A new iterative method for polynomial rootfinding based on the development of two novel recursive functions is proposed. In addition, the concept of
Perhaps the oldest problem in numerical analysis deals with the search of polynomials’ roots. Since Abel and Galois proved the nonexistence of radicalbased solutions for general polynomials or order higher than four, the only method to obtain the complete set of roots is numerical calculus and particularly the iterative methods. For any iterative method, a recursive function together with an initial guess is required. Most methods are focused on the local efficiency of the recursive schemes, convergence conditions, and velocity at the roots, whereas the study on where (and why) to start the iterative sequences is less considered. Hence, the challenge is to find reasonably efficient initial guesses, that is, starting points for the iterative sequences lying close to some of the roots.
The problem of searching zeros of functions has been extensively discussed in several books on numerical analysis; see, for instance, [
As mentioned, the location of initial approximations is of special relevance in iterative schemes so that the success or failure may largely depend on it. In the book of Kyurkchiev [
In this paper, we consider the
In order to validate the theoretical results, two numerical examples are analyzed. In the first two, the results of the proposed recursive functions are studied for single and multiple roots, respectively. In the third example, the influence of the pivots is discussed. In addition, in this last example the test of global convergence is applied, directly relating the proposed pivots to the success of the iteration scheme.
Based on the polynomial of (
Associated with the polynomial
As mentioned above,
The following proposition relates the polynomial roots with the fixed points of the defined functions.
If
Starting from the general form of the polynomial given by (
The following proposition presents a characterization of the multiple roots of
Let
If
If
(i) If
(ii) If
Since each root of the polynomial is a fixed point of one of the functions
Let
A complex number
Similarly, a
This subsection deals with the necessary conditions for the local convergence of the recursive sequences. The main result is presented in Theorem
Let
If
Consider
Let one assume that
the ball
the ball
The proof is developed for the first case; that is,
Theorem
The necessary condition imposed by (
In view of this reasoning, it seems that roots with
For any iterative numerical scheme, it is always desirable to provide
One defines the pivots of polynomial
The pivots of a polynomial have the property of lying close to some root when these pivots are relatively large (in absolute value) with respect to the rest of polynomial coefficients. This may be an important advantage because the pivots can be used as effective initial guesses in a recursive scheme. This behavior is explained in the results of this section. The first result (Proposition
Let
(i) From the definition of
This result justifies the use of the family of closed balls centered at
Let us assume that
Let
(i) From the definition of
(ii) Using the previous result
(iii) Following the same reasoning as that of (
(iv) From the bounds calculated in (i) and (ii)
(v) Let us define the function
The same conclusions of this lemma can easily be extrapolated for pivot
Under the same conditions of Lemma
(i) Evaluating the derivative of
(ii) From the definition of
With the help of the above lemmas, the main result on global convergence can already be presented.
Let
Let us demonstrate that
Therefore, the Banach contraction principle can be applied ensuring that there exists a unique fixed point
Lemma
Under the same conditions of Lemma
Let
From the definition of
The necessary conditions of this theorem are given in terms of the three numbers
Test of global convergence for polynomial
To conclude this section, we will make some remarks on the convergence to multiple roots. As proved in Proposition
Since the velocity of the recursive scheme given by the proposed functions
Let
As will be demonstrated, the introduction of the functions
The present subsection deals with the local convergence of sequences
Let
If
If
The previous theorem states that any root of the polynomial
As shown in Section
If
From the results obtained in Proposition
Differentiating now two times (
The previous proposition suggests that, under similar conditions as those imposed for
We think that the polynomial pivots
As proved in the theorems presented in this section, local convergence is always guaranteed for the corrected recursive functions. Therefore, choosing a starting point close enough to a root, the recursive sequence will converge. However, the following question arises: which complex number must be chosen as starting point? As well known, this is a key issue but very difficult to answer for any nonglobally convergent root finder. It was proved in the previous subsections that the pivots of the polynomial are a good choice under certain conditions. But how to relate the convergence region, that is, the set of valid initial points for which convergence holds, with the functions
To this end, firstly it will be demonstrated that if
From the definition of
Now, assuming that
Note that since
Secondly, let us consider the closed ball
Approximating
Approximating
For the
Intuitively, it can be generalized that the radius of convergence
Therefore, the higher the root in absolute value, the larger the convergence region, so that the proposed numerical method is expected to work better in the sense that there exist many more starting points from which convergence will hold. In the numerical examples this behavior will be validated showing that the proposed method presents convergence towards the largest roots. In practice, those polynomials with relatively high absolute values of the pivots with respect to the rest of the polynomial coefficients will present good behavior with respect to the convergence.
Let us consider the following
Results of local convergence for Example 1.
Fixed points of 
Fixed points of 






























































Example 1. Iteration errors. Pivots are
Some remarks on the influence of the pivots
In order to compare the proposed method with another one of quadratic convergence, the iteration error for Newton’s method has also been represented in Figure
In this example, the behavior of the method for multiple roots is examined using the polynomial
Note that it has the same roots as those of Example 1, though now
Results of local convergence for Example 2.
Fixed points of 
Fixed points of 











0.439  0.186  10.727 

2.569  4.062  0.492 

6.688  10.900  0.183 

1.000  1.037  0.964 

9.224  5.140  0.389 

17.410  24.286  0.082 

21.574  31.429  0.064 

100.890  221.590  0.009 

14.270  27.369  0.073 





48.698  101.740  0.020 








Example 2. Iteration errors. Pivots are
The results of the proposed method for functions
In this paper, a new method for polynomial rootfinding is presented. The key idea is the construction of two complex functions, called recursive functions, which can be used in a fixedpoint recursive scheme. Necessary conditions for local and global convergence are provided. The latter are studied in a closed ball centered at two certain characteristic points, called
The convergence of the recursive sequences is linear and is not guaranteed for any initial point in the complex plane. For these reasons, corrected recursive functions are constructed to accelerate the convergence. These functions are inspired in the wellknown Steffensen’s acceleration method and present the same properties: local convergence and the errordecay velocity quadratic for single roots and linear for multiple ones. The convergence region around the roots, that is, the set of complex values from which convergence holds, is studied. It is concluded that a direct relationship between the absolute value root and the convergence radius exists: the latter increases with the former.
Finally, the theoretical results are validated with two numerical examples. In the first, the convergence of the proposed recursive sequences for a singleroot polynomial is analyzed. This example clearly reveals the close connection between the pivots and the largest roots of the polynomial. In fact, under certain conditions, just evaluating the corrected recursive functions at the pivots can lead to a very accurate approximation of a root. In the second example the efficiency of the proposed method is studied for multiple roots. In them, as predicted by the theoretical results, the recursive sequences converge under a sublinear scheme, whereas the corrected sequences present now linear convergence. Further work is currently being developed to generalize the concept of polynomial pivots and to propose new more accurate initial approximations only based on the polynomial coefficients.
The authors declare that there is no conflict of interests regarding the publication of this paper.