In 2002, Dierk Schleicher gave an explicit estimate of an upper bound for the number of iterations of Newton's method it takes to find all roots of polynomials with prescribed precision. In this paper, we provide a method to improve the upper bound given by D. Schleicher. We give here an iterative method for finding an upper bound for the distance between a fixed point z in an immediate basin of a root α to α, which leads to a better upper bound for the number of iterations of Newton's method.

1. Introduction

Let P be a polynomial of degree d, and let Np(z)=z-P(z)/P′(z) be the Newton map induced by P. Let ℕ be the set of positive integers. For each k∈ℕ, let Npk denote the k-iterate of Np, that is, Np1=Np,Np2=Np∘Np, and Npk=Npk-1∘Np. For a root α of P, we say that a set U is the immediate basin of α if U is the largest connected open set containing α and Npk(z)→α, as k→∞, for all z∈U. Every immediate basin U is forward invariant, that is, Np(U)=U, and is simply connected (see [1, 2]). In 2002, Schleicher [3] provided an upper bound for the number of iterations of Newton's method for complex polynomials of fixed degree with a prescribed precision. More precisely, Schleicher proved that if all roots of P are inside the unit disc and 0<ɛ<1, there is a constant n(d,ɛ) such that for every root α of P, there is a point z with |z|=2 such that |Npn(z)-α|<ɛ for all n≥n(d,ɛ). Schleicher also showed that n(d,ɛ) can be chosen so thatn(d,ɛ)≤9πd4fd2ɛ2log2+|logɛ|+log13log2+1
withfd:=d2(d-1)2(2d-1)(2dd).

To obtain this estimate, Schleicher employed several rough estimates which cause the bound far from an efficient upper bound. The main point that causes the extremely inefficiency is the way Schleicher used to obtain fd which arose when he estimated an upper bound for the distance of a point z to a root α. Schleicher showed that if z is in the immediate basin of α and |Np(z)-z|=δ, then the distance between z and α is at most δfd.

In this paper, we give an algorithm to improve the value of fd. Even though, it is not an explicit formula, it can be easily computed. The following is our main result.

Main Theorem 1.

Let P(z) be a polynomial of degree d≥3, and let y be a positive number larger than 4d-3. If z0 is in an immediate basin of a root α and |Np(z0)-z0|=ɛ, then |z0-α|≤ɛM(d,y), where M(d,y):=max{y,Ad+y(d-1)/(y-1)} and Ad can be derived from the following iterative algorithm.

Let b=y(y-d)/(y-1), andA2=y(d-1)[2d(y-2d+3)-3y-1](y-1)(y-4d+3).
For k=2,…,d-1, set ak=1+∑j=2k-1(Ak/(Ak-Aj)).

If 2Ak<b then letAk+1=Ak((ak+d-k)Ak+b(k+1-ak-d)Ak(ak+1)-bak).
Otherwise let
Ak+1=Akak+d-kak.

Note that the value of M(d,y) in the main theorem depends only on the constant y and the degree d. Hence if we select y appropriately the value M(d,y) will be optimized under this method. However this estimate is still far away from the best possible one. We believe that this new upper bound M(d,y) is less than fd/2d/2 for all d≥10 when y=d1.52(4d/3)-2. We will discuss further about this matter in Section 4.

2. Preliminary Results

We will use B(a,r) for the open ball {z∈ℂ:|z-a|<r} and B¯(a,r) for the closed ball {z∈ℂ:|z-a|≤r}, where ℂ is the set of complex numbers. If S is a subset of ℂ, we denote the boundary of S by ∂S.

Lemma 2.1.

Let P be a polynomial. Let β be a complex number and r>0. Suppose that Re{(z-β)P′(z)/P(z)}≥1/2 whenever |z-β|=r and P(z)≠0. Let U be an immediate basin of a root α of P. If U∩B¯(β,r)≠∅, then α is in B(β,r).

Proof.

For |z-β|=r with P(z)≠0, we have
Np(z)-β=(z-β)(1-1g(z)),
where g(z)=(z-β)P′(z)/P(z). Hence, |Np(z)-β|≤|z-β| if and only if |(g(z)-1)/g(z)|≤1 which holds if Re{g(z)}≥1/2. It means that if z is a point in ∂B(β,r) and Re{g(z)}≥1/2, then the distance of Np(z) to β is at most the distance of z to β. In other words, the image of z under the map Np also lies inside B¯(β,r).

Let α be a root of P and U be its immediate basin. Suppose that α∉B¯(β,r) and z∈U∩B¯(β,r). Since U is forward invariant under Np, Np(z) still stays in U. Since U is connected, there is a curve γ0 connecting z to Np(z) and lying entirely in U. Since Npk(γ0) converges uniformly to α as k→∞, the set ⋃k=1∞Npk(γ0)∪{α} forms a continuous curve γ joining z and α. Note that γ is contained in U because Npk(γ0) lies inside U for all k∈ℕ.

Let w be the last intersection point of γ with ∂B(β,r) (i.e., the part of the curve γ that connects w to α stays outside B¯(β,r) except at w). So Np must send w to a point outside B¯(β,r), otherwise β is a fixed point of Np, which is impossible because all fixed points of Np are only the roots of P, and here P(z)≠0 on |z-β|=r. From the first paragraph, however, we also have Np(w)∈B¯(β,r). Hence we get a contradiction. Therefore if U∩B¯(β,r) is not empty, then α is in B(β,r), as desired.

Remark that, from the proof of Lemma 2.1, if β is a root of P and Re{(z-β)P′(z)/P(z)}≥1/2 for all |z-β|≤r, then the closed ball B¯(β,r) is contained in the immediate basin of β.

Lemma 2.2.

Let P be a polynomial of degree d≥3. Let α1 be a root of P and α2 the nearest root to α1. Let β=|α1-α2|, and let m be the multiplicity of α1. Suppose that there is a root α of P such that |α1-α|≥b for some positive number b≥β. Then the closed ball {z∈ℂ:|z-α1|≤δ} is contained entirely in the immediate basin of α1, where
δ=12(2d-1)[(2m+1)β+(2d-3)b-[(2m+1)β+b(2d-3)]2-4(2d-1)(2m-1)bβ[(2m+1)β+b(2d-3)]2-4(2d-1)(2m-1)bβ].

Proof.

Without loss of generality, we assume that α1=0. From the previous remark, it suffices to show that Re{zP′(z)/P(z)}≥1/2 for all |z|≤δ. Let P(z)=zm∏k=2d-m(z-αk). We have
zP′(z)P(z)=m+∑k=2d-mzz-αk.
Hence
Re{zP′(z)P(z)}=m+∑k=2d-mRe{zz-αk}≥m+r(d-m-1)r-β+rr-b,
where r=|z|. Note that β≤b. For r<β, we have
m+r(d-m-1)r-β+rr-b≥12,
if r≤δ. This shows that Re{zP′(z)/P(z)}≥1/2 for all |z|≤δ, as needed.

Note that if we set b=β in Lemma 2.2, then the closed ball centered at α1 of radius β(2m-1)/(2d-1) is contained in the immediate basin of α1. Furthermore, if m=1, the radius of the ball is β/(2d-1). (Schleicher [3, Lemma 4, page 938] made a small mistake about the radius of the ball. Indeed, he should get β/(2d-1) instead of β/2(d-1)).

Lemma 2.3.

Let P be a polynomial of degree d. For any complex number z and any positive number y>1, if |Np(z)-z|=ɛ and there is a root αd of P with |z-αd|≥yɛ, then there is a root α of P such that |z-α|≤y(d-1)ɛ/(y-1).

Proof.

Let α1,α2,…,αd be all roots of P. Suppose that |z-αd|≥yɛ. If |z-αj|>y(d-1)ɛ/y-1 for 1≤j≤d-1, then
|Np(z)-z|≥(∑j=1d1|z-αj|)-1>(y-1y(d-1)ɛ(d-1)+1yɛ)-1=ɛ,
a contradiction.

We are now ready to prove our main theorem.

3. Proof of Main Theorem

Let α1,α2,…,αd be all roots of P such that α1 is the nearest root to z0 and |α1-αk|≤|α1-αk+1| for k=2,…,d-1. Suppose that |z0-αd|≥yɛ. By Lemma 2.3, we have |z0-α1|≤y(d-1)ɛ/(y-1). Note that |α1-αd|≥bɛ. If α=α1, we are done. Otherwise, z is not in the immediate basin of α1; thus by Lemma 2.2 with m=1, we get that |z0-α1|>δ, where δ is defined in Lemma 2.2, that is,δ=3r2+bɛ(2d-3)-[3r2+bɛ(2d-3)]2-4(2d-1)bɛr22(2d-1),
where r2=|α1-α2|. Thus z0 satisfies the inequalitiesδ<|z0-α1|≤y(d-1)ɛy-1,
which holds if |α1-α2|<A2ɛ. If α=α2, we are done. Suppose next that α≠α2.

Now let |α1-αk|=ɛrk. If |z-α1|=A2ɛ and r3>A3, then Re{(z-α1)P′(z)P(z)}≥1+A2A2+r2+A2(d-3)A2-r3+A2A2-rd>1+12+A2(d-3)A2-r3+A2A2-b>12.
hence by Lemma 2.1α must be either α1 or α2 which is not the case. Therefore r3≤A3, and if α is α3 we are done. Otherwise, let |z-α1|=A3ɛ and suppose r4>A4; then Re{(z-α1)P′(z)/P(z)}>1/2, and by Lemma 2.1 we get a contradiction. Thus we obtain r4≤A4, and if α is α4 we are done. Continuing this process, finally we get rd≤Ad which gives |z0-αd|≤ɛ(Ad+y(d-1)/(y-1)).

Note that if Ad<b, it is a contradiction to the fact that ɛrd=|α1-αd|≥bɛ, which implies that assumption |z0-αd|≥yɛ is false. Hence in this case we have |z0-αd|<yɛ. The proof is now complete.

4. Discussion

For a fixed d, M(d,y) depends on only y. If we choose y too large (for instance, y≥fd), the value of M(d,y) is useless when it is compared to fd. So we have to choose y carefully so that M(d,y) is minimal as possible. We do not know yet whether there is an explicit formula for the value y that minimizes M(d,y). Table 1 below shows the values of M(d,y) where we set y=d1.524d/3-2. It seems that this method can reduce upper bounds for the distance of z0 to the root it converges to at least 2d/2 times compared to fd. If we replace fd in (1.1) by M(d,y), we derive a new upper bound for the number of iterations.

Examples of values of M(d,y) compared to fd when y=d1.524d/3-2.

d=

M(d,y) is less than

fd is greater than

fd/2d/2M(d,y) is greater than

10

1.3385×105

4.3758×106

1.0216

20

1.0131×1010

1.343×1013

1.2946

30

4.4559×1014

2.6158×1019

1.7915

40

1.5878×1019

4.2458×1025

2.5502

50

5.0059×1023

6.2420×1031

3.7162

60

1.1486×1028

8.6222×1037

5.4054

70

4.2054×1032

1.1410×1044

7.8967

80

1.1429×1037

1.4634×1050

11.6467

90

3.0424×1041

1.8327×1056

17.1212

100

7.9376×1045

2.2523×1062

25.2027

110

2.0274×1050

2.7262×1068

37.3244

120

5.1302×1054

3.2588×1074

55.0978

130

1.2839×1059

3.8546×1080

81.3792

140

3.1697×1063

4.5186×1086

120.7511

150

7.7889×1067

5.2563×1092

178.6315

160

1.8954×1072

6.0735×1098

265.0635

170

4.5932×1076

6.9764×10104

392.6175

180

1.1074×1081

7.9718×10110

581.5469

190

2.6450×1085

9.0669×10116

863.7282

200

6.3268×1089

1.0269×10123

1280.4536

Acknowledgment

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

PrzytyckiF.KrzyzewskiK.Remarks on the simple connectedness of basins of sinks for iterations of rational mapsShishikuraM.SchleicherD.The connectivity of the Julia set and fixed pointsSchleicherD.On the number of iterations of Newton's method for complex polynomials