Interval Linear Systems Research Articles

Interval Iterative Decreasing Dimension Method for Interval Linear Systems and Its Implementation to Analog Circuits

The iterative decreasing dimension method (IDDM) is an iterative method used to solve the linear algebraic system Ax=f. Such systems are important in modeling many problems in applied sciences. For a number of reasons, such as estimated measurements made for modeling, errors arising from floating point calculations, and approximation methods used for solutions, it becomes necessary to study intervals in the solutions of systems of linear equations. The objective of this paper is to utilize IDDM to achieve resolution in the interval linear system (ILS). During the calculations, the Kaucher space is considered an extended classical interval space. The solutions of Barth-Nuding and Hansen interval linear systems, which are commonly used in the literature to test the solutions of ILSs, are obtained with the interval iterative decreasing dimension method for interval linear systems (I-IDDM). Since IDDM is a variation method of Gaussian elimination, a comparative analysis of the results with the interval Gaussian elimination method (I-GEM) is performed. It has been demonstrated that our approach, I-IDDM, produces better outcomes than I-GEM. I-IDDM is also used to investigate the analog circuit problem, where interval analysis is crucial.

Transformations of solution semantics of interval linear equations system

A solution x to a system of interval linear equations Ax=b possesses its own semantics, which may involve combinations of tolerance, control, left- and right-localized semantics. In scenarios where the need arises to persist with a solution x despite changes in its semantics, corresponding adjustments in the interval information of the system become necessary. Our focus is on perturbing the original interval matrix A to produce a transformed matrix A′, ensuring that the solution x to A′x=b aligns with the new semantics. We present a series of theorems using quadratic programming concepts to determine A′ in a manner that closely approximates A. Several applications are provided to illustrate the practical utility of our approach.

Interval-based KKT framework for support vector machines and beyond

Our article proves inequalities for interval optimization and shows that feasible and descent directions do not intersect in constrained cases. Mainly, we establish some new interval inequalities for interval-valued functions by defining LC-partial order. We use LC-partial order to study Karush–Kuhn–Tucker (KKT) conditions and expands Gordan's theorems for interval linear inequality systems. By applying Gordan's theorem, we can determine the best outcomes for interval optimization problems (IOPs) that have constraints, such as Fritz John and KKT conditions. The optimality conditions are observed with inclusion relations rather than equality. We can use the KKT condition for binary classification with interval data and support vector machines(SVMs). We present some examples to illustrate our results.

A novel linear algebra-based method for complex interval linear systems in circuit analysis

In this paper, systems of multivariate interval linear equations with complex interval coefficients are examined, and a novel linear algebra-based approach for locating all of their solutions is proposed. The key concept is to convert the system into a crisp polynomial system that is equivalent and allows for the use of the innovative computational features of Gröbner bases. It is possible to calculate all of the system's precise solutions at once after an appropriate Gröbner basis has been determined. Design is a condition for the presence of a solution in complex interval linear systems. In addition, an algorithm is devised to retrieve all solutions using the eigenvalue approach. In addition, a proportional case is solved using the provided approach to demonstrate its efficiency and efficacy. The given approach can locate all solutions for linear systems with complex intervals. Additionally, it determines the presence or absence of a solution for the system. We use the aforementioned technique in the context of circuit analysis to demonstrate the effectiveness of the findings obtained.

Transformations of mixed solution types of interval linear equations system with boundaries on its left-hand side to linear inequalities with binary variables

To determine the solution set of an interval linear equations system Ax=b, it is necessary to know the semantics of the system. In addition to strong and weak semantics, there are four other important semantics: tolerance, control, left-localized, and right-localized. A solution x with one of these semantics provides that particular behavior in every equation of the system. It could happen that there is a solution x with one semantic in some equations and another semantic in the remaining equations of the system. This solution generates another solution type. We provide fifteen combinations of solution types generated by tolerance, control, left-localized, and right-localized semantics and prove that the solution set containing all four semantics is the weak (united) solution set. We show that each of these fifteen nonnegative solution types with some given boundaries on Ax can be transformed to be the solution set of linear inequalities with binary variables. The transformed system is deterministic and can be used in a linear programming problem with interval linear equation constraints to find an optimistic solution satisfying the semantics of the interval linear equations. Task management and course assignment examples are provided to show the necessity of these solution types.

Implementing Interval Linear Equations Systems for Enhanced Circuit Analysis

Implementing Interval Linear Equations Systems for Enhanced Circuit Analysis

Some Theoretical Aspects of the Neural Network Approach to Stabilization of Switched Interval Systems

We consider the problem of stabilization of a switched interval linear system with slow switchings that are inaccessible to observation. It is proposed to look for a solution in the class of variable structure controllers. To ensure the functionality of such a controller, it is necessary to construct an observer of the switching signal. This paper is devoted to some theoretical issues related to the period of quantization of the neural observer’s operating time.

Stability and stabilizability of linear time-invariant interval systems

In this paper, some new approaches for stability and stabilizability determination as well as state feedback stabilization controllers of linear time-invariant (LTI) interval systems are proposed. The presented stability conditions are less conservative than those of Kharitonov’s theorem, and Gerschgorin’s disc theorem methods. Moreover, some of the proposed stability, stabilizability, and feedback stabilization control methods for LTI interval systems are proved to be sufficient and necessary conditions. Compared with some traditional stability analysis and feedback stabilization design methods for LTI interval systems, these new approaches have lower computational complexity because of a special form of parameter vertex matrices developed in this work. Some numerical and practical examples are given to demonstrate the effectiveness and advantages of the proposed methods.

A Symbolic Algorithm for Solving Doubly Bordered k-Tridiagonal Interval Linear Systems

Doubly bordered k-tridiagonal interval linear systems play a crucial role in various mathematical and engineering applications where uncertainty is inherent in the system’s parameters. In this paper, we propose a novel symbolic algorithm for solving such systems efficiently. Our approach combines symbolic computation techniques with interval arithmetic to provide rigorous solutions in the form of tight interval enclosures. By exploiting the tridiagonal structure and employing a divide-and-conquer strategy, our algorithm achieves significantly reduced computational complexity compared to existing numerical methods. We also present theoretical analysis and provide numerical experiments to demonstrate the effectiveness and accuracy of our algorithm. The proposed symbolic algorithm offers a valuable tool for handling doubly bordered k-tridiagonal interval linear systems and opens up possibilities for addressing uncertainty in real-world problems with improved efficiency and reliability.

IntvalPy — a Python Interval Computation Library

The article presents the IntvalPy library which implements interval computations in Python. Unlike other existing interval libraries, IntvalPy allows one to work with both classical interval arithmetic and complete Kaucher interval arithmetic. In addition, the library was developed taking into account the IEEE 1788-2015 standard for interval arithmetic on digital computers, which guarantees high accuracy of the results and compatibility with the other existing software products. The top-level functionality of the IntvalPy library implements state-of-the-art methods for recognizing and estimating solution sets for interval linear systems of equations, computing their formal solutions, and visualizing solution sets for interval equations and systems of equations. As an example of the library application, we solve the practically important problem of estimating the parameters of the electrochemical process of the formation of loose metal powder precipitates. Additionally, numerical computation was carried out, as well as qualitative comparisons with other interval libraries, in order to demonstrate the functionality and optimality of implemented interval classes.

On unique solvability of the piecewise linear equation system

In this article, we take the piecewise linear equation system \(x-W|x|=b\), which is also known by absolute value equation, where \(W\in {\mathbb R}^ {n\times n}\), \(b\in {\mathbb R}^{n}\) are given and to undetermined the value of \(x\in {\mathbb R}^{n}\). The absolute value equation (AVE) has many applications in various fields of mathematics like bi-matrix games, linear interval systems, linear complementarity problems (LCP) etc. By the equivalence relation of AVE with LCP, some necessary and sufficient conditions proved the existence and unique solvability of the AVE. Some examples are also provided to highlight the current singular value conditions for a unique solution that may revise in the future. (small corrections operated in the pdf file on January 7, 2023)

Type‐2 fuzzy linear system of equations with application in static problem of structures

In this paper, a new method is proposed to solve type‐2 fuzzy linear system of equations by converting the system into three interval linear systems and one crisp linear system of equations. In this work, three possible cases of type‐2 fuzzy linear system is considered depending upon uncertainty (as type‐2 fuzzy number) in right‐hand side vector , in co‐efficient matrix , and both in co‐efficient matrix as well as right‐hand side vector . Triangular perfect quasi type‐2 fuzzy numbers are taken to propose the new method. The main significance of the proposed method is that it uses the arithmetic of interval theory, which makes this method easier to handle. Five numerical examples are solved to show the validation of the proposed technique. Further, two application problems related to 6‐bar truss and rectangular sheet are encountered in type‐2 fuzzy environment. Comparison among the results obtained by the present method and previously achieved results is also made in this work, which shows that the results are close enough and, in some cases, better bounds for the solution are obtained.

New Method for Solving Fuzzy LR Interval Linear Systems Using Least Squares Models

Recently, Ghanbari and Mahdavi-Amiri [2015, Soft Computing] have proposed a model to characterize exact solutions and presented an algorithm to compute approximate solutions. Here, inspired by Ghanbari and Mahdavi-Amiri, we show that solving fuzzy LR interval linear systems is equivalent to solving fuzzy LR linear systems (FLRLSs). Then we develop some necessary and sufficient conditions for the solvability of FLRLSs. We then provide a new general concept for an approximate solution when an FLRLS lacks a solution. To compute an approximate solution, we propose an algorithm using a least squares method. Finally, we show the appropriateness of our proposed approximate solution recently introduced numerically.

The Optimal Tolerance Solution of the Basic Interval Linear Equation and the Explanation of the Lodwick’s Anomaly

Determining the tolerance solution (TS) of interval linear systems (ILSs) has been a task under consideration for many years. It seems, however, that this task has not been fully and unequivocally solved. This is evidenced by the multiplicity of proposed methods (which sometimes provide different results), the existence of many questions, and the emergence of strange solutions provided by, for example, Lodwick’s interval equation anomaly (LIEA). The problem of solving ILEs is probably more difficult than we think. The article presents a new method of ILSs solving, but it is limited to the simplest, basic equation [a̲,a¯]X=[b̲,b¯], which is an element of all more complex forms of ILSs. The method finds the optimal TS for this equation by using multidimensional interval arithmetic (MIA). According to the authors’ knowledge, this is a new method and it will allow researchers to solve more complex forms of ILSs and various types of nonlinear interval equations. It can also be used to solve fuzzy linear systems (FLSs). The paper presents several examples of the method applications (including one real-life case).

Tolerance-localized and control-localized solutions of interval linear equations system and their application to course assignment problem

There are many approaches to solving a system of interval linear equations. Each of them has different semantics based on the context of the system. We define here two other types of solutions called ‘tolerance-localized’ and ‘control-localized’ solutions. A tolerance-localized solution means that it provides either tolerance, L-localized, or R-localized behavior in each equation of the system. The similar argument serves for a control-localized solution. Theorems are proved to obtain the characterizations of the new solutions. Both sets of tolerance-localized and control-localized solutions could be represented by a system of integer linear equations. An example of application to the course assignment problem is presented, where the teaching workload restriction has been considered as tolerance-localized or control-localized constraints.

Analytical descriptions of quantifier solutions to interval linear systems of relations

We study systems of relations of the form [Formula: see text], where [Formula: see text] is a vector of binary relations with the components “[Formula: see text]”, “[Formula: see text]”, and “[Formula: see text]”, while the parameters (elements of the matrix A and right-hand side vector b) are uncertain and can take values from prescribed intervals. What is considered to be the set of its solutions depends on which logical quantifier is associated with each interval-valued parameter and what is the order of the quantifier prefixes for specific parameters. For solution sets that correspond to the quantifier prefix of a general form, we present equivalent quantifier-free analytical descriptions in the classical interval arithmetic, in Kaucher complete interval arithmetic and in the usual real arithmetic.

Combined method for correction interval systems of linear algebraic equations

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.

Method for finding a solution to a linear nonstationary interval ОDЕ system

The article analyses a linear nonstationary interval system of ordinary differential equations so that the elements of the matrix of the system are the intervals with the known lower and upper bounds. The system is defined on the known finite time interval. It is required to construct a trajectory, which brings this system from the given initial position to the given final state. The original problem is reduced to finding a solution of the differential inclusion of a special form with the fixed right endpoint. With the help of support functions, this problem is reduced to minimizing a functional in the space of piecewise continuous functions. Under a natural additional assumption, this functional is Gateaux differentiable. For the functional, Gateaux gradient is found, necessary and sufficient conditions for the minimum are obtained. Оn the basis of these conditions, the method of the steepest descent is applied to the original problem. Some examples illustrate the constructed algorithm realization.

Solution types of two-sided interval linear system and their application on interval linear programming problems

Two-sided interval linear systems contain variables on both sides of equations or inequalities. We propose six types of solutions (weak, strong, tolerance, control, left-localized and right-localized solutions) to the system of equations together with four types of solutions (weak, strong, tolerance and control solutions) to the system of inequalities. We prove the necessary and sufficient conditions for checking their solvabilities in the form of systems of linear inequalities depending on the sign of variables. As a result, an interval linear programming problem with nonnegative variables and two-sided interval linear constraints can be solved by a standard linear programming approach.

Digital Superstabilization of a Switched Interval Linear System

We solve the problem of constructing a digital controller superstabilizing a continuous-time switched system whose operation modes are interval linear systems. The proposed approach to stabilization includes constructing a continuous/discrete-time closed-loop system with a digital controller, passing to its discrete-time model, and constructing a controller that simultaneously superstabilizes all the interval discrete-time systems in the finite family formed by the modes of the discrete-time model.