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

Abstract 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.


Introduction
The importance of the problem of finding polynomial zeros in the different branches of Science and Engineering has led to the development of many different methods for their numerical determination. There is also a significant interest in the development of new and efficient numerical iterative methods for determining the zeros of real and complex polynomials.
These methods can approximate the zeros of a polynomial either in a sequential or simultaneous manner. However, the determination of all zeros of a given polynomial by a sequential method involves repeated deflations, which can lead to very inaccurate results due to the problem of accumulating rounding errors when using finite precision floating-point arithmetic.
In turn, the simultaneous methods for polynomial zeros have the advantage of being inherently parallel and avoid the process of deflation, although these iterative methods usually need good initial approximations for all the zeros in order to converge.
In this work, a new family of numerical methods for the simultaneous approximation of simple complex polynomial zeros is presented. The proposed simultaneous methods are constructed on the basis of the well-known third order Ehrlich-Aberth iteration [1,6], combined with an iterative correction from the optimal fourth order two-step King's method for solving nonlinear equations [10].
2 A family of high order Ehrlich-type simultaneous methods with King's correction

The third order Ehrlich method
Let P be a monic polynomial of degree n, with (real or complex) simple zeros ζ 1 , . . . , ζ n , (1) Considering the correction term N (z) = P (z)/P (z) from the second order Newton's methodẑ = z − N (z) (where, for notation simplicity,ẑ represents a new approximation for a zero ζ) and taking its logarithmic derivative with respect to z, we obtain (2) 130 Ehrlich-type Methods with King's Correction for the Simultaneous Approximation of Polynomial Complex Zeros But we have and then From (4), we can easily derive the following fixed-point relation: Now, let z 1 , . . . , z n be distinct approximations to the zeros ζ 1 , . . . , ζ n of the polynomial P .
Putting ζ i ẑ i in (5), whereẑ i is a new approximation for the zero ζ i , setting z = z i , and substituting the zeros ζ j by their approximations z j (j = i) on the right-hand side of the above identity, we obtain the well-known Ehrlich method (also known as Ehrlich-Aberth method or Maehly-Ehrlich-Aberth method) for the simultaneous approximation of simple polynomial zeros.

Ehrlich-type simultaneous methods with King's correction
It is evident from the fixed-point relation (5) that better approximations z j will produce more accurate approximationsẑ i for the zeros ζ i .
In order to achieve this, and aiming to improve the convergence rate and efficiency of the iterative scheme due to Ehrlich, we propose a family of simultaneous iterative methods constructed on the basis of the third order Ehrlich iteration, combined with a correction based on the optimal fourth order twostep King's method for nonlinear equations [10].
The well-known King's family of fourth order multipoint iterative methods for determining a simple root of a nonlinear equation f (x) = 0 requires only two evaluations of the function and one of its first derivative per full iteration, and is defined by where β ∈ C is a parameter. The family of simultaneous iterative methods here proposed is obtained by combining the scheme of the third order Ehrlich iteration (6) with an iterative correction term obtained from King's fourth order iteration (7), given by where and β is a parameter.
Using the King approximation z j − C K (z j ) in (6) instead of z j , we obtain a new one-parameter family of Ehrlich-type simultaneous iterative methods with King's correction, defined bŷ where C K (z j ) is the iterative correction appearing in (8).

Order of convergence of the new family of combined methods
In this section, we analyze the order of convergence of the proposed family of Ehrlich-type simultaneous iterative methods with King's correction, which is established in the following theorem.
n be sufficiently close initial approximations to the simple zeros ζ 1 , . . . , ζ n of the polynomial P . Then, the order of convergence of the one-parameter family of iterative methods defined in (9) is six.
Proof. Consider the abbreviations and let i andˆ i be the errors of the numerical approximations defined as follows: Rearranging (9), we havê Substituting (2) into (14), we obtain and expanding this equation and simplifying, we get . (16) However, when i = j, we can write Substituting (17) into (16) yieldŝ and substituting (12) into this equation giveŝ But, by (12) and (13), we havê .

(20)
Simplifying and replacing the denominator of the second summation by (10), we get Multiplying and dividing the last equation by i , we havê and joining the two summations together, we obtain Considering (10) again, we havê and simplifying, we obtain where the summation that appears within parentheses in this equation is, according to (11), equal to −θ i . By making the proper substitution in (25), we get Reducing to the common denominator and simplifying, we finally obtainˆ By the assumption of the theorem, the errors are small enough in modulus, and thus it can be assumed . . , n}, where this notation means that the moduli of the pairs of complex numbers involved have the same order, that is, where the symbol O indicates the order of magnitude.
Analyzing (10) and (11), we observe that, for sufficiently close approximations to the corresponding zeros, the expression in the denominator of (11) is bounded and tends to (ζ i − ζ j ) 2 , for i = j.
Considering this, and taking into account that each method of the family of King's methods is of fourth order of convergence [10], i.e., for these methods,ẑ − ζ = O m ((z − ζ) 4 ), we find that

Numerical examples
In this section, some numerical examples are presented to illustrate the convergence behaviour and effectiveness of the proposed family of simultaneous iterative methods.
It is important to note here that not all values of the parameter β lead to convergence of the fourth order King's method (7) [4]. Some convenient values of this parameter can be found in the literature, including β = −1/2 [10], β = −9/2, β = 3.9 + 0.1i [4] and β = 3 − 2 √ 2i [12]. In the three examples here considered, we have adopted the value β = −7/10 for this parameter, which was determined through a computational parameter optimization.
In the examples presented, the initial approximations were chosen by using Aberth's approach [1]. The starting approximations to the zeros of each polynomial calculated by the Aberth's initialization procedure are equidistantly distributed along the circumference |z + a 1 /n| = r 0 , where a 1 is the coefficient of the second leading term of the monic polynomial considered, n is the polynomial degree, and r 0 is the radius of a disk centered at the origin of the complex plane containing all the zeros of the polynomial, obtained as usual (see, e.g., [13]) by applying the Guggenheimer's upper bound for the zeros [9].
The requested accuracy of 10 −12 was achieved in eigth iterations, while Ehrlich's method (6), on which the proposed method is based, required 14 iterations.
Example 2. In the second example, the proposed method (9), with β = −7/10, was applied to a real polynomial of degree 15, extracted from [14], which has only a real zero: The trajectories of approximations for the polynomial P 2 (z), generated by the method (9) with a 1 = 1, r 0 = 2 and β = −7/10, are presented in Figure 2.
In this case, the proposed combined iterative method required five iterations to obtain approximations to the zeros of the given polynomial with the requested accuracy.
For comparison, Ehrlich's method needed nine iterations to achieve the same accuracy.  Example 3. The last example here considered is a real polynomial of degree 20, extracted from [16], known as the scaled Wilkinson polynomial:  A more detailed view of the trajectories of approximations for the same polynomial corresponding to the last 11 iterations generated by the proposed simultaneous method is shown in Figure 4.
The third order Ehrlich method needed 45 iterations to reach the required accuracy, whereas the proposed method (9), with β = −7/10, performed only 15 iterations to reach the same desired accuracy. Table 1 contains a summary of the results obtained, showing the number of iterations required by the simultaneous iterative methods (6) and (9) to produce approximations to the zeros of the polynomials of Examples 1 to 3 with the requested accuracy of 10 −12 .

Conclusion
In this paper, a new sixth order family of combined iterative methods for the simultaneous approximation of simple complex zeros of a polynomial was introduced and discussed.
The proposed one-parameter family of simultaneous methods was constructed on the basis of the third order Ehrlich iteration, combined with an iterative correction based on King's family of optimal fourth order iterative methods for solving nonlinear equations.
It was proved that the use of King's correction allows to increase the order of convergence of the basic simultaneous method from three to six.
The results of some numerical examples corroborate the theoretical analysis and illustrate the effectiveness and rapid convergence of the proposed Ehrlich-type methods with King's correction for the simultaneous approximation of polynomial complex zeros.