Randomized approximation of bounded multicovering problems

The problem of finding approximate solutions for a subclass of multicovering problems denoted byILP(k, b) is considered. The problem involves findingx∈{0,1} ⁿ that minimizes ∑ _j x _j subject to the constraintAx≥b, whereA is a 0–1m×n matrix with at mostk ones per row,b is an integer vector, andb is the smallest entry inb. This subclass includes, for example, the Bounded Set Cover problem whenb=1, and the Vertex Cover problem whenk=2 andb=1. An approximation ratio ofk−b+1 is achievable by known deterministic algorithms. A new randomized approximation algorithm is presented, with an approximation ratio of (k−b+1)(1−(c/m)^1/(k−b+1)) for a small constantc>0. The analysis of this algorithm relies on the use of a new bound on the sum of independent Bernoulli random variables, that is of interest in its own right. The first author was supported in part by a Walter and Elise Haas Career Development Award and by a grant from the Israeli Science Foundation. This work was done white the third author was at the Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehovot 76100, Israel.     

Preconditioned conjugate gradient methods for the solution of indefinite least squares problems

The conjugate gradient (CG) method is considered for solving the large and sparse indefinite least squares (ILS) problem min  _x (b−Ax) ^T J(b−Ax) where J=diag (I _p ,−I _q ) is a signature matrix. However the rate of convergence becomes slow for ill-conditioned problems. The QR-based preconditioner is found to be effective in accelerating the convergence. Numerical results show that the sparse Householder QR-based preconditioner is superior to the CG method especially for sparse and ill-conditioned problems.     

Extended version with the analysis of dynamic system for iterative refinement of solution

The extended version with the analysis of dynamic system for Wilkinson's iteration improvement of solution is presented in this paper. It turns out that the iteration improvement can be viewed as applying explicit Euler method with step size h=1 to a dynamic system which has a unique globally asymptotically stable equilibrium point, that is, the solution x*=A ^?1 b of linear system Ax=b with non-singular matrix A. As a result, an extended iterative improvement process for solving ill-conditioned linear system of algebraic equations with non-singular coefficients matrix is proposed by following the solution curve of a linear system of ordinary differential equations. We prove the unconditional convergence and derive the roundoff results for the extended iterative refinement process. Several numerical experiments are given to show the effectiveness and competition of the extended iteration refinement in comparison with Wilkinson's.     

Numerical solution of fuzzy linear system

The aim of this paper is to present a numerical method for solving a general n×n fuzzy system of linear equations of the form Ax=b, where A is a crisp matrix and b an arbitrary fuzzy vector. We obtain the solution of n×n fuzzy linear systems by using Jacobi and Gauss-Seidel iterative methods and also show that the order of system will not be increased and the computing time will be shorter than other numerical methods. Finally, we illustrate this method by offering some numerical examples.     

New gauss–seidel like block iterative methods to solve discrete boundary value problems

The concept of new Gauss–Seidel like iterative methods, which was introduced in [3], will be extended so as to obtain a class of convergent Gauss–Seidel like block iterative methods to solve linear matrix equations Ax=b with an M-Matrix A. New block iterative methods will be applied to finite difference approximations of the Laplace's equation on a square (“model problem” [8]) which surpass even the block successive overrelaxation iterative method with optimum relaxation factor in this example.     

MRF: A FORTRAN IV computer program for the generation of univariant phase equilibria

A computer program, MRF, has been designed to generate all possible equations for chemical reactions between a chosen set of phases. All subsets of the phases are chosen and arranged into an equation and balanced using the formula Ax = b. A is a m × n matrix of rank n composed of n compositional vectors, b is a m dimensional vector and x is a (n + 1) dimensional vector of reaction coefficients. A m × n matrix also allows the solution to be independent of the components chosen. Provision is made for the exclusion of up to 5-phase assemblages.     

Convergence of the interval and point TOR method

In this paper, we present the interval version of the two parameter overrelaxation iterative (TOR) method and we obtain some convergence conditions when the matrix A of the linear system Ax?=?b belongs to some classes of matrice. Similar conditions were obtained for the point TOR method.

Some results for the accelerated overrelaxation interval and point iterative (AOR) method were also obtained, which coincides with those given by Martins in Ref. [7].     

Computation of eigenvalues of a real matrix

This paper presents a computational procedure for finding eigenvalues of a real matrix based on Alternate Quadrant Interlocking Factorization, a parallel direct method developed by Rao in 1994 for the solution of the general linear system Ax=b. The computational procedure is similar to LR algorithm as studied by Rutishauser in 1958 for finding eigenvalues of a general matrix. After a series of transformations the eigenvalues are obtained from simple 2×2 matrices derived from the main and cross diagonals of the limit matrix. A sufficient condition for the convergence of the computational procedure is proved. Numerical examples are given to demonstrate the method.     

Optimum extrapolated method in a special case with application to sor method

Given a linear system Ax = b and an iterative method x ^{(m + 1)} = Gx ^(m)+k, m = 0,1,2,…(1) to solve it, we determine analytically the optimum extrapolation factor of the extrapolated method of (1), when all the eigenvalues of G have the same modulus (Section 2). Then using the SOR theory in the case of consistently ordered matrices A and applying the results of Section 2 to the extrapolated SOR (ESOR) method, we show (Section 3), that the globally optimum parameters of it (and also of the AOR method) [2, 11, 14] are recovered.     

Generalized tridiagonal preconditioners for solving linear systems

The paper presents a type of tridiagonal preconditioners for solving linear system Ax=b with nonsingular M-matrix A, and obtains some important convergent theorems about preconditioned Jacobi and Gauss–Seidel type iterative methods. The main results theoretically prove that the tridiagonal preconditioners cannot only accelerate the convergence of iterations, but also generalize some known results.     

Algebraic approach to the interval linear static identification,tolerance, and control problems,or one more application of kaucher arithmetic

In this paper, theidentification problem, thetolerance problem, and thecontrol problem are treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution set Σ_??(A, b)={x ∈ ? ⁿ | (?A ∈ A)(Ax ∈ b)}, of thetolerable solution set Σ_??(A, b)={x ∈ ? ⁿ | (?A ∈ A)(Ax ∈ b)}, and of thecontrollable solution set Σ_??(A, b)={x ∈ ? ⁿ | (?b ∈ b)(Ax ∈b)} respectively. Analgebraic approach to their solution is developed in which the initial problem is replaced by that of finding analgebraic solution of some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its convergence and discuss numerical experiments.     

Convergence of bpsd method for T (q,r) matrix

In this paper, we obtain a necessary and sufficient condition, when the coefficient matrix A of the equation Ax = f considered is a T(l, 1) matrix, a sufficient condition, when A is a T(l, 2) or T(2, 1) matrix for the convergence of BPSD method. We also obtain the optimum parameters and the optimum rate of convergence of BPSD method, when A is T(l, 1) matrix and a necessary and sufficient condition, when A is positive definite and we point out that the necessary and sufficient condition in [1] and [9] is only sufficient.     

Further results on controllability and the linear equation Ax = b

The linear equation Ax = b, with A an n × n matrix and b an n × l matrix over a unique factorization domain R, is related to the controllability submodule U of the pair (A, b). It is shown that the above equation has a solution lying in V if, and only if, A is unimodular as an operator on U. An example is given of a matrix which is unimodular as an operator on the controllability submodule, but not as an operator on Rⁿ and sparseness of this occurrence is discussed.     

Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

For any A=A ₁+A ₂ j∈Q ^n×n and η∈<texlscub>i, j, k</texlscub>, denote A ^{η H} =?η A ^H η. If A ^{η H} =A, A is called an $\eta$-Hermitian matrix. If A ^{η H} =?A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ ^n×n and η AQ ^n×n , respectively.

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equation AXB+CYD=E over X∈η HQ ^n×n and Y∈η AQ ^n×n .     

On the solution of the fuzzy Sylvester matrix equation

In this paper, we consider the fuzzy Sylvester matrix equation AX+XB=C,AX+XB=C, where A ? \mathbbR^{n ×n}A\in {\mathbb{R}}^{n \times n} and B ? \mathbbR^{m ×m}B\in {\mathbb{R}}^{m \times m} are crisp M-matrices, C is an n×mn\times m fuzzy matrix and X is unknown. We first transform this system to an (mn)×(mn)(mn)\times (mn) fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.     

Optimization and pole placement for a single input controllable system

Given the system [xdot]=Ax+bu and the cost function J=dt, relations are to be determined among the open-loop characteristic polynomial, the closed-loop characteristic polynomial and the matrices A and Q. Those relations take a simple form if the system is in the standard controllable form. In this case the optimal control law can be found easily without solving the matrix Riccati equation while the minimum value of the cost function, if it is required, can be determined by solving a matrix equation of the form C ^T. X+XC= ?D     

Numerical solution of fully fuzzy linear systems by fuzzy neural network

In this paper, a new hybrid method based on fuzzy neural network (FNN) for approximate solution of fuzzy linear systems of the form Ax=d,Ax=d, where AA is a square matrix of fuzzy coefficients, xx and dd are fuzzy number vectors, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution of an n×nn\times n system of fuzzy linear equations that supposedly has a unique fuzzy solution, a simple algorithm from the cost function of the FNN is proposed. Finally, we illustrate our approach by some numerical examples.     

Incomplete oblique projections method for solving regularized least-squares problems in image reconstruction

In this paper we improve on the incomplete oblique projections (IOP) method introduced previously by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method uses IOP onto the set of solutions of the augmented system Ax?r=b, and converges to a weighted least‐squares solution of the system Ax=b. In image reconstruction problems, systems are usually inconsistent and very often rank‐deficient because of the underlying discretized model. Here we have considered a regularized least‐squares objective function that can be used in many ways such as incorporating blobs or nearest‐neighbor interactions among adjacent pixels, aiming at smoothing the image. Thus, the oblique incomplete projections algorithm has been modified for solving this regularized model. The theoretical properties of the new algorithm are analyzed and numerical experiments are presented showing that the new approach improves the quality of the reconstructed images.     

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.     

AN EXTENSION OF HILL CIPHER USING GENERALISED INVERSES AND mth RESIDUE MODULO n

Hill Cipher is a symmetric polyalphabetic block cipher that enciphers a string of letters into another string of the same length using the linear transformation y=xK. In deciphering, the determinant value must be less than 26 and relatively prime to 26 so that the matrix K of the linear transformation is invertible in modulo 26. The Affine Hill cipher extends the concept of Hill cipher by using the non-linear transformation y=xK+b. In this paper, we extend this concept to encrypt a plaintext of blocksize m to a ciphertext of blocksize n≥m using (a) affine transformation and (b) polynomial transformation to make this cipher more secure. Here the matrix K of the transformation need not be a square matrix. To enable decryption, we state the conditions to be satisfied by K which are as follows. Case (a): (i) For affine transformation, the generalised inverse K ⁺ of the matrix K corresponding to the transformation should satisfy the equation KK ⁺=I in modulo p where p is a chosen prime p>26. For m=n, K ⁺ is the usual inverse of the matrix K. Case(b): (i) For polynomial transformation, the generalised inverse K ⁺ should satisfy the above condition, (ii) If r is the degree of the polynomial, then choose those values of s≤r such that the sth root of modulo p exists for all elements in Z _p . In other words, choose those values of s that are relatively prime to Φ(p).