Nekrasov Matrices Research Articles

A note on infinity norm bounds for the inverses of Nekrasov matrices

In this paper, we present several new upper bounds for the infinity norm of the inverse of Nekrasov matrices by using scaling matrices with two values on their diagonals, which improve the corresponding results of Gao et al. and Kolotilina. Numerical examples and comparisons with the existing bounds are included.

Error Bounds for Linear Complementarity Problems of Nekrasov and Generalized Nekrasov Matrices

Error Bounds for Linear Complementarity Problems of Nekrasov and Generalized Nekrasov Matrices

Upper triangulation-based infinity norm bounds for the inverse of Nekrasov matrices with applications

Upper triangulation-based infinity norm bounds for the inverse of Nekrasov matrices with applications

Diagonal-Schur complements of Nekrasov matrices

The Schur and diagonal-Schur complements are important tools in many fields. It was revealed that the diagonal-Schur complements of Nekrasov matrices with respect to the index set $\{1\}$ are Nekrasov matrices by Cvetkovic and Nedovic [Appl. Math. Comput., 208:225-230, 2009]. In this paper, we prove that the diagonal-Schur complements of Nekrasov matrices with respect to any index set are Nekrasov matrices. Similar results hold for $\Sigma$-Nekrasov matrices. We also present some results on Nekrasov diagonally dominant degrees. Numerical examples are given to verify the correctness of the results.

Upper Bounds for ‖A−1Q‖∞

The paper suggests a general approach to deriving upper bounds for ‖A−1Q‖∞ from those for ‖A−1‖∞ for matrices A belonging to different subclasses of the class of nonsingular ℌ-matrices. The approach is applied to SDD, S-SDD, OBS, OB, and Nekrasov matrices.

Two infinity norm bounds for the inverse of Nekrasov matrices

Nekrasov matrices play an important role in various scientific disciplines. The estimation of infinity norm bounds for the inverse of Nekrasov matrices brings a lot of convinces in many fields. In this paper, we introduce two new bounds for the inverse of Nekrasov matrices. The advantages of our bounds and numerical examples are also presented.

On subdirect sums of Nekrasov matrices

Some new sufficient conditions are given, which ensure that the k-subdirect sum of Nekrasov matrices and strictly diagonally dominant matrices is in the class of Nekrasov matrices. In addition, several examples are also given to illustrate the conditions presented.

On π−nekrasov matrices

In this paper, we consider ??Nekrasov matrices, a generalization of {P1, P2}?Nekrasov matrices obtained by introducing the set ? = {P1, P2, ..., Pm} of m simultaneous permutations of rows and columns of the given matrix. For point-wise and block ??Nekrasov matrices we give infinity norm bounds for the inverse. For ??Nekrasov B?matrices, obtained through a special rank one perturbation, we present main results on infinity norm bounds for the inverse and error bounds for linear complementarity problems. Numerical examples illustrate the benefits of new bounds.

K-subdirect sums of Nekrasov matrices

In this paper, we give a sufficient and necessary condition for the subdirect sum of a Nekrasov matrix and a strictly diagonally dominant matrix being still a Nekrasov matrix. Adopting this sufficient and necessary condition, we present several simple sufficient conditions ensuring that the subdirect sum of Nekrasov matrices is in the same class. Examples are reported to illustrate the theoretical results.

Further Block Generalizations of Nekrasov Matrices

Further Block Generalizations of Nekrasov Matrices

Accurate determinants of some classes of matrices

We present parametrizations of B-matrices, Nekrasov matrices and B-Nekrasov matrices that allow us to compute their determinants with high relative accuracy. Numerical examples confirm the accuracy of the method.

A Block Generalization of Nekrasov Matrices

The paper introduces the so-called generalized Nekrasov (GN) matrices, which provide a block extension of the ordinary Nekrasov matrices. Basic properties of GN matrices are studied. In particular, it is proved that the GN matrices form a subclass of nonsingular H-matrices and this subclass is closed with respect to Schur complements obtained by eliminating leading principal block submatrices. Also an upper bound for the l∞-norm of the inverse to a GN matrix, which generalizes the known bound for Nekrasov matrices, is obtained. The case of block two-by-two GN matrices with scalar first diagonal blocks, which prove to be Dashnic–Zusmanovich matrices of the first type, is considered separately. The bounds obtained are applied to SDD matrices.

Note on subdirect sums of $ \{i_0\} $-Nekrasov matrices

<abstract><p>The concept of $ k $-subdirect sums of matrices, as a generalization of the usual sum and the direct sum, plays an important role in scientific computing. In this paper, we introduce a new subclass of $ S $-Nekrasov matrices, called $ \{i_0\} $-Nekrasov matrices, and some sufficient conditions are given which guarantee that the $ k $-subdirect sum $ A\bigoplus_k B $ is an $ \{i_0\} $-Nekrasov matrix, where $ A $ is an $ \{i_0\} $-Nekrasov matrix and $ B $ is a Nekrasov matrix. Numerical examples are reported to illustrate the conditions presented.</p></abstract>

Norm bounds for the inverse and error bounds for linear complementarity problems for {P1,P2}-Nekrasov matrices

{P1,P2}-Nekrasov matrices represent a generalization of Nekrasov matrices via permutations. In this paper, we obtained an error bound for linear complementarity problems for fP1; P2g-Nekrasov matrices. Numerical examples are given to illustrate that new error bound can give tighter results compared to already known bounds when applied to Nekrasov matrices. Also, we presented new max-norm bounds for the inverse of {P1,P2}-Nekrasov matrices in the block case, considering two different types of block generalizations. Numerical examples show that new norm bounds for the block case can give tighter results compared to already known bounds for the point-wise case.

Norm bounds for the inverse for generalized Nekrasov matrices in point-wise and block case

Lower-semi-Nekrasov matrices represent a generalization of Nekrasov matrices. For the inverse of lower-semi-Nekrasov matrices, a max-norm bound is proposed. Numerical examples are given to illustrate that new norm bound can give tighter results compared to already known bounds when applied to Nekrasov matrices. Also, we presented new max-norm bounds for the inverse of lower-semi-Nekrasov matrices in the block case. We considered two types of block generalizations and illustrated the results with numerical examples.

The closure property of the Schur complement for Nekrasove matrices and its applications in solving large linear systems with Schur-based method

In this paper, several conditions are presented to keep the Schur complement via a non-leading principle submatrix of some special matrices including Nekrasov matrices being a Nekrasov matrix, which is useful in the Schur-based method for solving large linear equations. And we give some infinity norm bounds for the inverse of Nekrasov matrices and its Schur complement to help measure whether the classical iterative methods are convergent or not. At last, in the applications of solving large linear equations by Schur-based method, some numerical experiments are presented to show the efficiency and superiority of our results.

Extended Irreducible Nekrasov Matrices as Subclasses of Irreducible H-Matrices

In this paper, we introduce the irreducible $$\alpha $$ -Nekrasov matrices and irreducible $$\alpha $$ -S-Nekrasov matrices as the extended irreducible Nekrasov matrices and we analyze the relationships among the involved matrices and irreducible H-matrices.

Nekrasov Type Matrices and Upper Bounds for Their Inverses

The paper considers the so-called P-Nekrasov and {P1, P2}-Nekrasov matrices, defined in terms of permutation matrices P, P1, P2, which generalize the well-known notion of Nekrasov matrices. For such matrices A, available upper bounds on ‖A−1‖∞ are recalled, and new upper bounds for the P-Nekrasov and {P1, P2}-Nekrasov matrices are suggested. It is shown that the latter bound generally improves the earlier bounds, as well as the bound for the inverse of a P-Nekrasov matrix and the classical bound for the inverse of a strictly diagonally dominant matrix. Bibliography: 12 titles.

New upper bounds for the infinity norm of Nekrasov matrices

New upper bounds for the infinity norm of Nekrasov matrices

An improvement of the infinity norm bound for the inverse of {P_{1},P_{2}}-Nekrasov matrices

A new upper bound for the infinity norm for the inverse of {P_{1},P _{2}}-Nekrasov matrices is given. It is proved that the upper bound is sharper than those in Cvetković et al. (Open Math. 13:96–105, 2015) and than well-known Varah’s bound for strictly diagonally dominant matrices. Numerical examples are given to illustrate the corresponding results.