Inverse Problem of the Interval Linear System of Equations

Given a nonsingular central matrix , a central vector and a prescribed interval solution vector , it is required to find the maximum allowable deviation or so that the solution of the interval linear system is contained in . Special cases for and are considered and bounds on the entries of and are obtained either in a closed form, whenever possible, or via solving a specially designed constrained optimization problem. Received: August 25, 1997; revised February 20, 1999     

The Optimal Solution of an Interval System of Linear Algebraic Equations

Both necessary and sufficient conditions for the coincidence of the interval hull of the united solution set and the algebraic solution for the arbitrary interval systems x = Mx + r satisfying the condition (|M |)< 1=" are=" proved=" in=" the=" paper.=" the=" necessary=" conditions=" are=" more=" restrictive=" than=" the=" sufficient=" ones,=" but=" almost=" always=" coincide=" with=">     

A Method for Outer Interval Solution of Systems of Linear Equations Depending Linearly on Interval Parameters

Consider the systems of linear interval equations whose coefficients are linear functions of interval parameters. Such systems, called parametrized systems of linear interval equations, are encountered in many practical problems, e.g in electrical engineering and 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.     

An Interval Linear Tolerance Problem

This paper is concerned with a tolerance problem for an interval linear system A x = b requiring inner estimation of the admissible solution set {x ⁿ | (A A)(Ax b)} formed by vectors x for which the product Ax remains within b for any possible A A. Methods for verifying the emptiness and nonemptiness of admissible solution sets are developed. Formulas for the dimensions of the interval solution of a tolerance problem with known center are derived.     

A Method for Outer Interval Solution of Linear Parametric Systems

The paper deals with the problem of determining an outer interval solution (interval enclosure of the solution set) of linear systems whose elements are affine functions of interval parameters. An iterative method for finding such a solution is suggested. A numerical example illustrating the new method is solved.     

Linear Interval Tolerance Problem and Linear Programming Techniques

In this paper, we consider the linear interval tolerance problem, which consists of finding the largest interval vector included in ([A], [b]) = {x R ⁿ | A [A], b [b], Ax = b}. We describe two different polyhedrons that represent subsets of all possible interval vectors in ([A], [b]), and we provide a new definition of the optimality of an interval vector included in ([A], [b]). Finally, we show how the Simplex algorithm can be applied to find an optimal interval vector in ([A], [b]).     

Solving Overdetermined Systems of Interval Linear Equations

An algorithm is developed to compute interval bounds on the set of all solutions to an overdetermined system of interval linear equations.     

The Hull of Preconditioned Interval Linear Equations

In Ph.D. thesis (Computer Methods for Design Automation, MIT, 1992), C. Bliek gave a method for computing the exact hull of a system of interval linear equations which had been preconditioned using an approximate inverse of the center of the coefficient matrix. In this paper, we simplify both the theoretical procedure and its practical implementation. We give easily verified conditions for regularity of the preconditioned matrix. We describe classes of problems for which preconditioning leaves some or all boundaries of the hull unchanged.     

A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations

Recently, Ning & Kearfott derived a formula for the interval enclosure of the solution set of linear systems of equations with uncertain data ranging in intervals, in the case when the coefficient matrix is an H-matrix. The enclosure is optimal when the midpoint matrix is diagonal, and when the midpoint is the identity, it reduces to the optimal method for enclosing preconditioned systems found by Hansen and Bliek and simplified by Rohn.An elementary proof of this formula is given using only simple properties of H-matrices and Schur complements. The new proof gives additional insight into why the theorem is true. It is also shown how to preserve rigor in the enclosure when finite precision arithmetic is used.     

Outer Estimation of Generalized Solution Sets to Interval Linear Systems

The work advances a numerical technique for computing enclosures of generalized AE-solution sets to interval linear systems of equations. We develop an approach (called algebraic) in which the outer estimation problem reduces to a problem of computing algebraic solutions of an auxiliary interval equation in Kaucher complete interval arithmetic.     

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λx, A(p) ∈ ℝ^n×n for the general case where the matrix elements a_ij(p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λ_μ, x^(μ)), μ = 1, ..., n, is suggested for the case where λ_μ is a simple eigenvalue. It is based on the use of an affine interval approximation of a_ij(p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.     

Formal Solution to Systems of Interval Linear or Non-Linear Equations

This is the first of two papers which present the Modal Interval Analysis as a framework where the search and interpretation of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this first paper is to show that the modal intervals are a suitable tool to approach problems where logical reference appear. Thus, it is possible to give a logical meaning to general solutions of the system of equations, obtained, in the linear case, by means of an algorithm whose convergence conditions are studied.     

线性方程组迭代法在流处理器上的映射与分析

斯坦福大学的Imagine流处理器具有很强的计算能力,如何将该体系结构应用在科学计算领域是当前研究的热点。解线性方程组的迭代法在工程和科学计算的各个领域中有着十分广泛的应用,该算法具有较好的计算密集性和并行性,十分适合流处理器的计算模型。本文分别针对系数矩阵的规模大小和稠密程度,介绍了Jacobi和Seidel迭代在流处
理器上的映射。实验结果表明,迭代算法能高效地开发Imagine的计算能力,取得较高的性能加速。     

Online Algebraic Identification Approach of Continuous Linear Time Delay Systems

In this paper, we propose an online algebraic method to identify linear continuous time delay process from step response, in presence of unknown initial state and constant load disturbance. The identification mechanism is split into two sequential steps. The time delay and dynamic of the plant is estimated firstly through a spectral formulation. In the next stage, the transient regime is deduced from a linear regression. To improve the robustness of the proposed method, a local convolution by sigmoid function is proposed. Simulation results are provided, at last, to show the effectiveness of our conceived approach.     

Outer Solution of Linear Systems Whose Elements Are Affine Functions of Interval Parameters

In this paper, linear systems whose elements are affine functions of a given set of independent intervals are considered. A direct method for computing an outer solution to such systems is suggested.     

Interval Arithmetic in Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition requires many very time consuming operations, including resultant computation, polynomial factorization, algebraic polynomial gcd computation and polynomial real root isolation. We show how the time for algebraic polynomial real root isolation can be greatly reduced by using interval arithmetic instead of exact computation. This substantially reduces the overall time for cylindrical algebraic decomposition.     

A Method for Solution of Systems of Linear Algebraic Equations with m-Dimensional @lambda;-Matrices

A system of linear algebraic equations with m-dimensional @lambda;-matrices is considered. The proposed method of searching for the solution of this system lies in reducing it to a numerical system of a special kind.     

The Control Problem for Stage-by-Stage Changing Linear Systems of Loaded Differential Equations

We consider the control problem for stage-by-stage changing linear differential equations and the optimal control problem with a quality criterion defined for the entire time interval. We formulate necessary and sufficient conditions for complete controllability and the existence of a program control and motion. We construct an explicit form of the control action for the control problem and propose a method for solving the optimal control problem. We give a solution of the control problem for a specific loaded system.     

A Comment on the Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients

This article is a short supplement to our previously published paper, in which we proved that each semialgebraic set can be represented as a projection of a solution set of some system of interval linear equations with dependent coefficients. The new result says that interval occurring can be chosen as narrow as wanted. The new result is proved by a simple linear transformation.     

Algebraic Pressure Segregation Methods for the Incompressible Navier-Stokes Equations

This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexact LU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled. The first author’s research was supported by the European Community through the Marie Curie contract NanoSim (MOIF-CT-2006-039522).