Interval System Of Linear Equations Research Articles

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.

Strong solvability of restricted interval systems and its applications in quadratic and geometric programming

We consider interval systems of linear equations and inequalities with a restriction to some a priori given set. We focus on a characterization of strong solvability, that is, solvability for each realization of interval values, and we compare this with an existence of a strong solution defined analogously. The motivation comes from the area of interval-valued optimization problems, where strong solvability means guaranteed feasibility of any realization of the problem. Strong solvability with strict inequalities implies the robust Slater condition, which ensures that standard optimality conditions can be used. We apply the issues particularly in two optimization classes, convex quadratic programming with quadratic constraints and posynomial geometric programming. For the former, we also utilize the presented result to improve a characterization of the worst case optimal value. Eventually, we state several open problems that emerged while deriving the results.

A realistic tolerant solution of a system of interval linear equations with the use of multidimensional interval arithmetic

A realistic tolerant solution of a system of interval linear equations with the use of multidimensional interval arithmetic

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.

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.

Solving System of Interval Linear Equations by Gauss Jordon Method Using Generalized Interval Arithmetic

We introduce an interval version of Gauss Jordon method for solving system of interval linear equations involving generalized interval numbers by applying the generalized interval arithmetic operations. The given interval system is solved without converting to equivalent crisp linear systems. Also numerical example is given to support the proposed algorithm.

Sign function and ANN based pole placement for computing interval controls

This paper deals with estimation of interval controls for interval Linear Time Invariant (LTI) plants using pole placement technique. Generally, the parameters associated with LTI plant system are assumed to be crisp values, but due to the occurrence of error in experimentation, measurement or due to maintenance induced errors, the parameters are imprecise. As such, the LTI problem reduces to interval LTI. The pole placement problem is then reduced to interval algebraic equation viz. interval Diophantine equation using proper transfer function. Further, the interval Diophantine equation is transformed to Interval System of Linear Equations (ISLE) viz. interval Sylvester matrix equation. Initially, using interval arithmetic an interval algebraic approach has been applied to solve the ISLE for computing non-negative or non-positive controls. Then, another approach using Artificial Neural Network (ANN) is proposed for computing the interval controls. Further, algorithms based on sign function and ANN procedures have been discussed. Finally, the efficiency of the proposed algorithms has been verified using different order interval plants.

Universal algebraic solution of the interval linear system: AX + BX = Y

The paper shows how to use the multidimensional interval RDM arithmetic to solve an interval system of linear equations. The main problems that arise when solving this type of equations using the standard interval arithmetic are discussed and the most important flaws of this type of solutions are listed. Next, an example is shown to demonstrate how to correctly solve systems of interval equations by using multidimensional interval RDM arithmetic.

Solving interval linear least squares problems by PPS-methods

In our work, we consider the linear least squares problem for m × n-systems of linear equations Ax = b, m ≥ n, such that the matrix A and right-hand side vector b can vary within an interval m × n-matrix A and an interval m-vector b, respectively. We have to compute, with a prescribed accuracy, outer coordinate-wise estimates of the set of all least squares solutions to Ax = b for A ∈A and b ∈b. Our article is devoted to the development of the so-called PPS-methods (based on partitioning of the parameter set) to solve the above problem. We reduce the normal equation system, associated with the linear lest squares problem, to a special extended matrix form and produce a symmetric interval system of linear equations that is equivalent to the interval least squares problem under solution. To solve such symmetric system, we propose a new construction of PPS-methods, called ILSQ-PPS, which estimates the enclosure of the solution set with practical efficiency. To demonstrate the capabilities of the ILSQ-PPS-method, we present a number of numerical tests and compare their results with those obtained by other methods.

ANN Based Solution of Uncertain Linear Systems of Equations

Linear systems of equations have many applications in the area of engineering sciences, mathematics, operations research and statistics. It is worth mentioning that the coefficient matrix of the linear systems of equations may not be always crisp due to various uncertainties. These uncertainties may be in the form of interval. Likewise, the solution set and right hand side vector may also be in interval. In this respect, a fully interval linear system of equations \( (\tilde{P}\tilde{z} = \tilde{q}) \) is one where the coefficient matrix, the unknown vector and the right hand side vector all are in the form of interval. Although various authors proposed different methods to handle the fully linear systems of equations but those are sometimes problem specific etc. As such, in this paper \( n \times n \) fully interval linear systems of equations has been solved based on artificial neural network (ANN) model. In this regard, step by step algorithm has been included. Further, a convergence theorem has also been discussed for choosing suitable learning parameter. Few numerical examples and an application problem related to electrical circuit have been solved using the proposed method. Detail procedure has been discussed with numerical results to show the efficacy and powerfulness of the method.

Relations between various methods for solving linear interval and parametric equations

In the first part of the paper, we consider standard systems of linear interval equations and we focus particularly on two solution methods, the Bauer–Skeel method and the Hansen–Bliek–Rohn method. We show relations between these two methods and between various modifications that are based on preconditioning of the system and on the residual form. We compare as well the quality of the bounds produced by the different variants and we show that for some variants, the Bauer–Skeel bounds naturally arise from other approaches such as Krawczyk of Jacobi iterations, too. In the second part of the paper, we consider interval parametric linear systems with affine-linear dependencies. We also investigate various forms of enclosures, and we not only compare them with each other, but we also show relations to some already known methods. As a consequence, we come up with novel and interesting relations between several algorithms.

Verification of Solutions of Equations Governing Fluid Flows

We have carried out the verifications of the solutions for the equations governing fluid flows. The equations solved are the similar equation of the boundary layer flow and the two-dimensional Navier-Stokes equations in the formulation of the stream function and the vorticity. The methods of the verification are the interval analysis and the Krawczyk operator based on the fixed point theorem. A modified Gaussian elimination method is introduced to solve the system of linear interval equations. The Krawczyk operator presents the results with a high accuracy. The modified elimination method gives the results with an accuracy of one or two order(s) of magnitude higher than those obtained by the original interval analysis.

New characterizations of tolerance-control and localized solutions to interval system of linear equations

In this paper, we propose new characterizations of tolerance-control and localized solutions to interval system of linear equations and apply our proposed results to represent their solution sets in terms of level sets.

Solving interval linear programming problems with equality constraints using extended interval enclosure solutions

This paper focuses on solving systems of interval linear equations and interval linear programming in a computationally efficient way. Since the computational complexity of most interval enclosure numerical methods is often prohibitive, a procedure to obtain a relaxation of the interval enclosure solution that is computationally tractable is proposed. We show that our approach unifies the four standard interval solutions—the weak, strong, control and tolerance solutions. The interval linear system methods require $$n\cdot 2^{n}$$ linear solutions. However, in the case of linear programming problems, we show that this requires just two optimization problem of the size of the problem itself. Numerical examples illustrate our results.

A New Method for Analyzing Three-phase Faults under Data Uncertainties

Fault analysis uses a given set of data regarding generations and line branches. The main objective is to calculate voltages and currents as a specific fault location is considered. As the input data are imprecise, this paper incorporates the interval mathematics into fault analysis problem in order to take into account data uncertainties. Therefore, this paper proposes a new method for modeling an interval three-phase fault analysis. At each iteration, the resulting interval linear system of equations is solved using Krawczyk method. The proposed method is implemented in the Matlab environment using the Intlab toolbox. The 14-bus test system is used. Results are later compared with the Monte Carlo Simulation.

Solving fully interval linear systems of equations using tolerable solution criteria

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.

Formal solution of an interval system of linear equations with an application in static responses of structures with interval forces

• Modelling uncertainties by interval methods. • Two approaches to solving interval linear equations. • Analysis of uncertain static responses of structures. We are concerned with the so called formal solution of an interval system of linear equations. We focus on the case where the coefficient matrix is deterministic (real) and the right-hand side is an interval vector. We show that the set of formal solutions represents a convex polyhedral set. We propose new properties of the formal solution related to its existence, uniqueness and robustness. As particular classes of problems we investigate also the situation where the coefficient matrix is an M-matrix or H-matrix. Example problems related to the structures, such as 6-bar truss and a rectangular sheet, are solved to illustrate the computational aspects of the methods.

A Sign Function Approach to Solve Algebraically Interval System of Linear Equations for Nonnegative Solutions

A Sign Function Approach to Solve Algebraically Interval System of Linear Equations for Nonnegative Solutions

Transformations of interval linear systems of equations and inequalities

Very recently, new results on transformation of interval linear systems and on generalizations of the Farkas lemma to interval systems appeared in the literature. They are by far not obvious since the standard transformations on linear systems are not easily adapted to interval system due to the dependency problem. The aim of this paper was to come up with new possible transformations and to extend the results to more general classes of AE interval systems and to linear parametric systems. We also show that the transformations can help in simplifying the proofs of some characterization theorems.

New characterizations for the solution set to interval linear systems of equations

New characterizations of the points from the solution set to interval linear systems of equations are proposed, alternatives to the well-known result by Oettli and Prager. We also introduce recognizing functionals of the solution sets that determine, at a given point, aggregated quantitative measures of compatibility (consistency) between the interval data of the system.