A modification of the generalized shift-splitting (GSS) method is presented for solving singular saddle point problems. In this kind of modification, the diagonal shift matrix is replaced by a block diagonal matrix which is symmetric positive definite. Semiconvergence of the proposed method is investigated. The induced preconditioner is applied to the saddle point problem and the preconditioned...

Several representations of the generalized Drazin inverse of an anti-triangular block matrix in Banach algebra are given in terms of the generalized Banachiewicz--Schur form.  

The reduction of the state-matrix of a linear time-invariant state-space model to a block-diagonal form by using a state coordinate transformation is equivalent with an additive decomposition of the corresponding transfer-function matrix. Computationally involved and large storage demanding algorithms for solving several systems modeling problems can be conveniently reformulated such that they ...

A polynomial matrix G(z) = Izm −∑m−1 i=0 Ciz with complex coefficients is called discrete-time stable if its characteristic values (i.e. the zeros of detG(z) ) are in the unit disc. A corresponding block companion matrix C is used to study discrete-time stability of G(z) . The main tool is the construction of block diagonal solutions P of a discrete-time Lyapunov inequality P−C∗PC 0 .

In this paper, we propose the block-diagonal matrix to approximate the Hessian matrix in the Levenberg Mar-quardt method in the training of neural networks. Two weight updating strategies, namely asynchronous and synchronous updating methods were investigated. Asyn-chronous method updates weights of one block at a time while synchronous method updates all weights at the same time. Variations of...

A fast approximate inversion method is proposed for the block lower triangular Toeplitz with tri-diagonal blocks (BL3TB) matrix. The BL3TB matrix is approximated by a block ε-circulant matrix, which can be efficiently inverted using the fast Fourier transforms. The error estimation is given to show the high accuracy of the approximation. In applications, the proposed method is employed to solve...

BLOCK TRIANGULAR PRECONDITIONERS FOR -MATRICES AND MARKOV CHAINS MICHELE BENZI AND BORA UÇAR Abstract. We consider preconditioned Krylov subspace methods for solving large sparse linear systems under the assumption that the coefficient matrix is a (possibly singular) -matrix. The matrices are partitioned into block form using graph partitioning. Approximations to the Schur complement are used t...

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diago...

In this paper we investigate whether matrices arising from linear or integer programming problems can be decomposed into so-called bordered block diagonal form. More precisely, given some matrix A, we try to assign as many rows as possible to some number β of blocks of size κ such that no two rows assigned to different blocks intersect in a common column. Bordered block diagonal form is desirab...

In this paper, based on the block (element)-wise inverse Jacket matrix, a unified fast hybrid diagonal block-wise transform (FHDBT) algorithm is proposed. A new fast diagonal block matrix decomposition is made by the matrix product of successively lower order diagonal Jacket matrix and Hadamard matrix. Using a common lower order matrix in the form of 1 1, a fast recursive structure can be devel...