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 t k diagnosability of star graph networks the t k diagnosis is a diagnostic strategy at system level that can significantly enhance the system s self diagnosing capability it can detect up to t faulty processors or nodes units which might include at most k misdiagnosed processors where k is typically a small number somani and peleg 1996 claimed that an n dimensional star graph denoted s n a well studied interconnection model for multiprocessor systems is k 1 n 3k 2 k diagnosable recently chen and liu 2012 found counterexamples for the diagnosability obtained in without further pursuing the cause of the flawed result in this paper we provide a new complete proof that an mbi n dimensional star graph is actually k 1 n 3k 1 k diagnosable where 1 leq k leq 3 and investigate the reason that caused the flawed result in based on our newly obtained fault tolerance properties we will also outline an o n log n diagnostic algorithm n n is the number of nodes in s n to locate all up to k 1 n 3k 1 faulty processors among which at most k 1 leq k leq 3 fault free processors might be wrongly diagnosed as faulty

comments on stability of an entire polytope of polynomials soh 1989 has slated that the stability of an entire polytope of polynomials is guaranteed if the zeros of the even and odd parts of the vertex polynomials have interlacing properties this result of soh is shown via a counter example in this note not to be correct in certain circumstances the flaw leading to this error is pointed out the remedy for son s error is provided

a note on diagnosability of large fault sets on star graphs diagnosability of a system directly refers to the maximum number of faulty vertices that can be identified by the system somani et al 2 proposed a generalized measure to increase the degree of diagnosability of the hypercubes and star graphs this paper provides counterexamples for the results of diagnosability of star graphs

comment on condition number of drazin inverse and their condition numbers of singular linear systems abstract in this paper we establish the condition number of drazin inverse of a singular matrix a where r a k r a k and k index a by the schur decomposition based on this form the spectral norm and frobenius norm of relative condition number for the drazin inverse and level 2 condition number of the drazin inverse are characterized the sensitivity for the drazin inverse solution of singular systems is also presented we obtain several formulas for the condition number of the drazin inverse by spectral norm instead of the p norm where p is a transformation matrix of the jordan canonical form of a thereby improving the earlier work y wei g wang d wang condition number of drazin inverse and their condition numbers of singular linear systems appl math comput 146 2003 455 467

on level 2 condition number for the weighted moore penrose inverse in this paper we present characterizations for the level 2 condition number of the weighted moore penrose inverse i e cond m n a cond m n 2 a cond m n a 1 where cond m n a is the condition number of the weighted moore penrose inverse of a rectangular matrix and cond m n 2 a is the level 2 condition number of this problem this paper extends the result by cucker diao and wei f cucker h diao y wei on the level 2 condition number for moore penrose inversion 2005 unpublished report and improves the results by wei and wang y wei d wang condition numbers and perturbation of weighted moore penrose inverse and weighted linear least squares problem appl math comput 145 2003 45 58

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

condition number of bott duffin inverse and their condition numbers various normwise relative condition numbers that measure the sensitivity of bott duffin inverse and the solution of constrained linear systems are characterized the sensitivity of condition number itself is then investigated finally upper bounds are derived for the sensitivity of componentwise condition numbers

diagnosability of hypercubes and enhanced hypercubes under the comparison diagnosis model a sengupta and a dahbura 1992 discussed how to characterize a diagnosable system under the comparison diagnosis model proposed by j maeng and m malek 1981 and a polynomial algorithm was given to identify the faulty processors provided that the system s diagnosability is known however for a general system the determination of its diagnosability is not algorithmically easy this paper proves that for the important hypercube structured multiprocessor systems n cubes the diagnosability under the comparison model is n when n spl ges 5 the paper also studies the diagnosability of enhanced hypercube which is obtained by adding 2 sup n 1 more links to a regular hypercube of 2 sup n processors it is shown that the augmented communication ability among processors also increases the system s diagnosability under the comparison model we prove that the diagnosability is n 1 for an enhanced hypercube when n spl ges 6

some topics on weighted moore penrose inverse weighted least squares and weighted regularized tikhonov problems it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems the classical normwise relative condition number formulas are given by higham linear algebra appl 214 1995 193 here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b moreover we consider the nearness to rank deficiency finally we give some decomposition tools and their application in the regularized weighted tikhonov problems

condition number of drazin inverse and their condition numbers of singular linear systems various normwise relative condition numbers that measure the sensitivity of drazin inverse and the solution of singular linear systems are characterized the sensitivity of condition number itself is then investigated finally upper bounds are derived for the sensitivity of structured perturbation

condition numbers and structured perturbation of the w weighted drazin inverse we give the explicit condition number formulas for the w weighted drazin inverse of a rectangular matrix we also present the structured perturbation of the w weighted drazin inverse and consider the perturbation for the w weighted drazin inverse of a rectangular matrix and give an explicit expression for the w weighted drazin inverse of a perturbed matrix under some certain conditions finally we discuss the minimum quality of the condition number

a note on the componentwise perturbation bounds of matrix inverse and linear systems in this note we present sharp bounds for the componentwise perturbation of matrix inverse and linear systems especially for nonsingular m matrix we also use matrix derivatives to deduce the matrix componentwise condition number and present some numerical tests to show our results

condition number for the drazin inverse and the drazin inverse solution of singular linear system with their condition numbers in this paper we investigate the condition number of drazin inverse and drazin inverse solution of singular linear system ax b where a is a nxn rank deficient matrix and b a real vector of size n x a real vector let a and b be two positive real numbers when we consider the weighted frobenius norm aa bb p q f on the data we get the formula of condition number of the drazin inverse solution of singular linear system for the normwise condition number the sensitivity of the relative condition number itself is studied the componentwise perturbation is also investigated

the t k diagnosability for regular networks the t k diagnosis strategy can significantly enhance the system s self diagnosing capability at the expense of no more than k fault free processors vertices being mistakenly diagnosed as faulty under the pmc model it is a generalization of the precise and pessimistic diagnosis strategies of system level diagnosis on multiprocessor systems it can detect up to t faulty processors vertices which might include at most k misdiagnosed processors vertices where k is typically a small number in the case k ge 1 to our knowledge there is no known t k diagnosis algorithm for general regular networks in this paper we first propose a general t k diagnosis k ge 1 algorithm for some m regular networks these m regular networks satisfying some conditions could establish the t k diagnosis algorithm say t k g diag to determine the t k diagnosability the complexity of this algorithm is only o n log n when n ge 2 m or o nm when n where n is the number of vertices in the network second we present a complete proof that the network g is actually t k diagnosable finally we establish the t k diagnosability 1 le k le 3 of some regular networks including an n dimensional alternating group graph an n dimensional split star network a l n hypermesh and an n l star graph which are well known interconnection networks proposed for multiprocessor systems

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

a note on condition numbers for generalized inverse at s 2 and constrained linear systems abstract in this paper we use the schur decomposition to measure the sensitivity of the general inverse a t s 2 and the constrained singular linear system ax b with regard to 2 norm and f norm rather than pq norm in 12 where p and q are nonsingular matrices the explicit forms of the condition numbers approximate the sensitivity of a t s 2 and the constrained singular linear system pretty well furthermore since schur decomposition is well posed the evaluation of the sensitivity can be numerically easy and stable

condition number related with w weighted drazin inverse and singular linear systems in this paper we introduce a pq norm and discuss the condition number of the w weighted drazin inverse we get some results on the condition number with respect to a d w and singular linear systems wawx b b r wa k 2 x r aw k 1 which are similar to the regular inverse a 1 and nonsingular linear systems ax b

structured perturbations of group inverse and singular linear system with index one in this paper we study the normwise perturbation theory of singular linear structured system with index one the structures under investigation are toeplitz circulant hankel matrices we show that the structured condition number is smaller than unstructured condition number for specific right hand side we present the condition number for structured system

the condition numbers for weighted moore penrose inverse and weighted linear least squares problem condition numbers play an important role in numerical analysis classical normise condition numbers are used to measure the size of both input perturbations and output errors in this paper we study the weighted normwise relative condition numbers for the weighted moore penrose inverse and the weighted linear least squares wls problems in the case of the full column rank matrix the bounds or formulas for the weighted condition numbers are presented the obtained results can be viewed as extensions of the earlier works studied by others

perturbation bounds for the generalized inverses at s 2 with application to constrained linear system in this paper the more general perturbation problem of the generalized inverse a t s 2 is considered we derive its perturbation bounds when the subspaces t and s also have perturbation and in special case when t and s have no perturbation we get some results on the condition numbers with respect to a t s 2 and constrained linear system ax b x t which are similar to the regular inverse a 1 and nonsingular linear system ax b

the sign real spectral radius and cycle products abstract the extension of the perron frobenius theory to real matrices without sign restriction uses the sign real spectral radius as the generalization of the perron root the theory was used to extend and solve the conjecture in the affirmative that an ill conditioned matrix is nearby a singular matrix also in the componentwise sense the proof estimates the ratio between the sign real spectral radius and the maximum geometric mean of a cycle product in this note we discuss bounds for this ratio including a counterexample to a conjecture about this ratio

on perturbation bounds of kronecker product linear systems and their level 2 condition numbers the classical perturbation theory of linear systems ax b is extended to kronecker product linear systems email protected b x d upper bounds are derived for the normwise and componentwise condition number the nearness to singularity and the sensitivity of the condition numbers are analyzed

on diagnosability of large fault sets in regular topology based computer systems the classical diagnosability approach has its limitation when dealing with large fault sets in large multiprocessor systems this is due to limited diagnosability of large multiprocessor systems connected using regular interconnection structures we propose an alternative approach to system diagnosis by allowing a few upper bounded number of units to be diagnosed incorrectly this measure is called t k diagnosability using this new measure it is possible to increase the degree of diagnosability of large system considerably the t k diagnosis guarantees that all the faulty units processors in a system are detected provided the number of faulty units does not exceed t while at most k units are incorrectly diagnosed we provide necessary and sufficient conditions for t k diagnosability and discuss their implication to demonstrate the power of this approach we analyze the diagnosability of large systems connected as hypercube star graph and meshes it is shown that a substantial increase in the degree of diagnosability of these structures is achieved compared with the degree of diagnosability achieved using the classic diagnosability approach at the cost of a comparably small number of incorrectly diagnosed units

condition numbers and perturbation of the weighted moore penrose inverse and weighted linear least squares problem it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of nonsingular linear systems here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b these explicit expressions extend the earlier work of several authors as we know the computed weighted least squares solution x of min email protected email protected m in the presence of the round off error satisfies the perturbed equation min email protected a e y b f m finally an upper bound of email protected n is derived for the case where the weighted matrix and vector norms and the assumption rank a e rank a are used

condition numbers and their condition numbers abstract various normwise relative condition numbers that measure the sensitivity of matrix inversion and the solution of linear systems are characterized new results are derived for the cases where two common noninduced matrix norms are used and where different vector norms are used for the domain and range of the matrix condition numbers that respect the structure of symmetric problems are also analyzed the sensitivity of the condition number itself is then investigated and we obtain sharp examples of demmel s general result that for certain problems in numerical analysis the condition number of the condition number is the condition number finally upper bounds are derived for the sensitivity of componentwise condition numbers

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

condition number related to the outer inverse of a complex matrix in this paper we obtain the formula for computing the condition number of a complex matrix which is related to the outer generalized inverse of a given matrix we use the schur decomposition of a matrix we characterize the spectral norm and the frobenius norm of the relative condition number of the generalized inverse and the level 2 condition number of the generalized inverse the sensitivity for the generalized drazin inverse solution of linear systems is presented we also present the structured perturbation of the generalized inverse

a contribution to the theory of condition in this paper the use of the condition number of a problem as defined by rice in 1966 is discussed for the eigenvalue eigenvector and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived

results on positive pairs of polynomials and their application to the construction of stability domains the real polynomial f s h s2 sg s2 is hurwitz if and only if h g forms a so called positive pair of polynomials in this paper conditions are given that ensure that some polynomials h and g form a positive pair and then it is shown how these results can be applied to the construction of some stability domains these hurwitz regions will be exactly described by linear inequalities

optimal a priori clipping estimation for wavelet based method of moments matrices wavelet bases are mainly used in the methods of moments mom to render the system matrix sparse clipping entries below a given threshold is an essential operation to obtain the desired sparse matrix in this paper we present a novel a priori way to estimate the clipping threshold that can be used if one wants an error on the solution below a desired accuracy the procedure has a very low computational complexity typically of o n

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

perturbation analysis for a class of fuzzy linear systems in this paper the perturbation analysis is presented for a class of fuzzy linear systems which can be solved by an embedding method we deduce the nonlinear upper perturbation bounds for the solutions and discuss the normwise mixed and componentwise condition numbers the results show how the perturbations of the coefficient matrix and the right hand vector impact the solutions to fuzzy linear systems

optimal two sided diagonal scaling for filter bank frames two sided diagonal scaling for transfer matrices is formulated an efficient algorithm is proposed to compute globally optimal solutions using the iterated bi section and linear matrix inequality lmi solver it is shown that the two sided scaling of filter bank fb frames can be implemented by the adjustment of channel gains and the periodic precoding of source signal and that the frame bound ratio of fb frames can be effectively improved by such scaling explicit formulas are established for both uniform and nonuniform fb frames including detail formulas for discrete weyl heisenberg frames and tree structured fbs discrete wavelets different examples show the effectiveness of the obtained results

componentwise perturbation theory for linear systems with multiple right hand sides existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right hand sides and to a general class of componentwise measure of perturbations involving holder p norms it is shown that for a system of order n with r right hand sides the componentwise backward error can be computed by finding the minimum p norm solutions to n underdetermined linear systems and an explicit expression is obtained in the case r 1 a perturbation bound is derived and from this the componentwise condition number is obtained to within a multiplicative constant applications of the results are discussed to invariant subspace computations quasi newton methods based on multiple secant equations and an inverse ode problem

structured perturbations and symmetric matrices abstract for a given n n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal bauer skeel condition number cannot be larger than 3 2 2 n in this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation for relative perturbations of the individual matrix components it is shown that a symmetric linear system can be arbitrarily ill conditioned while any symmetric and entrywise relative perturbation of the matrix of less than 100 does not produce a singular matrix that means that the inverse of the condition number and the distance to the nearest ill posed problem can be arbitrarily far apart finally we prove that restricting structured perturbations to symmetric entrywise perturbations cannot change the condition number by more than a factor 3 2 2 n

stability of an entire polytope of polynomials abstract stability is a central issue for polynomials with variable coefficients the stability of a class of polynomials whose coefficients are characterized by a polytope is studied we show that the stability of this whole family of polynomials is assured if the zeros of the even and odd parts of the vertex polynomials have interlacing properties

structured mixed and componentwise condition numbers of some structured matrices structured matrices such as cauchy vandermonde toeplitz hankel and circulant matrices are considered in this paper we apply a kronecker product based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems