulation for the elastodynamic half space green s tensor in this article a version of the frequency domain elastodynamic fast multipole boundary element method fm bem for semi infinite media based on the half space green s tensor and hence avoiding any discretization of the planar traction free surface is presented the half space green s tensor is often used in non multipole form until now for computing elastic wave propagation in the context of soil structure interaction with applications to seismology or civil engineering however unlike the full space green s tensor the elastodynamic half space green s tensor cannot be expressed using derivatives of the helmholtz fundamental solution as a result multipole expansions of that tensor cannot be obtained directly from known expansions and are instead derived here by means of a partial fourier transform with respect to the spatial coordinates parallel to the free surface the obtained formulation critically requires an efficient quadrature for the fourier integral whose integrand is both singular and oscillatory under these conditions classical gaussian quadratures would perform poorly fail or require a large number of points instead a version custom tailored for the present needs of a methodology proposed by rokhlin and coauthors which generates generalized gaussian quadrature rules for specific types of integrals has been implemented the accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single layer elastodynamic potentials involving up to about n 6 i 10 5 degrees of freedom in particular a complexity significantly lower than that of the non multipole version is shown to be achieved the 3d elastodynamic half space green s tensor is expressed as a 2d fourier integral it possesses the needed separated variable form allowing multipole like acceleration the 2d fourier integral is computed by a dedicated generalized gaussian quadrature efficiency and accuracy are assessed on comprehensive numerical tests a complexity significantly lower than using non multipole version is achieved

solution of a class of nonlinear matrix equations abstract in this paper we solve nonlinear matrix equations of the form x q i 1 p a i f i x a i r i and x q i 1 p a i f i x a i r i j 1 q b j g j x b j q j where 1 1 r i q j 1 1 q p n the collection of all n n hermitian positive definite matrices and a i b j s are n n matrices also f i g j s are monotone mappings from p n into p n examples are given to illustrate that the equations can not be solved by previously known theorems

on the first degree fejer riesz factorization and its applications to x a x 1 a q abstract given a laurent polynomial with matrix coefficients that is positive semi definite over the unit circle in the complex plane the fejer riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint this paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation x a x 1 a q in particular the nonlinear equation can be recast as a linear sylvester equation subject to unitary constraint the sylvester equation is readily obtainable from hermitian eigenvalue computation the unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate free newton iteration for high precision calculation this approach offers a complete parametrization of all solutions and in contrast to most existent algorithms makes it possible to find all solutions if so desired

verified computation for the hermitian positive definite solution of the conjugate discrete time algebraic riccati equation abstract a fast iterative algorithm is proposed for numerically computing an interval matrix containing the hermitian positive definite solution to the conjugate discrete time algebraic riccati equation this algorithm requires only cubic complexity per iteration and verifies the uniqueness of the contained solution numerical results show effectiveness and robustness of the algorithm

solving large scale nonlinear matrix equations by doubling we consider the solution of the large scale nonlinear matrix equation x bx 1a q 0 with a b q x cn n and in some applications b a or h the matrix q is assumed to be nonsingular and sparse with its structure allowing the solution of the corresponding linear system qv r in o n computational complexity furthermore b and a are respectively of ranks ra rb n the type 2 structure preserving doubling algorithm by lin and xu 2006 24 is adapted with the appropriate applications of the sherman morrison woodbury formula and the low rank updates of various iterates two resulting large scale doubling algorithms have an o ra rb 3 computational complexity per iteration after some pre processing of data in o n computational complexity and memory requirement and converge quadratically these are illustrated by the numerical examples

positive definite solutions of the matrix equations x a x q a q q 1 in this paper we investigate nonlinear matrix equations x a x qa q where q 1 we derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations we provide a sufficient condition for the equation x a x qa q to have two different positive definite solutions and several sufficient conditions for the equation x a x qa q to have a unique positive definite solution we also propose iterative methods for obtaining positive definite solutions of these equations numerical examples are given to illustrate the effectiveness of the methods

breadth first maximum likelihood sequence detection basics the problem of performing breadth first maximum likelihood sequence detection mlsd under given structural and complexity constraints is solved and results in a family of optimal detectors given a trellis with s states these are partitioned into c classes where b paths into each class are selected recursively in each symbol interval the derived result is to retain only those paths which are closest to the received signal in the euclidean hamming distance sense each member in the sa b c family of sequence detectors sa denotes search algorithm performs complexity constrained mlsd for the additive white gaussian noise awgn bsc channel the unconstrained solution is the viterbi algorithm va analysis tools are developed for each member of the sa b c class and the asymptotic snr probability of losing the correct path is associated with a new euclidean distance measure for the awgn case the vector euclidean distance ved the traditional euclidean distance is a scalar special case of this termed the scalar euclidean distance sed the generality of this ved is pointed out some general complexity reductions exemplify those associated with the va approach

on the hermitian positive definite solution of the nonlinear matrix equation x i 1mai x 1ai i abstract in this paper we study the matrix equation x i 1 m a i x 1 a i i where ai i 1 2 m is square matrices and obtain some conditions for the existence of positive definite solution of this equation two iterative algorithms to find the maximal positive definite solution are given some numerical results are reported to illustrate the effectiveness of the algorithms

reduced state sequence estimation with set partitioning and decision feedback a reduced state sequence estimator for linear intersymbol interference channels is described the estimator uses a conventional viterbi algorithm with decision feedback to search a reduced state subset trellis that is constructed using set partitioning principles the complexity of maximum likelihood sequence estimation mlse due to the length of the channel memory and the size of the signal set is systematically reduced an error probability analysis shows that a good performance complexity tradeoff can be obtained in particular the results indicate that the required complexity to achieve the performance of mlse is independent of the size of the signal set for large enough signal sets simulation results are provided for two partial response systems a simple technique for quadrature partial response signaling qprs is described that eliminates the quasicatastrophic nature of the ml trellis

an iterative algorithm for the solution of the discrete time algebraic riccati equation abstract the discrete time algebraic riccati equation is solved in this study by an iterative algorithm for the square root of a squared hamiltonian matrix which is obtained from the s 1 transformation of the symplectic pencil associated with the riccati equation the symplectic givens and n n block diagonal orthogonal transformations are used before the iterative process so that the iteration is structure preserving and can achieve on average 60 reduction of computation time compared with the qz algorithm a formal analysis for roundoff errors and some numerical examples are also given

complex symmetric stabilizing solution of the matrix equation x a x 1a q abstract we study the matrix equation x a x 1 a q where a is a complex square matrix and q is complex symmetric special cases of this equation appear in green s function calculation in nano research and also in the vibration analysis of fast trains in those applications the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators the stabilizing solution is the solution of practical interest in this paper we provide an elementary proof of the existence for the general matrix equation under an assumption that is satisfied for the two special applications moreover our new approach here reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part the unique stabilizing solution can be computed efficiently by the doubling algorithm