Interval Newton Method Research Articles

Solving Intuitionistic Fuzzy Unconstrained Optimization Problems Using Interval Newton’s Method

This research work studies the optimization of intuitionistic fuzzy valued functions in unconstrained problems. The Interval Newton's Method using an Intuitionistic approach addresses both single and multivariable optimization problems. The study incorporates a mathematical comprehension of interval intuitionistic fuzzy valued problems, as well as real-world examples to demonstrate their effectiveness. Furthermore, MATLAB code is provided to demonstrate the implementation of the Interval Newton's Method.

Feasibility Verification and Upper Bound Computation in Global Minimization Using Approximate Active Index Sets

We propose a new upper bounding procedure for global minimization problems with continuous variables and possibly nonconvex inequality and equality constraints. Upper bounds are crucial for standard termination criteria of spatial branch-and-bound (SBB) algorithms to ensure that they can enclose globally minimal values sufficiently well. However, whereas for most lower bounding procedures from the literature, convergence on smaller boxes is established, this does not hold for several methods to compute upper bounds even though they often perform well in practice. In contrast, our emphasis is on the convergence. We present a new approach to verify the existence of feasible points on boxes, on which upper bounds can then be determined. To this end, we resort to existing convergent feasibility verification approaches for purely equality and box constrained problems. By considering carefully designed modifications of subproblems based on the approximation of active index sets, we enhance such methods to problems with additional inequality constraints. We prove that our new upper bounding procedure finds sufficiently good upper bounds so that termination of SBB algorithms is guaranteed after a finite number of iterations. Our theoretical findings are illustrated by computational results on a large number of standard test problems. These results show that compared with interval Newton methods from the literature, our proposed method is more successful in feasibility verification for both, a full SBB implementation (42 instead of 26 test problems) and exhaustive sequences of boxes around known feasible points (120 instead of 29 test problems). History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0162 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0162 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A Special Interval Newton Step for Solving Nonlinear Equations

The problem of finding the zeros of a nonlinear equation has been extensively and thoroughly studied. Point methods constitute a significant and extensive category of techniques, allowing for the efficient finding of zeros with arbitrary precision under specific conditions. However, the limitation of these methods is that they typically yield a single zero. An alternative approach employs Interval Analysis, leveraging its properties to provide reliable and with certainty inclusions of all zeros within a given search interval. Interval methods, such as the Interval Newton method, exhibit quadratic convergence to the corresponding inclusions when monotonicity and simple zeros exist. Nonetheless, there exist pathological cases, like the existence of multiple zeros, where the obtained inclusions cannot be bounded with arbitrary precision, necessitating the adoption of bisection schemes to refine the search interval. These schemes not only increase computational time and cost but also result in a higher number of enclosures, enclosing sometimes the same zero more than once. The main objective of this work is to enhance the applicability of Interval Newton method in cases where no efficient alternative are available. Thus, in this paper, the Interval Newton method is studied and an adjusted perturbation technique is proposed to address the cases where multiple zeros exist. In particular, the given function is vertically shifted. Then, the Interval Newton operator is applied once to this shifted function. The resulting enclosures are then used to efficiently partition the search interval. The successful application of the Interval Newton method is expected to improve overall performance and reduce reliance on bisection schemes. Experimental results on a set of problems demonstrate the effectiveness of the proposed technique.

Rigorous numerical study of the Colpitts oscillator with an exponential nonlinearity.

The dynamics of the Colpitts oscillator with an exponential nonlinearity is investigated using rigorous interval arithmetic based tools. The existence of various types of periodic attractors is proved using the interval Newton method. The main results involve the chaotic case for which a trapping region for the associated return map is constructed and a rigorous lower bound for the value of the topological entropy is computed, thus proving that the system is chaotic in the topological sense. A systematic search for unstable periodic orbits embedded in the chaotic attractor is carried out and the results are used to obtain an accurate approximation of the topological entropy of the system.

Accuracy Bound for Positioning the 5R Planar Parallel Manipulator Subjected to Uncertainties in Dimensions and Actuations

This article is aimed to determine the accuracy bound for positioning the 5R planar parallel manipulator under the uncertainties in its dimensions and actuation. The accuracy bound is investigated numerically using the interval analysis and multi-dimensional bisection method (MDBM). These numerical methods can give multiple solutions within the workspace of the manipulator. Then, the exact solution is utilized for comparison with the numerical solutions. It is shown that the interval analysis via the interval Newton method gives the guaranteed bounds for positioning accuracy. Meanwhile, the MDBM and the exact solution coupling with the Monte Carlo simulation produce a scattered random value in a parallelogram shape inside the solution yielded by the interval Newton method. Overall, those three methods give nearly a good agreement in terms of that accuracy, but they are different at computation time.

A branch-and-prune method to solve closure equations in dual quaternions

Using dual quaternions, the closure equations of a kinematic loop can be expressed as a system of multiaffine equations. In this paper, this property is leveraged to introduce a branch-and-prune method specially tailored for solving such systems of equations. The new method is objectively simpler (in the sense that it is easier to understand and to implement) than previous approaches relying on general techniques such as interval Newton methods or methods based on Bernstein polynomials or linear relaxations. Moreover, it relies on two basic operations — linear interpolation and projection onto coordinate planes— that can be efficiently computed. The generality of the proposed method is evaluated on position analysis problems with 0-, 1-, and 2-dimensional solution sets, including the inverse kinematics of serial robots and the forward kinematics of parallel ones. The results obtained on these problems show that the efficiency of the method compares favorably to state-of-the-art alternatives.

A globally convergent interval Newton's method for computing and bounding real roots of a function with one variable

A globally convergent interval Newton's method for computing and bounding real roots of a function with one variable

Interval Arithmetic based Adaptive Filtering Technique for Removal of Noise in Audio Signal

Active noise cancellation is one of the fundamental problems in acoustic signal processing. The proposed work focuses on the enhancement of audio signal quality by cancelling the noise using interval analysis (arithmetic). An adaptive filters basically works on the concept of optimal weight calculations which is an optimization problem. This optimization problem can be more effectively solved using interval analysis. Interval analysis gives the boundary of the weight co officiants. Using interval Newton method, the weight co officiants are found. This algorithm is tested for noise cancellation of speech signal. The three adaptive filters algorithm used for comparison with the obtained results are Least Mean Square (LMS), Recursive Mean Square (RMS) filters and Kernel based filters. It is observed that the parameters mean square error is very less. The speed of convergence and signal to noise ratio is improved as compared to kernel methods. But processing time is very high and computational cost is doubled, as interval data includes infimum and supremum values. This algorithm can be used in noise cancelling headphones.

Interval extensions of the Halley method and its modified method for finding enclosures of roots of nonlinear equations

In this paper, interval extensions of the Halley method and its modified method for finding enclosures of roots of nonlinear equations are produced. Error analysis and convergence will be discussed. Also, these methods are compared together with the interval Newton method.

Existence proof of unimodal solutions of the Proudman–Johnson equation via interval analysis

The Proudman–Johnson equation is considered as a representative of the two-dimensional fluid flow. Its steady-states are computed and their existence and unimodality are proved rigorously via the interval analysis and the interval Newton method. By using both the multiple shooting method and multiple-precision arithmetic our verification succeeds up to the Reynolds number $$\le 5000$$ .

Fast and Rigorous Arbitrary-Precision Computation of Gauss--Legendre Quadrature Nodes and Weights

We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss--Legendre quadrature rules. Our focus is on making the evaluation practical for a wide range of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100000 bits. Our algorithm combines the summation by rectangular splitting of several types of expansions in terms of hypergeometric series with a fixed-point implementation of Bonnet's three-term recurrence relation. We then compute rigorous enclosures of the Gauss--Legendre nodes and weights using the interval Newton method. We provide rigorous error bounds for all steps of the algorithm. The approach is validated by an implementation in the Arb library, which achieves order-of-magnitude speedups over previous code for computing Gauss--Legendre rules suitable for precisions in the thousands of bits.

Numerical Study of Multiple Attractors in the Parallel Inductor–Capacitor–Memristor Circuit

Dynamical phenomena in the parallel inductor–capacitor–memristor circuit are studied numerically. A systematic search for coexisting attractors is carried out. The existence of multiple attractors is observed and bifurcation diagrams are constructed. Basins of attraction are computed. The coexistence of attractors is proved using interval analysis tools. The existence of periodic attractors is confirmed by applying the interval Newton method to prove the existence of stable periodic orbits of an associated return map. For numerically observed chaotic attractors the existence of attractors is proved by constructing trapping regions enclosing chaotic trajectories of the return map. The existence of topological chaos is proved using the method of covering relations.

Validated Computation of Heteroclinic Sets

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following an argument built on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method applied on short line segments. The definition of the heteroclinic sets involve compositions of the map, and we use a Lohner-type representation to overcome the accumulation of roundoff errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a posteriori error bounds on the truncation errors associated with the manifold approximations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map of $\mathbb{R}^3$.

Interval unions

This paper introduces interval union arithmetic, a new concept which extends the traditional interval arithmetic. Interval unions allow to manipulate sets of disjoint intervals and provide a natural way to represent the extended interval division. Considering interval unions lead to simplifications of the interval Newton method as well as of other algorithms for solving interval linear systems. This paper does not aim at describing the complete theory of interval union analysis, but rather at giving basic definitions and some fundamental properties, as well as showing theoretical and practical usefulness of interval unions in a few selected areas.

Numerical Method for Real Root Isolation of Semi-Algebraic System and Its Applications

In this paper, based on the homotopy continuation method and the interval Newton method, an efficient algorithm is introduced to isolate the real roots of semi-algebraic system. Tests on some random examples and a variety of problems including transcendental functions arising in many applications show that the new algorithm reduces the cost substantially compared with the traditional symbolic approaches. © 2016 American Scientific Publishers All rights reserved.

Optimal sixteenth order convergent method based on quasi-Hermite interpolation for computing roots.

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some other newly developed sixteenth-order methods. Interval Newton's method is also used for finding the enough accurate initial approximations. Some figures show the enclosure of finitely many zeroes of nonlinear equations in an interval. Basins of attractions show the effectiveness of the method.

New bifurcations in the simplest passive walking model

This paper uncovers several new stable periodic gaits in the simplest passive walking bipedal model proposed in the literature. It is demonstrated that the model has period-3 to period-7 gaits beside the period-1 gaits found by Garcia et al. By simulations, this paper shows that each of these new gaits leads to chaos via period-doubling bifurcation and loses its stability by cyclic-fold bifurcation. This interesting phenomenon suggests a series of new bifurcation scenarios that have not been observed before. To confirm the new gaits and their bifurcations, this paper presents computer assisted proofs on the existence and stability of each periodic gait and its period-doubling gaits with the interval Newton method. To verify that the routes indeed lead to chaos, computer-assisted proofs are also given by means of topological horseshoes theory.

Phase stability analysis using a modified affine arithmetic

Phase stability analysis is a crucial step in the determination of multiphase equilibrium. This analysis by the tangent plane distance (TPD) minimization is a well-known technique, as well as the difficulties in providing guarantees that the global minimum has been found. On this regard, interval methods are powerful tools since they provide such guarantees. In this work, an interval Newton method plus generalized bisection, based on a modified affine arithmetic, is used to reliably find all possible stationary points of the TPD function. Additionally, an improved convergence test is suggested as well as a special treatment for mole fraction weighted averages. Several mixtures with up to 5 components, including LLE island type ternary systems, were studied. Both activity coefficient models and cubic equations of state were considered. For all the cases tested, the proposed modified affine arithmetic method was superior to other interval-based methods.

A performance comparison of the zero-finding by extended interval Newton method for Peano monosplines

Peano monosplines are a special class of Peano kernels. Their zeros are essential for deriving multiple error bounds for numerical quadrature regarding the weight function w(x)≡1. As the performance of their zero-finding can influence the numerical results of Peano error constants, this article aims to give a concrete inspection of the improvements in deriving the zeros of Peano monosplines by the extended interval Newton method.

Interval Ostrowski-type methods with guaranteed convergence

In this paper, interval extensions of the classic Ostrowski and its modified method are introduced. Also, error analysis and convergence will be discussed. Some implemented examples with INTLAB are also included to illustrate the validity and applicability of the schemes. The results of proposed methods are compared with the results of the interval Newton method.