the bunch kaufman factorization of symmetric matrices signature similar to sign regular matrices abstract the bunch kaufman pivoting strategy is a most commonly used method in practice to factor symmetric indefinite matrices however this method in general may destroy the band structure of banded matrices and its growth factor bound of 2 57 n 1 can almost be attained in this paper we investigate the behavior of the method for factorizing symmetric indefinite matrices that are signature similar to sign regular matrices with signature the growth factor bound depending on flips in the signature is derived for factorizing such banded matrices if the bunch kaufman pivoting strategy is modified as suggested by sorensen and van loan then not only do the same growth factor bounds hold but also the bandwidth is nicely preserved

gaussian elimination is stable for the inverse of a diagonally dominant matrix let b e m n c be a row diagonally dominant matrix i e i b ii b ij i 1 n where 0 i 1 i 1 n with a max 1 i n i we show that no pivoting is necessary when gaussian elimination is applied to a b 1 moreover the growth factor for a does not exceed 1 the same results are true with row diagonal dominance being replaced by column diagonal dominance

quasi ldu factorization of nonsingular totally nonpositive matrices abstract let a a ij r n n be a nonsingular totally nonpositive matrix in this paper we describe some properties of these matrices when a11 0 and obtain a characterization in terms of the quasi ldu factorization of a where l is a block lower triangular matrix d is a diagonal matrix and u is a unit upper triangular matrix

total nonpositivity of nonsingular matrices in this paper nonsingular totally nonpositive matrices are studied and new characterizations are provided in terms of the signs of minors with consecutive initial rows or consecutive initial columns these characterizations extend an existing characterization that uses some restrictive hypotheses

on schur complements of sign regular matrices of order k abstract the issue regarding schur complements of sign regular matrices is rather subtle it is known that the class of totally nonnegative matrices is not closed under arbitrary schur complementation in this paper we demonstrate how schur complements of sign regular matrices of order k are sign regular of a certain order in particular some results for totally nonnegative and totally nonpositive matrices are provided as our corollaries

block sparsity induced adaptive filter for multi clustering system identification in order to improve the performance of least mean square lms based adaptive filtering for identifying block sparse systems a new adaptive algorithm called block sparse lms bs lms is proposed in this paper the basis of the proposed algorithm is to insert a penalty of block sparsity which is a mixed l2 0 norm of adaptive tap weights with equal group partition sizes into the cost function of traditional lms algorithm to describe a block sparse system response we first propose a markov gaussian model which can generate a kind of system responses of arbitrary average sparsity and arbitrary average block length using given parameters then we present theoretical expressions of the steady state misadjustment and transient convergence behavior of bs lms with an appropriate group partition size for white gaussian input data based on the above results we theoretically demonstrate that bs lms has much better convergence behavior than l0 lms with the same small level of misadjustment finally numerical experiments verify that all of the theoretical analysis agrees well with simulation results in a large range of parameters

a stable test for strict sign regularity a stable test to check if a given matrix is strictly sign regular is provided among other nice properties we prove that it has an optimal growth factor the test is compared with other alternative tests appearing in the literature and its advantages are shown

on the extension of some total positivity inequalities abstract some inequalities recently obtained for totally positive matrices are extended to more general classes of matrices among these larger classes of matrices sign consistent sign regular and matrices with signed bidiagonal decomposition are considered the class of matrices with a signed bidiagonal decomposition contains the nonsingular totally positive matrices and their inverses

region based wavelet packet adaptive algorithm for identification of sparse impulse responses identification of systems with sparse impulse response encounters large applicability numerous techniques have been proposed to identify such systems efficiently one strategy that leads to specially good results is to detect the active nonzero response samples and update only the corresponding adaptive coefficients wavelet based approaches have been shown to be specially effective to this end this paper proposes a new region based wavelet packet rbwp algorithm for efficient identification of systems with sparse impulse responses with arbitrary frequency spectra and with any delay of the effective response the discrete wavelet packet transform dwpt is adaptively tailored to the energy distribution of the unknown system s response spectrum the new algorithm leads to a reduced number of active weights and to a reduced computational complexity when compared with competing wavelet based algorithms monte carlo simulation results show good performances regarding convergence speed and robustness to design parameter choice

characterizations of jacobi sign regular matrices abstract several characterizations of jacobi nonsingular sign regular matrices are presented moreover a stable test of o n elementary operations is obtained to check if an n n jacobi nonsingular matrix is sign regular

zero loci of skew growth functions for dual artin monoids abstract we show that the skew growth function of a dual artin monoid of finite type p has exactly rank p l simple real zeros on the interval 0 1 the proofs for types a l and b l are based on an unexpected fact that the skew growth functions up to a trivial factor are expressed by jacobi polynomials due to a rodrigues type formula in the theory of orthogonal polynomials the skew growth functions for type d l also satisfy rodrigues type formulae but the relation with jacobi polynomials is not straightforward and the proof is intricate we show that the smallest root converges to zero as the rank l tends to infinity

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

required nonzero patterns for nonsingular sign regular matrices abstract an n n real matrix is called sign regular if for each k 1 k n all its minors of order k have the same nonstrict sign the zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero the patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

sign structure preserves for m banded factorizations of sign regular matrices abstract a square matrix a a ij is called m banded if a ij 0 for all i j m motivated by the results of rainey and habetler j w rainey g j habetler tridiagonalization of completely nonnegative matrices math comp 26 1972 121 128 we provide m banded factorizations of sign regular matrices that preserve their sign structures as applications some eigenvalue interlacing properties and majorization relations are obtained for certain sign regular matrices

inversion formula for the growth function of a cancellative monoid let m deg be a cancellative monoid m equipped with a degree map deg m r 0 and let pm deg t u m ltdeg u be its generating series here u lv for u v m if u lv and v lu see section 2 definition called the growth function of m deg in the present paper we introduce the skew growth function nm deg t of m deg and prove the inversion formula pm deg t nm deg t 1 where the skew growth function nm deg t is a signed generating series nm deg t 1 t t m i0 1 j1 jn n 1 mcm t tdeg of minimal common multiples mcm t of towers t t m i0 associated with the minimal set i0 of m if m z 0 is the multiplicative monoid of positive integers with deg log then pm deg exp s is riemann zeta function and the inversion formula turns out to be the euler product formula we discuss more examples on the behavior of the analytic function nm deg exp s on the boundary line s abscissa of absolute convergence of pm deg exp s

on nonsingular sign regular matrices abstract an m n matrix a is sign regular if for each k 1 k min m n all k k submatrices of a have determinant with the same nonstrict sign the zero pattern of nonsingular sign regular matrices is analyzed it is proved that the number of zero entries which can appear in a nonsingular sign regular matrix depends on its signature a matrix is totally nonpositive if all its minors are nonpositive a test for recognizing nonsingular totally nonpositive matrices is also provided

block sparse recovery via convex optimization given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks we study the problem of finding a block sparse representation of the signal i e a representation that uses the minimum number of blocks motivated by signal image processing and computer vision applications such as face recognition we consider the block sparse recovery problem in the case where the number of atoms in each block is arbitrary possibly much larger than the dimension of the underlying subspace to find a block sparse representation of a signal we propose two classes of nonconvex optimization programs which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks respectively since both classes of problems are np hard we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original nonconvex formulation our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary which are natural generalizations of existing notions of mutual and cumulative coherence we evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition we show that treating the face recognition problem as a block sparse recovery problem improves the state of the art results by 10 with only 25 of the training data

combined matrices of almost strictly sign regular matrices abstract the combined matrix is a very useful concept for many applications almost strictly sign regular assr matrices form an important structured class of matrices with two possible zero patterns which are either type i staircase or type ii staircase we prove that under an irreducibility condition the pattern of zero and nonzero entries of an assr matrix is preserved by the corresponding combined matrix without the irreducibility condition it is proved that type i and type ii staircases are still preserved illustrative numerical examples are included

sign regular matrices and neville elimination abstract a pivoting strategy of o n operations for the neville elimination of n n nonsingular sign regular matrices is introduced among other nice properties it is proved that it preserves sign regularity it is also shown its relationship with scaled partial pivoting strategies for neville elimination

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

variation diminishing transformations a direct approach to total positivity and its statistical applications abstract karlin has shown that the variation diminishing property possessed by totally positive distributions allows a unified and straightforward presentation of many basic properties of hypothesis tests his work in fact reveals that total positivity is equivalent to a suitably formulated variation diminishing property this largely expository article begins with the latter concept and consequently gives a more direct account of the theory this approach avoids the extensive mathematical preliminaries previously required and isolates the more important statistical property

a fast approach for overcomplete sparse decomposition based on smoothed ell 0 norm in this paper a fast algorithm for overcomplete sparse decomposition called sl0 is proposed the algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations and its applications include underdetermined sparse component analysis sca atomic decomposition on overcomplete dictionaries compressed sensing and decoding real field codes contrary to previous methods which usually solve this problem by minimizing the l 1 norm using linear programming lp techniques our algorithm tries to directly minimize the l 1 norm it is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state of the art interior point lp solvers while providing the same or better accuracy

monoids in the fundamental groups of the complement of logarithmic free divisors in c3 abstract we study monoids generated by certain zariski van kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in c 3 listed by sekiguchi these monoids admit positive homogeneous presentations theorem 1 five of them are artin monoids and eight of them are free abelian monoids the remaining four monoids are not gausian and hence are neither garside nor artin theorem 2 however we introduce the concept of fundamental elements for positive homogeneously presented monoids and we show that all 17 monoids possess fundamental elements theorem 3 as an application of the study of monoids we solve some decision problems for the fundamental groups except in three cases

a matricial description of neville elimination with applications to total positivity abstract the neville elimination process used by the authors in some previous papers in connection with totally positive matrices is studied in detail in the case of nonsingular matrices a wide class of matrices is found where neville elimination has a lower computational cost than gauss elimination finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices in terms of their neville elimination and that of their inverses

combined matrices of sign regular matrices abstract the combined matrix of a nonsingular matrix a is the hadamard entry wise product c a a a 1 t since each row and column sum of c a is equal to one the combined matrix is doubly stochastic when it is nonnegative in this work we study the nonnegativity of the combined matrix of sign regular matrices based upon their signature in particular a few coordinates of the signature e of a play a crucial role in determining whether or not c a is nonnegative