aliasing truncation errors in sampling approximations of sub gaussian signals this paper starts with new aliasing truncation error upper bounds in the sampling theorem for non bandlimited stochastic signals then it investigates l p 0 t approximations of the sub gaussian random signals explicit truncation error upper bounds are established the obtained rate of convergence provides a constructive algorithm for determining the sampling rate and the sample size in the truncated whittaker kotel nikov shannon expansions to ensure the approximation of the sub gaussian signals with given accuracy and reliability some numerical examples are presented

generalized schultz iterative methods for the computation of outer inverses abstract we consider a general matrix iterative method of the type x k 1 x k p a x k for computing an outer inverse a r g n g 2 for given matrices a c m n and g c n m such that a r g n g c m here p x is an arbitrary polynomial of degree d the convergence of the method is proven under certain necessary conditions and the characterization of all methods having order r is given the obtained results provide a direct generalization of all known iterative methods of the same type moreover we introduce one new method and show a procedure how to improve the convergence order of existing methods this procedure is demonstrated on one concrete method and the improvement is confirmed by numerical examples

characterizations and representations of the p q outer generalized inverse we present a new explicit expression for the p q outer generalized inverse based on the idempotent operators p and q we derive properties of the p q outer generalized inverse and analyse relations among the various p q outer generalized inverses

on integral representation of the generalized inverse at s 2 we present a general integral representation for the generalized inverse a t s 2 which extends earlier result on the moore penrose inverse weighted moore penrose inverse and drazin inverse

the iterative methods for computing the generalized inverse at s 2 of the bounded linear operator between banach spaces in this paper we present the iterative methods for computing the generalized inverse a t s 2 over banach spaces and also for computing the generalized drazin inverses of banach algebra elements moreover we estimate the error bounds of the iterative methods for approximating a t s 2

the 2 inverse with applications in statistics an inverse g of a given matrix a which satisfies the property gag g is known as a 2 inverse this paper presents a three phase inversion procedure for which the 2 inverse is a special case we present the geometry of 2 inverses and show that starting from 2 inverses various types of generalized inverses can be derived two examples of the occurrence of 2 inverses in statistics are given one concerning the constrained least squares estimator the other concerning a necessary and sufficient condition for a quadratic form of singular multivariate normal variates to follow a chi square distribution

representation and approximation for the drazin inverse a d in this paper we establish the representations for the drazin inverse a d of a square matrix a based on which we give the faddeev type algorithms for a d and for the weighted drazin inverse a d w of a rectangular matrix a the neumann type expansions and the iterative methods for a d and the iterative methods for a d based on functional interpolation

iterative methods for the drazin inverse of a matrix with a complex spectrum iterative methods for the drazin inverse of a square matrix with a real spectrum have been developed recently these methods are generalized in the case of matrices with complex spectra semiiterative method for the drazin inverse is also discussed numerical examples are given to illustrate the theoretic results

modified sms method for computing outer inverses of toeplitz matrices we introduce a new algorithm based on the successive matrix squaring sms method this algorithm uses the strategy of e displacement rank in order to find various outer inverses with prescribed ranges and null spaces of a square toeplitz matrix using the idea of displacement theory which decreases the memory space requirements as well as the computational cost our method tends to be very effective for toeplitz matrices

from zhang neural network to scaled hyperpower iterations abstract a class of scaled hyperpower iterative methods for computing outer inverses is considered this class appears during the construction of the discrete time zhang neural network for computing the usual matrix inverse the usual hyperpower iterative methods belong to this class additionally a more general class of scaled iterative methods which includes the scaled hyperpower method is defined and studied different values of the real scaling parameter are investigated both theoretically and numerically

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

on continuity of the generalized inverse at s 2 this paper studies the continuity of the generalized inverse a t s 2 since common important generalized inverses such as a a m n a d a a d w a l 1 and a l etc are all generalized inverse a t s 2 in fact we give a unified continuity of such important generalized inverses

an efficient computation of generalized inverse of a matrix abstract we propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop this yields a high efficiency index for that computational task the algorithm has a relatively mild numerical instability and we stabilize it at the price of adding two extra matrix multiplications per iteration loop this implies an efficiency index that exceeds the known record for numerically stable iterations for this task which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra our numerical tests cover a variety of examples in the category of generalized inverses such as drazin case rectangular case and preconditioning of linear systems the test results are in good accordance with our formal study

condition number related with generalized inverse a t s 2 and constrained linear systems in this paper we introduce a pq norm and discuss the condition number of the generalized inverse at s 2 we get some results on the condition numbers with respect to at s 2 and constrained linear systems ax b x t which are similar to the regular inverse a 1 and nonsingular linear systems ax b finally we investigate the structured perturbation of the generalized inverse at s 2

whittaker kotel nikov shannon approximation of φ sub gaussian random processes abstract the article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes then it investigates l p 0 t and uniform approximations of sub gaussian random processes by finite time sampling sums explicit truncation error upper bounds are established some specifications of the general results for which the assumptions can be easily verified are given direct analytical methods are employed to obtain the results

perturbation and expression for inner inverses in banach spaces and its applications abstract it is well known that if t is invertible and t 1 is its inverse then t 1 i t t t 1 1 is the inverse of t for t t t 1 1 the main purpose of this paper is to study the following problem if t is an inner inverse of t and t t t 1 is t i t t t 1 an inner inverse of t if the answer is no when is t i t t t 1 an inner inverse of t in this paper we give a complete answer to these problems and provide some characterizations for t i t t t 1 to be an inner inverse of t in banach spaces utilizing these results we investigate similar problems for the generalized inverse 1 3 inverse 1 4 inverse 1 5 inverse 1 2 3 inverse 1 2 4 inverse moore penrose inverse group inverse and drazin inverse the results obtained in this paper extend and improve many recent results in this area

minimal properties of moore penrose inverses abstract let ax y be consistent let x0 gy be any minimum norm solution satisfying ag ag and let a be the moore penrose inverse of a it is shown that g a for any in a class containing the unitarily invariant matrix norms the conditioning of the system ax y is studied via condition numbers c a g it is shown that c a g c a a for every moreover bounds on c a g are given in terms of singular values parallel results are found when a and g are symmetric with applications to linear models of less than full rank

iterative methods for computing the generalized inverses a 2 t s of a matrix a abstract in this paper the iterative methods for computing the generalized inverse a t s 2 the unique matrix x such that xax x r x t and n x s are established and the iterative methods for computing many important generalized inverses of matrices such as a a mn a d a a d w a l 1 a l and ap l are constructed

neural network for computing pseudoinverses and outer inverses of complex valued matrices we propose two continuous time neural networks for computing generalized inverses of complex valued matrices with rank deficient cases the first of them is applicable in the pseudoinverse computation and the second one is applicable in construction of outer inverses the proposed continuous time neural networks have a low complexity of implementation and they are proved to be globally convergent without any condition compared with the existing algorithms for computing the pseudoinverse and outer inverses of matrices the global convergence of the proposed continuous time neural networks is analyzed in the complex domain effectiveness of the proposed continuous time neural networks is evaluated numerically via examples

full rank representations of 2 4 2 3 inverses and successive matrix squaring algorithm abstract we present the full rank representations of 2 4 and 2 3 inverses with given rank as well as with prescribed range and null space as particular cases of the full rank representation of outer inverses as a consequence two applications of the successive matrix squaring sms algorithm from p s stanimirovic d s cvetkovic ilic successive matrix squaring algorithm for computing outer inverses appl math comput 203 2008 19 29 are defined using the full rank representations of 2 4 and 2 3 inverses the first application is used to approximate 2 4 inverses the second application after appropriate modifications of the sms iterative procedure computes 2 3 inverses of a given matrix presented numerical examples clarify the purpose of the introduced methods

matrix left symmetry factor and its applications in generalized inverses a 2 4 t s in this paper we propose some definitions about matrix symmetry factors and discuss the construction of matrix left symmetry factor in detail the explicit expression of the generalized inverses at s 2 4 which is a 2 4 inverses of a having the prescribed rang t and null space s is derived furthermore using the expression at s 2 4 we get the corresponding results for six important generalized inverses

the representation and approximation for the weighted moore penrose inverse we present a unified representation theorem for the weighted moore penrose inverse specific expressions and computational procedures for the weighted moore penrose inverse can be uniformly derived

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests

noncore square matrices miscellany a noncore square matrix is one whose unique core nilpotent decomposition does not exclusively consist of the core part if the core part is idempotent the matrix itself is called quasiidempotent this paper makes a brief survey of these and other noncore matrices and their properties

iterative methods for computing generalized inverses abstract in this article we extend the method of yong lin chen yong lin chen iterative methods for computing the generalized inverses a t s 2 of a matrix a appl math comput 75 2 3 1996 207 222 to infinite dimensional settings precisely we construct an iterative method for computing outer generalized inverses of operators on banach spaces and also for computing the generalized drazin inverse of banach algebra elements

neural network approach to computing outer inverses based on the full rank representation abstract the first intention of this paper is to emphasize equivalence between two well known general representations of outer inverses with prescribed range and null space of a given matrix two dynamical state equations corresponding to particular expressions which are included in these representations are defined within the second objective in this way two gradient based neural networks initiated by introduced dynamical equations are exploited in generating the class of outer inverses the results corresponding to most common generalized inverses are obtained in particular cases numerical results are presented

the analysis of panel data under a markov assumption abstract methods for the analysis of panel data under a continuous time markov model are proposed we present procedures for obtaining maximum likelihood estimates and associated asymptotic covariance matrices for transition intensity parameters in time homogeneous models and for other process characteristics such as mean sojourn times and equilibrium distributions generalizations to handle covariance analysis and to the fitting of certain nonhomogeneous models are presented and an example based on a longitudinal study of the smoking habits of school children is discussed questions of embeddability and estimation are examined

empirical bayes estimators for a finite markov chain summary using results of anderson goodman 1957 we develop empirical bayes estimates for the transition probabilities of a finite stationary markov chain some asymptotic properties of these estimators are also derived

representation and approximation of the outer inverse at s 2 of a matrix a abstract in this paper we establish a basic representation and a representation theorem for the outer inverse a t s 2 of a matrix a which is the matrix x satisfying xax x r x t and n x s we develop several specific representations and iterative methods for a t s 2 we show that this representation includes many of the traditional generalized inverses and outer inverses and the relation of our results to these outer inverses will be explored

computing generalized inverses of matrices by iterative methods based on splittings of matrices in this paper we study the iterative methods for computing the generalized inverses of the form a 2 t s we prove the result that if a has the splitting a m n and if the generalized inverses a 2 t s and m 2 t s exist then the iterationx j 1 m 2 t snx j m 2 t sconverges to a 2 t s for any x 0 if and only ift t s s email protected 2 t sn 1 or equivalently m 2 t email protected 2 t sn 1 the splitting a m n is called t s splitting of a if m satisfies email protected s c m where t and s are subspaces of c n and c m respectively with dimt dims we also present a specific method to construct convergent t s splitting of a apply these two results to the most of commonly used generalized inverses such as a a d a d w a h k we can get criteria of convergence for the corresponding iterations many old and new splittings and specific choice of the related convergent splittings

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

two improvements of the iterative method for computing moore penrose inverse based on penrose equations abstract two improvements of the iterative method for computing the moore penrose inverse introduced in petkovic and stanimirovic 2011 are introduced the first improvement defines new choice for the initial approximation and ensures better stability of the method the second one defines its extension to the set of outer inverses with prescribed range and null space numerical examples for both of these improvements are presented

method of elementary transformation to compute moore penrose inverse method of elementary transformation to compute moore penrose inverse is given by applying the rank equalities of matrix the inheritable properties of moore penrose inverse on rank are also discussed

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

supremum concentration inequality and modulus of continuity for sub nth chaos processes abstract this article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize gaussian processes sub gaussian processes and random fields that are in the nth chaos of a wiener process an upper bound of dudley type on the tail of the random field s supremum is derived using a generic chaining argument it implies similar results for the expected supremum and for the field s modulus of continuity we also utilize a sharp and convenient condition using iterated malliavin derivatives to arrive at similar conclusions for suprema via a different proof which does not require full knowledge of the covariance structure

truncation error bounds for the cardinal sampling expansion of band limited signals bounds are derived for the truncation error of the cardinal sampling expansion for a large class of band limited deterministic and random signals these bounds extend and improve upon the bounds available in the literature

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

generalized inverse a 2 t s and a rank equation in this paper the rank equationrankabcx rank a is considered in the sense that b and c are characterized when a 2 t s is solution to this equation as the special cases similar results for a a m n a d and a g are also derived this contributes to certain recent results in the literature including that obtained by grosz linear algebra appl 289 1999 127 and thome and wei appl math comput 141 2003 471

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses

computing outer inverses by scaled matrix iterations convergence analysis for a class of iterative methods for computing outer inverses with prescribed range and null space is studied furthermore several heuristics for accelerating iterative methods via scaling are proposed in fact we are motivated by the fact that scaling of iterative methods for computing generalized inverses is investigated so far only on low order methods our intention is to test the scaling on higher order iterative methods here we also propose a new higher order iteration although the introduced method is efficient in terms of computational efficiency index we test its acceleration through several experiments

a note on the representation and approximation of the outer inverse at s 2 of a matrix a in this note we present a unified representation and an approximation theorem for the outer inverse a t s 2 of a matrix a which is the matrix x satisfying xax x r x t n x s based on the new approximation theorem we can develop several expressions and computational procedures for the generalized inverse a t s 2 which improve the recent work on this topic

condition numbers for the outer inverse and constrained singular linear system in this paper we investigate the condition number of the outer inverse at s 2 and outer inverse at s 2 solution of constrained linear system ax b x t where a is a real m n matrix b and x are real vectors t is a subspace let and be two positive real numbers when we consider the weighted frobenius norm a b qp f on the data we get the formula of condition number for the outer inverse at s 2 solution of constrained linear system for the normwise condition number the sensitivity of the relative condition number itself is studied and the componentwise perturbation is also investigated

partial orderings preorderings and the polar decomposition of matrices abstract when a unique decomposition of a complex rectangular matrix into two components is given then a partial ordering on the domain of one of the components induces a preordering on the whole set of matrices and partial orderings on the domains of both components induce a partial ordering on the whole set in this not we consider the three main partial orderings lo wner star and rank subtractivity on the respective domains of the two components of matrices subjected to polar decomposition a ben israel t n e greville generalized inverses theory and applications wiley new york 1974 p 255 and investigate the resulting pre and partial orderings

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

iterative methods for computing generalized inverses and splittings of operators abstract in this paper we consider an iterative method for computing generalized inverses of linear operators based on splittings of operators thus the result from yong lin chen xue yuan tan computing generalized inverses of matrices by iterative methods based on splitting of matrices appl math comp 163 2005 309 325 is extended to infinite dimensional settings

an application of φ subgaussian technique to fourier analysis abstract we present an application to random fourier series for subgaussian random variables in particular we generalize previously known notions of dependence for random variables by introducing the concept of f manageable random variables and consider fourier series of f manageable subgaussian random variables gaussian series are then a particular case of the series considered in the paper conditions for the uniform convergence of such series in probability are established and a rate of convergence is investigated moreover the results are new even for the case of independent random variables

the hyperpower iteration revisited the hyperpower method is generalized by inseting an idempotent matrix p it is shown that most of the iterations that can be used for computing generalized inverses can be recast in this form by suitable choice of the idempotent the limit or the initial values the convergence behavior of bqk is analyzed and the results are applied to give a detailed investigation of this iteration

partial orders based on outer inverses classes of outer inverses are used to create new partial orders and unify the classification of some of the old partial orders these classes are then compared with those obtained by using g inverses inner inverses

limit representations of generalized inverses and related methods we generalize the iterative method for implementation of the limit representation of the moore penrose inverse lim a email protected a a 1a introduced by zukovski and lipcer z vicisl mat i mat fiz 12 1972 843 15 1975 489 more precisely we define an iterative process for implementing the general limit formula lim a 0 ai r s 1r in a certain case the presented method gives an iterative process which implements the known limit representation of the drazin inverse introduced by meyer siam j appl math 26 1974 469 also the known limit representation of the generalized inverse a t s 2 is a special case of the general limit formula in the case of a full rank matrix a we define a limit representation and an iterative method for computation of the left or right inverses of a also we introduce two alternative limit representations for the set of 2 and 1 2 inverses and develop the corresponding iterative processes moreover we propose a finite algorithm for computation of the generalized inverses contained in the limit formula lim a 0 ai r s 1r using the method developed by ji appl math comput 61 1994 151

displacement structure of the generalized inverse a t s 2 it is well known that matrices with a uv displacement structure possess generalized inverse with a vu displacement structure estimation for the displacement rank of a t s 2 u va t s 2 are presented where a t s 2 is the 2 inverse of a we extend the results due to heinig and hellinger in 1994

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

full rank representation of generalized inverse at s 2 and its application in this paper we introduce a full rank representation of the generalized inverse a t s 2 of a given complex matrix a which is based on an arbitrary full rank decomposition of g where g is a matrix such that r g t and n g s using this representation we introduce the minor of the generalized inverse a t s 2 as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained as an application we use an example to demonstrate that this representation is correct

the representation of generalized inverse at s 2 3 and its applications this paper presents the explicit expression for matrix right symmetry factor with prescribed rang and null space moreover the explicit expression for generalized inverse a t s 2 3 which is a 2 3 inverse of a having the prescribed rang t and null space s is derived as an application two numerical examples are given

on the perturbation and subproper splittings for the generalized inverse at s 2 of rectangular matrix a abstract in this paper the perturbation and subproper splittings for the generalized inverse at s 2 the unique matrix x such that xax x r x t and n x s are considered we present lower and upper bounds for the perturbation of at s 2 convergence of subproper splittings for computing the special solution at s 2 b of restricted rectangular linear system ax b x t are studied for the solution at s 2 b we develop a characterization therefore we give a unified treatment of the related problems considered in literature by ben israel berman hanke neumann plemmons etc

an empirical bayes model for markov dependent binary sequences with randomly missing observations abstract we develop an improved empirical bayes estimation methodology for the analysis of two state markov chains observed from heterogeneous individuals first the two transition probabilities corresponding to each chain are assumed to be drawn from a common bivariate distribution that has beta marginals second randomly missing observations are incorporated into the likelihood for the hyperparameters by efficiently summing over all possible values for the missing observations a likelihood ratio test is used to test for dependence between the transition probabilities posterior distributions for the transition probabilities are also derived as is an approximation for the equilibrium probabilities the proposed procedures are illustrated in a numerical example and in an analysis of longitudinal store display data

full rank representations of outer inverses based on the qr decomposition abstract an efficient algorithm for computing a t s 2 inverses of a given constant matrix a based on the qr decomposition of an appropriate matrix w is presented correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full rank factorization are derived in particular cases we derive representations of 2 4 and 2 3 inverses numerical examples on different test matrices dense or sparse are presented as well as the comparison with several well known methods for computing the moore penrose inverse and the drazin inverse

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

further results on iterative methods for computing generalized inverses in this paper we reconsider the iterative method x k x k email protected i ax k 1 k 1 2 email protected email protected 0 for computing the generalized inverse a t s 2 over banach spaces or the generalized drazin inverse a d of a banach algebra element a reveal the intrinsic relationship between the convergence of such iterations and the existence of a t s 2 or a d and present the error bounds of the iterative methods for approximating a t s 2 or a d moreover we deduce some necessary and sufficient conditions for iterative convergence to a t s 2 or a d

gauss jordan elimination method for computing outer inverses this paper deals with the algorithm for computing outer inverse with prescribed range and null space based on the choice of an appropriate matrix g and gauss jordan elimination of the augmented matrix g i the advantage of such algorithms is the fact that one can compute various generalized inverses using the same procedure for different input matrices in particular we derive representations of the moore penrose inverse the weighted moore penrose inverse the drazin inverse as well as 2 4 and 2 3 inverses numerical examples on different test matrices are presented as well as the comparison with well known methods for generalized inverses computation

core ep decomposition and its applications abstract in this paper we introduce a new decomposition called the core ep decomposition in the present paper for square matrices and give some of its applications by applying the decomposition we derive several characterizations of the core ep inverse introduce two orders the core ep order and the core minus partial order and characterize their properties we also give some properties of the decomposition

a new representation for at s 2 3 a new representation of a t s 2 3 of a having prescribed range space t and null space s is derived using this representation the well known generalized inverses such as the moore penrose inverse the group inverse the drazin inverse the bott duffin inverse and the generalized bott duffin inverse are computed three numerical examples are worked out to demonstrate the computation of the moore penrose inverse the drazin inverse and the 2 3 inverse to show the efficacy of our approach

assessing the order of dependence for partially exchangeable binary data abstract the problem we consider is how to assess the order of serial dependence within partially exchangeable binary sequences we obtain exact conditional tests comparing any two orders by finding the conditional distribution of data given certain transition counts these tests are facilitated with a new monte carlo scheme asymptotic tests are also discussed in particular we show that the likelihood ratio tests have an asymptotic chi squared distribution thus generalizing the results of billingsley for the particular case of markov chains we apply these methods to several datasets and perform a simulation to study their properties

innovation based on gaussian elimination to compute generalized inverse at s 2 abstract in this paper we execute elementary row and column operations on the partitioned matrix g a g g g 0 into i s 0 0 0 0 0 a t s 2 to compute generalized inverse a t s 2 of a given complex matrix a where g is a matrix such that r g t and n g s the total number of multiplications and divisions operations is t m n s 2 m n 2 4 m s 1 2 n s m s n s m n s and the upper bound of t m n s is less than 6 m n 2 3 2 n 3 1 2 n 2 when n m a numerical example is shown to illustrate that this method is correct

partial orders based on core nilpotent decomposition abstract in this paper we study partial orders in terms of the core nilpotent decomposition we derive some characterizations of the c n partial ordering create a new partial ordering the g drazin partial ordering which is a generalization of c n partial ordering and give some properties and characterizations of the g drazin partial ordering

g outer inverse of banach spaces operators abstract the aim of this paper is to present a new generalized inverse which is called a g outer inverse of operators between two banach spaces as a generalization of g drazin inverses of square matrices and banach space operators several characterizations of this new inverse are given based on the g outer inverse we define and characterize the g outer partial order extending the g drazin partial order to operators between two banach spaces we also relate this new partial order to the minus partial order in the end we study an outer binary relation which is an equivalence relation on the corresponding set

the representation and approximation for drazin inverse we present a unified representation theorem for drazin inverse specific expression and computational procedures for drazin inverse can be uniformly derived numerical examples are tested and the application to the singular linear equations is also considered

a note on the rank subtractivity ordering abstract some well known matrix partial orderings can be created by supplementing the so called rank subtractivity partial ordering by an appropriate preordering in this note a new partial ordering with the same property is introduced this new ordering is seen to be weaker than the other orderings but still preserving some inheritance properties

a geometrical approach on generalized inverses by neumann type series the convergence of the neumann type series to 1 2 inverses has been shown by k tanabe linear algebra appl 10 1975 163 in this paper these results indicating conditions characterizing the convergence of this series to different generalized inverses are extended in addition these results for obtaining different generalized inverses from the hyperpower method are applied finally generalized involutory matrices are introduced and characterized using the obtained results

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

an iterative method for solving general restricted linear equations the newton iterative method for computing outer inverses with prescribed range and null space is used in the non stationary richardson iterative method to develop an iterative method for solving general restricted linear equations starting with any suitably chosen initial iterate our method generates a sequence of iterates converging to the solution the necessary and sufficient conditions for the convergence along with the error bounds are established the applications of the iterative method for solving some special linear equations are also discussed a number of numerical examples are worked out they include singular square rectangular randomly generated rank deficient matrices full rank matrices and a set of singular matrices given in matrix computation toolbox mctoolbox with the condition numbers ranging from order 1016 to 1050 the mean cpu time mct and the error bounds are the performance measures used our results when compared with the results obtained by chen 1997 leads to substantial improvement in terms of both computational speed and accuracy

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

successive matrix squaring algorithm for computing outer inverses abstract in this paper we derive a successive matrix squaring sms algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix a c r m n we generalize the results from the papers l chen e v krishnamurthy i macleod generalized matrix inversion and rank computation by successive matrix powering parallel computing 20 1994 297 311 y wei successive matrix squaring algorithm for computing drazin inverse appl math comput 108 2000 67 75 y wei h wu j wei successive matrix squaring algorithm for parallel computing the weighted generalized inverse a mn appl math comput 116 2000 289 296 and obtain an algorithm for computing various classes of outer generalized inverses of a instead of particular matrices used in these articles we use an appropriate matrix r c s n m s r numerical examples are presented

time shifted aliasing error upper bounds for truncated sampling cardinal series abstract time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific l p p 1 classes analogous results are obtained in multidimensional regular sampling truncation error analysis is provided in all cases considered moreover sharpness of bounding inequalities convergence rates and various sufficient conditions are discussed

a markov mixture model for magazine exposure abstract a magazine exposure model that mixes klotz s 1973 dependent bernoulli trials model for nonsubscribers with a degenerate distribution for subscribers is proposed let xi 1 if a person reads an issue of a particular magazine and 0 otherwise klotz s parameterization is pr xi 1 p and pr xi 1 x i 1 1 for i 1 k using the markov assumption he obtains the joint distribution of r k i 2 x i 1 xi s k i 1 xi and t x 1 xk of which we are interested in the marginal distribution of s the total number of issues a person reads it is expected that p will be low for nonsubscribers but high for subscribers so this heterogeneity is modeled by mixing klotz s markov model with a point mass of magnitude at the point s k maximum likelihood estimates of p and are used to fit the markov mixture model to 40 magazines from a large print media survey the proposed model is shown to give a much better fit to these data than the beta binomial model the most popular nonp

the representation and computation of generalized inverse at s 2 this paper presents a novel representation for the generalized inverse a t s 2 based on this we give an algorithm to compute this generalized inverse as an application we use gauss jordan elimination to compute the weighted moore penrose inverse a m n and the drazin inverse a d

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived