On a second order method for the simultaneous inclusion of polynomial complex zeros in rectangular arithmetic

Starting from separated rectangles in the complex plane which contain polynomial complex zeros, an iterative method of second order for the simultaneous inclusion of these zeros is formulated in rectangular arithmetic. The convergence and a condition for convergence are considered. Applying Gauss-Seidel approach to the proposed method, two accelerated interval methods are formulated. TheR-order of convergence of these methods is determined. An analysis of the convergence order is given in the presence of rounding errors. The presented methods are illustrated numerically in examples of polynomial equations.     

Weierstrass-like Methods with Corrections for the Inclusion of Polynomial Zeros

In this paper, we present iterative methods of Weierstress’ type for the simultaneous inclusion of all simple zeros of a polynomial. The main advantage of the proposed methods is the increase of the convergence rate attained by applying suitable correction terms. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis for the total-step and the single-step methods. Numerical examples are given.     

Inclusion Isotonicity of Convex–Concave Extensions for Polynomials Based on Bernstein Expansion

In this paper the expansion of a polynomial into Bernstein polynomials over an interval I is considered. The convex hull of the control points associated with the coefficients of this expansion encloses the graph of the polynomial over I. By a simple proof it is shown that this convex hull is inclusion isotonic, i.e. if one shrinks I then the convex hull of the control points on the smaller interval is contained in the convex hull of the control points on I. From this property it follows that the so-called Bernstein form is inclusion isotone, which was shown by a longish proof in 1995 in this journal by Hong and Stahl. Inclusion isotonicity also holds for multivariate polynomials on boxes. Examples are presented which document that two simpler enclosures based on only a few control points are in general not inclusion isotonic. Received September 12, 2002; revised February 5, 2003 Published online: April 7, 2003     

On the convergence condition of generalized root iterations for the inclusion of polynomial zeros

Using a fixed point relation based on the logarithmic derivative of the k-th order of an algebraic polynomial and the definition of the k-th root of a disk, a family of interval methods for the simultaneous inclusion of complex zeros in circular complex arithmetic was established by Petković [M.S. Petković, On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic, Computing 27 (1981) 37–55]. In this paper we give computationally verifiable initial conditions that guarantee the convergence of this parallel family of inclusion methods. These conditions are significantly relaxed compared to the previously stated initial conditions presented in literature.     

Slope Methods of Higher Order for the Inclusion of Complex Roots of Polynomials

We consider iterative methods for the inclusion of a simple complex root of a polynomial based on a slope in circular complex arithmetic. Combining the Newton method with the interval slope, two new algorithms of higher order in circular complex arithmetic are developed. The convergence analysis and numerical examples are included.     

The root and Bell’s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial,Part II: Round-off error analysis by use of interval arithmetic

In Part I (Ikhile, 2008) [4], it was established that the root and Bell’s disk/point iteration methods with or without correction term are of the same asymptotic error propagation characteristics in the simultaneous determination of the zeros of a polynomial. This concluding part of the investigation is a study in round-offs, its propagation and its effects on convergence employing interval arithmetic means. The purpose is to consequently draw attention on the effects of round-off errors introduced from the point arithmetic part, on the rate of convergence of the generalized root and Bell’s simultaneous interval iteration algorithms and its enhanced modifications introduced in Part I for the numerical inclusion of all the zeros of a polynomial simultaneously. The motivation for studying the effects of round-off error propagation comes from the fact that the readily available computing devices at the moment are limited in precision, more so that accuracy expected from some programming or computing environments or from these numerical methods are or can be machine dependent. In fact, a part of the finding is that round-off propagation effects beyond a certain controllable order induces overwhelmingly delayed or even a severely retarded convergence speed which manifest glaringly as poor accuracy of these interval iteration methods in the computation of the zeros of a polynomial simultaneously. However, in this present consideration and even in the presence of overwhelming influence of round-offs, we give conditions under which convergence is still possible and derive the error/round-off relations along with the order/$R$-order of convergence of these methods with the results extended to similar interval iteration methods for computing the zeros of a polynomial simultaneously, especially to Bell’s interval methods for refinement of zeros that form a cluster. Our findings are instructive and quite revealing and supported by evidence from numerical experiments. The analysis is preferred in circular interval arithmetic.     

A family of simultaneous methods for the determination of polynomial complex zeros

Using the development of a rational function by elementary fractions, a family of methods for the simultaneous determination of polynomial complex zeros is derived. All the methods of the family are cubically convergent for simple zeros. The known simultaneous procedures of the third order are included. The presented class of iteration functions is suitable for the parallel inclusion of polynomial complex zeros by circular regions. The family of methods, defined in complex circular arithmetic, gives a new interval method with cubic convergence. Numerical example is given.     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

Halley-Like Method with Corrections for the Inclusion of Polynomial Zeros

In this paper we present iteration methods of Halley's type for the simultaneous inclusion of all zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis for the total-step and the single-step methods with Newton's corrections. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. A numerical example is given. Received: June 23, 1998     

Schröder-like algorithms for multiple complex zeros of a polynomial

Using the iterative method of Newton's type in circular arithmetic, introduced in [14], a new iterative method for finding a multiple complex zero of a polynomial is derived. This method can be regarded as a version of classical Schröder's method. Initial conditions which guarantee a safe convergence of the proposed method are stated. The increase of the computational efficiency is achieved by a combination of the complex approximation methods of Schröder's type with some interval methods. The presented algorithms are analysed in view of their efficiency and illustrated numerically in the example of a polynomial equation.     

Spectral Galerkin Discretization for Hydrodynamic Stability Problems

A spectral Galerkin discretization for calculating the eigenvalues of the Orr-Sommerfeld equation is presented. The matrices of the resulting generalized eigenvalue problem are sparse. A convergence analysis of the method is presented which indicates that a) no spurious eigenvalues occur and b) reliable results can only be expected under the assumption of scale resolution, i.e., that Re/p ² is small; here Re is the Reynolds number and p is the spectral order. Numerical experiments support that the assumption of scale resolution is necessary in order to obtain reliable results. Exponential convergence of the method is shown theoretically and observed numerically. Received November 11, 1998; revised March 1, 2000     

Polynomial root finding by means of continuation

In this paper we deal with the problem of locating all the zeros of a given polynomialp(x) and approximating them to any degree of precision: by combining classical iterative methods with homotopy path tracking techniques, we introduce a new algorithm for polynomial root finding, prove its convergence and estimate its computational cost.     

Ostrowski-Like Method with Corrections for the Inclusion of Polynomial Zeros

In this paper we construct iterative methods of Ostrowski's type for the simultaneous inclusion of all zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods with Newton and Halley's corrections. The case of multiple zeros is also considered. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. Numerical examples and an analysis of computational efficiency are given.     

Numerical Stability of Path Tracing in Polyhedral Homotopy Continuation Methods

The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotopy continuation parameter s and various scaling strategies to enhance the numerical stability. Advantages of employing the new homotopy parameter s are discussed. Numerical results are included to illustrate the improved performance of the presented techniques.A considerable part of this work was conducted while this author was visiting Tokyo Institute of Technology. Research supported by Kosef R004-000-2001-00200.     

The convergence of Euler-like method for the simultaneous inclusion of polynomial zeros

In this paper, we give the convergence analysis of the Euler-like iterative method for the simultaneous inclusion of all simple real or complex zeros of a polynomial. The established initial conditions provide the safe convergence of the considered method and the fourth order of convergence. These conditions are computationally verifiable, which is of practical importance. A procedure for the choice of initial inclusion disks is also given.     

Empirical Investigation of the Convergence Speed of Inclusion Functions in a Global Optimization Context

This paper deals with the empirical convergence speed of inclusion functions applied in interval methods for global optimization. According to our experience the natural interval extension of a given function can be as good as a usual quadratically convergent inclusion function, and although centered forms are in general only of second-order, they can perform as one of larger convergence order. These facts indicate that the theoretical convergence order should not be the only indicator of the quality of an inclusion function, it would be better to know which inclusion function can be used most efficiently in concrete instances. For this reason we have investigated the empirical convergence speed of the usual inclusion functions on some test functions.This work has been supported by the Grants OTKA T 034350 and T 032118, OMFB D–30/2000, and OMFB E–24/2001.The authors are grateful for the anonymous referees for their suggestions.     

Stormer-Numerov HDG Methods for Acoustic Waves

We introduce and analyze the first energy-conservative hybridizable discontinuous Galerkin method for the semidiscretization in space of the acoustic wave equation. We prove optimal convergence and superconvergence estimates for the semidiscrete method. We then introduce a two-step fourth-order-in-time Stormer-Numerov discretization and prove energy conservation and convergence estimates for the fully discrete method. In particular, we show that by using polynomial approximations of degree two, convergence of order four is obtained. Numerical experiments verifying that our theoretical orders of convergence are sharp are presented. We also show experiments comparing the method with dissipative methods of the same order.     

On the Suboptimality of the <Emphasis Type="Italic">p</Emphasis>-Version Interior Penalty Discontinuous Galerkin Method

We address the question of the rates of convergence of the p-version interior penalty discontinuous Galerkin method (p-IPDG) for second order elliptic problems with non-homogeneous Dirichlet boundary conditions. It is known that the p-IPDG method admits slightly suboptimal a-priori bounds with respect to the polynomial degree (in the Hilbertian Sobolev space setting). An example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated in practice through numerical experiments is presented. Moreover, the performance of p-IPDG on the related problem of p-approximation of corner singularities is assessed both theoretically and numerically, witnessing an almost doubling of the convergence rate of the p-IPDG method.     

Neural Network Output Optimization Using Interval Analysis

The problem of output optimization within a specified input space of neural networks (NNs) with fixed weights is discussed in this paper. The problem is (highly) nonlinear when nonlinear activation functions are used. This global optimization problem is encountered in the reinforcement learning (RL) community. Interval analysis is applied to guarantee that all solutions are found to any degree of accuracy with guaranteed bounds. The major drawbacks of interval analysis, i.e., dependency effect and high-computational load, are both present for the problem of NN output optimization. Taylor models (TMs) are introduced to reduce these drawbacks. They have excellent convergence properties for small intervals. However, the dependency effect still remains and is even made worse when evaluating large input domains. As an alternative to TMs, a different form of polynomial inclusion functions, called the polynomial set (PS) method, is introduced. This new method has the property that the bounds on the network output are tighter or at least equal to those obtained through standard interval arithmetic (IA). Experiments show that the PS method outperforms the other methods for the NN output optimization problem.     

Descent Iterations for Improving Approximate Eigenpairs of Polynomial Eigenvalue Problems with General Complex Matrices

In this paper, we consider descent iterations with line search for improving an approximate eigenvalue and a corresponding approximate eigenvector of polynomial eigenvalue problems with general complex matrices, where an approximate eigenpair was obtained by some method. The polynomial eigenvalue problem is written as a system of complex nonlinear equations with nondifferentiable normalized condition. Convergence theorems for iterations are established. Finally, some numerical examples are presented to demonstrate the effectiveness of the iterative methods. Received April 9, 2001; revised October 2, 2001 Published online February 18, 2002