On the monotonicity of the interval versions of Schulz's method II

In this note we are considering the interval versions of Schulz's method for bounding the inverse of a matrix. We prove a necessary and sufficient criterion for the existence of an interval inclusion such that the iterates are strictly monotonic in both bounds. In particular, such an initial interval-matrix is constructed.     

On the monotonicity of higher-order interval iterations for bounding the inverse matrix

We are considering modifications of the higher-order interval Schulz methods for improving an initial inclusion for the inverse of a matrix. It is proved that the methods are strictly monotone in both bounds if this is the case for the first iteration step. A necessary and sufficient condition for the existence of such an initial inclusion is given as well as an easy-to-compute formula for it.     

Accelerating Krawczyk-like interval algorithms for the solution of nonlinear systems of equations by using second derivatives

Recently the properties of Krawczyk-like iterative interval methods for the solution of systems of nonlinear equations have been discussed in several papers (e.g. [2], [3], [5], [6]). These methods converge to a solution under relatively weak conditions provided an initial inclusion vector is known. In the present paper we describe a method that improves the convergence speed for an important class of problems by using second partial derivatives. This method is particularly interesting for large systems with a Jacobi matrix whose off-diagonal coefficients are all constant.     

Inner estimation of the united solution set of interval linear algebraic system

It is shown that an algebraic interval solution of interval linear algebraic systems with matrix composed of “reverse” interval elements of the input matrix is a maximum inner estimation for the mited solution set in the sense of inclusion.     

Exact Maximum Singular Value Calculation of an Interval Matrix

In this note, we present a method for calculating the maximum singular value of an interval matrix. First, we provide an algorithm for calculating the maximum singular value of a square interval matrix. Then, based on this algorithm, we extend the result to non-square interval matrix case and to the case of computing the minimum singular value. Through numerical examples, the validity of the suggested methods is illustrated. Particularly, we compare the newly-proposed method with an existing method to show that the new method finds the correct bound of the maximum singular value with no exception     

A new approach to obtain algebraic solution of interval linear systems

In this paper, an algebraic solution of interval linear system involving a real square matrix and an interval right-hand side vector is obtained. A new approach to solve such systems based on the new concept “inclusion linear system” is proposed. Moreover, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Furthermore, based on our method, an algorithm is proposed and numerically demonstrated. Finally, we compare the result obtained by our method with that obtained by interval Gauss elimination procedure.     

On the range of eigenvalues of an interval matrix

We describe a method for enclosing the set of real eigenvalues of an interval matrix pertaining to eigenvectors of a given sign pattern.     

On the convergence condition of generalized root iterations for the inclusion of polynomial zeros

Using a fixed point relation based on the logarithmic derivative of the k-th order of an algebraic polynomial and the definition of the k-th root of a disk, a family of interval methods for the simultaneous inclusion of complex zeros in circular complex arithmetic was established by Petković [M.S. Petković, On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic, Computing 27 (1981) 37–55]. In this paper we give computationally verifiable initial conditions that guarantee the convergence of this parallel family of inclusion methods. These conditions are significantly relaxed compared to the previously stated initial conditions presented in literature.     

Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

By exploiting generalized error-bounds for the well-known hyperpower methods for approximating the inverse of a matrix we derive inclusion methods for the inverse matrix. These methods make use of interval operations in order to give guaranteed inclusions whenever the convergence of the applied hyperpower method can be shown. The efficiency index of some of the new methods is greater than that of the optimal methods in [2] or [5]. A numerical example is given.     

Inverting an interval Hessian of a factorable function

For twice-differentiable factorable functions, a new computer code provides the Hessian matrix as the sum of outer products of vectors and an interval Hessian as the sum of outer products of interval vectors. A practical method for inverting an interval Hessian of a factorable function which exploits this special structure is presented. Computational experience with this method and other inversion techniques is reported.     

Estimation of algebraic solution by limiting the solution set of an interval linear system

In this paper, we introduce a new method for estimating the algebraic solution of an interval linear system (ILS) whose coefficient matrix is real-valued and right-hand side vector is interval-valued. In the proposed method, we first apply the interval Gaussian elimination procedure to obtain the solution set of an interval linear system and then by limiting the solution set of related ILS by the limiting factors, we get an algebraic solution of ILS. In addition, we prove that the obtained solution by our method satisfies the related interval linear system. Finally, based on our method, an algorithm is proposed and numerically demonstrated.     

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.     

An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation

To date, the only effective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary differential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval Hermite-Obreschkoff (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and order, our IHO scheme has a smaller truncation error, better stability, and requires fewer Taylor coefficients and high-order Jacobians.The stability properties of the ITS and IHO methods are investigated. We show as an important by-product of this analysis that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.     

Derivative Free Inclusion Methods for Polynomial Zeros

Two iterative methods for the simultaneous inclusion of complex zeros of a polynomial are presented. Both methods are realized in circular interval arithmetic and do not use polynomial derivatives. The first method of the fourth order is composed as a combination of interval methods with the order of convergence two and three. The second method is constructed using double application of the inclusion method of Weierstrass’ type in serial mode. It is shown that its R-order of convergence is bounded below by the spectral radius of the corresponding matrix. Numerical examples illustrate the convergence rate of the presented methods     

Fast Inclusion of Interval Matrix Multiplication

This paper is concerned with interval matrix multiplication.New algorithms are proposed to calculate an inclusion of the product of interval matrices using rounding mode controlled computation. Thecomputational cost of the proposed algorithms is almost the same as that for calculating an inclusion of the product of point matrices.Numerical results are presented to illustrate that the new algorithms are much faster than the conventional algorithms and that the guaranteed accuracies obtained by the proposed algorithms are comparable to those of the conventional algorithms.     

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.     

On the solution of interval linear systems

In the literature efficient algorithms have been described for calculating guaranteed inclusions for the solution of a number of standard numerical problems [3,4,8,11,12,13]. The inclusions are given by means of a set containing the solution. In [12,13] this set is calculated using an affine iteration which is stopped when a nonempty and compact set is mapped into itself. For exactly given input data (point data) it has been shown that this iteration stops if and only if the iteration matrix is convergent (cf. [13]). In this paper we give a necessary and sufficient stopping criterion for the above mentioned iteration for interval input data and interval operations. Stopping is equivalent to the fact that the algorithm presented in [12] for solving interval linear systems computes an inclusion of the solution. An algorithm given by Neumaier is discussed and an algorithm is proposed combining the advantages of our algorithm and a modification of Neumaier's. The combined algorithm yields tight bounds for input intervals of small and large diameter. Using a paper by Jansson [6,7] we give a quite different geometrical interpretation of inclusion methods. It can be shown that our inclusion methods are optimal in a specified geometrical sense. For another class of sets, for standard simplices, we give some interesting examples.     

一种改进的区间数DEMATEL决策方法

针对由区间数构建初始矩阵的DEMATEL决策问题,提出了一种基于态度的决策方案,首先分析了现有区间数DEMATEL决策方法的欠缺之处,指出了规范化过程中规范因子的片面性和在求取原因度时左右区间数不合理的问题,然后给出了解决问题的计算步骤。最后利用案例进行了分析和应用,通过对比现有的决策方案,表明提出的方法简单可行,具有一定的参考价值。     

Snap-buckling of nonlinearly-elastic finite element bars

In this study, the material nonlinearities are added to the classical elastica problem of large deflections of buckled bars. This class of problems is treated by the finite-element matrix displacement method. The formulations developed are appropriate for a midpoint-tangent incremental procedure and coordinate transformation. The material nonlinearity is incorporated in the stiffness matrix explicitly for any nonlinearly-elastic material if its stress-strain or moment-curvature relation is known. An example of the postbuckling of a nonlinearly-elastic cantilever bar is illustrated with results in agreement with an alternative numerical solution obtained by Oden and Childs [1]. The example is further illustrated with the inclusion of various initial deflections to the bar.     

Improved results for linear discrete-time systems with an interval time-varying input delay

This paper addresses the problem of delay-dependent stability analysis and controller synthesis for a discrete-time system with an interval time-varying input delay. By dividing delay interval into multiple parts and constructing a novel piecewise Lyapunov–Krasovskii functional, an improved delay-partitioning-dependent stability criterion and a stabilisation criterion are obtained in terms of matrix inequalities. Compared with some existing results, since a tighter bounding inequality is employed to deal with the integral items, our results depend on less number of linear matrix inequality scalar decision variables while obtaining same or better allowable upper delay bound. Numerical examples show the effectiveness of the proposed method.