Convergence and Dynamics of Schröder’s Method for Zeros of Analytic Functions with Unknown Multiplicity

Abstract

In this paper, we investigate the local convergence of Schröder’s method for finding zeros of analytic functions with unknown multiplicity. Thus, we obtain a convergence theorem that provides exact domains of initial points together with error estimates to ensure the Q-quadratic convergence of Schröder’s method right from the first step. A comparison with the famous Newton’s method, based on the convergence and dynamics when it is applied to some polynomial and non-polynomial equations, is also provided.

1. Introduction

One of the main problems arising from mathematics, natural sciences, engineering and economics is solving the nonlinear equation

$$f\left(x\right)=0.$$

It is well known that iteration methods are among the most efficient tools for solving the above equation, and the most famous among them is Newton’s method

$${\displaystyle x_{k+1}=x_k-\frac{f\left(x_k\right)}{f^{\prime }\left(x_k\right)}.}$$

(1)

Newton’s method (1) is quadratically convergent for simple zeros but converges only linearly to multiple zeros. To overcome this problem, in 1870, Schröder [1] proposed the following variant of Newton’s method:

$${\displaystyle x_{k+1}=x_k-m\frac{f\left(x_k\right)}{f^{\prime }\left(x_k\right)}}$$

(2)

which preserves the quadratic convergence to zero with known multiplicity $m\ge 1$. Obviously, the main disadvantage of the method (2) is the requirement of knowledge of the zero’s multiplicity in advance. In the same paper [1], Schröder presented the following iterative method as well:

$$x_{k+1}=x_k-{\displaystyle \frac{f\left(x_k\right)f^{\prime }\left(x_k\right)}{f^{\prime }{\left(x_k\right)}^2-f\left(x_k\right)f^{″}\left(x_k\right)}},k=0,1,2,\dots$$

(3)

This method is also quadratically convergent but does not require prior knowledge of the multiplicity of the zeros. We note that a comprehensive local convergence analysis of Schröder’s method (3) for individually and simultaneously finding polynomial zeros can be found in [2,3], respectively, while a detailed dynamical analysis and some more historical notes are available in [4,5].

In this paper, using a general convergence theory about the Picard iteration [6,7,8]

$$x_{k+1}=T{x}_k(k=0,1,2,\dots ),$$

(4)

we conduct a detailed analysis of the local convergence of Schröder’s method (3) for finding the zeros of analytic functions with unknown multiplicity. A new local convergence theorem (Theorem 1) is proven, which provides exact domains of initial conditions and error estimates to ensure the Q-quadratic convergence of the method right from the first iteration. As consequences, we obtained a ball convergence theorem (Theorem 2) and a Smale-type theorem (Corollary 1) that gives a ball of approximate zeros of the second kind for Schröder’s method. As far as we are aware, there is no such result regarding Schröder’s method (3) in the literature.

At the end of the paper, some numerical examples are performed to confirm the theoretical results. Numerical comparisons with Newton’s methods (1) and (2) are also provided based on their convergence and dynamics when applied to some polynomial and non-polynomial equations.

2. Main Results

Let $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$ be analytic in a neighborhood of $\xi \in \mathcal{D}$. We define the function $E:\mathbb{C}\to {\mathbb{R}}_{+}$ by

$$E\left(x\right)=\gamma |x-\xi |,$$

(5)

where, for an arbitrary integer $n\ge 1$, the quantity $\gamma$ is defined by

$$\gamma =\gamma (f,\xi )=\underset{k>n}{sup}{\left|{\displaystyle \frac{n!f^{\left(k\right)}\left(\xi \right)}{k!\mathcal{M}\left(\xi \right)}}\right|}^{1/(k-n)},$$

(6)

where $\mathcal{M}\left(\xi \right)={min}_{1\le j\le n}\left|f^{\left(j\right)}\left(\xi \right)\right|\ne 0$.

In what follows, we study the local convergence of Schröder’s method applied for the approximation of the zeros of f, requiring no information about their multiplicities but only about their number. For this purpose, we define Schröder’s iterative sequence as follows:

$$x_{k+1}=S_f\left(x_k\right),$$

(7)

where Schröder’s iteration function $S_f:D\subset \mathbb{C}\to \mathbb{C}$ is defined by

$$S_f\left(x\right)=\left\{\begin{array}{cc}{\displaystyle x-{\left(\frac{f^{\prime }\left(x\right)}{f\left(x\right)}-\frac{f^{″}\left(x\right)}{f^{\prime }\left(x\right)}\right)}^{-1}}\hfill & \mathrm{if}f\left(x\right)\ne 0,\hfill \\ x\hfill & \mathrm{if}f\left(x\right)=0.\hfill \end{array}\right.$$

(8)

Obviously, the domain of $S_f$ is the set

$$D=\left\{x\in \mathbb{C}:f\left(x\right)\ne 0\Rightarrow f^{\prime }\left(x\right)\ne 0\mathrm{and}{\displaystyle \frac{f^{\prime }\left(x\right)}{f\left(x\right)}}-{\displaystyle \frac{f^{″}\left(x\right)}{f^{\prime }\left(x\right)}}\ne 0\right\}.$$

(9)

Note that the analyticity of f assures the existence of all derivatives in $S_f$.

Before stating our main result, for $n\ge 1$ we define the number $\tau$ by

$$\tau ={\displaystyle \frac{2}{n+3+\sqrt{n^2+6n+1}}}$$

(10)

and the real functions $g,h$ and $\varphi$ by

$$\begin{array}{ccc}\hfill g\left(t\right)& =& (3-2n)t^4+(3n^2-7)t^3-(5n^2-3n-5)t^2+(3n^2+n-1)t,\hfill \\ \hfill h\left(t\right)& =& n{t}^4-(2n^2+n+1)t^3+(3n^2-n+2)t^2-(2n^2+3n+1)t+1\hfill \end{array}$$

(11)

and

$$\varphi \left(t\right)=g\left(t\right)/h\left(t\right).$$

(12)

We note that $\tau$ is the minimal solution of $2t^2-(n+3)t+1=0$ and that the function $\varphi$ is quasi-homogeneous of the exact first degree ([9], Definition 8) in the interval $[0,\mu )$, where $\mu$ is the unique solution of the equation $h\left(t\right)=0$ in the interval $[0,\tau ]$. Observe that $\mu$ exists since h is continuous and strictly decreasing on t, $h\left(0\right)=1$ and $h\left(\tau \right)<0$ for all $n\ge 1$.

The following is our main convergence result in this paper:

Theorem 1. 

Let $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$ has n zeros in $\mathbb{C}$, $\xi \in \mathcal{D}$ be a zero of f and let f be analytic in a neighborhood of ξ. Suppose $x_0\in \mathcal{D}$ is an initial guess satisfying

$$\Phi \left(E\left(x_0\right)\right)>0,$$

(13)

where E is defined by (5) and $\Phi =h-g$ with functions h and g defined by (11). Then, Schröder’s iterative sequence (7) is well defined and converges Q-quadratically to ξ with the following error estimates for all $k\ge 0$:

$$|x_{k+1}-\xi |\le {\lambda }^{2^k}|x_k-\xi \left|\mathit{and}\right|x_k-\xi |\le {\lambda }^{2^k-1}|x_0-\xi |,$$

(14)

where $\lambda =\varphi \left(E\left(x_0\right)\right)$ and the function ϕ is defined by (12). Additionally, the following estimate of the asymptotic error constant holds:

$$\underset{k\to \infty }{lim\; sup}{\displaystyle \frac{|x_{k+1}-\xi |}{|x_k{-\xi |}^2}}\le (3n^2+n-1)\gamma .$$

(15)

Let us formulate Theorem 1 in the following equivalent form, which gives exact convergence balls of Schröder’s method (7).

Theorem 2. 

Let $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$ has n zeros in $\mathbb{C}$ and f be analytic in a neighborhood of its zero ξ. If R is the unique solution of the equation $\Phi \left(t\right)=0$ in the interval $(0,1)$, where Φ is the same function as in Theorem 1, then $B(\xi ,R/\gamma )$ is a convergence ball that provides Q-quadratic convergence of Schröder’s method (7) with estimates (14) and (15).

In light of Smale’s concept of approximate zero of second kind [10,11], we give the following general definition associated with the Picard iteration (4).

Definition 1. 

Let $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$ and $\xi \in \mathcal{D}$ be a zero of f. If the Picard iteration (4) is well defined and converges to ξ with order $r>1$, then $x_0\in \mathcal{D}$ is called an approximate zero of the second kind of f for iteration (4) if the following estimate holds:

$$|x_k-\xi |\le {\left({\displaystyle \frac{1}{2}}\right)}^{(r^k-1)/(r-1)}|x_0-\xi |\mathit{for}\mathit{all}k>0.$$

(16)

ξ shall be called an associated zero of $x_0$.

In line with Definition 1, the following consequence of Theorem 2 provides a ball of approximate zeros of the second kind for Schröder’s method (7).

Corollary 1. 

Let $\xi \in \mathcal{D}$ be one of the n-th zeros of a function $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$ which is supposed to be analytic in a neighborhood of ξ. Let also, $\mathcal{R}$ be the unique solution of the equation $\Psi \left(t\right)=0$ in the interval $(0,1)$, where $\Psi =h-2g$, with h and g defined by (11). Then, any $x_0\in B(\xi ,\mathcal{R}/\gamma )$ is an approximate zero of the second kind for Schröder’s method (7).

3. Proofs of the Main Results

To make the paper self-contained, we start this section by stating some previous definitions and results from [8] that are crucial for the proofs of our main results.

We start with the following extension of ([8], Definition 3.1) which involves an arbitrary integer n instead of the multiplicity of the root of f:

Definition 2. 

A function $T:D\subset \mathbb{C}\to \mathbb{C}$ is said to be a gamma-like iteration function at a point $\xi \in \mathbb{C}$ if there is a quasi-homogeneous function $\varphi :J\to {\mathbb{R}}_{+}$ of degree $p\ge 0$ such that for each $x\in \mathbb{C}$ with $E\left(x\right)\in J$ it follows that

$$x\in D\mathit{and}\left|T\right(x)-\xi |\le \varphi \left(E\right(x\left)\right)|x-\xi |,$$

(17)

where the function E is defined by (5). The function ϕ is called a control function of T.

We use the following variant of ([8], Theorem 3.4) to proof our main theorem:

Theorem 3 

([8] Theorem 3.4). Let $T:D\subset \mathbb{C}\to \mathbb{C}$ be a gamma-like iteration function at a point $\xi \in D$ with control function ϕ of exact degree $p\ge 0$ and $x_0\in \mathbb{C}$ be an initial approximation that satisfies the condition

$$E\left(x_0\right)\in \{t\in J:\varphi \left(t\right)<1\},$$

(18)

then ξ is a fixed point of T, the Picard iteration (4) is well defined and converges to ξ with Q-order $r=p+1$, and for all $k\ge 0$, the following error estimates hold:

$$|x_{k+1}-\xi |\le {\lambda }^{r^k}|x_k-\xi \left|\mathit{and}\right|x_k-\xi |\le {\lambda }^{(r^k-1)/(r-1)}|x_0-\xi |,$$

(19)

where $\lambda =\varphi \left(E\left(x_0\right)\right)$. In addition, the following estimate of the asymptotic error constant holds:

$$\underset{k\to \infty }{lim\; sup}{\displaystyle \frac{|x_{k+1}-\xi |}{|x_k{-\xi |}^r}}\le {\gamma }^p\underset{t\to 0^{+}}{lim}{\displaystyle \frac{\varphi \left(t\right)}{t^p}}.$$

The following technical lemmas play an important role in the proofs of our results.

Lemma 1 

([8] Lemma 4.1). Let $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$, $\xi \in \mathcal{D}$ be a zero of f with known multiplicity $m\ge 1$ and $\mathcal{E}\left(x\right)<1$, where $\mathcal{E}:\mathbb{C}\to {\mathbb{R}}_{+}$ is defined by

$$\mathcal{E}\left(x\right)={\gamma }_m|x-\xi |,$$

where ${\gamma }_m$ is defined by

$${\gamma }_m={\gamma }_m(f,\xi )=\underset{k>m}{sup}{\left|{\displaystyle \frac{m!f^{\left(k\right)}\left(\xi \right)}{k!f^{\left(m\right)}\left(\xi \right)}}\right|}^{1/(k-m)}.$$

(20)

Then, the following estimates hold true:

$$\left|a\right|\le {\displaystyle \frac{\mathcal{E}\left(x\right)}{1-\mathcal{E}\left(x\right)}},\left|b\right|\le {\displaystyle \frac{(m+1)\mathcal{E}\left(x\right)-m\mathcal{E}{\left(x\right)}^2}{{(1-\mathcal{E}\left(x\right))}^2}},\left|c\right|\le {\displaystyle \frac{m(m-1){(1-\mathcal{E}\left(x\right))}^2-2(m-1)\mathcal{E}\left(x\right)+2m}{{(1-\mathcal{E}\left(x\right))}^3}}\mathcal{E}\left(x\right),$$

(21)

where $a,b$ and c are defined by

$$a=\sum _{k=m+1}^{\infty }{A}_k,b=\sum _{k=m+1}^{\infty }k{A}_k\mathit{and}c=\sum _{k=m+1}^{\infty }k(k-1)A_k$$

(22)

with $A_k$ defined by

$$A_k=\left({\displaystyle \frac{m!f^{\left(k\right)}\left(\xi \right)}{k!f^{\left(m\right)}\left(\xi \right)}}\right){(x-\xi )}^{k-m}.$$

Remark 1. 

It is worth noting that the quantity ${\gamma }_m$ defined by (20) has been used by Proinov [6], Ivanov [8] and Kostadinova and Ivanov [12] to study the convergence of the famous Newton, Halley and Chebyshev’s methods when applied to zeros of analytic functions with known multiplicity m. In fact, this quantity was first used by Smale [10], in the case $m=1$, and by Yakoubsohn [13] for $m\ge 1$.

The next lemma is an immediate consequence of Lemma 1 owing to the inequalities $1\le m\le n$ and $\mathcal{E}\left(x\right)\le E\left(x\right)$.

Lemma 2. 

Let $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$ have n zeros in $\mathbb{C}$ counted with their multiplicities, $\xi \in \mathcal{D}$ be a zero of f and $E\left(x\right)<1$, where $E:\mathbb{C}\to {\mathbb{R}}_{+}$ is defined by (5). Then, the following inequalities hold:

$$\left|a\right|\le {\displaystyle \frac{E\left(x\right)}{1-E\left(x\right)}},\left|b\right|\le {\displaystyle \frac{(n+1)E\left(x\right)-E{\left(x\right)}^2}{{(1-E\left(x\right))}^2}}\mathit{and}\left|c\right|\le {\displaystyle \frac{n(n-1){(1-E\left(x\right))}^2+2n}{{(1-E\left(x\right))}^3}}E\left(x\right),$$

(23)

where the numbers $a,b$ and c are defined by (22).

Lemma 3 

([8] Lemma 4.2). Let $f:\mathcal{D}\subset \mathbb{C}\to \mathbb{C}$ have n zeros in $\mathbb{C}$ and $\xi \in \mathcal{D}$ be a zero of f. If f is analytic around ξ, then there is $m\in \mathbb{N}$ such that $a,b$ and c can be defined by (22) and the following claims hold true:

(i)

If $x\in \mathcal{D}$ is such that $f\left(x\right)\ne 0$ and $a\ne -1$, then we have

$${\displaystyle \frac{f^{\prime }\left(x\right)}{f\left(x\right)}}={\displaystyle \frac{m+b}{(1+a)(x-\xi )}}.$$

(ii)

If $x\in \mathcal{D}$ is such that $f^{\prime }\left(x\right)\ne 0$ and $b\ne -m$, then we have

$${\displaystyle \frac{f^{″}\left(x\right)}{f^{\prime }\left(x\right)}}={\displaystyle \frac{m^2-m+c}{(m+b)(x-\xi )}}.$$

Proof of Theorem 1. 

First, we shall prove that Schröder’s iteration Function (8) is a gamma-like iteration function at a point $\xi$ with the control function $\varphi$ defined by (12).

Define E by (5) and the number $\mu$ as the unique solution of the equation $h\left(t\right)=0$ in the interval $[0,\tau ]$. Let $x\in \mathcal{D}$ be such that $E\left(x\right)<\mu$, then we have

$$h\left(E\right(x\left)\right)>0.$$

(24)

Now, we have to prove that

$$|S_f\left(x\right)-\xi |\le \varphi \left(E\left(x\right)\right)|x-\xi |,$$

(25)

where $\varphi$ is defined by (12). Let $f\left(x\right)\ne 0$ and $\xi$ be a zero of f with multiplicity $m\ge 1$. Define the numbers a, b and c by (22). Then, from the triangle inequality, the second inequality of (23) and $E\left(x\right)<\mu <\tau$, we obtain

$$|m+b|\ge |1+b|\ge 1-\left|b\right|\ge 1-{\displaystyle \frac{(n+1)E\left(x\right)-E{\left(x\right)}^2}{{(1-E\left(x\right))}^2}}={\displaystyle \frac{2E{\left(x\right)}^2-(n+3)E\left(x\right)+1}{{(1-E\left(x\right))}^2}}>0$$

(26)

which means that $b\ne -m$, and therefore, by Lemma 3 (i), we get that $f^{\prime }\left(x\right)\ne 0$.

Further, from the definition of Schröder’s iteration Function (8) and Lemma 3, we get

$$S_f\left(x\right)-\xi =x-\xi -{\displaystyle \frac{(x-\xi )(1+a)(m+b)}{{(m+b)}^2-(m^2-m+c)(1+a)}}=\sigma (x-\xi ),$$

(27)

where

$$\sigma ={\displaystyle \frac{{(m+b)}^2-(1+a)(m^2+b+c)}{{(m+b)}^2-(m^2-m+c)(1+a)}}={\displaystyle \frac{b^2+2(m-1)b-(m^2+b+c)a-c}{b^2+2mb-(m^2-m+c)a-c+m}}.$$

(28)

From this and $1\le m\le n$, we get

$$\sigma \le {\displaystyle \frac{b^2+2(n-1)b-(1+b+c)a-c}{b^2+2b-(n^2-1+c)a-c+1}}.$$

(29)

So, from (29), the triangle inequality and the estimates (23) and (26), we obtain

$$\begin{array}{ccc}\hfill \left|\sigma \right|& \le & {\displaystyle \frac{{\left|b\right|}^2+2(n-1)\left|b\right|+(1+|b|+|c\left|\right)\left|a\right|+\left|c\right|}{1-{\left|b\right|}^2-2\left|b\right|-(n^2-1+\left|c\right|\left)\right|a|-\left|c\right|}}\hfill \\ & \le & {\displaystyle \frac{B^2+2B(n-1){(1-E)}^2+E{(1-E)}^3+BE(1-E)+C{E}^2+C(1-E)E}{{(1-E)}^4-B^2-2B{(1-E)}^2-((n^2-1){(1-E)}^3+EC)E-C(1-E)E}}\hfill \\ & =& {\displaystyle \frac{g\left(E\right)}{h\left(E\right)}}=\varphi \left(E\right),\hfill \end{array}$$

(30)

where, for the sake of brevity, we use the following denotations $E=E\left(x\right)$, $B=(n+1)E\left(x\right)-E{\left(x\right)}^2$ and $C=n(n-1){(1-E\left(x\right))}^2+2n$. The last inequality together with (27) proves (25), which, according to Definition 2, means that $S_f$ is a gamma-like iteration function at a point $\xi$ with a control function $\varphi$ defined by (12). On the other hand, since $g\left(t\right)>0$ for all $t\in (0,1)$ and $n\ge 1$, then from $\Phi \left(t\right)>0$ we get $h\left(t\right)>g\left(t\right)>0$ which implies $t<\mu$ and $\varphi \left(t\right)<1$, where $\mu$ is the unique solution of the equation $h\left(t\right)=0$ in the interval $[0,\tau ]$ with $\tau$ defined by (10). Consequently, the proof follows from Theorem 3 because the initial condition (13) implies (18) with $J=[0,\mu )$. □

Proof of Corollary 1. 

Since $g\left(t\right)>0$ for all $t\in (0,1)$ and $n\ge 1$, then $\Psi \left(t\right)<\Phi \left(t\right)$ which means that $\mathcal{R}<R$ and so $B(\xi ,\mathcal{R}/\gamma )\subset B(\xi ,R/\gamma )$. Hence, by Theorem 2, we conclude that Schröder’s iteration (7) is Q-quadratically convergent to $\xi$ with error estimate

$$|x_k-\xi |\le {\left({\displaystyle \frac{1}{2}}\right)}^{2^k-1}|x_0-\xi |\mathrm{for}\mathrm{all}k>0$$

which, according to Definition 1, means that $\xi$ is an approximate zero of the second kind of f for Schröder’s method (7), which completes the proof. □

4. Numerical Experiments

In this section, we apply Schröder’s method (3) and Newton’s methods (1) and (2) for the approximation of simple and multiple roots of some polynomial and non-polynomial equations. In order to stress on the convergence and the dynamics of the mentioned methods in different scenarios, we conducted four examples. In the first one, we emphasize the influence of the multiplicity of the roots on the convergence and dynamics of Newton’s method (N) (1), Newton’s method for multiple zeros (NMZ) (2) and Schröder’s method (S) (3), applying them to some polynomial equations with roots of different multiplicity.

To show the behavior of the methods when applied to polynomials with close zeros and non-polynomial equations, in the other examples, we apply the methods N and S to a polynomial describing a physics problem and to two transcendental equations arising from chemical and biomedical engineering, each possessing only simple zeros.

Let $f:\mathbb{C}\to \mathbb{C}$ and $x_0$ be an initial guess. In the present examples, we apply the methods N, NMZ and S to find a root $\xi$ of f and to obtain their basins of attraction. The dynamical planes are obtained using a mesh of $400\times 400$ points and 50 iterations. The yellow points in the figures depict the exact roots of the tested function/equation. For the purposes of the convergence tests, we define the following quantity for all $k\ge 0$:

$${\epsilon }_k=|x_k-x_{k+1}|$$

and we use ${\epsilon }_k<{10}^{-5}$ as a stop criterion. In the tables, we present the approximated root $\xi$ with its multiplicity and the values of ${\epsilon }_k$ and ${\epsilon }_{k+1}$. All results are presented with at least six decimal places.

Example 1 

(Polynomials with multiple zeros). In this example, we apply the methods N, NMZ and S for the approximation of the zeros of the following polynomials:

$$f_1={(x^3-1)}^2,f_2={(x^3-1)}^3,f_3={(x^3-1)}^5,f_4={(x^4-1)}^2,f_5={(x^4-1)}^3\mathit{and}{f}_6={(x^4-1)}^5.$$

The basins of attraction of the roots of $f_1-f_6$ are obtained by the N, NMZ and S methods in the square

$$\{x\in \mathbb{C}:|Re\left(x\right)|\le 2\mathit{and}|Im\left(x\right)|\le 2\}$$

and the initial guess $x_0=-1.432374+1.600693i$ is randomly generated in the same square.

Regardless of the ‘rough’ choice of the initial approximation, it can be seen in

Table 1. Convergence results for Example 1.

Polynomial	Method	Root	Multiplicity	k	${\mathit{\epsilon }}_{\mathit{k}}$
$f_1$	N	$-0.5+0.866i$	2	19	$5.623\times {10}^{-6}$
	NMZ	$-0.5+0.866i$	2	5	$4.016\times {10}^{-7}$
	S	$-0.5+0.866i$	2	6	$1.713\times {10}^{-8}$
$f_2$	N	$-0.5+0.866i$	3	30	$7.524\times {10}^{-6}$
	NMZ	$-0.5+0.866i$	3	5	$3.979\times {10}^{-7}$
	S	$-0.5+0.866i$	3	6	$8.343\times {10}^{-9}$
$f_3$	N	$-0.5+0.866i$	5	51	$8.336\times {10}^{-6}$
	NMZ	$-0.5+0.866i$	5	5	$4.015\times {10}^{-7}$
	S	$-0.5+0.866i$	5	6	$1.683\times {10}^{-8}$
$f_4$	N	i	2	28	$6.366\times {10}^{-6}$
	NMZ	i	2	19	$8.837\times {10}^{-6}$
	S	1	2	10	$3.293\times {10}^{-9}$
$f_5$	N	i	3	34	$8.839\times {10}^{-6}$
	NMZ	i	3	19	$8.864\times {10}^{-6}$
	S	1	3	10	$4.185\times {10}^{-10}$
$f_6$	N	i	5	110	$2.458\times {10}^{-7}$
	NMZ	Diverges
	S	1	5	11	$6.168\times {10}^{-6}$

that S has a good convergence behavior, which is not influenced by increasing the multiplicity of the root, while N gets slower as the multiplicity of the root increases. Also, one can observe that NMZ and S behave quite similarly for the polynomials $f_1$–$f_3$, but this is not the case for $f_4$–$f_6$, where NMZ starts lagging behind or diverges (for $f_6$).

As for the dynamics, it can be seen in Figure 1 and Figure 2 that the higher is the multiplicity of the root, the more unstable the methods are; this mostly applies to NMZ. In particular, the divergent areas of the methods are colored white. In fact, N demonstrates the highest stability, at the expense of its linear convergence. What is interesting is that for $f_4$–$f_6$, Schröder’s method does not follow the ‘nearest root principal’ (see [14] and references therein); this is easily seen from Figure 2, where the white points depict the initial approximation $x_0$ and the black ones denote the approximated zeros.

Example 2 

(Van der Waals equation). A well-known equation describing the behavior of real gases is Van der Waals equation (see e.g., [15]). In the case of oxygen, it reads as follows:

$$100,000x^3-5622.002x^2+1.382x-0.04403052=0.$$

(31)

This equation has simple roots ${\xi }_{1,2}=0.000053\pm 0.002800i$ and ${\xi }_3=0.056113$. The basins of attraction for (31) are obtained in the square

$$\{z\in \mathbb{C}:|Re\left(x\right)|\le 0.15\mathit{and}|Im\left(x\right)|\le 0.15\}$$

while the initial guess $x_0=0.028450-0.136954i$ is randomly taken from the same square.

It can be seen from

Table 2. Convergence results for Example 2.

Method	Root	k	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\epsilon }}_{\mathit{k}+1}$
N	$0.000053-0.002800i$	11	$4.097\times {10}^{-8}$	$3.026\times {10}^{-13}$
S	$0.000053+0.002800i$	6	$5.335\times {10}^{-8}$	$5.132\times {10}^{-13}$

that for this example, S behaves much better than N, which could be due to the very close distance between the roots. One can see that the root is found by S only at the sixth iteration with ${\epsilon }_k$ less than ${10}^{-7}$, and at the next iteration, the root is found with an accuracy of ${10}^{-12}$. The basins of attraction for this example are plotted in Figure 3. As shown in the figure, starting from same initial point, the two methods converge to different roots. In this example, Schröder’s method again does not follow the ‘nearest root principal’.

Example 3 

(Batch distillation at infinite reflux [16]). In this example, we consider the following equation that describes the batch distillation of a binary mixture at infinite reflux

$$ln(1-x)-lnx-63ln(0.95-x)-6.6377125=0,$$

(32)

where x represents the mole fraction of one component of the mixture in the still. The equation (32) possesses the following simple roots ${\xi }_{1,2}=-0.0032378\pm 0.02603831i$. The basins of attraction of the roots of (32) are obtained by the methods N and S in the square

$$\{z\in \mathbb{C}:|Re\left(x\right)|\le 0.5\mathit{and}|Im\left(x\right)|\le 0.5\}$$

and the initial guess $x_0=0.169084-0.288865i$ is randomly taken from the same square.

One can see from

Table 3. Convergence results for Example 3.

Method	Root	k	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\epsilon }}_{\mathit{k}+1}$
N	$-0.003237-0.026038i$	4	$8.529\times {10}^{-8}$	$6.297\times {10}^{-14}$
S	$-0.003237+0.026038i$	5	$4.783\times {10}^{-7}$	$1.978\times {10}^{-12}$

that for this example, N has better convergence behavior than S. For instance, the root is found by N only at the fourth iteration with ${\epsilon }_k$ less than ${10}^{-7}$, and at the next iteration the root is found with accuracy ${10}^{-13}$. The basins of attraction for this example are plotted in Figure 4. Ones again, the two methods converge to different roots starting from same initial point and S does not follow the ‘nearest root principal’.

In order to show some limitations of Schröder’s method, in the following example, we consider a nonlinear equation arising from a known biomedical engineering problem ([17], Chapter 4).

Example 4 

(Blood rheology [18,19]). Consider the irrational equation

$${\displaystyle \frac{1}{21}}{x}^4-{\displaystyle \frac{4}{3}}x+{\displaystyle \frac{16}{7}}\sqrt{x}+0.6=0$$

(33)

with zeros ${\xi }_{1,2}=-2.400167\pm 2.205360i$ and ${\xi }_{3,4}=1.919730\pm 1.470124i$. As in [12], the basins of attraction for this example are obtained in the square

$$\{z\in \mathbb{C}:|Re\left(x\right)|\le 3\mathit{and}|Im\left(x\right)|\le 3\}$$

and the initial approximation $x_0=-0.2+0.4i$ is chosen to show a weakness of Schröder’s method.

Table 4. Convergence results for Example 4.

Method	Root	k	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\epsilon }}_{\mathit{k}+1}$
N	$1.919730+1.470124i$	17	$5.266164\times {10}^{-7}$	$1.702875\times {10}^{-13}$
S	Diverges

and Figure 5 exhibit the numerical results and the basins of attraction for this example. It is seen that S diverges while N shows good convergence and stability, no matter the ‘rough’ choice of the initial guess. In fact, the divergence areas of Schröder’s method are colored in dark blue and orange. The white points again depict the initial guess $x_0$ and the black one is the approximated zero.

5. Conclusions

An analysis of the local convergence of Schröder’s method (7) for finding the zeros of analytic functions with unknown multiplicity has been provided in this paper. A local convergence result (Theorem 1) that ensures exact domains of initial approximations with a priori and a posteriori error estimates to provide the Q-quadratic convergence of Schröder’s method (7) right from the first step has been presented. To the best of our knowledge, there is no such result regarding Schröder’s method in the literature. As consequences of our main theorem, a ball convergence theorem (Theorem 2) and a Smale-type theorem (Corollary 1) that provides a ball of approximate zeros of the second kind for Schröder’s method have been obtained. Note that, using Proinov’s approach [6], we can further extend our study to functions in Banach spaces.

Four numerical examples have been conducted to emphasize the dynamics and convergence of Newton and Schröder’s methods when applied to some polynomials with multiple zeros (Example 1), to a polynomial with close simple zeros (Example 2) and to two transcendental equations arising from chemical and biomedical engineering problems (Examples 3 and 4). The obtained numerical results (Table 1, Table 2, Table 3 and Table 4) and the generated dynamical planes (Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5) have shown that Schröder’s method (3) has better convergence and dynamical behavior than Newton’s method for multiple zeros (2), especially when the multiplicity of the zeros increases. On the other hand, Newton’s method (1) demonstrates the best dynamical behavior at the expense of its linear convergence. In addition, the examples have shown that Schröder’s method does not satisfy the ‘nearest root principal’ when applied to transcendental equations and even to some polynomial ones.