AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation49531210.1155/2011/495312495312Research ArticleOn the Distance to a Root of PolynomialsChaiyaSomjate1, 2AtakishiyevNatig1Department of MathematicsFaculty of ScienceSilpakorn UniversityNakorn Pathom 73000Thailandsu.ac.th2Centre of Excellence in MathematicsCommission on Higher EducationSi Ayutthaya RoadBangkok 10400Thailand201127102011201129072011010920112011Copyright © 2011 Somjate Chaiya.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 be the set of positive integers. For each k, let Npk denote the k-iterate of Np, that is, Np1=Np,Np2=NpNp, and Npk=Npk-1Np. 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 zU. Every immediate basin U is forward invariant, that is, Np(U)=U, and is simply connected (see [1, 2]). In 2002, Schleicher  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 nn(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 d3, 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 d10 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:|z-a|<r} and B¯(a,r) for the closed ball {z:|z-a|r}, where is the set of complex numbers. If S is a subset of , 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 UB¯(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 αB¯(β,r) and zUB¯(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=1Npk(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.

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 UB¯(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 d3. 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 bβ. Then the closed ball {z:|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)=zmk=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 βb. For r<β, we have m+r(d-m-1)r-β+rr-b12, if rδ. 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 1jd-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 αα2.

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 r3A3, 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 r4A4, and if α is α4 we are done. Continuing this process, finally we get rdAd 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, yfd), 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 thanfd is greater thanfd/2d/2M(d,y) is greater than
101.3385×1054.3758×106 1.0216
201.0131×10101.343×1013 1.2946
304.4559×10142.6158×1019 1.7915
401.5878×10194.2458×1025 2.5502
505.0059×10236.2420×1031 3.7162
601.1486×10288.6222×1037 5.4054
704.2054×10321.1410×1044 7.8967
801.1429×10371.4634×1050 11.6467
903.0424×10411.8327×1056 17.1212
1007.9376×10452.2523×1062 25.2027
1102.0274×10502.7262×1068 37.3244
1205.1302×10543.2588×1074 55.0978
1301.2839×10593.8546×1080 81.3792
1403.1697×10634.5186×1086 120.7511
1507.7889×10675.2563×1092 178.6315
1601.8954×10726.0735×1098 265.0635
1704.5932×10766.9764×10104 392.6175
1801.1074×10817.9718×10110 581.5469
1902.6450×10859.0669×10116 863.7282
2006.3268×10891.0269×10123 1280.4536
Acknowledgment

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

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