moving horizon estimation for markov jump systems in this paper we study the state estimation problem for a class of nonlinear markov jump systems with the moving horizon estimation algorithm which is an optimization based filtering method the optimal estimate is obtained by minimizing a quadratic estimation cost function defined on fixed sliding horizon measurements for this purpose the quadratic estimation cost function is formulated from the negative logarithm of the joint states distribution whose particular factorization is represented by a bayesian network by analyzing the corresponding full information estimation as a basic problem for nonlinear markov jump systems the formulation of the moving horizon estimation is developed an example is presented to compare the proposed new estimation technique with classical interacting multiple model particle filtering and show the effectiveness and advantages of the new estimation scheme

on multivariate rayleigh and exponential distributions in this paper expressions for multivariate rayleigh and exponential probability density functions pdfs generated from correlated gaussian random variables are presented we first obtain a general integral form of the pdfs and then study the case when the complex gaussian generating vector is circular we consider two specific circular cases the exchangeable case when the variates are evenly correlated and the exponentially correlated case expressions for the multivariate pdf in these cases are obtained in integral form as well as in the form of a series of products of univariate pdfs we also derive a general expression for the multivariate exponential characteristic function cf in terms of determinants in the exchangeable and exponentially correlated cases cf expressions are obtained in the form of a series of products of univariate gamma cfs the cf of the sum of exponential variates in these cases is obtained in closed form finally the bivariate case is presented mentioning its main features while the integral forms of the multivariate pdfs provide a general analytical framework the series and determinant expressions for the exponential cfs and the series expressions for the pdfs can serve as a useful tool in the performance analysis of digital modulation over correlated rayleigh fading channels using diversity combining

on the additive structure of the inverses of banded matrices the additive structure of the inverses of banded matrices is investigated under certain conditions the inverse of a 2k 1 diagonal symmetric banded matrix can be expressed as a sum of k symmetric matrices belonging to the class of inverses of symmetric irreducible tridiagonal matrices in the nonsymmetric case a more complicated structure is obtained applications are mentioned for the resolution of constant coefficient banded linear systems in vlsi models

on the reachability in any fixed time for positive continuous time linear systems this paper deals with the reachability of continuous time linear positive systems the reachability of such systems which we will call here the strong reachability amounts to the possibility of steering the state in any fixed time to any point of the positive orthant by using nonnegative control functions the main result of this paper essentially says that the only strongly reachable positive systems are those made of decoupled scalar subsystems moreover the strongly reachable set is also characterized

inverses of quasi tridiagonal matrices abstract discretizations in various types of problems lead to quasi tridiagonal matrices in this paper the inverse of a nonsingular quasi tridiagonal matrix is obtained in addition a necessary and sufficient condition for a block matrix to have a quasi tridiagonal inverse is derived

a hidden markov model with dependence jumps for predictive modeling of multidimensional time series abstract hidden markov models hmms are a popular approach for modeling sequential data typically based on the assumption of a first or moderate order markov chain however in many real world scenarios the modeled data entail temporal dynamics the patterns of which change over time in this paper we address this problem by proposing a novel hmm formulation treating temporal dependencies as latent variables over which inference is performed specifically we introduce a hierarchical graphical model comprising two hidden layers on the first layer we postulate a chain of latent observation emitting states the temporal dependencies between which may change over time on the second layer we postulate a latent first order markov chain modeling the evolution of temporal dynamics dependence jumps pertaining to the first layer latent process as a result of this construction our method allows for effectively modeling non homogeneous observed data where the patterns of the entailed temporal dynamics may change over time we devise efficient training and inference algorithms for our model following the expectation maximization paradigm we demonstrate the efficacy and usefulness of our approach considering several real world datasets

analysis of hybrid selection maximal ratio combining in correlated nakagami fading we present an analysis of a hybrid selection maximal ratio combining diversity system over an evenly correlated slow frequency nonselective nakagami fading channel where the correlation coefficient between any pair of the diversity branch gain amplitudes is the same and all average branch signal to noise ratios snrs are equal in this system the l branches with the largest instantaneous snr out of n available branches are selected and combined using maximal ratio combining from the joint characteristic function cf of the instantaneous branch snrs we obtain an expression for the cf of the combiner output snr as a series of elementary cfs the expression can be conveniently used to obtain the symbol error probability of coherent detection of different m ary modulation schemes we illustrate our methodology using m ary phase shift keying as an example

reachability observability and realizability of continuous time positive systems this paper discusses reachability observability and realizability of single input single output linear time invariant systems in which state variables and or input output functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems we define a set reachable from the origin with nonnegative inputs and also a set observable with nonnegative outputs we investigate geometrical structures of the sets through convex analysis and a duality relation between them is established next we consider positive realization of a given transfer function using the reachable set and the observable set we give a necessary and sufficient condition for positive realizability an example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization

bounds for entries of the inverse matrix of a tridiagonal matrix inequalities for the moduli of entries of the inverse matrix of a tridiagonal matrix are obtained these results are applicable to a relatively large family of matrices which occur in many branches of numerical analysis applied mathematics and engineering

on the solution of a second order linear homogeneous difference equation with variable coefficients abstract a closed form solution of a second order linear homogeneous difference equation with variable coefficients is presented as an application of this solution we obtain expressions for cos n and sin n 1 sin as polynomials in cos

advances on the reachability index of positive 2 d systems this work was supported by dgi grant mtm2010 18228 recommended by associate editor z wang

determinantal formulae for matrices with sparse inverses ii asymmetric zero patterns abstract in an earlier paper formulae for det a as a ratio of products of principal minors of a were exhibited for any given symmetric zero pattern of a 1 these formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of a 1 however just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries the symmetry of the zero pattern is not essential for these formulae we describe here how analogous formulae for det a may be obtained in the asymmetric zero pattern case by introducing a directed spanning tree we also examine the converse question of determining all possible zero patterns of a 1 which guarantee that a certain determinantal formula holds

the method of polarized traces for the 2d helmholtz equation we present a solver for the 2d high frequency helmholtz equation in heterogeneous acoustic media with online parallel complexity that scales optimally as o n l where n is the number of volume unknowns and l is the number of processors as long as l grows at most like a small fractional power of n the solver decomposes the domain into layers and uses transmission conditions in boundary integral form to explicitly define polarized traces i e up and down going waves sampled at interfaces local direct solvers are used in each layer to precompute traces of local green s functions in an embarrassingly parallel way the offline part and incomplete green s formulas are used to propagate interface data in a sweeping fashion as a preconditioner inside a gmres loop the online part adaptive low rank partitioning of the integral kernels is used to speed up their application to interface data the method uses second order finite differences the complexity scalings are empirical but motivated by an analysis of ranks of off diagonal blocks of oscillatory integrals they continue to hold in the context of standard geophysical community models such as bp and marmousi 2 where convergence occurs in 5 to 10 gmres iterations while the parallelism in this paper stems from decomposing the domain we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines

the green s function for the huckel tight binding model applications of the huckel tight binding model are ubiquitous in quantum chemistry and solid state physics the matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph which is also the laplacian matrix plus twice the identity in this paper we analytically calculate the determinant and when it exists the inverse of this matrix in connection with the green s function g of the n n huckel matrix a corollary is a closed form expression for a harmonic sum eq 12 we then extend the results to d dimensional lattices whose linear size is n the existence of the inverse becomes a question of number theory we prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if n 1 and d are odd and d is smaller than the smallest divisor of n 1 we corroborate our results by demonstrating the entry patterns of the green s function and discuss applications related to transpo

a backward three term recurrence relation for vector valued continued fractions and its applications this paper presents a backward three term recurrence relation which is a solution of an open problem in graves morris numer math 42 1983 331 by means of this relation we succeed in establishing some conclusions about the two backward successive convergents of vector valued continued fractions moreover an expression for the vector valued continued fraction in series form is obtained and some theorems of the convergence of the vector valued continued fraction are proved by virtue of the recurrence relation

variable length markov chains we study estimation in the class of stationary variable length markov chains vlmc on a finite space the processes in this class are still markovian of high order but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high order markov chains from an algorithmic view the vlmc model class has attracted interest in information theory and machine learning but statistical properties have not yet been explored provided that good estimation is available the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest the latter is exemplified with some dna data a version of the tree structured context algorithm proposed by rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of vlmcs this remains true even when the underlying model increases in dimensionality fur thermore consistent estimation of minimal state spaces and mixing properties of fitted models are given we also propose a new bootstrap scheme based on fitted vlmcs we show its validity for quite general stationary categorical time series and for a broad range of statistical procedures

parallel solution of block tridiagonal linear systems abstract the explicit structure of the inverse of block tridiagonal matrices is presented in terms of blocks defined by linear recurrence relations parallel algorithms are shown which solve block second order linear recurrences without using commutativity moreover we investigate the parallel solution of the associated block tridiagonal linear system using this theoretical background the implementation of the algorithms is analyzed both on a small number of processors and on a hypercube the resulting complexity is given in terms of parallel steps each consisting of block operations and the cost due to interprocessor communications is taken into account too

a fast algorithm for evaluating n th order tri diagonal determinants the cost of all existing algorithms for evaluating the nth order determinants numerical analysis 7th edition brooks cole publishing pacific grove ca 2001 is at most o n3 in the current article we present a new efficient computational algorithm for evaluating the nth order tri diagonal determinants with cost o n only the algorithm is suited for implementation using computer algebra systems such as maple and macsyma some examples are given to illustrate the algorithm

analytical inversion of general periodic tridiagonal matrices in this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed toeplitz for both periodic and non periodic tridiagonal matrices an efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given it is suited for implementation using computer algebra systems such as maple matlab macsyma and mathematica an example is also given to illustrate the algorithm

a simple graph theoretic characterization of reachability for positive linear systems in this paper we consider discrete time linear positive systems that is systems defined by a pair a b of non negative matrices we study the reachability of such systems which in this case amounts to the freedom of steering the state in the positive orthant by using non negative control sequences this problem was solved recently canonical forms for positive discrete time linear control systems linear algebra appl 310 2000 49 however we derive here necessary and sufficient conditions for reachability in a simpler and more compact form these conditions are expressed in terms of particular paths in the graph which is naturally associated with the system

toeplitz matrices with banded inverses abstract given a toeplitz matrix t with banded inverse i e t 1 ij 0 for j i p we show that the elements of t can be expressed in terms of the roots of a polynomial then using properties we have previously established we generalize this result appropriately to allow singular t and show that the converse also holds finally we give a sufficient condition for the decay of the elements of t as one moves away from the diagonal

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

the multivariable servomechanism problem for positive lti systems in this technical note we study the servomechanism problem for positive open loop stable lti systems under constant tracking signals and constant disturbances in particular we show that in general the mimo servomechanism problem for positive systems is not solvable but is solvable for various classes of mimo systems and for the class of siso systems in which the disturbances are small compared to the tracking signal in particular we give existence results as to what subclass of reference and disturbance signals can be considered for the case when either the mathematical model of the plant is known or unknown and we provide the controller structure that solves the problem for the case of measurable or unmeasurable disturbance signals

vanishing minor conditions for inverse zero patterns abstract we study the relationship between the zero nonzero pattern of an invertible matrix and the vanishing minors of the matrix and of its inverse in particular we show how to determine when a matrix b could be the inverse of a matrix a with a given zero nonzero pattern in fact there is always a set of almost principal minors of b in one to one correspondence with the set of zero entries of a whose vanishing implies that b 1 has zeros everywhere that a does provided certain principal minors of b do not vanish

the inverse of banded matrices the inverses of r banded matrices for r 1 2 3 have been thoroughly investigated as one can see from the references we provide let b r n email protected email protected n be an nxn matrix of entries a j i email protected email protected r email protected email protected r with the remaining un indexed entries all zeros in this paper generalizing a method of mallik 1999 5 we give the lu factorization and the inverse of the matrix b r n if it exists our results are valid for an arbitrary square matrix taking r n and so we will give a new approach for computing the inverse of an invertible square matrix our method is based on hessenberg submatrices associated to b r n

an inversion algorithm for a banded matrix in this paper an inversion algorithm for a banded matrix is presented the n twisted decompositions of a banded matrix are given first then the inverse of the matrix is obtained one column at time the method is about two times faster than the standard method based on the lu decomposition as is shown with the analysis of computing complexity and the numerical experiments

canonical forms for positive discrete time linear control systems abstract in this paper the properties of reachability controllability and essential reachability of positive discrete time linear control systems are studied these properties are characterized in terms of the directed graph of the state matrix from these characterizations canonical forms of those properties are deduced

on the eigenvalues of some tridiagonal matrices a solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by william trench the method presented can be generalizable to other problems

explicit inverse of a tridiagonal p r toeplitz matrix abstract we have named tridiagonal p r toeplitz matrix to those tridiagonal matrices in which each diagonal is a quasi periodic sequence d p j r d j so with period p n but multiplied by a real number r we present here the necessary and sufficient conditions for the invertibility of this kind of matrices and explicitly compute their inverse the techniques we use are related with the solution of boundary value problems associated to second order linear difference equations these boundary value problems can be expressed throughout the discrete schrodinger operator and their solutions can be computed using recent advances in the study of linear difference equations with quasi periodic coefficients the conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix whereas the solutions of the boundary value problems provides the entries of the inverse matrix

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

structural and computational properties of possibly singular semiseparable matrices a classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers for establishing recurrence relations among its columns or rows and moreover for efficiently evaluating its characteristic polynomial in this paper we provide sparse structured representations of a semiseparable matrix a which hold independently of the fact that a is singular or not these relations are found by pointing out the band structure of the inverse of the sum of a plus a certain sparse perturbation of minimal rank further they can be used to determine in a computationally efficient way both a reflexive generalized inverse of a and its characteristic polynomial

on a constant diagonals matrix in this paper we consider a constant diagonals matrix the matrix was discussed in witula and slota r witula d slota on computing the determinants and inverses of some special type of tridiagonal and constant diagonals matrices appl math comput 189 1 2007 514 527 the authors gave some results on determinant and the inverse of the matrix for some special cases we give lu factorization and then compute determinant of the matrix we determine eigenvalues of the matrix further we obtain some relationships between permanent of the matrix and terms of a certain recurrence relation

connecting optimization with spectral analysis of tri diagonal matrices we show that the global minimum resp maximum of a continuous function on a compact set can be approximated from above resp from below by computing the smallest rest largest eigenvalue of a hierarchy of r times r tri diagonal matrices of increasing size equivalently it reduces to computing the smallest resp largest root of a certain univariate degree r orthonormal polynomial this provides a strong connection between the fields of optimization orthogonal polynomials numerical analysis and linear algebra via asymptotic spectral analysis of tri diagonal symmetric matrices

transformations to rank structures by unitary similarity we study transformations by unitary similarity of a nonderogatory matrix to certain rank structured matrices this kind of transformations can be described in a unified framework that involves krylov matrices the rank structures here addressed are preserved by qr iterations and every iterate can be associated with a suitable krylov matrix

determinantal formulae for matrices with sparse inverses abstract the determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be read off from the graph of the inverse of the matrix the only information used is the zero pattern of the inverse and each zero pattern yields one or more corresponding formulae for the determinant

inverses of banded matrices abstract we establish a correspondence between the vanishing of a certain set of minors of a matrix a and the vanishing of a related set of minors of a 1 in particular inverses of banded matrices are characterized we then use our results to find patterns for toeplitz matrices with banded inverses finally we give an interesting determinant formula for inverses of banded matrices and show that in general a banded partial matrix may be completed in a unique way to give a banded inverse of the same bandwidth

the minimal eigenvalues of a class of block tridiagonal matrices in this correspondence we study the minimal eigenvalues of a class of block tridiagonal matrices from telecommunication system analysis we present an eigenvalue analysis for two user systems and efficient estimates for m user systems

on the closed representation for the inverses of hessenberg matrices the general representation for the elements of the inverse of any hessenberg matrix of finite order is here extended to the reduced case with a new proof those entries are given with proper hessenbergians from the original matrix it justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order and some elementary properties of the inverse these results are applied on the resolvent matrix associated to a finite hessenberg matrix in standard form two examples on the unit disk are given

on band matrices and their inverses abstract structural properties of the inverses of band matrices are discussed the definition of semiseparable matrices is given and the theorem is proved that the inverse of a strict band matrix is a semiseparable matrix and vice versa finally a recurrence algorithm is recommended for computing the blocks of the inverses of strict band matrices

improving formulas for the eigenvalues of finite block toeplitz tridiagonal matrices abstract after a short overview improvements based on the kronecker product are proposed for the eigenvalues of n n block toeplitz tridiagonal block tt matrices with k k matrix entries common in applications some extensions of the spectral properties of the toeplitz tridiagonal matrices are pointed out the eigenvalues of diagonalizable symmetric and skew symmetric block tt matrices are studied besides if certain matrix square root is well defined it is proved that every block tt matrix with commuting matrix entries is isospectral to a related symmetric block tt one further insight about the eigenvalues of hierarchical hermitian block tt matrices of use in the solution of pdes is also achieved

reachability properties of continuous time positive systems in this technical note reachability properties of continuous time positive systems are introduced and characterized specifically first it is shown that reachability and strong reachability are equivalent properties and thus the characterization of strong reachability derived in is extended to the weaker notion of reachability in the second part of the technical note essential reachability is introduced and necessary and sufficient conditions for this property to hold are provided

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

invertible completions of partial operator matrices the nonsymmetric case abstract we consider nonsymmetric partial operator matrices r whose directed graphs belong to a certain class and for which certain key fully specified principal submatrices are invertible under these circumstances we prove the existence of a unique invertible completion f of r such that f 1 ij is block zero whenever rij is unspecified in this way the existing results of johnson and lundquist are generalized to the nonsymmetric case

estimates for the inverse of tridiagonal matrices arising in boundary value problems by considering tridiagonal matrices as three term recurrence relations with dirichlet boundary conditions one can formulate their inverses in terms of green s functions this analysis is applied to three point difference schemes for 1 d problems and five point difference schemes for 2 d problems we derive either an explicit inverse of the jacobian or a sharp estimate for both uniform and nonuniform grids

epidemic spreading by indirect transmission in a compartmental farm abstract in this paper we present a discrete dynamic system which describes an epidemic spreading within a single farm where animals are separated into batches in this model we consider an indirect transmission of the disease coming from the bacteria remaining in the reservoir and taking into account the transfer of bacteria between adjacent compartments in our model tridiagonal matrices of non negative blocks are involved the development of the matrix spectral properties allows us to improve our understanding of the epidemic spreading within a farm with the above mentioned characteristics based on the results obtained we have determined some bounds to obtain the maximum number of batches and the maximum population in each batch to ensure that the disease dies out

positive semidefinite matrices with a given sparsity pattern abstract this paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries one of the main purposes is to obtain information and if possible describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros in particular we study those patterns for which the ranks of extremal matrices are less than a given number

local positive reachability of nonlinear continuous time systems local positive reachability of positive nonlinear continuous time systems is investigated it is shown that the system is locally positively reachable from the origin if and only if in some neighborhood of the origin for each subface of the nonnegative cone of the state space there is a control value such that the vector field corresponding to this control value is tangent to the subface and nonzero this is a natural generalization of a characterization of positive reachability of positive linear continuous time systems contrary to discrete time systems positive reachability of continuous time systems requires that very restrictive conditions be satisfied

storage compression of inverses of band matrices abstract two algorithms are presented which reduce the total amount of storage locations needed by the inverse of a band matrix the first compresses the existing inverse of a general band matrix the second computes a compressed form of the inverse of tridiagonal block matrix both algorithms have been numerically tested over significant classes of matrices

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

a study on quaternion blockquasi tridiagonal systems the objective of the present work is to study the solution of quaternion block quasi tridiagonalsystems kershaw and rozsa and romani have proposed a method for calculating matrix inverses using second kind chebyshev polynomials this method has been generalized later to block tridiagonal matrices with quaternionic entries by costa and serodio in the present work we make use of this method to solve block quasi tridiagonal systems of the same kind the computational effort to obtain the solution is evaluated and future more efficient strategies are proposed

factorizations and representations of the backward second order linear recurrences we show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second order linear recurrences also considering how to the negatively subscripted terms of second order linear recurrences can be connected to chebyshev polynomials by determinants of these matrices we give factorizations and representations of these numbers

positivity of continuous time descriptor systems with time delays this technical note is concerned with positivity characteristic of continuous time descriptor systems with time delays first a set of necessary and sufficient conditions is presented to check the property then considering a descriptor time delay system with two assumptions a new time delay system is established whose positivity is equivalent to that of the original system furthermore a set of necessary and sufficient conditions is provided to check the positivity of the new system finally a numerical example is given to illustrate the validity of the results obtained

an analytical approach explicit inverses of periodic tridiagonal matrices abstract we derive an explicit formula for the inverse of a general periodic tridiagonal matrix our approach is to derive its l u factorization using backward continued fractions bcf which are an essential tool in number theory we then use these formulae to construct an algorithm for inverting a general periodic tridiagonal matrix which we implement in maple 1 finally we present the results of testing the efficiency of our new algorithm against another published implementation and against the library procedures available within maple to invert a general matrix and to compute its determinant

low grade matrices and matrix fraction representations abstract the lower grade lgrade of an n n matrix a is the largest rank of any subdiagonal block of a symmetric partition of a a number of algebraic results on lgrade are given when a has lgrade d it can be approximately decomposed as a u v where u is an upper triangular matrix and v has rank d if a satisfies ga n with g and n having lower bandwidths d g and d n then the decomposition is exact a u v where u is an upper triangular matrix with lower bandwidth equal to d n d g and v has low rank generically d g this result generalizes the well known representations of a when a g 1 and g is banded a generalization of the givens rotation product decomposition of unitary hessenberg matrices is given and its structure is analyzed these consecutive subblock products are used to construct a representation of an lgrade d matrix a of the form ga n with g and n having lower bandwidth d g can be chosen to be lower triangular or unitary

two sided bounds on the inverses of diagonally dominant tridiagonal matrices abstract we establish upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices these bounds improve the bounds recently given by shivakumar and ji moreover we show how to improve our bounds iteratively for an n n m matrix this iterative refinement yields the exact inverse after n 1 steps

positive input reachability and controllability of positive systems abstract we study controllability and reachability for discrete linear control systems x k 1 ax k bu k in which the state vector x k the control vector u k the n n matrix a and the n m matrix b have nonnegative entries for an unconstrained system controllability is equivalent to reachability from the zero state in n steps furthermore a b is controllable if and only if the matrix c n b ab a 2 b a n 1 b has rank n we show that these properties are not equivalent for positive systems this has important implications for control strategy the timing of control inputs and the size and structure of the control matrix b are much more critical the zero nonzero pattern of c n plays a crucial role we illustrate these fundamental differences with models of a two species reversible chemical reaction and with two pharmacokinetic models of drug distribution in the human body

a new recursive algorithm for inverting general tridiagonal and anti tridiagonal matrices in the current article the authors present a new recursive symbolic computational algorithm for inverting general tridiagonal and anti tridiagonal matrices an illustrative example is given

a note on generalized hessenberg matrices two different generalizations of the concept of hessenberg matrices have appeared recently in the literature in this note we study their connections in more detail and give a general structure theorem for these classes of matrices

a fast numerical algorithm for the inverse of a tridiagonal and pentadiagonal matrix in the current paper we present a computationally efficient algorithm for obtaining the inverse of a tridiagonal matrix and a pentadiogonal matrix few conditions are required and the algorithm is suited for implementation using computer algebra systems

local observability of nonlinear positive continuous time systems abstract two observability properties of continuous time nonlinear positive systems are studied local observability and local positive observability an algebraic necessary and sufficient condition for local observability is proved it is expressed with the aid of ideals and cones in the ring of germs of analytic functions local positive observability is characterized in a more geometric fashion this characterization naturally extends the known characterization of positive observability of positive linear systems as for linear systems criteria for local positive observability for nonlinear systems are very restrictive this result is parallel to characterization of local positive reachability of nonlinear systems

controllability and reachability criteria for discrete time positive systems discrete time positive systems have been the object of widespread interest in the literature in the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which in this context make their appearance in this paper the properties of ordinary and essential controllability and reachability are analysed based on a graph theoretic approach we provide practical criteria to verify whether a given positive system f g is endowed with these properties together with canonical forms for describing reachable controllable pairs

robust sequential data modeling using an outlier tolerant hidden markov model hidden markov chain models using finite gaussian mixture models as their hidden state distributions have been successfully applied in sequential data modeling and classification applications nevertheless gaussian mixture models are well known to be highly intolerant to the presence of untypical data within the fitting data sets used for their estimation finite student s t mixture models have recently emerged as a heavier tailed robust alternative to gaussian mixture models overcoming these hurdles to exploit these merits of student s t mixture models in the context of a sequential data modeling setting we introduce in this paper a novel hidden markov model where the hidden state distributions are considered to be finite mixtures of multivariate student s t densities we derive an algorithm for the model parameters estimation under a maximum likelihood framework assuming full diagonal and factor analyzed covariance matrices the advantages of the proposed model over conventional approaches are experimentally demonstrated through a series of sequential data modeling applications

the interplay of ranks of submatrices a banded invertible matrix t has a remarkable inverse all upper and lower submatrices of t 1 have low rank depending on the bandwidth in t the exact rank condition is known and it allows fast multiplication by full matrices that arise in the boundary element method we look for the right proof of this property of t 1 ultimately it reduces to a fact that deserves to be better known complementary submatrices of any t and t 1 have the same nullity the last figure in the paper when t is tridiagonal shows that on and above the diagonal of t 1 all rows are proportional

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

transformations of matrices into banded form abstract a class of full or triangular matrices r is described for which there exist banded matrices b such that the product br is also banded the banded matrices yield recursion relations for solving systems of linear equations examples of such matrices arising from second derivative operators acting on orthogonal function expansions are used to illustrate the main theorem and its application practical considerations in efficient implementation are discussed

a note on the nullity theorem in this paper we take a closer look at the nullity theorem as formulated by markham and fiedler in 1986 the theorem is a valuable tool in the computations with structured rank matrices it connects ranks of subblocks of an invertible matrix a with ranks of other subblocks in his inverse a 1 a little earlier barrett and feinsilver 1981 proved a theorem very close to the nullity theorem but restricted to semiseparable and tridiagonal matrices which are each others inverses we will adapt the ideas of barrett and feinsilver to come to a new alternative proof of the nullity theorem based on determinantal formulas in the second part of the paper we extend the nullity theorem to make it suitable for two types of decompositions namely the lu and the qr decomposition these theorems relate the ranks of subblocks of the factors l u and q to the ranks of subblocks of the factored matrix it is shown that a combination of the nullity theorem and his extended versions is suitable to predict in an easy manner the structure of decompositions and or of inverses of structured rank matrices e g higher order band higher order semiseparable hessenberg and many other types of matrices as examples to show the power of the nullity theorem and the related theorems we apply them to semiseparable and related matrices

controllability of multi input positive discrete time systems linear discrete time positive systems arise in various fields of science such as economic modelling behavioural science stochastic processes etc recently some authors have examined the problem of the controllability of such systems for the single input case this paper provides necessary and sufficient conditions for characterizing the controllability properties of multi input systems the results here developed show that only a narrow class of these systems enjoys the properties of reachability and controllability

on periodic block tridiagonal matrices abstract periodic block tridiagonal matrices are defined and conditions are given for factorizing their characteristic polynomial by means of the zeros of chebyshev polynomials of the second kind these conditions are expressed by the block centrosymmetry of certain submatrices along the main diagonal the main theorems can be considered as generalizations of an earlier result published by elsner and redheffer and by rozsa

a fast preconditioned conjugate gradient toeplitz and toeplitz like solvers abstract for a toeplitz or toeplitz like matrix t we define a preconditioning applied to the symmetrized matrix t h t which decreases the condition number compared to the one of t h t and even the one of t this enables us to accelerate the conjugate gradient algorithm for solving toepiltz and toeplitz like linear systems thus extending the previous results of 1 restricted to the hermitian positive definite case the extension relies on some recent formulae of gohberg and olshevsky for the inverses of toeplitz like matrices

explicit formula for the inverse of a tridiagonal matrix by backward continued fractions in this paper we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse for this purpose considering usual continued fraction we define backward continued fraction for a real number and give some basic results on backward continued fraction we give the relationships between the usual and backward continued fractions then we reobtain the lu factorization and determinant of a tridiagonal matrix furthermore we give an efficient and fast computing method to obtain the elements of the inverse of a tridiagonal matrix by backward continued fractions comparing the earlier result and our result on the elements of the inverse of a tridiagonal matrix it is seen that our method is more convenient efficient and fast