Polynomial and Rational Approximations and the Link between Schröder’s Processes of the First and Second Kind

Abstract

We show that Schröder’s processes of the first kind and of the second kind to obtain a simple root of a nonlinear equation are related by polynomial and rational approximations.

1. Introduction

In [1, 2], Schröder proposed two fixed point processes to find a simple root 𝛼 of a nonlinear equation 𝑓(𝑥)=0. These two processes have been reconsidered in Kalantari et al. [3] and Kalantari [4] from a modern point of view. Iteration functions (IF) of arbitrary order 𝑝≥2 associated with the two processes will be noted 𝐸_𝑝(𝑥) for Schröder’s process of the first kind and 𝑆_𝑝(𝑥) for Schröder’s process of the second kind.

Order 2 processes are 𝐸₂(𝑥)=𝑆₂(𝑥)=𝑁_𝑓(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥), (1) where 𝑁_𝑓(𝑥) is the Newton's IF, and order 𝑝>2 processes can be expressed as 𝐸_𝑝(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Λ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)},𝑆_𝑝(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Γ_𝑝−3(𝜉)Γ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}, (2) where Λ_𝑞(𝜉) and Γ_𝑞(𝜉) are polynomials of degree 𝑞≥0, such that Λ_𝑞(0)=1=Γ_𝑞(0).

A question raised and discussed in [3–6] is to find and explain a possible link between the two processes. The main result of this paper is to show that 𝐸_𝑝(𝑥) is a polynomial approximation of 𝑆_𝑝(𝑥), and 𝑆_𝑝(𝑥) is a rational approximation of 𝐸_𝑝(𝑥). More precisely, we explain the relation between the polynomials Λ_𝑝−2(𝜉), Γ_𝑝−3(𝜉), and Γ_𝑝−2(𝜉) in (2). This paper completes the work done previously in [3–6].

Other links or comparisons could be established between these two families, for example, between their basins of attraction, their asymptotic constants, and their complexities. First results in these directions appeared in [7, 8], for example, but are not the object of the present paper.

In the next section we present notations and definitions used in this paper. Sections 3 and 4 present the two processes of Schröder and their corresponding polynomials Λ_𝑞(𝜉) and Γ_𝑞(𝜉). We prove the main result in the last section.

2. Preliminaries

Througout the paper we consider real valued functions which are regular enough to be differentiated sufficiently many times. The 𝑙th derivative will be noted 𝑓^(𝑙)(𝑥) for 𝑙=1,2,…. We will use the notation 𝑔(𝑥)=𝑂(𝑓(𝑥)) (and 𝑔(𝑥)=𝑜(𝑓(𝑥))) for two functions 𝑓(𝑥) and 𝑔(𝑥) defined around 𝑥=𝛼, when the following limit exists (and is finite): lim_𝑥→𝛼𝑔(𝑥)𝑓(𝑥)=𝑐 (3) for 𝑐≠0 (for 𝑐=0, resp.).

Let 𝛼 be a fixed point of an IF Φ(𝑥), and let the sequence {𝑥_𝑘+1=Φ(𝑥_𝑘)}^+∞_𝑘=0 converge to 𝛼. Let 𝑝 be a positive integer such that the following limit exists (and is finite): 𝑥_𝑘+1−𝛼𝑥_𝑘−𝛼^𝑝−−−−−−−−→_𝑘→+∞𝐾_𝑝(𝛼;Φ). (4) We say that the convergence of the sequence to 𝛼 is of (integer) order 𝑝 if and only if 𝐾_𝑝(𝛼;Φ)≠0. We also say that Φ(𝑥) is of order 𝑝.

Let 𝛼 be a simple root of 𝑓(𝑥), which means that 𝑓(𝛼)=0 and 𝑓⁽¹⁾(𝛼)≠0; then 𝑔(𝑥)=𝑂((𝑥−𝛼)^𝑝) is equivalent to 𝑔(𝑥)=𝑂(𝑓^𝑝(𝑥)) or 𝑔(𝑥)=𝑂((𝑓/𝑓⁽¹⁾)^𝑝(𝑥)). Moreover if 𝛼 is a fixed point of an IF Φ_𝑝(𝑥) of order 𝑝, then we can write [9] Φ_𝑝(𝑥)=𝛼+𝑂(𝑥−𝛼)^𝑝=𝛼+𝑂𝑓^𝑝(𝑥)𝑓=𝛼+𝑂𝑓^(1)𝑝(𝑥). (5)

3. Schröder’s Process of the First Kind

3.1. The Process

Schröder’s process of the first kind, proposed to increase the order of convergence of a fixed-point method [1–3, 9, 10], can be obtained by considering Taylor’s expansion of the inverse 𝑔(𝑦) of 𝑓(𝑥) around 𝑦=0 [9]. It is also associated with Chebyshev and Euler [11–13]. The IF 𝐸_𝑝(𝑥) of order 𝑝 is defined by the series 𝐸_𝑝(𝑥)=𝑥+^𝑝−1_𝑙=1𝑐_𝑙(𝑥)𝑓^𝑙(𝑥)=^𝑝−1_𝑙=0𝑐_𝑙(𝑥)𝑓^𝑙(𝑥), (6) where 𝑐₀(𝑥)=𝑥 and 𝑐_𝑙(𝑥)1=−𝑙1𝑓⁽¹⁾(𝑥)𝑑𝑐𝑑𝑥^𝑙−1(𝑥)=(−1)^𝑙1𝑙!𝑓⁽¹⁾(𝑥)𝑑𝑑𝑥^𝑙𝑐₀(𝑥) (7) for 𝑙=1,2,….

Relation (7) implies that 𝑐_𝑙(𝑥) is a rational function of the form 𝑐_𝑙(𝑥)=𝑓polynomialofdegree𝑙−1w.r.t.₍₁₎(𝑥),𝑓⁽²⁾(𝑥),…,𝑓^(𝑙)(𝑥)×𝑓⁽¹⁾(𝑥)^{2𝑙−1−1}. (8) Then we can write 𝐸_𝑝(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Λ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}, (9) where Λ_𝑝−2(𝜉)=^𝑝−2_𝑙=0𝜆_𝑙(𝑥)𝜉^𝑙,𝜆_𝑙(𝑥)=−𝑓1for𝑙=0,₍₁₎(𝑥)^𝑙𝑑𝑙+1𝜆𝑑𝑥^𝑙−1(𝑥)𝑓⁽¹⁾(𝑥)^𝑙for𝑙≥1. (10)(11) Consequently 𝜆_𝑙(𝑥)=𝑓polynomialofdegree𝑙w.r.t.₍₁₎(𝑥),𝑓⁽²⁾(𝑥),…,𝑓^(𝑙+1)(𝑥×𝑓₍₁₎)(𝑥)^𝑙−1. (12)

3.2. Examples

The first 4 IFs are given here. For(a)𝑝=2: 𝐸₂(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥), (13) which corresponds to Newton’s IF of order 2;(b)𝑝=3: 𝐸₃(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)−1𝑓2!⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)², (14) which corresponds to Chebyshev’s IF order 3 [11];(c)𝑝=4: 𝐸₄(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)−1𝑓2!⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)²−13𝑓3!⁽²⁾(𝑥)²−𝑓⁽¹⁾(𝑥)𝑓⁽³⁾(𝑥)𝑓⁽¹⁾(𝑥)²×𝑓(𝑥)𝑓⁽¹⁾(𝑥)³; (15)(d)𝑝=5: 𝐸₅(𝑥)1=𝑥−𝑓⁽¹⁾(𝑥)𝑓(𝑥)−1𝑓2!⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)²−13𝑓3!⁽²⁾(𝑥)²−𝑓⁽¹⁾(𝑥)𝑓⁽³⁾(𝑥)𝑓⁽¹⁾(𝑥)²×𝑓(𝑥)𝑓⁽¹⁾(𝑥)³−1𝑓4!⁽¹⁾(𝑥)²𝑓⁽⁴⁾(𝑥)𝑓⁽¹⁾(𝑥)³+15𝑓⁽²⁾(𝑥)³𝑓⁽¹⁾(𝑥)³−10𝑓⁽¹⁾(𝑥)𝑓⁽²⁾(𝑥)𝑓⁽³⁾(𝑥)𝑓⁽¹⁾(𝑥)³×𝑓(𝑥)𝑓⁽¹⁾(𝑥)⁴. (16)

4. Schröder’s Process of the Second Kind

4.1. The Process

Different equivalent formulations exist for Schröder’s process of the second kind [4, 6, 13, 14]. One such form is based on a determinental identity. Let Δ₀(𝑥)=1 and for 𝑝≥1Δ_𝑝(𝑥)=𝑓⁽¹⁾(𝑥)𝑓⁽²⁾(𝑥)𝑓2!⋯⋯⋯^(𝑝)(𝑥)𝑓(𝑥)𝑓𝑝!₍₁₎(𝑥)𝑓⁽²⁾(𝑥)𝑓2!⋯⋯^(𝑝−1)(𝑥)!(𝑥)𝑓𝑝−10𝑓⁽¹⁾(𝑥)𝑓⁽²⁾(𝑥)⋯𝑓2!^(𝑝−2)(𝑥)!(𝑥)𝑓𝑝−2⋮⋱⋱⋱⋱⋮0⋯0𝑓⁽¹⁾(𝑥)𝑓⁽²⁾(𝑥)(𝑥)𝑓2!0⋯⋯0𝑓⁽¹⁾(𝑥). (17) Expanding this determinant along the first line, we obtain Δ_𝑝(𝑥)=^𝑝_𝑗=1(−1)^𝑗+1𝑓^(𝑗)(𝑥)𝑓𝑗!^𝑗−1(𝑥)Δ_𝑝−𝑗(𝑥) (18) for 𝑝≥1. Using 𝑅_𝑝(𝑥)=Δ_𝑝(𝑥)/𝑓^𝑝+1(𝑥), Schröder’s process of the second kind [1, 2] of order 𝑝 is defined by 𝑆_𝑝(𝑥)𝑅=𝑥−^𝑝−2(𝑥)𝑅_𝑝−1(𝑥)(𝑥)Δ=𝑥−𝑓^𝑝−2(𝑥)Δ_𝑝−1(𝑥). (19)

We prove by mathematical induction that Δ_𝑝(𝑥)=(−1)^𝑝𝑓^𝑝+1(𝑥)1𝑝!𝑓(𝑥)^(𝑝),𝑓(𝑥)Δ⁽¹⁾_𝑝−1(𝑥)=𝑝𝑓⁽¹⁾(𝑥)Δ_𝑝−1(𝑥)−𝑝Δ_𝑝(𝑥). (20)

Using 1𝑓(𝑥)^(𝑝)=^𝑝+1_𝑙=2𝑑_𝑝,𝑙(𝑥)1𝑓(𝑥)^𝑙, (21) then we have 𝑑_1,2(𝑥)=−𝑓⁽¹⁾(𝑥), and we can obtain recursively 𝑑_𝑝,𝑙(𝑥) for 𝑝≥2 by the formulae 𝑑_𝑝,𝑙(𝑥)=𝑑⁽¹⁾_{𝑝−1,𝑙}(𝑥)𝑑for𝑙=2,_{(1)𝑝−1,𝑙}(𝑥)−(𝑙−1)𝑑_{𝑝−1,𝑙−1}(𝑥)𝑓⁽¹⁾(𝑥)for𝑙=3,…,𝑝,−𝑝𝑑_{𝑝−1,𝑝}(𝑥)𝑓⁽¹⁾(𝑥)for𝑙=𝑝+1. (22) Consequently 𝑑_𝑝,𝑙(𝑥)=−𝑓^(𝑝)(𝑥),for𝑙=2,polynomialofdegree𝑙−1w.r.t.𝑓⁽¹⁾(𝑥),𝑓⁽²⁾(𝑥),…,𝑓^{(𝑝+2−𝑙)}(𝑥())for𝑙=3,…,𝑝,−1^𝑝𝑝!𝑓⁽¹⁾(𝑥)^𝑝for𝑙=𝑝+1. (23) It follows that Δ_𝑝(𝑥) is a polynomial of degree 𝑝−1 with respect to 𝑓(𝑥) and the term not depending on 𝑓(𝑥) is [𝑓⁽¹⁾(𝑥)]^𝑝. Hence Δ_𝑝(𝑥)=𝑓⁽¹⁾(𝑥)^𝑝Γ_𝑝−1(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}, (24) where Γ_𝑝−1(𝜉) is a polynomial of degree 𝑝−1 such that Γ_𝑝−1(0)=1. Let us set Γ_𝑝−1(𝜉)=^𝑝−1_𝑙=0𝛾_{𝑝−1,𝑙}(𝑥)𝜉^𝑙, (25) where the coefficients 𝛾_{𝑝−1,𝑙}(𝑥) are rational functions of the form 𝛾_{𝑝−1,𝑙}(𝑥)=(−1)^𝑝𝑑_{𝑝,𝑝+1−𝑙}(𝑥)𝑝!𝑓⁽¹⁾(𝑥)^𝑝−𝑙=𝑓polynomialofdegree𝑝−𝑙w.r.t.₍₁₎(𝑥),𝑓⁽²⁾(𝑥),…,𝑓^(𝑙+1)(𝑥×𝑓₍₁₎)(𝑥)^{𝑝−𝑙−1}. (26)(27) Then we obtain 𝑆_𝑝(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Γ_𝑝−3(𝜉)Γ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}. (28)

4.2. Examples

The first 4 IFs are presented here. For(a)𝑝=2: 𝑆₂(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥), (29) which corresponds to Newton’s IF of order 2;(b)𝑝=3: 𝑆₃(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)×11−(1/2)𝑓⁽²⁾(𝑥)/𝑓⁽¹⁾(𝑥)𝑓(𝑥)/𝑓⁽¹⁾(𝑥), (30) which is Halley’s IF of order 3 [15];(c)𝑝=4: 𝑆₄(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)×11−2𝑓⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)×𝑓1−⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)+1𝑓3!⁽³⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)²⁻¹; (31)(d)𝑝=5: 𝑆₅(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)×𝑓1−⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)+1𝑓3!⁽³⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)²×31−2𝑓⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)+13𝑓⁽³⁾(𝑥)𝑓⁽¹⁾(𝑥)+14𝑓⁽²⁾(𝑥)𝑓⁽¹⁾(𝑥)²𝑓(𝑥)𝑓⁽¹⁾(𝑥)²−1𝑓4!⁽⁴⁾(𝑥)𝑓⁽¹⁾(𝑥)𝑓(𝑥)𝑓⁽¹⁾(𝑥)³⁻¹. (32)

5. Proof of the Main Result

Since both processes 𝐸_𝑝(𝑥) and 𝑆_𝑝(𝑥) are of order 𝑝, following [9], the next result holds.

Lemma 1. Let 𝛼 be a simple root of 𝑓(𝑥); then 𝐸_𝑝(𝑥)−𝑆_𝑝(𝑥)=𝑂𝑓^𝑝(𝑥)𝑓(𝑥)=𝑂𝑓⁽¹⁾(𝑥)^𝑝. (33)

From this lemma 𝑓(𝑥)𝑓⁽¹⁾(𝑥)Λ_𝑝−2(𝜉)−Γ_𝑝−3(𝜉)Γ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}𝑓(𝑥)=𝑂𝑓⁽¹⁾(𝑥)^𝑝, (34) and so Λ_𝑝−2(𝜉)−Γ_𝑝−3(𝜉)Γ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}𝑓(𝑥)=𝑂𝑓⁽¹⁾(𝑥)^𝑝−1. (35)

The next step is to consider the following basic result about polynomial and rational approximations.

Lemma 2. Let us consider the expression Λ_𝑝(𝜉)=Γ_𝑝−1(𝜉)Γ_𝑝(𝜉)+𝑂𝜉^𝑝+1, (36) where Λ_𝑝(𝜉) and Γ_𝑝(𝜉) are polynomials of degree 𝑝 such that Λ_𝑝(0)=1=Γ_𝑝(0).(a)If Γ_𝑝−1(𝜉) and Γ_𝑝(𝜉) are given, there exists one and only one polynomial Λ_𝑝(𝜉) such that (36) holds.(b)If Γ_𝑝−1(𝜉) and Λ_𝑝(𝜉) are given, there exists one and only one polynomial Γ_𝑝(𝜉) such that (36) holds.

Proof. This result is based on the following identity: Γ_𝑝−1(𝜉)Γ_𝑝(𝜉)=Γ_𝑝−1(𝜉)1−1−Γ_𝑝(𝜉)=Γ_𝑝−1(𝜉)^+∞_𝑙=01−Γ_𝑝(𝜉)^𝑙=Γ_𝑝−1(𝜉)^𝑝_𝑙=01−Γ_𝑝(𝜉)^𝑙+𝑂𝜉^𝑝+1, (37) and we would like to have Λ_𝑝(𝜉)=Γ_𝑝−1(𝜉)^𝑝_𝑙=01−Γ_𝑝(𝜉)^𝑙+𝑂𝜉^𝑝+1. (38) We know that 𝜆₀=𝛾_𝑝−1,0=𝛾_𝑝,0=1. Moreover the coefficient of 𝜉^𝑙 on the left-hand side is 𝜆_𝑙 and on the right-hand side is 𝛾_𝑝,𝑙𝛾+anexpressionintermsof^{𝑝−1,𝑗}𝛾for𝑗=0,…,𝑙,_𝑝,𝑗for𝑗=0,…,𝑙−1 (39) for 𝑙=1,…,𝑝. This expression shows that if the 𝛾_𝑝’s and the 𝛾_𝑝−1’s are given, we can obtain 𝜆_𝑝’s, and conversely if the 𝜆_𝑝’s and the 𝛾_𝑝−1’s are given, we can obtain the 𝛾_𝑝’s.

In view of these two lemmas we obtain the main result of this paper.

Theorem 3. 𝐸_𝑝(𝑥) and 𝑆_𝑝(𝑥) are related as follows.(a)For 𝑆_𝑝(𝑥) given by (28), one can obtain the form (9) of 𝐸_𝑝(𝑥) by expanding the denominator in (28), multiplying, and truncating to keep powers of 𝑓(𝑥)/𝑓⁽¹⁾(𝑥) up to 𝑝−1.(b)Since 𝑆₂(𝑥)=𝐸₂(𝑥)=𝑁_𝑓(𝑥), one can obtain recursively 𝑆_𝑝(𝑥) given by (28) from 𝐸_𝑝(𝑥) given by (9).

Proof. (a) Indeed, if 𝑆_𝑝(𝑥) is given, which means we know Γ_𝑝−3(𝜉) and Γ_𝑝−2(𝜉), we can write 𝑆_𝑝(𝑥)𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Γ_𝑝−3(𝜉)Γ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Γ_𝑝−3(𝜉)1−1−Γ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Γ_𝑝−3(𝜉)^𝑝−2_𝑙=01−Γ_𝑝−2(𝜉)^𝑙+𝑂𝜉^𝑝−1_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}𝑓(𝑥)=𝑥−𝑓⁽¹⁾(𝑥)Λ_𝑝−2(𝜉)_{𝜉=𝑓(𝑥)/𝑓(1)(𝑥)}+𝑂𝑓^𝑝(𝑥)=𝐸_𝑝(𝑥)+𝑂𝑓^𝑝(𝑥), (40) which follows from part (a) of Lemma 1.
(b) As already observed, 𝐸₂(𝑥)=𝑆₂(𝑥)=𝑁_𝑓(𝑥). If, for 𝑝>2, we have 𝑆_𝑝−1(𝑥), and we know Γ_𝑝−4(𝜉) and Γ_𝑝−3(𝜉), and 𝐸_𝑝(𝑥), and we know also Λ_𝑝−2(𝜉), then we can determine Γ_𝑝−2(𝜉) from part (b) of Lemma 1. Consequently we obtain 𝑆_𝑝(𝑥).

The computation of the polynomials Λ_𝑝(𝜉) and Γ_𝑝(𝜉), and their coefficients 𝜆_𝑝’s and the 𝛾_𝑝’s, can be done explicitely using (11) for the 𝜆_𝑝’s and (22) and (26) for the 𝛾_𝑝’s. The verification of the link between the two Schröder’s processes has already been done using symbolic computation up to order 20 [5, 6].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work has been financially supported by an individual discovery grant from NSERC (Natural Sciences and Engineering Research Council of Canada).