Aberth Method Research Articles

Ehrlich-type Methods with King's Correction for the Simultaneous Approximation of Polynomial Complex Zeros

There are many simultaneous iterative methods for approximating complex polynomial zeros, from more traditional numerical algorithms, such as the well-known third order Ehrlich–Aberth method, to the more recent ones. In this paper, we present a new family of combined iterative methods for the simultaneous determination of simple complex zeros of a polynomial, which uses the Ehrlich iteration and a correction based on King's family of iterative methods for nonlinear equations. The use of King's correction allows increasing the convergence order of the basic method from three to six. Some numerical examples are given to illustrate the convergence behaviour and effectiveness of the proposed sixth order Ehrlich-like family of combined iterative methods for the simultaneous approximation of simple complex polynomial zeros.

Efficient high degree polynomial root finding using GPU

Polynomials are mathematical algebraic structures that play a great role in science and engineering. Finding the roots of high degree polynomials is computationally demanding. In this paper, we present the results of a parallel implementation of the Ehrlich–Aberth algorithm for the root finding problem for high degree polynomials on GPUs using CUDA and on multi-core processors using OpenMP. The main result we achieved is to solve high degree polynomials (up to 1,000,000) efficiently. We also compare the Ehrlich–Aberth method and the Durand–Kerner one on both full and sparse polynomials. Accordingly, our second result is that the first method is much faster and more efficient. Last, but not least, an original proof of the convergence of the asynchronous implementation for the EA method is produced.

On the simultaneous refinement of the zeros of H-palindromic polynomials

In this paper we propose a variation of the Ehrlich–Aberth method for the simultaneous refinement of the zeros of H-palindromic polynomials.

Solving polynomial eigenvalue problems by means of the Ehrlich–Aberth method

Given the n×n matrix polynomial P(x)=∑i=0kPixi, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial detP(x), is treated in polynomial form rather than in matrix form by means of the Ehrlich–Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich–Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich–Aberth iteration to structured matrix polynomials is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.

Locating the Eigenvalues of Matrix Polynomials

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by Pellet [Bull. Sci. Math. (2), 5 (1881), pp. 393--395], some results from Bini [Numer. Algorithms, 13 (1996), pp. 179--200] based on the Newton polygon technique, and recent results from Gaubert and Sharify (see, in particular, [Tropical scaling of polynomial matrices, Lecture Notes in Control and Inform. Sci. 389, Springer, Berlin, 2009, pp. 291--303] and [Sharify, Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues, Ph.D. thesis, Ecole Polytechnique, Paris, 2011]). These extensions are applied to determine effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich--Aberth method. Numerical experiments that show the computational advantage of these results are presented.

The Ehrlich–Aberth method for palindromic matrix polynomials represented in the Dickson basis

An algorithm based on the Ehrlich–Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is introduced in order to exploit the symmetry of the roots by halving the total number of the required approximations. The rank structure properties of the linearization allow the design of a fast and numerically robust implementation of the root-finding iteration. Numerical experiments that confirm the effectiveness and the robustness of the approach are provided.

Point estimation of simultaneous methods for solving polynomial equations: A survey (II)

The construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root-finding algorithm is one of the most important problems in solving nonlinear equations. Smale's “point estimation theory” from 1981 was a great advance in this topic; it treats convergence conditions and the domain of convergence in solving an equation f ( z ) = 0 using only the information of f at the initial point z 0 . The study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult, even in the case of algebraic polynomials. In the light of Smale's point estimation theory, an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials. We state new, improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations: Ehrlich–Aberth's method, Ehrlich–Aberth's method with Newton's correction, Börsch-Supan's method with Weierstrass’ correction and Halley-like (or Wang–Zheng) method. The introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods, but also explicitly gives their order of convergence. The stated initial conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.

On a modification of the Ehrlich–Aberth method for simultaneous approximation of polynomial zeros

A modified method of the fourth order for the simultaneous determination of simple complex zeros of a polynomial, which may be regarded as an extension of the Ehrlich–Aberth method, is given. This method is derived using a very simple procedure which is also applicable for the construction of a whole class of simultaneous methods. The convergence analysis of the presented method is performed under computationally verifiable initial conditions, which is of significant practical importance. Numerical results obtained by several iterative methods of the fourth order are also given.

Point estimation of simultaneous methods for solving polynomial equations: a survey

One of the most important problems in solving nonlinear equations is the construction of such initial conditions which provide both the guaranteed and fast convergence of the considered numerical algorithm. Smale's approach from 1981, known as “point estimation theory”, treats convergence conditions and the domain of convergence in solving an equation f( z)=0 using only the information of f at the initial point z 0. A procedure of this type is applied in this paper to iterative methods for the simultaneous approximation of simple zeros of polynomial equations. We have stated new, refined initial conditions which ensure the guaranteed convergence of the most frequently used simultaneous methods for solving algebraic equations: the Durand–Kerner method, Börsch-Supan method, the Ehrlich–Aberth method and the square-root family of one parameter methods. The stated initial conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.

Finding roots of a real polynomial simultaneously by means of Bairstow's method

Aberth's method for finding the roots of a polynomial was shown to be robust. However, complex arithmetic is needed in this method even if the polynomial is real, because it starts with complex initial approximations. A novel method is proposed for real polynomials that does not require any complex arithmetic within iterations. It is based on the observation that Aberth's method is a systematic use of Newton's method. The analogous technique is then applied to Bairstow's procedure in the proposed method. As a result, the method needs half the computations per iteration than Aberth's method. Numerical experiments showed that the new method exhibited a competitive overall performance for the test polynomials.

Numerical computation of polynomial zeros by means of Aberth's method

An algorithm for computing polynomial zeros, based on Aberth's method, is presented. The starting approximations are chosen by means of a suitable application of Rouche's theorem. More precisely, an integerq ≥ 1 and a set of annuliAi,i=1,...,q, in the complex plane, are determined together with the numberki of zeros of the polynomial contained in each annulusAi. As starting approximations we chooseki complex numbers lying on a suitable circle contained in the annulusAi, fori=1,...,q. The computation of Newton's correction is performed in such a way that overflow situations are removed. A suitable stop condition, based on a rigorous backward rounding error analysis, guarantees that the computed approximations are the exact zeros of a “nearby” polynomial. This implies the backward stability of our algorithm. We provide a Fortran 77 implementation of the algorithm which is robust against overflow and allows us to deal with polynomials of any degree, not necessarily monic, whose zeros and coefficients are representable as floating point numbers. In all the tests performed with more than 1000 polynomials having degrees from 10 up to 25,600 and randomly generated coefficients, the Fortran 77 implementation of our algorithm computed approximations to all the zeros within the relative precision allowed by the classical conditioning theorems with 11.1 average iterations. In the worst case the number of iterations needed has been at most 17. Comparisons with available public domain software and with the algorithm PA16AD of Harwell are performed and show the effectiveness of our approach. A multiprecision implementation in MATHEMATICA™ is presented together with the results of the numerical tests performed.

Aberth's method for the parallel iterative finding of polynomial zeros

Aberth's method for the iterative finding of polynomial zeros using simultaneous interacting guesses is seen to be the systematic use of Newton's method under Parallel Anticipatory Implicit Deflation (PAID), changing the single polynomial problem into that for a system of rational functions. It offers the prospect of parallel execution, discourages coincident convergences, and subjects all iterates to the same algorithm, without the uneven roundoff error accumulation due to deflation typical in serial iterative methods. Yet it converges slowly to multiple zeros, and consumes far more total computing power than serial methods. Factors hindering convergence include problem symmetry and clustering of iterates, alleviated by asymmetric initial guesses and an immediate updating strategy, as seen through the speedup in the experimental APL program PAIDAX.

Solving certain queueing problems modelled by toeplitz matrices

By using the concept of generating function associated with a Toeplitz matrix, we analyze existence conditions for the probability invariant vector π of certain stochastic semi-infinite Toeplitz-like matrices. An application to the shortest queue problem is shown. By exploiting the functional formulation given in terms of generating functions, we devise a weakly numerically stable algorithm for computing the probability invariant vector π. The algorithm is divided into three stages. At the first stage the zeros of a complex function are numerically computed by means of an extension of the Aberth method. At the second stage the first k components of π are computed by solving an interpolation problem, where k is a suitable constant associated with the matrix. Finally, at the third stage a triangular Toeplitz system is solved and its solution is refined by applying the power method or any other refinement method based on regular splittings. In the solution of the triangular Toeplitz system and at each step of the refinement method, special FFT-based techniques are applied in order to keep the arithmetic cost within the O(n log n) bound, where n is an upper bound to the number of the computed components. Numerical comparisons with the available algorithms show the effectiveness of our algorithm in a wide set of cases.

Improvement on Aberth's method for choosing initial approximations to zeros of polynomials

Concerns a method for choosing the initial approximations to the zeros of a polynomial. Methods for finding all the zeros of a polynomial simultaneously, which can be considered modified Newton Methods, have been reported by quite a few authors. The initial approximations for these simultaneous methods take much CPU-time to find all the zeros of a polynomial, since CPU time depends mostly on the initial distribution of zeros. A new method for choosing the initial approximations is proposed, which simplifies Aberth's method (Math. Comput., vol.27, p.339-44, 1973) for choosing the initial approximations, and reduces the CPU time to obtain initial approximations and find all the zeros of a polynomial. In this method, the quasi-standard deviation obtained by the degree and coefficients of a polynomial is defined, and this gives the initial approximations. A comparison between the method is made through numerical experiments. >