Efficient approaches for enclosing the united solution set of the interval generalized Sylvester matrix equations (original) (raw)
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International Journal of Systems Science, 2019
In this paper, we investigate the parametric generalised Sylvester matrix equation A(p)XB(p) + C(p)XD(p) = F(p), whose elements are linear functions of uncertain parameters varying within intervals. This model generalises both interval matrix equations and parametric interval linear systems, so it is a quite general model. First, we give some sufficient conditions under which the solution set of this parametric equation is bounded. We then propose several approaches for enclosing the solution set that acquire tighter enclosures than those obtained by relaxing the parametric system to an interval system. Some special cases of interval systems that inherently are parametric (have dependent structure) are considered, too. Finally, numerical experiments are given to illustrate the effectiveness of the proposed methods.
Enclosing the solution set of overdetermined systems of interval linear equations
2014
We describe two methods to bound the solution set of full rank interval linear equation systems Ax = b where A ∈ IRm×n, m ≥ n is a full rank interval matrix and b ∈ IRm is an interval vector. The methods are based on the concept of generalized solution of overdetermined systems of linear equations. We use two type of preconditioning the m × n system: multiplying the system with the generalized inverse of the midpoint matrix or with the transpose of the midpoint matrix. It results an n × n system which we solve using Gaussian elimination or the method provided by J. Rohn in [8]. We give some examples in which we compare the efficiency of our methods and compare the results with the interval Householder method [11]. Mathematics Subject Classification (2010): 65G06.
International Journal of Nonlinear Analysis and Applications, 2017
This paper introduces the emph{interval unilateral quadratic matrix equation}, IUQeIUQeIUQe and attempts to find various analytical results on its AE-solution sets in which A,BA,BA,B and CCCCCCCCC are known real interval matrices, while XXX is an unknown matrix. These results are derived from a generalization of some results of Shary. We also give sufficient conditions for non-emptiness of some quasi-solution sets, provided that AAA is regular. As the most common case, the united solution set has been studied and two direct methods for computing an outer estimation and an inner estimation of the united solution set of an interval unilateral quadratic matrix equation are proposed. The suggested techniques are based on nonlinear programming as well as sensitivity analysis.
Reliable Computing, 2006
Consider the systems of linear interval equations whose coefficients are affine-linear functions of interval parameters. Such systems, called parametrized systems of linear interval equations, are encountered in many practical problems, e.g in structure mechanics. A direct method for computing a tight enclosure for the solution set is proposed in this paper. It is proved that for systems with real matrix and interval right-hand vector the method generates the hull of the solution set. For such systems an explicit formula for the hull is also given. Finally some numerical examples are provided to demonstrate the usefulness of the method in structure mechanics.
A method for outer interval solution of parametrized systems of linear equations
Consider the systems of linear interval equations whose coefficients are affine-linear functions of interval parameters. Such systems, called parametrized systems of linear interval equations, are encountered in many practical problems, e.g in structure mechanics. A direct method for computing a tight enclosure for the solution set is proposed in this paper. It is proved that for systems with real matrix and interval right-hand vector the method generates the hull of the solution set. For such systems an explicit formula for the hull is also given. Finally some numerical examples are provided to demonstrate the usefulness of the method in structure mechanics.
Computing the spectral decomposition of interval matrices and a study on interval matrix powers
Applied Mathematics and Computation, 2021
We present an algorithm for computing a spectral decomposition of an interval matrix as an enclosure of spectral decompositions of particular realizations of interval matrices. The algorithm relies on tight outer estimations of eigenvalues and eigenvectors of corresponding interval matrices. We present a method for general interval matrices as well as its modification for symmetric interval matrices. As an illustration, we apply the spectral decomposition to computing powers of interval matrices. Numerical results suggest that a simple binary exponentiation is more efficient for smaller exponents, but our approach becomes better when computing higher powers or powers of a special type of matrices. In particular, we consider symmetric interval and circulant interval matrices. In both cases we utilize some properties of the corresponding classes of matrices to make the power computation more efficient.
Method for the Solution of Interval Systems of Linear Equations
International Journal of Advances in Applied Sciences, 2013
In this paper, we discuss the solution of interval system of linear equations and proposed a new method for handling this type of system of linear equations. In this model we consider the coefficient matrix and the right vector as interval. Example problems are given to have the efficiency and powerfulness of the proposed method.
A method to rigorously enclose eigendecompositions of interval matrices
In this paper, a rigorous computational method to enclose eigendecompositions of complex interval matrices is proposed. Each eigenpair x=(lambda,v)x=(\lambda,v)x=(lambda,v) is found by solving a nonlinear equation of the form f(x)=0f(x)=0f(x)=0 via a contraction argument. The set-up of the method relies on the notion of radii polynomials, which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.
Solving fully interval linear systems of equations using tolerable solution criteria
Soft Computing, 2017
A new method has been proposed here for solving fully interval linear systems of equations where both coefficient matrix and the right-hand side vector are intervals. In this method, center solution has been used with the tolerable solution criteria to compute the inner solution set. A few example problems are solved to demonstrate the proposed method. Numerical results are compared with existing methods and are found to be in good agreement.
A Comparison of some Methods for Solving Linear Interval Equations
SIAM Journal on Numerical Analysis, 1997
Certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined. We extend a proposed technique of Hansen and Rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix A = [A, A] is an H-matrix; when A is centered about a diagonal matrix, these bounds are the smallest possible (i.e., the bounds are then the solution hull). Hansen's scheme also computes the solution hull when the linear interval system Ax = b = [b, b] is such that A is inverse positive and b = −b = 0. Earlier results of others also imply that, when A is an M-matrix and b ≥ 0, b ≤ 0, or 0 ∈ b, interval Gaussian elimination (IGA) computes the hull. We also give a method of computing the solution hull inexpensively in many instances when A is inverse positive, given an outer approximation such as that obtained from IGA. Examples are used to compare these schemes under various conditions.