On the global convergence of Schröder’s iteration formula for real zeros of entire functions

Journal of Computational and Applied Mathematics 358 (2019) 136–145

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Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam

On the global convergence of Schröder’s iteration formula for real zeros of entire functions ∗

Hiroshi Sugiura a , , Takemitsu Hasegawa b a b

Department of Mechatronics, Nanzan University, Showa-ku, Nagoya, 466-8673, Japan Department of Information Science, University of Fukui, Fukui, 910-8507, Japan

article

info

a b s t r a c t Schröder’s formula of the second kind of order m of convergence (S2-m formula) is a generalization of Newton’s (m = 2) and Halley’s (m = 3) iterative formulae for finding zeros of functions. The authors showed that the S2-m formula of every odd order m ≥ 3 converges globally and monotonically to real zeros of polynomials on the real line. For Halley’s formula, such the convergence property for real zeros of entire functions had been shown by Davies and Dawson. In this paper, we extend both results by showing that the S2-m formula of every odd order m ≥ 5 has the same convergence property for real zeros of entire functions. By numerical examples we illustrate the monotonic convergence of the formula of odd order m = 3, 5 and 7 and the non-monotonic convergence of even order m = 2, 4 and 6. Further, we compare several formulae of both first and second kinds in performance. © 2019 Elsevier B.V. All rights reserved.

Article history: Received 20 September 2018 Received in revised form 7 January 2019 MSC: 65H04 65H05 Keywords: Root finding Nonlinear equation Schröder’s method Global and monotonic convergence Entire function

1. Introduction For finding zeros of functions f (z), many iterative methods have been developed, such as well-known Newton’s and Halley’s formulae (cf. [1, p. 246], [2, p. 230, p. 257]) with the iteration function (IF) K (f ; z) given, respectively, by K (f ; z) = x −

f (z) f ′ (z)

,

K (f ; z) = x −

2f (z)f ′ (z) 2{f ′ (z)}2

− f (z)f ′′ (z)

.

With an initial value z0 , the iteration formula is given by zk+1 = K (f ; zk )

(k = 0, 1, . . . ).

(1.1)

In a sufficiently close neighbourhood of a simple zero of f (z), Newton’s formula converges quadratically and Halley’s formula cubically. For entire functions of genus ≤ 2 with only real zeros, Davies and Dawson [3] give a remarkable result that Halley’s formula converges globally and monotonically (GM convergence) to zeros of the entire functions. Generalizations of both formulae to an arbitrary order of convergence are Schröder’s formulae of the first and second kinds [4] (cf. [5–8]), classic iteration formulae. Suppose that f (z) is a function analytic on the neighbourhood of a zero of f (z). Let ˆ f (z) = 1/f (z) and ˆ f (κ ) (z) the κ -fold derivative of ˆ f (z). Schröder’s formula of the second kind of order m (S2-m formula) or König’s formula (cf. [9]) has the IF Km (f ; z) (m ≥ 2) given by Km (f ; z) = z + ∆m (f ; z),

∆m (f ; z) := (m − 1)

ˆ f (m−2) (z) , ˆ f (m−1) (z)

∗ Corresponding author. E-mail addresses: [email protected] (H. Sugiura), [email protected] (T. Hasegawa). https://doi.org/10.1016/j.cam.2019.02.035 0377-0427/© 2019 Elsevier B.V. All rights reserved.

(1.2)

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137

Fig. 1. The iteration functions K4 (f ; x) (left) and K5 (f ; x) (right) for f (x) = sin π x. The curves y = K4 (f ; x) and y = x intersect at the zeros x = ξl = l (−∞ < l < ∞) of f (x). The curves y = K5 (f ; x) and y = x intersect at the zeros x = ξl and at the extraneous points ηl = l + 1/2 (−∞ < l < ∞), where ˆ f (3) (ηl ) = 0. They are interlaced with each other.

(cf. [10, p125]). See [8] for details on Schröder’s formulae. The S2-m formula with Km (f ; z) given by (1.2) is identical to Newton’s formula for m = 2 and Halley’s formula for m = 3. The authors [11] proved that for zeros of polynomials with only real zeros, the S2-m formula of every odd order m ≥ 3 enjoys the feature of the GM convergence. The two figures in Fig. 1 depict the behaviours of the S2-m formulae with the IFs K4 (f ; x) and K5 (f ; x) for f (x) = sin π x. The right figure suggests the GM convergence of the sequence {xk+1 = K5 (f , xk )}∞ k=0 of odd order m = 5. The left shows no GM convergence for the S2-m formula of even order m = 4. The remaining problem is to show theoretically, for zeros of entire functions with only real zeros, the GM convergence of the S2-m formula of every odd order m ≥ 3. In this note, we achieve this ultimate goal in Theorem 4.7. We establish Theorem 4.7 by exploiting the result on the GM convergence for polynomials proved in [11] in two stages. In the first stage, we prove the GM convergence for some analytic functions; see Theorem 3.7 in Section 3.1 and its proof in Section 3.2. In the second stage, we prove Theorem 4.7 by showing that the entire functions are contained in the family of the analytic functions. The rest of the paper is as follows. Section 2 reviews some results in [11] used to show the GM convergence for polynomials with only real zeros. In Section 3.1, we define a family of polynomials that yields a family of analytic functions as limiting functions of the polynomials. For the analytic functions, we state a theorem on the GM convergence of the S2-m formula of odd order m. In Section 3.2, we prove the theorem by applying the results in Section 2. In Section 4, we prove our main Theorem 4.7. In Section 5, numerical examples illustrate the GM convergence of the S2-m formula of odd order m = 3, 5 and 7. Further, we compare several formulae in performance, the rate of success in convergence, the number of iterations required and execution time on a computer. 2. Review of convergence results for polynomials In this section, we review some definitions and lemmas proved in [11] used to show the GM convergence of the S2-m formula of odd order m ≥ 3 to zeros of polynomials with only real zeros. Definition 2.1 (Global and Monotonic Convergence). An iteration method is said to converge globally and monotonically to zeros of a function f (x) on the real line, if the following condition is satisfied. Let · · · < αi−1 < αi < αi+1 < · · · be the sequence of all different zeros of f (x). Then, there exists an interlacing sequence {βi }, where βi−1 < αi < βi , with property that every iteration sequence {xk } starting from the initial value x0 ∈ (βi−1 , βi ) converges monotonically to αi . We define a family of polynomials p(x) with only real zeros. Let n and ν be positive integers such that 1 ≤ ν ≤ n and p(x) a polynomial of degree n with only real zeros α1 , . . . , αν given by p(x) = c

ν ∏

(x − αi )µi ,

i=1

ν ∑

µ i = n,

c ∈ R \ {0},

(2.1)

i=1

with positive integers µi (1 ≤ i ≤ ν ). We assume that α1 < α2 < · · · < αν . Let ˆ p(x) = 1/p(x). For an integer m ≥ 2 we consider the IF Km (p; x) (1.2) with p instead of f . Lemma 2.2.

The zero αi of p(x) in (2.1) is a stationary point of Km (p; x), namely

Km (p; αi ) = αi Lemma 2.3.

(1 ≤ i ≤ ν ).

Let κ be an even integer > 0. For p(x) in (2.1) we have

p(x) ˆ p(κ ) (x) > 0

(x ̸ ∈ {α1 , . . . , αν }).

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Lemma 2.4. Let m be an odd integer ≥ 3. Let βi be a zero of ˆ p(m−2) (x) on (αi , αi+1 ) for 1 ≤ i ≤ ν − 1. Further, set β0 = −∞ and βν = ∞. Then, for 1 ≤ i ≤ ν , we have x < Km (p; x) < αi

(βi−1 < x < αi ),

αi < Km (p; x) < x

(αi < x < βi ).

Remark 2.5. The zeros βi of ˆ p(m−2) (x) for i = 1, . . . , ν − 1 satisfying Km (p; βi ) = βi and p(βi ) ̸ = 0 are said to be extraneous fixed-points of Km (p; x); see [12]. Further, these fixed-points are said to be repulsive since dKm (p; βi )/dx > 1; see [9,12,13]. 3. Preliminaries: convergence for analytic functions In Section 2, we reviewed the GM convergence of the S2-m formula of odd order m ≥ 3 for real zeros of polynomials p(x) (2.1). Davies and Dawson [3] prove that Halley’s method (S2-m with m = 3) enjoys the same property for real zeros of the entire functions of genus 2 given by f (z) = z n exp(a + bz − cz 2 )

∞ ∏

(1 − z /ξl ) ez /ξl ,

(3.1)

l=1

n ∈ N0 := N ∪ {0},

ξl ∈ R (1 ≤ l < ∞),

a, b ∈ R , ∞ ∑

c ≥ 0,

|ξl |2 < ∞,

l=1

where the infinite product in (3.1) is locally uniformly convergent (cf. [14, p. 21]). Note that in (3.1) we denote by a and b constants and later by (a, b) an interval on the real line. This section gives preliminaries to achieve our goal that for real zeros of the entire functions (3.1), the S2-m formula of odd order m ≥ 5 shares with Halley’s method the common property, the GM convergence. We outline the present and next sections. In Section 3.1, we define a family of polynomials that leads to a family of analytic functions on a domain ⊂ C as limiting functions of the polynomials. In Section 3.2, using the lemmas given in Section 2, we prove the GM convergence of the S2-m formula of odd order m ≥ 3 for the analytic functions. In Section 4, we show that the family of analytic functions contains the entire functions (3.1), so achieve our goal. 3.1. A family of analytic functions We begin with a family of polynomials P (3.2) that is instrumental in defining a family of analytic functions F(D) on a domain D ⊂ C such that F(C) contains the entire functions given by (3.1). Let a family of polynomials P of the complex variable z ∈ C be defined by n { ∏

⏐ ⏐

}

(z − ζi )⏐ c ∈ R, ζi ∈ R, n ≥ 1 .

P= c

(3.2)

i=1

If for a polynomial p ∈ P with c ̸ = 0 , we restrict z ∈ R, then p can be expressed by (2.1) and satisfies all the results in Section 2. Using the polynomials ∈ P we define a family of analytic functions F(D) on a domain D ⊂ C as follows. Definition 3.1 (A Family of Complex Functions F(D)). For a complex domain D ⊂ C, let F(D) be a family of complex functions f on D such that there exists a sequence of polynomials pj ∈ P (j ≥ 0) locally uniformly convergent to f on D. Remark 3.2. The function f ∈ F(D) is an analytic function on the domain D. H. Cartan [15, p. 41] gives an important lemma on zeros of analytic functions. Lemma 3.3 (Cartan). If f is an analytic function on a domain D and if f is not identically zero, then the set of zeros of f is a discrete set (in other words, all the points of this set are isolated). Next, we give several notations and definitions followed by Lemma 3.4 on the number of zeros of functions. Assume that ε is an arbitrary positive number and α a point ∈ C. Suppose that A and B are sets ⊂ C. Then, we define some symbols as follows: 1. 2. 3. 4.

Denote Denote Denote Denote

by by by by

N(α, ε ) = {z | |z − α| < ε} a ε -neighbourhood of α (cf. [16, p. 73]). cl(A) the closure of A and by ∂ A the boundary of A. dist(α, B) = infβ∈B |α − β| the distance of α and B. dist(A, B) = infα∈A,β∈B |α − β| the distance of A and B.

Lemma 3.4 gives the relation of the number of zeros of fj and that of f = limj→∞ fj .

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139

Fig. 2. Arrangement of the zeros ξ and ξl (l = 1, 2, . . . ) and the boundaries ∂ D and ∂ U of the domains D and U = N(ξ , ε ), respectively.

Lemma 3.4. Let f be an analytic function defined on the domain D ⊂ C and not identically zero. Let ξ be a zero of multiplicity µ of f . Assume that a sequence of functions fj (j ≥ 0) analytic on D is locally uniformly convergent to f . Then, there exists a positive ε0 such that for arbitrary ε satisfying 0 < ε < ε0 there exists a positive integer M such that for every integer j > M the number of zeros of fj in the ε -neighbourhood N(ξ , ε ) of ξ is µ, counted with multiplicity. Proof. Let Z = {z |f (z) = 0} and set ε0 = dist(ξ , ∂ D ∪ (Z \ {ξ })). From Lemma 3.3 we see that ε0 > 0. For arbitrary ε satisfying 0 < ε < ε0 let U = N(ξ , ε ). Then, cl(U) ⊂ D since ε < ε0 ≤dist(ξ , ∂ D). So, f is an analytic function on cl(U). Further, we have Z ∩ ∂ U = ∅, since dist(Z , ∂ U) = min{dist(ξ , ∂ U), dist(Z \ {ξ }, ∂ U)} ≥ min{ε, ε0 − ε} > 0. Therefore, r := minz ∈∂ U |f (z)| > 0. Fig. 2 depicts the arrangement of the zeros ξ and ξl (l = 1, 2, . . . ) and the boundaries ∂ D and ∂ U of the domains D and U, respectively. From the assumption that {fj } (j ≥ 0) converges uniformly to f on the compact set ∂ U, we see that there exists a positive integer M such that for every j > M we have |fj (z) − f (z)| < r for z ∈ ∂ U. In view of this fact and Rouché’s theorem [16, p. 280], the number of zeros of fj in U is the same as those of f , that is, µ. □ Lemma 3.5.

The zeros of a function f ∈ F(D) not identically zero are all real.

Proof. The proof is by contradiction. Assume that f has an imaginary zero ξ of multiplicity µ ≥ 1. From Definition 3.1 and Lemma 3.4 there exists a positive ε0 and for arbitrary ε such that 0 < ε < min(ε0 , ℑξ ) and sufficiently large j, the number of zeros of pj on the ε -neighbourhood N(ξ , ε ) is µ. Since ε < ℑξ , we have N(ξ , ε ) ∩ R = ∅. This means that pj has a non-real zero, contradicting the fact that pj ∈ P. □ 3.2. Convergence for analytic functions In this subsection, we assume the following property on the real function f (x) on a real interval (a, b). Property 3.6. The real function f (x) can be analytically continued to f (z) on a domain D including (a, b) and f ∈ F(D). Further, f (x) has zeros on (a, b) but is not identically zero. We now state the primary result of this paper. Theorem 3.7. For functions f ∈ F(D) satisfying Property 3.6, the S2-m formula of every odd order m ≥ 3 in (1.1) with Km (f ; x) (1.2) converges globally and monotonically to a zero of f on (a, b). Proving Theorem 3.7 requires some preliminaries. Let {pj } ⊂ P be a sequence that converges to f locally uniformly on D. Lemma 3.3 implies that the set of zeros of f on an interval (a, b) is countable; denote the zeros on (a, b) by ξl , where

· · · < ξ−1 < ξ0 < ξ1 < · · · , and {ξl } has no accumulation point in (a, b). Therefore, when ξL = min{ξl } does not exist, inf{ξl } = liml→−∞ ξl = a. Similarly, when ξR = max{ξl } does not exist, sup{ξl } = liml→∞ ξl = b. Now, we summarize notations on the zeros of functions. Recall that we denote by ζi a zero of a polynomial p in (3.2), by {αi } (α1 < α2 < · · ·) mutually distinct zeros of p as defined in (2.1) and by {ξl } mutually distinct zeros of an analytic function f . Proving Theorem 3.7 needs the following lemmas and definition. Lemma 3.8 is crucial.

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Fig. 3. The interlacing of ξl and ηl when ηL−1 is a zero of ˆ f (m−2) (x) and sup{ξl } = b with the sign ± of f (x)ˆ f (m−2) (x).

Lemma 3.8.

Let κ be an even integer > 0. Then, f (x)ˆ f (κ ) (x) ≥ 0 for x ∈ (a, b) \ {ξl }. (κ )

(κ )

(κ )

Proof. In view of Lemma 2.3 we have pj (x)ˆ pj (x) > 0 or pj (x)ˆ pj (x) = ∞ for x ∈ R and since {pj (x)ˆ pj (x)} (j ≥ 0) converges to f (x)ˆ f (κ ) (x) on (a, b) \ {ξl }, we establish the lemma. □ Lemma 3.9 follows from Lemma 3.8 and suggests that ˆ f (m−2) (x) has zeros on (a, b). We prepare some definitions for Lemma 3.9. Let m be an odd integer ≥ 3. For the function f we define the lower and upper bounds ˆ a and ˆ b by

ˆ a = inf{x ∈ (a, b) \ {ξl } | f (x)ˆ f (m−2) (x) > 0}, ˆ b = sup{x ∈ (a, b) \ {ξl } | f (x)ˆ f (m−2) (x) < 0}, respectively. Lemma 3.9.

Let m be an odd integer ≥ 3. Then, the followings hold:

1. On (ξl , ξl+1 ) ⊂ (a, b), the function ˆ f (m−2) (x) has a unique zero ηl and f (x)ˆ f (m−2) (x) < 0

(ξl < x < ηl ), (m−2) ˆ f (x)f (x) > 0 (ηl < x < ξl+1 ).

(3.3) (3.4)

2. If there exists the minimum zero ξL = min{ξl }, then ˆ a < ξL and f (x)ˆ f (m−2) (x) > 0

(ˆ a < x < ξL ).

(3.5)

3. If there exists the maximum zero ξR = max{ξl }, then ξR < ˆ b and f (x)ˆ f (m−2) (x) < 0

(ξR < x < ˆ b).

(3.6)

Proof. We start by verifying (3.3) and (3.4). On the interval (ξl , ξl+1 ) the function f (x) > 0 or f (x) < 0. We now verify the case of f (x) > 0; the case of f (x) < 0 is verified similarly. In view of Lemma 3.8 and m − 1 being even we have ˆ f (m−1) (x) ≥ 0, therefore ˆ f (m−2) (x) is increasing on (ξl , ξl+1 ). Since ˆ f (m−2) (x) diverges at ξl and ξl+1 , we see that lim ˆ f (m−2) (x) = −∞,

x→ξl +0

lim

x→ξl+1 −0

ˆ f (m−2) (x) = ∞,

and ˆ f (m−2) (x) has a zero ηl on (ξl , ξl+1 ). In view of Lemma 3.3 and fˆ (m−2) (x) being not identically zero, ηl is isolated and a unique zero on (ξl , ξl+1 ). Thus, we verified (3.3) and (3.4). Next, we prove (3.5); similar arguments hold for (3.6). Assume that f (x) > 0 on the interval (a, ξL ); similar arguments hold for f (x) < 0. Then, in view of Lemma 3.8, ˆ f (m−1) (x) ≥ 0 and ˆ f (m−2) (x) is increasing on the interval. Since ˆ f (m−2) (x) (m−2) (m−2) ˆ ˆ (x) = ∞. It follows that there exists ε > 0 such that f (x)f (x) > 0 for diverges at ξL , we have limx→ξL −0 f x ∈ (ξL − ε, ξL ). Therefore, ˆ a ≤ ξL − ε < ξL and (3.5) is valid since ˆ f (m−2) (x) is increasing on (ˆ a, ξL ). □ Remark 3.10. If there exists the minimum zero ξL = min{ξl }, then ˆ f (m−2) (x) is increasing (if f (x) > 0) or decreasing (if f (x) < 0) on (a, ξL ). Hence, ˆ f (m−2) (x) has at most one zero on (a, ξL ). If there exists the unique zero ηL−1 , then ˆ a = ηL−1 , otherwise ˆ a = a. If ξL does not exist, then liml→−∞ ξl = a since {ξl } has no accumulation point on (a, b). This implies that ˆ a = a from (3.4). Similarly, if there exists the maximum zero ξR = max{ξl }, then ˆ f (m−2) (x) has at most one zero on (ξR , b). If there exists the unique zero ηR , then ˆ b = ηR , otherwise ˆ b = b. If ξR does not exist, then liml→∞ ξl = b and ˆ b = b from (3.3). Fig. 3 depicts the arrangement of {ηl } and {ξl } for the case a < ˆ a = ηL−1 < ξL < · · · < ˆ b = b and the sign (±) of f (x)ˆ f (m−2) (x). Corollary 3.11 follows immediately from Lemmas 2.2 and 2.4. Corollary 3.11. followings hold:

Let m be an odd integer ≥ 3. For a polynomial p ∈ P of degree n (n ≥ 1) let α be a zero of p. Then, the

Km (p; x) ≤ α

(a < x ≤ α ),

Km (p; x) ≥ α

(α ≤ x < b).

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141

Lemmas 3.12 and 3.13 are key lemmas to prove Theorem 3.7. Lemma 3.12. Let m be an arbitrary integer ≥ 2. Then, a zero ξ of f (z) is a removable singularity of Km (f ; z) and Km (f ; ξ ) = ξ . Proof. Assume that ξ is a zero of multiplicity µ ≥ 1 of f (z). Then, ξ is a pole of order µ + k of ˆ f (k) (z). Therefore, ξ is (m−2) (m−1) (m−2) (m−1) ˆ ˆ ˆ ˆ a removable singularity of f (z)/f (z) and f (ξ )/f (ξ ) = 0. It follows that ξ is a removable singularity of Km (f , z) and Km (f ; ξ ) = ξ . □ Lemma 3.13. Let m be an odd integer ≥ 3 and ηl the unique zero of ˆ f (m−2) (x) on (ξl , ξl+1 ). Further, set ηL−1 = ˆ a if ξL = min{ξl } exists and ηR = ˆ b if ξR = max{ξl } exists. Then, for every ξl and for Km (f , x) in (1.2) we have x < Km (f ; x) ≤ ξl

(ηl−1 < x < ξl ),

(3.7)

ξl ≤ Km (f ; x) < x

(ξl < x < ηl ).

(3.8)

Proof. We prove only (3.7). The proof of (3.8) is similar. First, we verify the relation given in (3.7) for x ∈ (ηl−1 , ξl ) \ˆ Z (m−1) , (m−1) (m−1) ˆ ˆ where Z = {z ∈ D | f (z) = 0}. Next, we show that (3.7) holds for x ∈ (ηl−1 , ξl ). In view of Lemmas 3.8 and 3.9 and (1.2) we have Km (f ; x) − x m−1

=

ˆ f (m−2) (x) f (x)ˆ f (m−2) (x) = > 0 (x ∈ (ηl−1 , ξl ) \ ˆ Z (m−1) ). (m − 1) ˆ ˆ f (x) f (x)f (m−1) (x)

It follows that x < Km (f ; x)

(x ∈ (ηl−1 , ξl ) \ ˆ Z (m−1) ).

(3.9)

Next, we verify that Km (f ; x) ≤ ξl . In view of Lemma 3.4 we see that there exists a constant ε0 > 0 such that for arbitrary ε satisfying 0 < ε < ε0 there exists an integer M > 0 such that for every j > M the polynomial pj (z) has zeros on the ε -neighbourhood N(ξl , ε ). Therefore, pj (x) has a zero αj,l on the interval (ξl − ε, ξl + ε ). So, from Corollary 3.11 we see that Km (pj ; x) ≤ αl,j < ξl + ε

(ηl−1 < x < ξl − ε ).

(3.10)

Since {Km (pj ; x)} (j ≥ 0) converges to Km (f , x) on (ηl−1 , ξl ) \ ˆ Z (m−1) , from (3.10) we have Km (f ; x) ≤ ξl + ε

(x ∈ (ηl−1 , ξl − ε ) \ ˆ Z (m−1) ).

(3.11)

Letting ε → 0 in (3.11) and using (3.9) we obtain, for x ∈ (ηl−1 , ξl ) \ ˆ Z (m−1) , x < Km (f ; x) ≤ ξl .

(3.12)

This implies that Km (f ; x) is bounded by the functions u(x) = x and v (x) = ξl on (ηl−1 , ξl ) \ ˆ Z (m−1) . Therefore, Km (f ; x) has no pole on (ηl−1 , ξl ) and (ηl−1 , ξl ) ∩ ˆ Z (m−1) = ∅. It follows that (3.12) holds for x ∈ (ηl−1 , ξl ). Thus, we verified (3.7). □ Remark 3.14. From the proof of Lemma 3.13, we see that if x ∈ (ξl−1 , ξl ) satisfies ˆ f (m−1) (x) = 0, then x = ηl−1 . If ηl−1 is a zero of multiplicity µ of ˆ f (m−2) (x), then it is a zero of multiplicity µ − 1 of ˆ f (m−1) (x). Therefore, x = ηl−1 is a removable singularity of Km (f ; x), that is, Km (f ; x) is analytic for x ∈ (ξl−1 , ξl ). Further, clearly Km (f ; ηl−1 ) = ηl−1 . Proof of Theorem 3.7. Lemmas 3.12 and 3.13 imply that for an arbitrary zero ξl of f (x), every iteration sequence {xk } of the IF Km (f ; x) starting from the initial value x0 ∈ (ηl−1 , ηl ) converges monotonically to ξl . So, Theorem 3.7 is established. □ 4. Main theorem: convergence for entire functions In Section 3, we proved the GM convergence of the S2-m formula of odd order m ≥ 3 for real zeros of the analytic functions in F(D). Here, we give two theorems on entire functions. Theorem 4.1, together with Theorem 3.7, immediately implies our main Theorem 4.7. Theorem 4.1.

Let E2 be the family of entire functions f (z) of genus 2 given by (3.1). Then, E2 ⊂ F(C).

Before proving Theorem 4.1, we give some examples of entire functions. Example 4.2.

The following functions are entire functions ∈ E2 ,

sin π z = π z

∞ ( ∏

1−

l=1

z2 ) l2

,

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Jn (z) =

∞ (z /2)n ∏(

n!

1/Γ (z) = eγ z z

1−

l=1

∞ ( ∏

1+

l=1

z2 ) j2n,l

,

z ) −z /l e , l

where Jn (z) is the Bessel function of order n with zeros jn,l and Γ (z) the gamma function with Euler’s constant γ (cf. [14, p. 25]). The following Lemmas 4.3∼4.6 are instrumental in proving Theorem 4.1. Lemma 4.3 is trivial. Lemma 4.3.

Let f and g be functions such that f , g ∈ F(C). Then, the product fg ∈ F(C) as well.

Lemma 4.4.

For a real α let f (z) = eα z . Then, f ∈ F(C).

Proof. It is trivial when α = 0. Assume that α ̸ = 0 and let pn (z) = (1 + α z /n)n (n ≥ 1). Then, we have pn ∈ P since zeros of pn are real. Further, since {pn } (n ≥ 1) converges locally uniformly on C, limn→∞ pn = f . □ Lemma 4.5.

2

For a, b ∈ R and c ≥ 0 let f (z) = ea+bz −cz . Then, f ∈ F(C).

Proof. If c = 0, then Lemma 4.5 is reduced to Lemma 4.4. For c > 0 let pn (z) = {1 + (bz − cz 2 )/n}n (n ≥ 1). Then, the zeros of pn are all real since the discriminant b2 + 4nc ≥ 0. So we have pn ∈ P. Further, since {pn } (n ≥ 1) converges 2 locally uniformly to ebz −cz on C, we have limn→∞ ea pn = f . □ Lemma 4.6.

If {fn } ⊂ F(C) (n ≥ 0) is locally uniformly convergent to a function f in the entire C, then f ∈ F(C).

Proof. Let {pn,k } ⊂ P (k ≥ 0) be a sequence of polynomials locally uniformly convergent to fn . For arbitrary integer m ≥ 0, let ν (m) be an integer satisfying max |fν (m) (z) − f (z)| ≤ m−1 .

|z |≤m

Further, for arbitrary integers m ≥ 0 and n ≥ 0 let κ (n, m) ≥ 0 be an integer such that max |pn,κ (n,m) − fn (z)| ≤ m−1 .

|z |≤m

Let ˜ pm be denoted by

˜ pm = pν (m),κ (ν (m),m) ∈ P (m ≥ 0). For arbitrary R > 0 let m ≥ R. If |z | ≤ R, then we have |z | ≤ m and

|˜ pm (z) − f (z)| = |pν (m),κ (ν (m),m) (z) − f (z)| ≤ |pν (m),κ (ν (m),m) (z) − fν (m) (z)| + |fν (m) − f (z)| ≤ m−1 + m−1 = 2m−1 . Consequently, max|z |≤R |˜ pm (z) − f (z)| → 0 as m → ∞.

□

Proof of Theorem 4.1. We use Lemmas 4.3∼4.6. Let gl (z) = ez /ξl and Gm (z) = z n

m ∏

(1 − z /ξl )ez /ξl ,

G∞ (z) = z n

l=0

∞ ∏

(1 − z /ξl )ez /ξl .

l=0

Then, since gl ∈ F(C) from Lemma 4.4, in view of Lemma 4.3, Gm ∈ F(C). Therefore, in view of Lemma 4.6, G∞ ∈ F(C). Further, for h(z) = exp(a + bz − cz 2 ), from Lemma 4.5 we see that h ∈ F(C). Finally, for f given by (3.1), in view of Lemma 4.3, f = hG∞ ∈ F(C). □ We conclude with the main theorem. Theorem 4.7. For the entire functions f (x) in (3.1) the S2-m formula of every odd order m ≥ 3 in (1.1) converges globally and monotonically to their zeros. Proof. Theorem 4.7 immediately follows from Theorems 3.7 and 4.1.

□

H. Sugiura and T. Hasegawa / Journal of Computational and Applied Mathematics 358 (2019) 136–145

143

Table 1 Values of ∆m (f ; xk ) for m = 2, . . . , 7 in (1.2) with the initial value x0 = 2.1 to the zero 2 of f (x) = sin π x. k

0 1 2 3 4 k

0 1 2 3 4

m 2

4

6

−0.10342515 3.43 × 10−3 −1.32 × 10−7 7.60 × 10−21 −1.44 × 10−60

−0.10007984 7.98 × 10−5 −2.46 × 10−20 6.79 × 10−98 −1.09 × 10−485

−0.10000130 1.30 × 10−6 −7.40 × 10−41 1.44 × 10−280 −1.48 × 10−1958

m 3

5

7

−0.098239444 −1.76 × 10−3 −8.98 × 10−9 −1.19 × 10−24 −2.77 × 10−72

−0.099977959 −2.22 × 10−5 −9.85 × 10−24 −1.76 × 10−115 −3.18 × 10−574

−0.099999743 −2.57 × 10−7 −1.47 × 10−46 −2.94 × 10−321 −3.73 × 10−2244

Table 2 Values of ∆m (J0 ; xk ) for m = 2, . . . , 7 in (1.2) with the initial value x0 = 5 to the zero j0,2 = 5.520078 · · · of the Bessel function J0 (x) of order 0. k

0 1 2 3 4 k

0 1 2 3 4

m 2

4

6

0.54214921 −2.21 × 10−2 4.77 × 10−5 2.06 × 10−10 3.83 × 10−21

0.52191376 −1.84 × 10−3 1.76 × 10−13 1.47 × 10−53 7.09 × 10−214

−1.00 × 10−4 1.88 × 10−27 7.82 × 10−164 4.10 × 10−982

3

5

7

0.49613381 2.39 × 10−2 2.24 × 10−6 1.83 × 10−18 1.01 × 10−54

0.51921494 8.63 × 10−4 9.12 × 10−18 1.21 × 10−87 4.84 × 10−437

0.52004712 3.10 × 10−5 5.54 × 10−35 3.21 × 10−243 7.15 × 10−1701

0.52017880

m

5. Numerical examples In this section, we give numerical examples of two types. First, using three test functions, we demonstrate the GM convergence of the S2-m formula. Second, using a test function f (x) = 1/Γ (x), we compare the Schröder formulae of the first and second kinds in performance, namely, the numbers of successes in convergence and failures and the execution time on a computer. Numerical computations are carried out by using Mathematica (ver.11.3.0.0) on the iMac (3.2 GHz, Intel Core i5) with ten thousand significant digits in Section 5.1 and the double precision arithmetic (unit roundoff u = 2−53 ≈ 1.11 × 10−16 [17, p. 41]) in Section 5.2. 5.1. GM convergence of the S2-m formula Using three test functions, we give numerical results to demonstrate the GM convergence of the S2-m formula of every odd order m ≥ 3. Our test functions are the entire functions f (x) in Example 4.2: f (x) = sin π x,

f (x) = J0 (x),

f (x) = 1/Γ (x),

(5.1)

where J0 (x) is the Bessel function of order 0 and Γ (x) the gamma function. ∑sThe values of ∆m (f ; xk ) (m = 2, . . . , 7) in Table 1 for f (x) = sin π x show the convergence behaviours x0 + k=0 ∆m (p; xk ) → 2 (s → ∞) of the IF (1.2) of order m = 2, . . . , 7 with the initial value x0 = 2.1 to the zero 2. We observe the monotonic convergence of the iterations of odd order 3,5 and 7, as expected from Theorem 4.7, and the alternating convergence of those of even order 2,4 and 6. Table 2 shows the convergence behaviours of ∆m (J0 ; x) (m = 2, . . . , 7) with x0 = 5 to the zero j0,2 = 5.520078 . . . of the Bessel function J0 (x) of order 0. Fig. 4 depicts the IFs K5 (J0 ; x) (left) and K4 (J0 ; x) (right) for J0 (x) in (5.1). The right figure suggests the monotonic convergence of the S2-m formula of order m = 5, while the left shows no monotonic convergence

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H. Sugiura and T. Hasegawa / Journal of Computational and Applied Mathematics 358 (2019) 136–145

Fig. 4. The iteration functions K4 (J0 ; x) (left) and K5 (J0 ; x) (right) for the Bessel function J0 (x) of order 0. The zeros of J0 (x) give the intersections of y = Km (J0 ; x) for m = 4 and 5 and y = x.

Fig. 5. The iteration functions K4 (1/Γ ; x) (left) and K5 (1/Γ ; x) (right) for the reciprocal gamma function 1/Γ (x).

(ν )

of the S2-m formula of even order m = 4. Computing Km (J0 ; x) requires the derivatives J0 (x). These can be obtained from J−ν+2i (x) (i = 0, 1, . . . ); see [18, p. 361: 9.1.31]. Fig. 5 depicts the behaviours of the IFs K4 (1/Γ ; x) and K5 (1/Γ ; x) for the reciprocal gamma function 1/Γ (x). The right figure suggests that the S2-m formula of order m = 5 converges monotonically to the zeros of 1/Γ (x) with the starting value x0 ∈ (−∞, ηR ) \ {ηl }−∞
H. Sugiura and T. Hasegawa / Journal of Computational and Applied Mathematics 358 (2019) 136–145

145

Fig. 6. The reciprocal gamma function 1/Γ (x). Table 3 Numbers of success and failure, total numbers of iterations and execution time. Formula

Success

Failure

Iterations

0

−1

−2

Newton S2-3 -4 -5

246 252 255 250

419 535 454 522

186 213 212 228

89 0 51 0

60 0 28 0

S1-3 -4 -5

184 163 139

368 303 296

138 122 110

117 134 159

193 278 296

−ℓ

Time T [s]

t [µs]

10 541 3 744 5 693 2 740

0.05551 0.03337 0.1809 0.2147

5.27 8.91 31.8 78.3

18 395 24 696 25 258

0.3334 1.003 1.936

18.1 40.6 76.6

F

odd ∏ order ≥ 3 converges globally and monotonically to zeros of the entire functions such as f (x) = xn exp(a + bx − ∞ cx2 ) l=1 (1 − x/ξl ) ex/ξl with only real zeros in the sense of Definition 2.1. Several numerical examples illustrated the global and monotonic convergence behaviours of the formula of the second kind of odd order ≥ 3. It is an open problem to elucidate the relation between the family of analytic functions F(C) and that of the entire functions C2 , or whether C2 = F(C) or C2 ⫋ F(C). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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