On some interval methods for algebraic, exponential and trigonometric polynomials (original) (raw)
On an Interval Method for the Inclusion of One Polynomial Zero
2013
In this paper we construct a new interval method for the inclusion of one simple or multiple complex polynomial zero in circular complex arithmetic. We present the convergence analysis starting from the computationally verifiable initial condition that guarantees the convergence of this inclusion method. We also give two numerical examples in order to demonstrate convergence behavior of the proposed method.
Symmetry
This paper describes the extended method of solving real polynomial zeros problems using the single-step method, namely, the interval trio midpoint symmetric single-step (ITMSS) method, which updates the midpoint at each forward-backward-forward step. The proposed algorithm will constantly update the value of the midpoint of each interval of the previous roots before entering the preceding steps; hence, it always generate intervals that decrease toward the polynomial zeros. Theoretically, the proposed method possesses a superior rate of convergence at 16, while the existing methods are known to have, at most, 9. To validate its efficiency, we perform numerical experiments on 52 polynomials, and the results are presented, using performance profiles. The numerical results indicate that the proposed method surpasses the other three methods by fine-tuning the midpoint, which reduces the final interval width upon convergence with fewer iterations.
International Journal of Mathematical Analysis
A new modified interval symmetric single-step procedure ISS1-5D which is the extension from the previous ISS1 is proposed. In procedure ISS1 we define informational efficiency of a method as the higher R-order of convergence evaluation. The procedure is tested on five test polynomials and the results are obtained using MATLAB 2007 software in association with IntLab V5.5 toolbox to record the CPU times and the number of iterations.
On the efficiency of some combined methods for polynomial complex zeros
Journal of Computational and Applied Mathematics, 1990
Interval methods for the simultaneous inclusion of polynomial zeros produce the approximations that contain the exact zeros providing not only error bounds automatically but also take into account rounding errors without altering the fundamental structure of the interval formula. However, at present, the computational costs of most interval methods are still great, in general. In this paper several effective algorithms which preserve the inclusion property concerning the complex zeros and which have a high computational efficiency are constructed. These algorithms combine the efficiency of ordinary floating-point iterations with the accuracy control that may be obtained by the iterations in interval arithmetic. Several examples are included to illustrate the efficiency and some advantages of the proposed combined methods.
Some improved inclusion methods for polynomial roots with Weierstrass' corrections
Computers & Mathematics with Applications, 1993
One decade ago, the third order method without derivatives for the simultaneous inclusion of simple zeros of a polynomial was proposed in [1]. Following Nourein's idea [2], some modifications of this method with the increased convergence are proposed. The acceleration of convergence is attained by using Weierstrass' corrections without additional calculations, which provides a high computational efficiency of the modified methods. It is proved that their R-orders of convergence are asymptotically greater than 3.5. The presented interval methods are realized in circular complex arithmetic.
Interval Root Finding and Interval Polynomials: Methods and Applications in Science and Engineering
Polynomial Paradigms: Trends and Applications in Science and Engineering, 2022
Polynomial systems are at the heart of a wide range of scientific fields. Multiple forms of polynomials are often involved in attempts to reduce currently open or hard problems to problems already solved. Because of their ever-increasing importance in many practical applications, polynomials span the entire spectrum of various disciplines of mathematics, science and engineering. In practice, a recurring problem is to determine the roots of polynomials under parametric uncertainty or imprecision. Interval mathematics is a subtle body of principles and methods for manipulating systems involving quantifiable uncertainties. Interval methods have proven successful and reliable for computing guaranteed enclosures of roots of polynomials under parametric uncertainty. This "self-validating" feature of interval analysis makes it competitive and preferable to the ordinary approximation methods in many practical applications. In this connexion, by the pursuit of the quest of reliable knowledge amidst uncertainty, the raison d'être of the present work is a systematic investigation of uncertain polynomials. We begin by laying out a rigorous algebraic foundation for ordinary polynomials and some of their generalizations, and then, we go further to define two new generalizations of ordinary and trigonometric polynomials, namely, generalized n-adic polynomials and n-adic S-polynomials. These two generalizations are so framed as to provide a rigorous mathematical foundation for interval and set-valued polynomials. Next, we formalize the theories of interval algebra and interval polynomials and show how to obtain guaranteed interval enclosures of families of generalized real polynomials. Afterwards, we describe how to use infinite polynomials with Taylor models to compute finer interval enclosures. Moreover, we examine some of the interval methods for root finding, with special attention is paid to the interval branch and prune method. Finally, we put forward a more refined interval branch and prune algorithm. In order to illustrate the significance of our algorithm, we compute a finer interval enclosure for an example introduced by Eldon Hansen and G. William Walster in 2003. We show that our interval enclosure is sharper than the optimal result given by Hansen and Walster. The algorithms of this work are coded using the software package, InCLosure. Many numerical examples are given, showing that all the results are guaranteed interval enclosures.
On iteration methods without derivatives for the simultaneous determination of polynomial zeros
Journal of Computational and Applied Mathematics, 1993
Carstensen, C. and M.S. PetkoviC, On iteration methods without derivatives for the simultaneous determination of polynomial zeros, Journal of Computational and Applied Mathematics 45 (1993) 251-266. Several algorithms for simultaneously approximating simple complex zeros of a polynomial are presented. These algorithms use Weierstrass' corrections and do not require any polynomial derivatives. It is shown that Nourein's method is, actually, regula falsi for Weierstrass' corrections. Convergence analysis and computational efficiency are given for the considered methods in complex and circular arithmetic. Special attention is paid to hybrid methods that combine the efficiency of floating-point arithmetic and the inclusion property of interval arithmetic.
Symmetry
In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method.