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

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

the expression of the generalized inverse of the perturbed operator under type i perturbation in hilbert spaces let h1 h2 be two hilbert spaces over the complex field c and let t h1 h2 be a bounded linear operator with the generalized inverse t let t t t be a bounded linear operator with t t 1 suppose that dim ker t dim ker t or r t r t 0 then t has the generalized inverse t i i t t 1 i t t i t i t t 1 1 1 tt 1t x i tt tt tt i tt 1 i tt ast 1tt i tt i 1 with t t 1 t t this result gives an analogue of theorem 3 9 of m z nashed generalized inverses and applications academic press new york 1976 in hilbert spaces

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

some equivalent conditions of stable perturbation of operators in hilbert spaces let h 1 h 2 be two hilbert spaces over the complex field c and let t email protected h 1 h 2 be two bounded linear operators with the m p generalized inverse t if r t r t 0 we say that t is the stable perturbation of t in this paper we give five equivalent conditions that make t being the stable perturbation of t under the assumption t email protected email protected email protected 1 these equivalent conditions generalize not only the notation of rank preserving perturbations of matrices but also the notation of acute perturbations of matrices as a result we obtain the following suppose that r t r t 0 and t email protected email protected email protected 1 email protected t t email protected email protected t email protected email protected t email protected email protected email protected this result generalizes corresponding result when h 1 h 2 are all finite dimensional

representations of error terms in jensen s and some related inequalities with applications abstract we give integral representations of error terms in the classical jensen s inequality as well as in four related inequalities starting from these results it is possible to derive various lower and upper asymptotic expansions for solutions of the functional equation f x 1 f x g x in particular we present several results concerning the gamma function

on the drazin inverse of the sum of two operators and its application to operator matrices abstract given two bounded linear operators f g on a banach space x such that g 2 f g f 2 0 we derive an explicit expression for the drazin inverse of f g for this purpose firstly we obtain a formula for the resolvent of an auxiliary operator matrix in the form m f i g f g from the provided representation of f g d several special cases are considered in particular we recover the case g f 0 studied by hartwig et al r e hartwig g wang y wei some additive results on drazin inverse linear algebra appl 322 2001 207 217 for matrices and by djordjevic and wei d s djordjevic y wei additive results for the generalized drazin inverse j aust math soc 73 1 2002 115 126 for operators finally we apply our results to obtain representations for the drazin inverse of operator matrices in the form m a b c d which are extensions of some cases given in the literature

sharp norm estimations for moore penrose inverses of stable perturbations of hilbert c module operators it is well known that norm upper bound vert bar t dag vert leq frac vert t dag vert 1 vert t dag vert cdot vert delta t vert plays a fundamental role in stable resp rank preserving perturbation analysis for moore penrose inverses of hilbert space operators resp matrices in this paper in the general setting of hilbert c module operators we provide a new approach to the study of norm upper bounds of bar t dag and obtain a sharp estimation for vert bar t dag vert this obtained estimation is applied to the study of the linear least squares problem and of perturbation analysis for the schur complement of a positive semidefinite operator matrix some new norm upper bounds for schur complements of positive semidefinite operator matrices are derived

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

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

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

on stable perturbations for outer inverses of linear operators in banach spaces abstract in this paper we investigate stable perturbations and their characterizations for various types of outer inverses such as generalized 2 3 2 4 2 5 1 2 3 1 2 4 moore penrose group drazin and generalized drazin inverses some known results are improved and extended

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

an improvement on the perturbation of the group inverse and oblique projection the perturbations of the group inverse a and oblique projection aa of a square matrix a have been previously studied under certain assumptions on the matrix a and a perturbation matrix e upper bounds for b bb b a a and bb aa aa where b a e are given in the literature recently upper bounds for the general case have been published by y wei appl math comp 98 1999 29 however the special cases in the literature and the continuity of the group inverse do not follow from the general upper bounds in this paper we derive new general upper bounds which not only cover all the special cases but also are sharper than wei s results such that the continuity of the group inverse directly follows a numerical example is given to illustrate the sharpness of the new general upper bounds

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

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

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 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

perturbations and expressions for generalized inverses in banach spaces and moore penrose inverses in hilbert spaces of closed linear operators abstract the problems of perturbation and expression for the generalized inverses of closed linear operators in banach spaces and for the moore penrose inverses of closed linear operators in hilbert spaces are studied we first provide some stability characterizations of generalized inverses of closed linear operators under t bounded perturbation in banach spaces which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression t i tt 1 utilizing these results we investigate the expression for the moore penrose inverse of the perturbed operator in hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation an explicit representation for the moore penrose inverse of the perturbation is also given moreover we give an equivalent condition for the moore penrose inverse to have the simplest expression t i tt 1 the results obtained in this paper extend and improve many recent results in this area

new perturbation results on pseudo inverses of linear operators in banach spaces abstract let x and y be banach spaces let t x y be a bounded linear operator with closed range and let s x y be another bounded linear operator we study conditions on s t that guarantee the closeness of the range of s and obtain some new bounds on the pseudo inverse of s

logarithmic convexity and inequalities for the gamma function abstract we propose a method based on logarithmic convexity for producing sharp bounds for the ratio x x as an application we present an inequality that sharpens and generalizes inequalities due to gautschi chu boyd lazarevic lupas and kershaw

on the continuity of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range we study the continuity problem of the generalized inverse of t and related least squares solutions to the operator equation tx y

on the perturbation of the least squares solutions in hilbert spaces abstract let h1 and h2 be hilbert spaces and let t h1 h2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

gurland s ratio for the gamma function we consider the ratio t x y x y 2 x y 2 and its properties related to convexity logarithmic convexity schur convexity and complete monotonicity several new bounds and asymptotic expansions for t are derived sharp bounds for the function x x 1 e x are presented as well as bounds for the trigamma function the results are applied to a problem related to the volume of the unit ball in r n and also to the problem of finding the inverse of the function x t 1 x 3 x which is of importance in applied statistics

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

krull s theory for the double gamma function abstract the barnes g function g x 1 2 satisfies the functional equation log g x 1 log g x log x we complement w krull s work in bemerkungen zur differenzengleichung g x 1 g x x math nachrichten 1 1948 365 376 with additional results that yield a different characterization of the function g new expansions and sharp bounds for g on x 0 in terms of gamma and digamma functions a new expansion for the gamma function and summation formulae with polygamma functions

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

operators with equal projections related to their generalized inverses in this article we characterize operators on banach spaces which have the same projections related to their outer or inner generalized inverses as corollaries we obtain well known results for the drazin inverse of bounded operators

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

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

perturbation of least squares problem in hilbert spaces let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

perturbation analysis of generalized inverses of linear operators in banach spaces abstract let x 1 x 2 be two banach spaces and t x 1 x 2 be a bounded linear operator with a bounded generalized inverse t we study the perturbation problem of generalized inverse t and provide two new stability characterizations of generalized inverse we also give a continuity characterization of moore penrose inverse in hilbert spaces our results are new even in the case of matrices

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

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

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

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

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

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

a globally convergent method based on fischer burmeister operators for solving second order cone constrained variational inequality problems the karush kuhn tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of fischer burmeister operators over second order cones the clarke generalized differential of the semismooth mapping is presented a modified newton method with armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem an illustrative example is given to show how the globally convergent method works

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

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

perturbation analysis for the operator equationtx bin banach spaces letx1 x2be two banach spaces over the complex fieldcand lett x1 x2be a bounded linear operator with the generalized inverset lett t tbe a bounded linear operator with t t 1 suppose thatdimkert dimkert orr t kert 0 thenthas the generalized inverset i t t 1t with t t 1 t t this result generalizes theorem 3 9 ofna using this result we give some results of the perturbation analysis for the operator equationtx b

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

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

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

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

perturbation analysis for the reduced minimum modulus of bounded linear operator in banach spaces let x and y be two banach spaces over the complex field c and let t x y be a bounded linear operator with the generalized inverse t let t t dt be a bounded linear operator with t dt 1 2 let c t denote the reduced minimum modulus of t we first establish the upper semi continuity theorem for c t furthermore if dimkert dimkert or r t kert 0 then c t is continuous under the operator norm also we have a corollary in hilbert spaces which improves the result in linear algebra appl 262 1997 229 242 theorem 4 1

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

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

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

perturbation analysis for oblique projection generalized inverses of closed linear operators in banach spaces in this paper we study the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces by the method of the perturbation analysis of linear operators we obtain an explicit perturbation theorem and error estimates for the oblique projection generalized inverse of closed linear operators under the t bounded perturbation which extend the known results on the perturbation of the oblique projection generalized inverse of bounded linear operators in banach spaces

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

some new perturbation theorems for generalized inverses of linear operators in banach spaces abstract et x y be banach spaces and let t b x y be a linear operator with closed range r t we consider the perturbation problem for the bounded outer inverse the bounded 2 3 inverse and the bounded generalized inverse of t with respect to the projectors some new perturbation theorems are given by the generalized neumman lemma and the concept of stable perturbations in banach spaces

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

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

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

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

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

on the perturbation of the group generalized inverse for a class of bounded operators in banach spaces abstract given a bounded operator a on a banach space x with drazin inverse a d and index r we study the class of group invertible bounded operators b such that i a d b a is invertible and r b n a r 0 we show that they can be written with respect to the decomposition x r a r n a r as a matrix operator b b 1 b 12 b 21 b 21 b 1 1 b 12 where b 1 and b 1 2 b 12 b 21 are invertible several characterizations of the perturbed operators are established extending matrix results we analyze the perturbation of the drazin inverse and we provide explicit upper bounds of b a d and b b a d a we obtain a result on the continuity of the group inverse for operators on banach spaces

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

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