An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

Abstract

In this note a study of the convergence properties of some starters $E_0 = E_0(e,M)$ in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter $E_0(e,M)$ a set of values $(e,M) \in [0, 1) \times [0, \pi ]$ that lead to the q-convergence in the sense that Newton’s sequence $(E_n)_{n \ge 0}$ generated from $E_0 = E_0(e,M)$ is well defined, converges to the exact solution $E^* = E^*(e,M)$ of KE and further $\vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert$ holds for all $n \ge 0$. This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s $\alpha$-test with $q=1/2$. Also since in KE the convergence rate of Newton’s method tends to zero as $e \rightarrow 0$, we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with $q = e^k \; \widetilde{q}$ for all $e \in [0,1)$ and a fixed $\widetilde{q} \le 1$. Some remarks on the use of this theorem to derive a priori estimates of the error $\vert E_n - E^* \vert$ after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.

1 Introduction

The iterative solution of elliptic Kepler’s equation

$$\begin{aligned} E- e \; \sin E = M, \end{aligned}$$

(1)

where M is the mean anomaly (for periodicity in $[0,\pi ]$), $e \in [0,1)$ the eccentricity of an elliptic orbit and the unknown E the eccentric anomaly, has been the subject of an extensive list of references for a long time. Here we mention just a few of them (Smith 1979; Broucke 1980; Mikkola 1987; Colwell 1993; Markley 1995; Odell and Gooding 1986; Feinstein and McLaughlin 2006; Charles and Tatum 1998; Palacios 2002; Mortari and Clochiatti 2007; Dubinov and Galidakis 2007; Davis et al. 2010; Calvo et al. 2013; Avendaño et al. 2015; Calvo et al. 2017).

In many cases, taking a suitable starter $E_0 = E_0(e,M)$, Newton’s method is applied to the equation $f_e(E ; M) \equiv E- e \sin E - M = 0$ with the recursion

$$\begin{aligned} E_{k+1} = N (E_k) \equiv E_k - \dfrac{f_e (E_k; M) }{ f_e'(E_k; M)}, \quad k=0,1, \ldots \end{aligned}$$

(2)

where $N (\cdot ) = N_{f_e} ( \cdot )$ is the Newton’s iterator of $f_e$, to get a Newton’s sequence $(E_k)_{k \ge 0}$ that converges to the unique solution $E^* = E^*(e,M)$ of $f_e(E ; M)=0$. Then our aim is to make the small number n of iterations so that $E_n$ provides an approximation to $E^*$ with the desired accuracy. Observe that for $e=1$ and $E_k=0$ the recursion (2) becomes singular and therefore we expect to find some difficulties with the convergence when (e, M) is close to (1, 0).

First of all observe that the Newton’s iterator $N(E_k)$ defined in (2) is well defined for all $M \in [0,\pi ]$ and $e \in [0,1)$ for any $E_k$, then for all starter $E_0=E_0(e,M)$ the corresponding sequence $( E_n )_{n \ge 0}$ is well defined. However we cannot ensure their convergence to some $E^*$ that would be the unique solution $E^* = E^*(e,M)$ of $f_e(E ; M)=0$. In fact it has been shown in Charles and Tatum (1998) that with the standard choice $E_0(e,M)=M$, while there is convergence for $(e,M)= (0.991, 0.13 \pi )$ and for $(e,M)= (0.993, 0.13 \pi )$, Newton’s method seems not converge for $(e,M)= (0.992, 0.13 \pi )$. This implies that for a given starter $E_0(e,M)$ it is important to identify the set of $(e,M) \in [0,1) \times [0, \pi ]$ that ensures the convergence of Newton’s iterates.

There are some particular starters that lead to convergent Newton’s sequences. Thus if $0 \le f_e( E_0(e,M)) \le \pi$, taking into the monotonic increasing and convexity of $f_e(E ; M)$ with respect to $E \in [0, \pi ]$ for all $M \in [0,\pi ]$ and $e \in [0,1)$ we have $f_e(E^*(e,M) ; M )=0 \le f_e(E_0(e,M) ; M )$ which by monotony of $f_e$ implies $E^*(e,M) \le E_0(e,M)$. Now by the convexity of $f_e$ the sequence $(E_k)_{k \ge 0}$ decreases monotonically to $E^*$. Therefore $0 \le f_e( E_0(e,M)) \le \pi$ is a sufficient condition to ensure the convergence of Newton’s sequence. In particular for the simplest guess $E_0 = E_0(e,M) = \pi$ we have the desired monotone decreasing convergence.

A second point concerns the rate of convergence of a convergent Newton’s sequence $(E_k)_{k \ge 0}$ starting from $E_0 = E_0(e,M)$ as a function of $(e,M) \in [0,1) \times [0, \pi ]$. Of course Newton’s method is a second-order method and we have

$$\begin{aligned} \lim _{k \rightarrow \infty } \dfrac{| E_{k+1} - E^*|}{| E_{k} - E^*|^2} = \dfrac{ e \vert \sin (E^*) \vert }{ 2 ( 1 - e \cos (E^*))}, \end{aligned}$$

(3)

and this shows that the so-called quotient convergence factor (Ortega and Rheinboldt 1970, p. 282) given by the right-hand side of (3) may become arbitrary large for $(e,E^*)$ close to (1, 0), i.e., the singular point of the Newton’s iteration. Moreover this quadratic convergence only holds for $E_k$ in a sufficiently small neighborhood of $E^*$ that is not known in advance.

In general to study the convergence of Newton iterations starting from $x_0$ for solving a nonlinear system $F(x)=0$ with exact solution $x^*$, there are several results depending on the assumptions of F around $x_0$. Some well known are the Newton–Kantorovich’s (Kantorovich and Akilov 1982) and Smale’s (1986) theorems revised more recently in Dedieu (2006) and Argyros et al. (2014). These results depend on the measure of the error in the approximation to the exact solution $x^*$. Some frequently used measures are

$$\begin{aligned} ed(x_n) = \Vert x_{n+1}- x_n \Vert , \end{aligned}$$

(4)

that appears, e.g., in the original paper of Smale (1986) and

$$\begin{aligned} eg(x_n) = \Vert x^*- x_n \Vert , \qquad es(x_n) = \Vert F'(x_0)^{-1} F(x_n) \Vert , \end{aligned}$$

(5)

that are two natural measures: the first one in the sense of the global error and the second one considered by Smale in Blum et al. (2010).

Here for a starter $E_0 = E_0(e,M)$ of KE $f_e(E ; M) \equiv E- e \sin E - M = 0$ well defined for all $(e,M) \in [0,1) \times [0, \pi ]$ we introduce the following

Definition 1

The starter $E_0$ of KE is q-convergent for $(e,M) \in [0,1) \times [0, \pi ]$ if the Newton’s sequence $(E_n)_{n \ge 0}$ defined by (2) converges to some $E^*$, the unique solution of $f_e(E ; M)=0$, and there exists some $q \in [0,1)$ such that the following estimate holds

$$\begin{aligned} \vert E_n - E^* \vert \le q^{2^n -1} \; \vert E_0 - E^* \vert \quad n=0,1,2, \ldots \end{aligned}$$

(6)

Moreover for a given $q \in [0,1)$ the set of q-convergent values $(e,M) \in [0,1) \times [0, \pi ]$ will be called the q-convergence set of the starter $E_0 = E_0(e,M)$.

Note that for $q=1/2$, $E_0$ is an approximate zero of KE in Smale’s notation (Smale 1986; Avendaño et al. 2014).

The aim of this paper is to use a version of Newton–Kantorovich theorem given by Xinghua (1999) that obtains optimal error estimates of Newton’s sequence to derive sufficient conditions of q-convergence of a starter $E_0 = E_0(e,M)$ for KE and the corresponding q-convergence set. Note a similar study with $q=1/2$ has been carried out in Avendaño et al. (2014) by using the so-called $\alpha$-test based on Smale’s theorem (Blum et al. 2010). Here it will be seen that for many starters the q-factor can be written in the form $q = e^k \; \widetilde{q}$ with some $0 \le \widetilde{q} < 1$ and $k \ge 0$ which, in view of (6), imply sharper error bounds for small eccentricities, whereas in Avendaño et al. (2014) the bound is uniform in the eccentricity. Further, other estimates (4)–(5) are derived that allow us a priori and a posteriori estimates of the error after n iterations. The paper is organized as follows: In Sect. 2 the original Wang Xinghua version of Newton–Kantorovich theorem as well as their particular case for Kepler’s equation are presented. In Sect. 3 the q-convergence properties of some well-known starters are studied including other error estimates. The paper ends with some conclusions on the use of these results in practical applications.

2 Error estimates in Newton’s iteration of Kepler’s equation

First of all we recall the version of Newton–Kantorovich theorem given by Xinghua (1999) in Theorem 4.3.

Theorem 1

Let X and Y be Banach spaces, D an open convex subset of X and $F: D \subset X \rightarrow Y$ be a Frechet differentiable nonlinear operator. Assume that $F'(x_0)$ is invertible for a given starting point $x_0 \in D$ and that there exist positive constants $\beta , L$ and $\lambda$ so that

• 1.
  $$\begin{aligned} \beta = \Vert F'(x_0)^{-1} \; F(x_0) \Vert . \end{aligned}$$

  (7)
• 2.
  $$\begin{aligned} \Vert F'(x_0)^{-1} \left( F'(y) - F'(z) \right) \Vert \le L \Vert y - z \Vert \qquad \hbox {for all} \; y,z \in D. \end{aligned}$$

  (8)
• 3.
  $$\begin{aligned} \lambda = L \beta < 1/2. \end{aligned}$$

  (9)

• 4. The closed ball of center $x_0$ and radius $r=( 1- \sqrt{1- 2 \lambda } )/L$, $B(x_0,r)= \{ x ; \Vert x - x_0 \Vert \le r \}$ satisfies $B (x_0,r) \subset D$.

Then $F'(x)$ is invertible in $B (x_0,r)$ and the sequence $\left( x_k \right) _{k \ge 0}$ given by

$$\begin{aligned} x_{k+1} = x_k - F'(x_k)^{-1} \; F(x_k), \quad k=0,1, \ldots \end{aligned}$$

remains in $B (x_0,r)$ and converges to some $x^* \in B (x_0,r)$ with $F(x^*) = 0$ and the following (optimal) estimates hold

$$\begin{aligned} eg(x_n) \le \dfrac{ q^{2^n -1}}{ \mu _n (q)} \; eg (x_0) \le \dfrac{ q^{2^n -1}}{ \mu _{n-1} (q^2)} \; ed (x_0) \end{aligned}$$

(10)

and

$$\begin{aligned} \begin{aligned} \dfrac{ ed (x_n)}{ \xi (q^{2^n}) } \le eg (x_n)&\le \dfrac{ q}{ \beta }\; \mu _{n-1} (q^2) \; ed (x_{n-1})^2 \le q^{2^n -1} \; ed (x_{n-1}) \end{aligned} \end{aligned}$$

(11)

where q, and the functions $\mu _n$ and $\xi$ are given by

$$\begin{aligned} q= & {} q ( \lambda ) =\dfrac{ 1- \sqrt{1- 2 \lambda }}{1 + \sqrt{1 - 2 \lambda }}\nonumber \\= & {} \dfrac{ 2 \lambda }{(1 + \sqrt{1 - 2 \lambda })^2},\; \lambda = \dfrac{ 2 q}{(1+q)^2}, \end{aligned}$$

(12)

$$\begin{aligned} \mu _n (\omega )= & {} \sum _{i=0}^{2^n -1} \omega ^i = \dfrac{ 1 - \omega ^{2^n}}{1- \omega } \ge 1 \quad (0 \le \omega <1),\nonumber \\ \xi (\omega )= & {} \dfrac{1 + \sqrt{ 1 + 4 \omega /(1+ \omega )^2}}{2}. \end{aligned}$$

(13)

Note that according to Eq. (12) when $\lambda \in (0, 1/2)$ the corresponding $q \in (0,1)$. Hence to have $q=1/2$ we must take $\lambda = 4/9$ and for $q=1/10$ we must take $\lambda = 20/121$. Further observe that for $\omega \in [0,1]$ the function $\xi (\omega )$ is monotonic increasing and therefore

$$\begin{aligned} 1= \xi (0)< \xi ( \omega )< \xi (1) = \dfrac{1+ \sqrt{2}}{2}, \quad \hbox {for} \quad 0< \omega < 1. \end{aligned}$$

In the case of KE, $D = X = Y = \mathbb {R},$ $x = E$ and

$$\begin{aligned} F(x) = f_e(E ;M) \equiv x - e \sin x - M , \quad \hbox {with} \; e \in [0,1), \; M \in [0, \pi ] \end{aligned}$$

(14)

is analytic for all $x \in \mathbb {R}$.

Denoting the starting value for the solution of (14) by $x_0= E_0 =E_0(e ,M)$ the constants $\beta$ and L of (7), (8) are

$$\begin{aligned} \beta = \dfrac{ \vert E_0 - e \sin E_0 - M \vert }{ 1 - e \cos E_0}, \quad L = \dfrac{e}{ 1 - e \cos E_0 }, \end{aligned}$$

(15)

and therefore for each $E_0$, $\lambda = \lambda ( E_0)$ is a function of e and M defined by (9) and given by

$$\begin{aligned} \lambda =\lambda (E_0) = \dfrac{ e \vert E_0 - e \sin E_0 - M \vert }{ (1 - e \cos E_0)^2} = \dfrac{ e \vert E_0(e, M) - e \sin (E_0(e, M)) - M \vert }{ (1 - e \cos (E_0(e, M)) )^2}. \end{aligned}$$

(16)

Hence applying Theorem (1) to the KE: $f_e(E ; M)=0$ we have

Proposition 1

Let $E_0 = E_0(e,M)$, be a given starter for solving KE , then for all $\nu \in (0,1/2)$ the set

$$\begin{aligned} \varSigma _q (E_0) = \left\{ (e,M)\in [0,1)\times [0,\pi ] ; \lambda (E_0)\le \nu <1/2 \right\} \end{aligned}$$

(17)

with $q = q(\nu )$ given by (12) is contained in the q-convergence set of $E_0$. In addition the following estimates hold

$$\begin{aligned}&\displaystyle | E^* - E_n | \le \dfrac{q^{2^n -1} }{ \mu _n(q)} \; | E^* - E_0 | \le q^{2^n -1} \; | E^* - E_0 |, \end{aligned}$$

(18)

$$\begin{aligned}&\displaystyle | E^* - E_n | \le \dfrac{ q^{2^n -1}}{ \mu _{n-1}(q^2)} \; | E_1 - E_0 | \le q^{2^n -1} \; | E_1 - E_0 |, \end{aligned}$$

(19)

$$\begin{aligned}&\displaystyle | E^* - E_n | \le q^{2^n -1} \; | E_n - E_{n-1} | , \end{aligned}$$

(20)

$$\begin{aligned}&\displaystyle | E_{n+1} - E_n | \le \xi \left( q^{2^n}\right) \; | E^*- E_{n} | \le \left( \dfrac{1+ \sqrt{2}}{2} \right) \; | E^*- E_{n} |. \end{aligned}$$

(21)

Next we will derive some consequences that follow from the above Proposition: First of all for a given starter $E_0$ the set $\varSigma _{q}$ ($E_0$) gives a useful information about the values of eccentricity and mean anomaly that satisfy the above bounds (18) to (21) and therefore on the q-convergence with this factor.

Secondly in view of (16) there is a $k \ge 1$ such that

$$\begin{aligned} \lambda (E_0) = e^k \; \lambda _k (E_0), \quad \hbox {with} \; \left. \lambda _k (E_0)\right| _{e\rightarrow 0} \ne 0. \end{aligned}$$

(22)

Then $q(\lambda )$, for $\lambda \in [0, 1/2]$ given by (12) can be written as

$$\begin{aligned} q ( \lambda )= & {} \dfrac{ 2 \lambda }{ ( 1 + \sqrt{1- 2 \lambda })^2} = e^k \; \dfrac{ 2 \lambda _k }{ ( 1 + \sqrt{1- 2 e^k \; \lambda _k })^2}\nonumber \\\le & {} e^k \; \dfrac{ 2 \lambda _k }{ ( 1 + \sqrt{1- 2 \; \lambda _k })^2} = e^k \; q(\lambda _k). \end{aligned}$$

(23)

Thus for $0< \rho <1$ we may define the sets

$$\begin{aligned} \widetilde{\varSigma }_{\rho } (E_0) = \left\{ (e,M)\in [0,1)\times [0,\pi ] ; \; q ( \lambda _k ) \le \rho < 1 \right\} , \end{aligned}$$

(24)

so that for the starter $E_0 = E_0( e , M)$ if $(e,M) \in \widetilde{\varSigma }_{\rho } (E_0)$ we have q-convergence with the factor $q = e^k \; \rho$ for all $e \in [0,1)$ and now the q-convergence factor tends to zero as $e^k$, in contrast with the information derived from the sets $\varSigma _q (E_0)$ in which the convergence is uniform in the whole set.

Further observe that the inequality (18) bounds the global error in the nth iteration $eg (E_n) = | E^* - E_n |$ in terms of the global error in the starting value $eg (E_0) = | E^* - E_0 | \le \pi$. Clearly the factor $q^{2^n -1}$ with $0< q <1$ implies the desired q-convergence and it can be considered as an a priori error bound after a given number n of iterations. Also if we want $| E^* - E_n | \le Etol$, it is enough to select $n>0$ so that $\delta _n = \delta _n (e,M)= q^{2^n -1} \; | E_1 - E_0| \le Etol$. For example, since $| E^* - E_0 | \le \pi$ for all $E_0$, if $(e;M) \in \varSigma _{1/10} (E_0),$ after four Kepler’s iterations ($n=4$) we can ensure a global error $|E^*-E_4| \le \pi \times 10^{-15}$.

The inequality (20) provides an additional information with a more practical value because $| E_n - E_{n-1} |$ is available along the iterative process and then allows us to choose dynamically the number of iterations depending on our accuracy requirements. Thus if we want $|E^* - E_n| \le Etol$, it is enough to check $q^{2^n -1} |E_n - E_{n-1} | \le Etol$. Finally, (21) provides a two-sided error bound of the global error $| E^* - E_n |$.

3 The convergence of some starters

Next we examine the convergence properties of some well-known starters in the literature $E_0 = E_0(e;M)$ obtaining the sets $\varSigma _q (S_0) \subset [0,1) \times [0, \pi ]$ that ensure the convergence bounds (18)–(21) for $\nu \le 1 /2.$ We have included the value $\nu = 1/2$ corresponding to $q(1/2)=1$ because this is the boundary of q-convergence ensured by Proposition 1. In particular we consider the values of $\nu = 1/2, 4/9, 8/25, 20/121$ that correspond to the q-factors $q=1, 1/2, 1/4, 1/10$, respectively. Note that $q=1/2$ is the corresponding to Smale $\alpha$-test. For some starters we have used the notations $S_j$ employed by Odell and Gooding (1986) in Table I.

3.1 The starter $E_{0}= \pi$

This is one of the simplest starters considered in the literature and by (2) the first Newton’s iteration gives $N_{f_e} (\pi ) = ( M + \pi e)/(1+e) \in [0,\pi ]$ that has a low computational cost (does not require the computation of trigonometric functions or roots). Because of this we will study this starter from the first iteration, i.e.

$$\begin{aligned} E_0 = E_0(e;M) = \dfrac{ M+ \pi e}{ 1+e}, \end{aligned}$$

(25)

instead of the above $E_{0}= \pi$. Now the function $\lambda (E_0)$ of (16) becomes

$$\begin{aligned} \lambda (E_{0}) = \lambda (E_{0}) (e,M) = \dfrac{ e^2 \left[ \pi - (1+e) \sin (( M + \pi e)/(1+e)) - M \right] }{ (1+e) (1- e \cos (( M + \pi e)/(1+e) )^2}, \end{aligned}$$

and for $\nu <1/2$ the set $\varSigma _q(E_0)$ is defined implicitly by

$$\begin{aligned} \varSigma _q (E_{0}) = \left\{ (e,M) \in [0,1) \times [0, \pi ] \; ; \; \lambda (E_{0})(e,M) \le \nu < 1/2 \right\} , \end{aligned}$$

(26)

where $q = q(\nu )$ is given by (12).

In Fig. 1 we display the boundaries of the sets $\varSigma _q (E_{0})$ for $\nu =1/2, 4/9, 8/25, 20/121$ corresponding to the q-factors $q=1, 1/2, 1/4, 1/10$. Obviously $q_1 < q_2 \le 1$ implies that $\varSigma _{q_1} (E_{0}) \subset \varSigma _{q_2} (E_{0})$. Here the sets $\varSigma _q (E_{0})$ have the points $(e,M) \in [0,1) \times [0,\pi ]$ above the corresponding boundaries.

From the Proposition 1 it follows the q-convergence of starter (25) with factor $q < 1$ for all (e, M) above the boundary of $\varSigma _1 (E_{0})$. In particular for all $e < 0.461359= e_1$ this property holds for all M and this implies that solving KE along an orbit with eccentricity $e < e_1$ we have q-convergence for all values of the mean anomaly. Here $e_1$ is the unique root of $\lambda (E_0) (e, M=0) =1/2$ in (0, 1). Similar conclusions follow from the boundaries of the other sets $\varSigma _q (E_{0})$. Thus, in the case of $q=1/2$ for $e < 0.42019$ (the root of $\lambda (E_0) (e, M=0) =4/9$) we have q-convergence with factor $q \le 1/2$. On the other hand for $M > 0.155763$ we have q-convergence with factor $q < 1$ for all values of the eccentricity.

Fig. 1

Boundaries of the $\varSigma _q(E_0)$ sets of starter (25) for the values of factor $q=1,1/2,1/4,1/10$. The sets $\varSigma _q$ are the points $(e,M) \in [0,1) \times [0,\pi ]$ above the corresponding boundaries

Some relevant consequences for small eccentricities can be derived from the scaled q-convergence sets

$$\begin{aligned} \widetilde{\varSigma }_{\rho } ( E_0) = \Big \{ (e,M) \in [0,1) \times [0,\pi ] \; ; \; \widetilde{q} ( \lambda _2 (E_0)) \le \rho \le 1 \Big \}. \end{aligned}$$

Here

$$\begin{aligned} \lambda _2 (E_0)= \lambda _2(E_0) (e,M) = \dfrac{ \left[ \pi - (1+e) \sin (( M + \pi e)/(1+e)) - M \right] }{ (1+e) (1- e \cos (( M + \pi e)/(1+e) )^2}. \end{aligned}$$

In Fig. 2 we display the boundaries of the scaled $\rho$-convergence sets for $\rho = 1, 1/2,1/4,1/10$. These scaled sets include the points of $\widetilde{\varSigma }_{\rho } (E_{0})$, i.e., $(e,M) \in [0,1) \times [0,\pi ]$ above the corresponding boundaries in which $q = e^2 \; \rho$. From this Fig. 2 it follows that in the case of $\rho =1/2$ for $M > 1.2$ we have q-convergence with factor $q < (1/2) \;e^2$.

Fig. 2

Boundaries of the scaled sets $\widetilde{\varSigma }_{\rho }$ of starter (25) for the values $\rho =1,1/2,1/4,1/10$. The $\rho$-convergence scaled sets $\widetilde{\varSigma }_{\rho }$ are the points $(e,M) \in [0,1) \times [0,\pi ]$ above the corresponding boundaries in which $q = e^2 \; \rho$

To illustrate the upper bound (20) of the error $|E^* - E_n|$ in the nth Newton’s iteration as a function of the last two iterations in a region of q-convergence, we consider the solution of an elliptic orbit with eccentricity $e=0.3$. In Fig. 3 we display for $M \in [0, \pi ]$ the values of Ex of the upper bound $q^{2^n-1} |E_n- E_{n-1}|= 10^{-Ex}$ for eccentricity $e=0.3$ and $n=2,3,4,5$ (here Ex is limited between 0 and 60). Observe that for eccentricity $e=0.3$ all values of M are included in the q-convergence region $\varSigma _{1/4} (S_0)$; therefore, we take $q=1/4$ in the above upper bound.

Fig. 3

Bounds $Etol = 10^{- Ex}$ of the errors $| E^* -E_n|$ given by Prop. 1 for $e=0.3$ and $n=2,3,4,5$

Fig. 4

Level curves indicate the positive decimal logarithm of the error bound $q^{2^n -1} \; | E_1 - E_0 |$ of $\vert E^*-E_n\vert$ for the starter (25) for $n=2,\ldots ,4$ iterations of Newton’s method. Here q is defined by (12) with $\lambda$ given by (16)

Finally, in Fig. 4, we display the number of correct figures in the error after $n=1,2,3,4$ iterations of Newton’s method. The quadratic convergence of Newton’s method can be appreciated because from an iteration to the next one it can be noted that approximately the same level curve duplicates their value. These figures show the regions in $(e,M) \in [0,1)\times [0,\pi ]$ with a number of correct digits in the error $\vert E^*-E_n\vert$ after n iterations.

3.2 The starter $E_{0}= 0$

Now the first Newton’s iteration gives $N_{f_e} (0) = M/(1-e)$ that has a low computational cost. Then, as in the previous starter, we will consider instead of $E_0=0$ the starter after the first iteration given by

$$\begin{aligned} E_0 = \min \{ M/(1-e) , \pi \} = \left\{ \begin{array}{ll} M/(1-e) &{} \quad \hbox {for} \; M < \pi (1-e) ,\\ \pi &{} \quad \hbox {otherwise} \end{array} \right. \end{aligned}$$

(27)

and the q-convergence regions are defined by (17)

In Fig. 5 we display the boundaries of the q-convergence regions $\varSigma _q (E_{0})$ for $\nu =1/2, 4/9, 8/25, 20/121$ corresponding to the q-factors $q=1, 1/2, 1/4, 1/10$. Here the convergence regions are the sets of $(e,M) \in [0,1) \times [0,\pi ]$ outside the corresponding boundaries. A study of scaled q-convergence sets and error bounds as in the previous starter is skipped for brevity.

Fig. 5

Boundaries of the $\varSigma _q(E_0)$ sets of starter (27) for $q= 1, 1/2, 1/4, 1/10$. The sets $\varSigma _q$ are the points of $(e,M) \in [0,1) \times [0,\pi ]$ on the left of the corresponding boundaries

3.3 The starter $S_1$: $E_0= M$

Now the function $\lambda (S_1)$ of (17) becomes

$$\begin{aligned} \lambda (S_1) = \dfrac{ e^2 \sin M }{ (1-e \cos M )^2}, \end{aligned}$$

and the sets of q-convergence derived from Proposition 1 can be defined explicitly as a function of $\nu < 1/2$ by

$$\begin{aligned} \varSigma _q (S_1) = \Big \{ (e,M) ; 0 \le e \le \min \left\{ 1, \sqrt{\nu }/(\sqrt{\sin M} + \sqrt{\nu }\cos M) \right\} ,\quad M \in [0, \pi ] \Big \}. \end{aligned}$$

(28)

In Fig. 6 we display the boundaries of the sets $\varSigma _q (E_{0,1})$ for $\nu =1/2, 4/9, 8/25, 20/121$ corresponding to the q-factors $q=1, 1/2, 1/4, 1/10$. Here the convergence regions are the sets of $(e,M) \in [0,1) \times [0,\pi ]$ between the corresponding two boundaries.

Fig. 6

Boundaries of the $\varSigma _q(E_0)$ sets for several values of q of starter $E_0 = M$

Note that as remarked in Odell and Gooding (1986), for $e>0.9733$ there is a range of values of M for which Newton’s iterations diverge.

3.4 The starter $S_2$: $E_0= M + e \sin M$

Now the function $\lambda (S_2)$ of (17) becomes

$$\begin{aligned} \lambda (S_2) = \dfrac{ e^2 | \sin M - \sin E_0 |}{ (1-e \cos E_0 )^2}, \end{aligned}$$

and for $\nu < 1/2$ the set $\varSigma _q(E_0)$ is defined implicitly by

$$\begin{aligned} \varSigma _q (S_2) = \left\{ (e,M) \in [0,1) \times [0, \pi ] \,\,\, \Big | \,\,\, \dfrac{ e^2 | \sin M - \sin E_0 |}{ (1-e \cos E_0 )^2} \le \nu \right\} \end{aligned}$$

(29)

with $E_0=M + e \sin M$. In Fig. 7 we display the boundaries of the sets of (29) for $q=1, 1/2, 1/4, 1/10$.

Fig. 7

$\varSigma _q(E_0)$ sets of starter $S_2$: $E_0 = M + e \sin M$ for $q=1,1/2,1/4,1/10$. Here the sets $\varSigma _q$ include the points $(e,M) \in [0,1) \times [0,\pi ]$ under the upper boundary and on the left of the lower boundary

3.5 The starter $S_4$: $E_0= M + e$

Now the function $\lambda (S_4)$ of (17) becomes

$$\begin{aligned} \lambda (S_4) = \dfrac{ e^2 ( 1 - \sin (M+e) )}{ (1-e \cos (M+e) )^2}, \end{aligned}$$

and the set $\varSigma _q$ is defined implicitly by

$$\begin{aligned} \varSigma _q (S_4) = \left\{ (e,M) \in [0,1) \times [0, \pi ] \,\,\, \Big | \,\,\, \dfrac{ e^2 ( 1 - \sin (M+e) )}{ (1-e \cos (M+e) )^2} \le \nu <1/2 \right\} . \end{aligned}$$

(30)

In Fig. 8 we display the boundaries of the q-convergence sets (30) for $q=1, 1/2, 1/4, 1/10$.

Fig. 8

Boundaries of the $\varSigma _q(E_0)$ sets of starter $S_4$: $E_0 = M + e$ for $q=1,1/2,1/4,1/10$. Here the sets $\varSigma _q$ include the points $(e,M) \in [0,1) \times [0,\pi ]$ under the upper boundary and above of the lower boundary

3.6 The starter $S_7$

Here the function $E_0= E_0(e,M)$ is given by

$$\begin{aligned} E_0(e,M)= \hbox {Min}\;\left\{ \dfrac{M}{1-e}, \quad M+e , \quad \dfrac{M + e \pi }{1+e} \right\} , \end{aligned}$$

(31)

and it is a piecewise linear function of the mean anomaly.

Taking into account that $\pi \ge E_0(e,M) \ge E^* (e,M)$ the Newton’s sequence starting from $E_0$ is monotonically decreasing and convergent for all $(e,M) \in [0,1) \times [0 , \pi ]$.

Concerning the rate of convergence, in Fig. 9 we display the boundaries of the q-convergence sets $\varSigma _q(S_7)$ for $q=1, 1/2, 1/4, 1/10$. Here the corresponding sets are limited between the upper and lower lines.

Fig. 9

Boundaries of $\varSigma _q(E_0)$ sets of starter $S_7$: (31) for $q=1, 1/2, 1/4, 1/10$ . The corresponding sets are limited between the upper and lower boundaries

3.7 The starter of Charles and Tatum (1998)

This starter is given by

$$\begin{aligned} E_0= E_0(e,M) = M + e \left( ( \pi ^2 M)^{1/3} - \left( \dfrac{\pi }{15}\right) \sin M - M \right) . \end{aligned}$$

(32)

It is an empirical starter that has been obtained by means of some systematic trial and error and correction by graphical means. The authors claim that in the range $e \in [0.991,1]$ and $M \in [0.001, 0.1]$ convergence to nine decimal places was achieved in four Newton’s iterations. In Fig. 10 we display the $\varSigma _q(E_0)$ sets corresponding to the values $\nu = 4/9$ $(q=1/2)$ and $\nu = 20/121$ in which $q=1/10$.

Fig. 10

Boundaries of $\varSigma _q(E_0)$ sets of starter (32) for $q=1, 1/2, 1/4, 1/10$. Here the $\varSigma _q$ sets include all points $(e,M) \in [0,1) \times [0, \pi ]$ outside the corresponding boundaries

3.8 Broucke’s starter (1980)

This is a piecewise linear function in the mean anomaly M with coefficients depending on the eccentricity that is given by

$$\begin{aligned} E_0(e,M) = \left\{ \begin{array}{ll} \gamma (-e) M, &{} \quad \hbox {for} \; M \in [0, 1-e),\\ \xi (-e) M + (e/2), &{}\quad \hbox {for}\; M \in [1-e, \pi /2-e),\\ \gamma (e) M + \nu _1 (e), &{}\quad \hbox {for}\; M \in [\pi /2-e, \pi -e-1),\\ \xi (e) M + \nu _2 (e) , &{}\quad \hbox {for}\; M \in [\pi -e-1, \pi ],\\ \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} \begin{array}{ll} \gamma (e) = \dfrac{ 2 \pi + (2+ \pi ) e}{ (e+1)(4 e + 2 \pi )}, &{} \qquad \xi (e) = \dfrac{ \pi + e}{ \pi + 2 e}, \\ \nu _1 (e) = \dfrac{ e ( 3 \pi + 2 e)}{ 2 \pi + 4 e}, &{} \qquad \nu _2 (e)= \dfrac{ e \pi ( 2 + 4 e + \pi )}{ (e+1)(4 e + 4 e + 2 \pi )} . \end{array} \end{aligned}$$

As it has been shown in our paper Calvo et al. (2013), this is globally a good starter in the sense of the $\Vert \cdot \Vert _2$ and $\Vert \cdot \Vert _1$ norms of the error in $e \in [0,1),$ and $M \in [0, \pi ]$.

In Fig. 11 we display the q-convergence sets for $q=1, 1/2, 1/4, 1/10$.

Fig. 11

Boundaries of $\varSigma _q(E_0)$ of Broucke’s starter for $q=1, 1/2, 1/4, 1/10$. Here the $\varSigma _q$– sets include all points $(e,M) \in [0,1) \times [0, \pi ]$ outside the corresponding boundaries

3.9 The starter of Calvo et al. (2013)

The authors of the present paper have derived a starter that minimizes in a global sense the error for $e \in [0,1)$, $M \in [0, \pi ]$. This starter is a piecewise linear function in the mean anomaly given by

$$\begin{aligned} E_0(e,M)= \left\{ \begin{aligned}&p_1 M + q_1, \quad M \in [M_0,M_1] ,\\&p_2 M + q_2, \quad M \in [M_1,M_2] ,\\&p_3 M + q_3, \quad M \in [M_2,M_3], \end{aligned} \right. \end{aligned}$$

(33)

with $M_0=0$, $M_1= 1 - e$, $M_2 = \pi -1 -e$, $M_3 = \pi$ and

$$\begin{aligned} \begin{aligned}&p_1 = \dfrac{\eta _1}{1-e}, \quad p_2 =\dfrac{ \eta _2}{ \pi -2}, \quad p_3= \dfrac{ \pi - \eta _1 - \eta _2}{1+e}, \\&q_1 = 0, \quad q_2= \eta _1 + \dfrac{ (e-1) \eta _2}{ (\pi -2)}, \quad q_3 = \dfrac{ \pi (\eta _1 + \eta _2) + \pi (1-e - \pi )}{ 1+e}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\eta _1 = \dfrac{a_0 + a_1 e}{1+ a_2 e},\quad a_0=1,\quad a_1=-0.633589, \quad a_2=-0.564096, \\&\eta _2 = \dfrac{ b_0 + b_1 e}{ 1+ b_2 e},\quad b_0 = \pi -2,\quad b_1= -0.860154,\quad b_2 = -0.777978. \end{aligned} \end{aligned}$$

Next, in Fig. 12 we display the q-convergence sets for $q=1, 1/2, 1/4, 1/10$.

Fig. 12

Boundaries of $\varSigma _q(E_0)$ sets of CEMR’s starter in the (e, M) plane for $q=1, 1/2, 1/4, 1/10$

3.10 The starter of Markley

Markley (1995) proposed a very accurate starter $S_{M}$ with a quite large computational cost that combined with a fifth-order iterative method gives in one iteration a maximum relative error less than $10^{-18}$ when roundoff errors and double precision are properly addressed.

In this starter $E = E_0$ is the exact solution of the cubic equation

$$\begin{aligned} \left[ 3 (1-e) + \alpha e \right] \; E^3 - 3 M \; E^2 + 6 \alpha (1-e) \; E - 6 \alpha M = 0, \end{aligned}$$

(34)

where $\alpha$ is, in principle, a specified constant. Here we have chosen the value

$$\begin{aligned} \alpha = \dfrac{ 3 \pi ^2}{ (\pi ^2 -6)}, \end{aligned}$$

(35)

although a non-constant improved value of $\alpha$ is

$$\begin{aligned} \alpha (e, M) = \dfrac{ 3 \pi ^2 + 1.6\, \pi ( \pi - M)/(1+e)}{ \pi ^2 - 6}. \end{aligned}$$

(36)

Note that (34) arises by substituting in KE the function $\sin E$ by the Padé approximation

$$\begin{aligned} \dfrac{ E( 6 \alpha - ( \alpha -3) E^2)}{ 6 \alpha + 3 E^2}. \end{aligned}$$

From equation (34) it follows that

$$\begin{aligned} M = \dfrac{ ( 3 ( 1 - e) + \alpha e ) E_0^3 + 6 \alpha (1-e) E_0}{6 \alpha + 3 E_0^2}, \end{aligned}$$

(37)

and this implies that in the case of Markley starter the function $\lambda = \lambda (S_{M})$ of (16) can be written explicitly as a function of $(e, E_0) \in [0,1) \times [0, \pi ]$ instead of the variables $(e, M) \in [0,1) \times [0, \pi ]$ and is given by

$$\begin{aligned} \lambda (S_{M})= & {} \dfrac{ e | E_0 - e \sin E_0 -M |}{(1 - e \cos E_0)^2} = - e^2 \; \dfrac{ - 6 \pi ^2 E_0 + 6 E_0^3 + ( - 6 E_0^2 + \pi ^2 ( 6 + E_0^2)) \sin E_0}{( - 6 E_0^2 + \pi ^2 ( 6 + E_0^2))(1- e \cos E_0)^2}.\nonumber \\ \end{aligned}$$

(38)

Note that $E_0(e, \cdot ){:}\, [0,\pi ] \rightarrow [0, \pi ]$ is a one-to-one map. Because of this we will consider $\lambda (S_{M}) = \lambda (S_{M}) ( e, E_0)$ .

By using (37) we may check that $f_e (E_0;M) \ge 0$ for all $e \in [0,1)$, $E_0 \in [0, \pi ]$. Hence $E_0(e,M) \ge E^*(e,M)$ and Markley starter is a monotonic starter for all $e \in [0,1)$, $E_0 \in [0, \pi ]$.

It can be seen that

$$\begin{aligned} \sup \left\{ \lambda (S_{M}) ; \; (e, E_0) \in [0,1) \times [0, \pi ] \right\} = \lambda (S_{M}) (1, 2.20982) = 0.0171415 = \nu _{M}, \end{aligned}$$

and $q ( \nu _{M} ) = 0.00872089$; therefore, this starter is q-convergent with a factor $q < q ( \nu _{M} )$ for all $(e, M) \in [0,1) \times [0, \pi ]$.

In Fig. 13 we display the boundaries of the sets $\varSigma _q(S_M)$ for $q=5/1000,1/1000, 1/10000$. Here we have chosen much smaller values of q than in the above starters because in this case the starter is very accurate and Newton’s convergence is very fast.

Clearly the above behavior of the q-convergence together with the fact that

$$\begin{aligned} \sup \left\{ | N{f_e}(E_0) - E_0 |; \; (e, E_0) \in [0,1) \times [0, \pi ] \right\} = 0.0136618, \end{aligned}$$

implies a very fast convergence for all $(e, M) \in [0,1) \times [0, \pi ]$ for this starter. The calculation of scaled q-convergence regions and error bounds can be carried out as in the first starter.

Fig. 13

Boundaries of $\varSigma _q(E_0)$ sets $\varSigma _q(S_M)$ of Markley’s starter for $q=5/1000, 1/1000, 1/10000$. The corresponding regions are on the left side of the boundaries

3.11 The starter $S_{10}$

By substituting in KE $\sin E$ by their third-order truncated expansion at $E=0$ we have the cubic equation in the eccentric anomaly

$$\begin{aligned} P_e(E;M) = (1-e) E + e \; (E^3 / 6) - M = 0. \end{aligned}$$

(39)

Now in the starter $S_{10}$, $E_0 = E_0(e,M)$ is defined as the solution of (39) for $e \in [0,1)$,$M \in [0,\pi ]$.

First of all observe that $P_e(0;M)= -M \le 0$ and $P_e(\pi ;M) >0$ together with the fact that $P_e'(E,M) = (1-e)+ e E^2/2 >0$ implies that for all $e \in [0,1)$, $M \in [0,\pi ]$ the equation (39) has a unique solution. Further, for such a solution $E = E_0$ we have

$$\begin{aligned} f_e (E_0 ; M ) = e \Big [ \left( E_0 - \dfrac{E_0^3}{6} \right) - \sin E_0 \Big ] < 0, \end{aligned}$$

from the monotonic increasing and the convexity of $f_e$ with respect to E it follows that $E_0(e,M) < E^* (e,M)$ and the first Newton’s iteration $E_1 = N_f (E_0)> E^*$. Also it can be seen that $E_1 \le \pi$, so the remaining iterations decrease montoniquely to the exact solution $E^*(e,M)$.

Taking into account (39) for the $\lambda$-function (16) we get

$$\begin{aligned} \lambda (S_{10}) = - \dfrac{ e^2 ( - 6 E_0 + E_0^3 + 6 \sin E_0)}{ 6 ( 1 - e \cos E_0)^2}, \end{aligned}$$

(40)

where $E_0 = E_0(e,M)$ is defined by (39). In Fig. 14 we display the boundaries of the q-convergence sets $\varSigma _q ( S_{10})$ for $q=1, 1/2$

Fig. 14

Boundaries of $\varSigma _q(E_0)$ sets (in white) $\varSigma (S_{10)}$ around the point $(e,M) = (1,\pi )$

A remarkable fact already noted in Avendaño et al. (2014) is that these q-convergence regions include a neighborhood of the singular point $(e,M)=(1,0)$ in the Newton’s iteration in contrast with some other previous starters. This fact has been used by Avendaño et al. (2014) to construct a globally simple starter in the whole set $(e,M) \in [0,1) \times [0, \pi ]$ that satisfies the $\alpha$-test, i.e., is q-convergent with factor $q=1/2$.

Also it must be noticed that in Avendaño et al. (2014) it has been proved that $S_{10}$ satisfies the $\alpha$-test for all $(e,M) \in [0,1) \times [0, \pi ]$. However as can be seen in Fig. 14 the sufficient condition of q-convergence derived from Proposition 1 does not guarantee this property for $q=1/2$. In fact

$$\begin{aligned} \lambda (S_{10}) (1, \pi ) = \dfrac{ \pi ( \pi ^2 -6)}{ 24} > \dfrac{4}{9}, \end{aligned}$$

and by continuity the set $\varSigma _{1/2} ( S_{10})$ does not include a neighborhood of $(e,M)=(1, \pi )$. This implies that the sufficient conditions of q-convergence of $\alpha$-test and Proposition 1 can be different for the same starter.

4 Conclusions

By using a version of the Newton–Kantorovich theorem on the convergence of Newton’s method for the solution of implicit equations adapted to the KE, we show that it is possible to study the q-convergence properties of some well-known starters for solving KE. In particular for a given starter S we may determine the sets $\varSigma _q (S)$ of $(e,M) \in [0,1) \times [0, \pi ]$ in which we may ensure q-convergence with rate $q<1$. Also we may compute the scaled convergence sets in which the convergence rate has the form $q = e^k \; \widetilde{q}$ for $k \ge 1$ and therefore tends to zero as the eccentricity tends to zero. The theorem allows us to derive for each starter a priori and a posteriori error bounds on the iteration error depending on the number of iterations and (e, M). Thus we present here a number of tools that allow a practical user to choose the most efficient and accurate starter (or some combination of them) according to their particular requirements.