complex valued matrix differentiation techniques and key results a systematic theory is introduced for finding the derivatives of complex valued matrix functions with respect to a complex valued matrix variable and the complex conjugate of this variable in the framework introduced the differential of the complex valued matrix function is used to identify the derivatives of this function matrix differentiation results are derived and summarized in tables which can be exploited in a wide range of signal processing related situations

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

neural network approach to computing matrix inversion a neural network for matrix inversion is proposed we show both analytically and by simulations that this network is guaranteed to be stable and to provide results arbitrarily close to the accurate inversion of a matrix within an elapsed time of only a few characteristic time constants of the network the parameters such as the interconnection strengths of this network can be obtained from the known matrix without any computations

a family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non square matrix based on a quadratical convergence method a family of iterative methods to compute the approximate inverse of square matrix are presented the theoretical proofs and numerical experiments show that these iterative methods are very effective and more importantly these methods can be used to compute the inner inverse and their convergence proofs are given by fundamental matrix tools

jordan algebraic approach to convexity theorems for quadratic mappings we describe a jordan algebraic version of results related to convexity of images of quadratic mappings as well as related results on exactness of symmetric relaxations of certain classes of nonconvex optimization problems the exactness of relaxations is proved based on rank estimates our approach provides a unifying viewpoint on a large number of classical results related to cones of hermitian matrices over real and complex numbers we describe apparently new results related to cones of hermitian matrices with quaternion entries and to the exceptional 27 dimensional euclidean jordan algebra

computational algorithms for computing the inverse of a square matrix quasi inverse of a non square matrix and block matrices in this paper we have written out some methods to find the inverse of a non singular matrix and also the quasi inverse of a singular matrix the corresponding algorithms have been used to solve the linear system of equations ax b as x a 1b a comparison table of the numerical results has been designed in each case to compare the implementation of the algorithms with each other the important part of the paper is presenting a block matrix method for finding the inverse matrix the block method finds the inverse matrix with a simple and easy algorithm

searching for shortest and least programs abstract the kolmogorov complexity of a string x is defined as the length of a shortest program p of x for some appropriate universal machine u that is u p x and p is a shortest string with this property neither the plain nor the prefix free version of kolmogorov complexity are recursive but for both versions it is well known that there are recursive exact solovay functions that is recursive upper bounds for kolmogorov complexity that are infinitely often tight let a coding function for a machine m be a function f such that f x is always a program of x for m from the existence of exact solovay functions it follows easily that for every universal machine there is a recursive coding function that maps infinitely many strings to a shortest program extending a recent line of research in what follows it is investigated in which situations there is a coding function for some universal machine that maps infinitely many strings to the length lexicographically least program the main results which hold in the plain as well as in the prefix free setting are the following for every universal machine there is a recursive coding function that maps infinitely many strings to their least programs there is a partial recursive coding function defined in the natural way for some universal machine that for every set maps infinitely many prefixes of the set to their least programs exactly for every set that is bennett shallow not deep there is a recursive coding function for some universal machine that maps all prefixes of the set to their least programs differences between the plain and the prefix free frameworks are obtained by considering effective sequences i 1 i 2 of mutually disjoint finite sets and asking for a recursive coding function for some universal machine that maps at least one string in each set i n to its least code such coding functions do not exist in the prefix free setting but exist in the plain setting in case the sets i n are not too small

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

recursive computational depth in the 1980 s bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object s structure in particular bennett identified the classes of weakly deep and strongly deep sequences and showed that the halting problem is strongly deep juedes lathrop and lutz subsequently extended this result by defining the class of weakly useful sequences and proving that every weakly useful sequence is strongly deep

on languages reducible to algorithmically randomlanguages in this paper languages bounded reducible to algorithmically random languages are studied these are the languages whose characteristic sequences are algorithmically random as defined by martin lof inform and control 9 1966 pp 602 619 here rand denotes the class of algorithmically random languages the reducibilities le cal r are very general but are defined so that if a in r b then there is a machine m with the properties that l m b a and every computation of m relative to any oracle halts book lutz and wagner math systems theory 27 1994 pp 201 209 studied sf almost r defined to be a mid for almost every b a le cal r b they showed that sf almost r r rand cap rec where rec denotes the class of recursive languages so that sf almost r is the recursive part of r rand in this paper this characterization is strengthened by showing that for every b in rand sf almost r r b cap rec a pair a b of languages is an independent pair of algorithmically random languages if a oplus b in rand in this paper it is shown that for every r and for every independent pair a b sf almost r r a cap r b

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

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

feasible reductions to kolmogorov loveland stochastic sequences for every binary sequence a there is an infinite binary sequence s such that a ttp s and s is stochastic in the sense of kolmogorov and loveland

weakly useful sequences an infinite binary sequence x is defined to be i strongly useful if there is a computable time bound within which every decidable sequence is turing reducible to x and ii weakly useful if there is a computable time bound within which all the sequences in a non measure 0 subset of the set of decidable sequences are turing reducible to x juedes lathrop and lutz theorectical computer science 132 1994 37 proved that every weakly useful sequence is strongly deep in the sense of bennett the universal turing machine a half century survey 1988 227 and asked whether there are sequences that are weakly useful but not strongly useful the present paper answers this question affirmatively the proof is a direct construction that combines the martingale diagonalization technique of lutz siam journal on computing 24 1995 1170 with a new technique namely the construction of a sequence that is computably deep with respect to an arbitrary given uniform reducibility the abundance of such computably deep sequences is also proven and used to show that every weakly useful sequence is computably deep with respect to every uniform reducibility

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

recursively enumerable reals and chaitin ω numbers a real is called recursively enumerable if it is the limit of a recursive increasing converging sequence of rationals following solovay unpublished manuscript ibm thomas j watson research center yorktown heights new york may 1975 215 pp and chaitin ibm j res develop 21 1977 350 359 496 we say that an r e real dominates an r e real if from a good approximation of from below one can compute a good approximation of from below we shall study this relation and characterize it in terms of relations between r e sets solovay s unpublished manuscript ibm thomas j watson research center yorktown heights new york may 1975 215 pp like numbers are the maximal r e real numbers with respect to this order they are random r e real numbers the halting probability of a universal self delimiting turing machine chaitin s number j assoc comput mach 22 1975 329 340 is also a random r e real solovay showed that any chaitin number is like in this paper we show that the converse implication is true as well any like real in the unit interval is the halting probability of a universal self delimiting turing machine

on the polynomial depth of various sets of random strings we introduce a general framework for defining the depth of an infinite binary sequence with respect to a class of observers we show that our general framework captures all depth notions introduced in computability complexity theory so far we review most such notions show how they are particular cases of our general depth framework and review some classical results about the different depth notions we use our framework to define new notions of polynomial depth called monotone poly depth based on a polynomial version of monotone kolmogorov complexity we show that monotone poly depth satisfies all desirable properties of depth notions we give two natural examples of deep sets by showing that both the set of levin random strings and the set of kolmogorov random strings are monotone poly deep

an improved method for the computation of the moore penrose inverse matrix abstract in this article we provide a fast computational method in order to calculate the moore penrose inverse of singular square matrices and of rectangular matrices the proposed method proves to be much faster and has significantly better accuracy than the already proposed methods while works for full and sparse matrices

computational depth and reducibility this paper reviews and investigates bennett s notions of strong and weak computational depth also called logical depth for infinite binary sequences roughly an infinite binary sequence x is defined to be weakly useful if every element of nonnegligible set of decidable sequences is reducible to x in recursively bounded time it is shown that every weakly useful sequence is strongly deep this result which generalizes bennett s observation that the halting problem is strongly deep implies that every high turing degree contains strongly deep sequences it is also shown that in the sense of baire category almost every infinite binary sequence is weakly deep but not strongly deep

complex valued zhang neural network for online complex valued time varying matrix inversion abstract in this paper a new complex valued recurrent neural network cvrnn called complex valued zhang neural network cvznn is proposed and simulated to solve the complex valued time varying matrix inversion problems such a cvznn model is designed based on a matrix valued error function in the complex domain and utilizes the complex valued first order time derivative information of the complex valued time varying matrix for online inversion superior to the conventional complex valued gradient based neural network cvgnn and its related methods the state matrix of the resultant cvznn model can globally exponentially converge to the theoretical inverse of the complex valued time varying matrix in an error free manner moreover by exploiting the design parameter 1 superior convergence can be achieved for the cvznn model to solve such complex valued time varying matrix inversion problems as compared with the situation without design parameter involved i e the situation with 1 computer simulation results substantiate the theoretical analysis and further demonstrate the efficacy of such a cvznn model for online complex valued time varying matrix inversion

the global power of additional queries to p random oracles we consider separations of reducibilities by random sets first we show a result on polynomial time bounded reducibilities that query their oracle nonadaptively for every p random set r there is a set that is reducible to r with k 1 queries but is not reducible to any other p random set with at most k queries this result solves an open problem stated in a recent survey paper by lutz and mayordomo eatcs bulletin 68 1999 pp 64 80 second we show that the separation result above can be transferred from the setting of polynomial time bounds to a setting of rec random sets and recursive reducibilities this extends the main result of book lutz and martin inform and comput 120 1995 pp 49 54 who by using different methods showed a similar separation with respect to martin lof random sets moreover in both settings we obtain similar separation results for truth table versus bounded truth table reducibility

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

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

a novel iterative method for computing generalized inverse in this letter we propose a novel iterative method for computing generalized inverse based on a novel kkt formulation the proposed iterative algorithm requires making four matrix and vector multiplications at each iteration and thus has low computational complexity the proposed method is proved to be globally convergent without any condition furthermore for fast computing generalized inverse we present an acceleration scheme based on the proposed iterative method the global convergence of the proposed acceleration algorithm is also proved finally the effectiveness of the proposed iterative algorithm is evaluated numerically

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

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

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

every sequence is reducible to a random one every infinite sequence is turing reducible to an infinite sequence which is random in the sense of martin lof

iterative method for computing the moore penrose inverse based on penrose equations an iterative algorithm for estimating the moore penrose generalized inverse is developed the main motive for the construction of the algorithm is simultaneous usage of penrose equations 2 and 4 convergence properties of the introduced method as well as their first order and second order error terms are considered numerical experiment is also presented