A new method for solving symmetric circulant tridiagonal systems of linear equations

In this paper a new method for computing the solution of a linear system having a symmetric circulant tridiagonal matrix is presented. This special kind of system appears in many applications. After an appropriate partition of the system and the elimination of the last unknown, we apply the Woodbury formula to arrive at a very efficient and stable method, which is quite competitive with Gaussian elimination and with the modified double sweep method.     

Efficient preconditioned NHSS iteration methods for solving complex symmetric linear systems

Based on the new HSS (NHSS) iteration method introduced by Pour and Goughery (2015), we propose a preconditioned variant of NHSS (P*NHSS) and an efficient parameterized P*NHSS (PPNHSS) iteration methods for solving a class of complex symmetric linear systems. The convergence properties of the P*NHSS and the PPNHSS iteration methods show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess when the parameters are properly chosen. Moreover, we discuss the quasi-optimal parameters which minimize the upper bounds for the spectral radius of the iteration matrices. Numerical results show that the PPNHSS iteration method is superior to several iteration methods whether the experimental optimal parameters are used or not.     

Classical foundations of algorithms for solving positive definite Toeplitz equations

The ongoing development and analysis of efficient algorithms for solving positive definite Toeplitz equations is motivated to a large extent by the importance of these equations in signal processing applications. The role of positive definite Toeplitz matrices in this and other areas of mathematics and engineering stems from Schur's study of bounded analytic functions on the unit disk, and Szegő's theory of polynomials orthogonal on the unit circle. These ideas underlie several Toeplitz solvers, and provide a useful framework for understanding the relationships among these algorithms. In this paper we give an overview of several direct algorithms for solving positive definite Toeplitz systems of linear equations from this classical viewpoint.     

The AOR method for solving linear interval equations

This paper is motivated by the paper [7], where the SOR method for solving linear interval equations was considered. It is known that sometimes the AOR method for systems of linear (“point”) equations converges faster than the SOR method. We give some sufficient conditions for the convergence of the interval AOR method for the same class of interval matrices which are considered in [7].     

Iterative methods for systems of equations with interval coefficients and linear form

We introduce iterative methods for systems of equations with interval coefficients and linear form by suitable matrix splittings. When compared to the iterative methods for systems amenable to iteration introduced in [1], improved convergence and inclusion properties can be proved under suitable conditions. The method can also be used in the solution of specific nonlinear systems of equations by interval arithmetic methods.     

A direct method for solving linear systems

In this work we propose a direct method for solving systems of linear equations which is based on a successive LU-decomposition of matrices of the form l + uv ^T . Simultaneously, the factors of an LU-decomposition of the coefficient matrix are obtained. A specific choice of the “rank-one decomposition” of the given matrix leads to a variant of the Gauss elimination process.     

Two-parameter TSCSP method for solving complex symmetric system of linear equations

We introduce a two-parameter version of the two-step scale-splitting iteration method, called TTSCSP, for solving a broad class of complex symmetric system of linear equations. We present some conditions for the convergence of the method. An upper bound for the spectral radius of the method is presented and optimal parameters which minimize this bound are given. Inexact version of the TTSCSP iteration method (ITTSCSP) is also presented. Some numerical experiments are reported to verify the effectiveness of the TTSCSP iteration method and the numerical results are compared with those of the TSCSP, the SCSP and the PMHSS iteration methods. Numerical comparison of the ITTSCSP method with the inexact version of TSCSP, SCSP and PMHSS are presented. We also compare the numerical results of the BiCGSTAB method in conjunction with the TTSCSP and the ILU preconditioners.     

A symmetric iterative interval method for systems of nonlinear equations

An iterative method for nonlinear systems of equations is presented that is based on the idea of symmetric methods known from linear systems. Due to the use of interval arithmetic the convergence to a solution can be proved under relatively weak conditions provided an initial inclusion of that solution is known. The concept of symmetry leads to a reduction of computation time compared to some well-known methods.     

A new splitting and preconditioner for iteratively solving non-Hermitian positive definite systems

A new iteration method for solving a linear system with coefficient matrix being non-Hermitian positive definite is presented in this note. We study the spectral radius and contraction properties of the iteration matrix and then analyze the best possible choice of the parameter. With the results obtained, we show that the new method is convergent for a non-Hermitian positive definite linear system and propose a preconditioner to improve the condition number of the system. The numerical examples show that the new method is much more efficient than the HSS (or PSS) iteration method.     

A symmetric positive definite preconditioner for saddle-point problems

For the classical saddle-point problem, we present precisely two intervals containing the positive and the negative eigenvalues of the preconditioned matrix, respectively, when the inexact version of the symmetric positive definite preconditioner introduced in Section 2.1 of Gill et al. [Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl. 13 (1992), pp. 292–311] is employed. The model of Stokes problem is used to test the effectiveness of the presented bounds as well as the quality of the symmetric positive definite preconditioner.     

Efficient Jarratt-like methods for solving systems of nonlinear equations

We present the iterative methods of fourth and sixth order convergence for solving systems of nonlinear equations. Fourth order method is composed of two Jarratt-like steps and requires the evaluations of one function, two first derivatives and one matrix inversion in each iteration. Sixth order method is the composition of three Jarratt-like steps of which the first two steps are that of the proposed fourth order scheme and requires one extra function evaluation in addition to the evaluations of fourth order method. Computational efficiency in its general form is discussed. A comparison between the efficiencies of proposed techniques with existing methods of similar nature is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are confirmed in the examples. It is shown that the present methods are more efficient than their existing counterparts, particularly when applied to the large systems of equations.     

Quotient convergence and multi-splitting methods for solving singular linear equations

Abstract   In this paper, we use the group inverse to characterize the quotient convergence of an iterative method for solving consistent singular linear systems, when the matrix index equals one. Next, we show that for stationary splitting iterative methods, the convergence and the quotient convergence are equivalent, which was first proved in [7]. Lastly, we propose a (multi-)splitting iterative method A=F–G, where the splitting matrix F may be singular, endowed with group inverse, by using F ^# as a solution tool for any iteration. In this direction, sufficient conditions for the quotient convergence of these methods are given. Then, by using the equivalence between convergence and quotient convergence, the classical convergence of these methods is proved. These latter results generalize Cao’s result, which was given for nonsingular splitting matrices F. Keywords: Group inverse, singular linear equations, iterative method, P-regular splitting, Hermitian positive definite matrix, multi-splitting, quotient convergence AMS Classification: 15A09, 65F35     

Explicit isomorphic iterative methods for solving arrow-type linear systems

A new class of approximate inverse arrow-type matrix techniques based on the concept of sparse approximate LU-type factorization procedures is introduced for computing explicitly approximate inverses without inverting the decomposition factors. Isomorphic methods in conjunction with explicit preconditioned schemes based on approximate inverse matrix techniques are presented for the efficient solution of arrow-type linear systems. Applications of the proposed method on linear systems is discussed and numerical results are given     

The preconditioned variational methods for solving large linear systems

The implementation of the Preconditioned Conjugate Gradient method for the solution of large linear systems arising from the discretization of differential operators, requires the predetermination of only one iteration parameter. The numerical determination of the optimal value of this constant parameter, involve the spectral bounds of some matrices and can be obtained in O(N²) sine function evaluations, where 1/N is the discretization mesh size. It is shown that this parameter can be chosen in a stable manner in O(1) operations per iteration, if it is allowed to vary with the iteration index from information derived from the gradient parameters.     

Rational Runge-Kutta methods for solving systems of ordinary differential equations

Some nonlinear methods for solving single ordinary differiential equations are generalized to solve systems of equations. To perform this, a new vector product, compatible with the Samelson inverse of a vector, is defined. Conditions for a given order are derived.     

Enclosing solutions of overdetermined systems of linear interval equations

A method for enclosing solutions of overdetermined systems of linear interval equations is described. Several aspects of the problem (algorithm, enclosure improvement, optimal enclosure) are studied.     

Algorithms for solving systems of linear diophantine equations in residue rings

Algorithms are proposed that construct the basis of the set of solutions to a system of homogeneous or inhomogeneous linear Diophantine equations in a residue ring modulo n when the prime factors of n are known. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 27–40, November–December 2007.     

A parallel partition method for solving banded systems of linear equations

The partition method of Wang for tridiagonal equations is generalized to the arbitrary band case. A stability criterion is given. The algorithm is compared to Gaussian elimination and cyclic reduction.     

Nonequidistant modified predictor-corrector methods for solving systems of differential equations

This paper shows that it is possible to develop nonequidistant predictor-corrector formulae with minimum error bounds for solving systems of differential equations such that the tedious difficulties which arise in practical applications can be overcome. General predictor-corrector formulae with variable steps are constructed. Explicit third order- and fourth order-two points formulae are derived. Also fourth order-three points formulae are represented. Two theorems are given. A flow chart for general nonequidistant predictor-corrector methods using automatic control for the step length is compactly represented for solving systems of differential equations. These methods are recommended to be used widely in practice because of many advantages.