# The Point Zoro Symmetric Single-Step Procedure for Simultaneous Estimation of Polynomial  Zeros

Authors: Mansor Monsi; Nasruddin Hassan; Syaida Fadhilah Rusli
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The point symmetric single step procedure PSS1 hasR-order of convergence at least 3. This procedure is modified by adding another single-step, which is the third step in PSS1. This modified procedure is called the point zoro symmetric single-step PZSS1. It is proven that the R-order of convergence of PZSS1 is at least 4 which is higher than the R-order of convergence of PT1, PS1, and PSS1. Hence, computational time is reduced since this procedure is more efficient for bounding simple zeros simultaneously.


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## Body

## 1. Introduction

The iterative procedures for estimating simultaneously the zeros of a polynomial of degreen were discussed, for example, in Ehrlich [1], Aberth [2], Alefeld and Herzberger [3], Farmer and Loizou [4], Milovanović and Petković [5] and Petković and Stefanović [6]. In this paper, we refer to the methods established by Kerner [7], Alefeld and Herzberger [3], Monsi, and Wolfe [8], Monsi [9] and Rusli et al. [10] to increase the rate of convergence of the point zoro symmetric single-step method PZSS1. The convergence analysis of this procedure is given in Section 3. This procedure needs some preconditions for initial points xi(0) (i=1,…,n) to converge to the zeros xi* (i=1,…,n), respectively, as shown subsequently in the sequel. We also give attractive features of PZSS1 in Section 3.


## 2. Methods of Estimating polynomial zeros

Letp:C→C be a polynomial of degree n defined by(2.1)p(x)=∑i=0naixi,ai∈C (i=1,…,n) and an≠0. Let x*=(x1*,x2*,…,xn*)T be the distinct zeros of p(x)=0, expressed in the form:(2.2)p(x)=∏i=1n(x-xi*)=0,
with an=1. Suppose that, for j=1,…,n,xj is an estimate of xj*, and let q:C→C be defined by(2.3)q(x)=∏j=1n(x-xj).
Then, (2.4)q′(xi)=∏j≠in(xi-xj),(i=1,…,n).
By (2.2), if for i=1,…,n, xi≠xj (j=1,…,n;j≠i), then(2.5)xi*=xi-p(xi)∏j≠i(xi-xj*).
Now,  xj≈xj* (j=1,…,n) so by (2.5),(2.6)xi*≈xi-p(xi)∏j≠i(xi-xj)(i=1,…,n).
An iteration procedure  PT1  of (2.6)  is defined by (2.7)xi(k+1)=xi(k)-p(xi(k))∏j≠i(xi(k)-xj(k))(i=1,…,n)(k≥0),
which has been studied by Kerner [7]. Furthermore, the following procedure PS1(2.8)xi(k+1)=xi(k)-p(xi(k))∏j=1i-1(xi(k)-xj(k+1))∏j=i+1n(xi(k)-xj(k))
has been studied by Alefeled and Herzberger [3].The symmetric single-step idea of Aitken [11] and the procedure PS1 of Alefeld and Herzberger [3] are  used to derive the  point symmetric single-step procedure PSS1(Monsi [9]). The procedure PSS1 is defined by(2.9)xi(k,0)=xi(k)(i=1,…,n),xi(k,1)=xi(k)-p(xi(k))∏j=1i-1(xi(k)-xj(k,1))∏j=i+1n(xi(k)-xj(k,0))(i=1,…,n),xi(k,2)=xi(k)-p(xi(k))∏j=1i-1(xi(k)-xj(k,1))∏j=i+1n(xi(k)-xj(k,2))(i=1,…,n),xi(k+1)=xi(k,2)(i=1,…,n).
The following definitions and theorem (Alefeld and Herzberger [12], Ortega and Rheinboldt [13]) are very useful for evaluation of  R-order of convergence of an iterative procedure I.Definition 2.1.
If there exists ap≥1 such that for any null sequence {w(k)}generated from {x(k)},then the R-factor of the sequence {w(k)}is defined to be
(2.10)Rp(w(k))={lim⁡k→∞⁡sup⁡⁡‖w(k)‖1/k,p=1,lim⁡k→∞⁡sup⁡⁡‖w(k)‖1/pk,p>1,
where Rpis independent of the norm ∥·∥.Definition 2.2.
We next define theR-order of the procedure I in terms of the R-factor as
(2.11)OR(I,x*)={+∞ifRp(I,x*)=0,forp≥1,inf⁡⁡{p∣p∈[1,∞),Rp(I,x*)=1},otherwise.
Suppose that Rp(w(k))<1,then it follows from Ortega and Rheinboldt [13] that the R-order of I satisfies the inequality OR(I,x*)≥p.Theorem 2.3.
LetI be an iterative procedure and let Ω(I,x*) be the set of all sequences {x(k)}generated by I which converges to the limit x*. Suppose that there exists a p≥1 and a constant γ such that for any {x(k)}∈Ω(I,x*),
(2.12)‖x(k+1)-x*‖≤γ‖x(k)-x*‖p,k≥k0=k0({x(k)}).
Then, it follows that R-order of I satisfies the inequality OR(I,x*)≥p.We will use this result in order to calculate theR-order of convergence of PZSS1 in the subsequent section.For comparison, the procedure (2.7) has R-order of convergence at least 2 or OR(PT1,x*)≥2, while the R-order of convergence of (2.8) is greater than 2 or OR(PS1,x*)>2. However, the R-order of convergence of PSS1 is at least 3 or OR(PSS1,x*)≥3.


## 3. The Point Zoro Symmetric Single-Step Procedure PZSS1

The value ofxi(k,2) which is computed from (3.1c) requires (n-i) multiplications, one division, and (n-i+1) subtractions, increasing the lower bound on the R-order by unity compared with the R-order of  PS1.  Furthermore, the value of xi(k,3) which is computed from (3.1d)  requires (n-i) multiplications, one division, and (n-i+1) subtractions, increasing the lower bound on the R-order by unity compared with the R-order of  PSS1. This observation gives rise to the idea that it might be advantageous to add another step in PSS1. This leads to what is so called the  point zoro symmetric single-step procedure PZSS1 which consists of generating the sequences {xi(k)} (i=1,…,n) from(3.1a)xi(k,0)=xi(k)(i=1,…,n),(3.1b)xi(k,1)=xi(k)-p(xi(k))∏j=1i-1(xi(k)-xj(k,1))∏j=i+1n(xi(k)-xj(k,0))(i=1,…,n),(3.1c)xi(k,2)=xi(k)-p(xi(k))∏j=1i-1(xi(k)-xj(k,1))∏j=i+1n(xi(k)-xj(k,2))(i=1,…,n),(3.1d)xi(k,3)=xi(k)-p(xi(k))∏j=1i-1(xi(k)-xj(k,3))∏j=i+1n(xi(k)-xj(k,2))(i=1,…,n),(3.1e)xi(k+1)=xi(k,3)(i=1,…,n)(k≥0).The procedure PZSS1 has the following attractive features.From (3.1b), (3.1c),  and (3.1d), it follows that for k≥0,     (i)
the valuesp(xi(k)) (i=1,…,n)  which are computed for use in (3.1b) are  reused in  (3.1c) and (3.1d).(ii)
xn(k,2)=xn(k,1) and  x1(k,3)=x1(k,2), so that  xn(k,2) and x1(k,3) need not be  computed.(iii)
The product(3.2)∏j=1i-1(xi(k)-xj(k,1))(i=2,…,n),
which are computed for use in (3.1b) are reused in (3.1c).(iv)
The product(3.3)∏j=i+1n(xi(k)-xj(k,2))(i=n-1,…,1)
which are computed for use in (3.1c) are reused in (3.1d).The following lemmas (Monsi [9]) are required in the proof of Theorem 3.4.Lemma 3.1.
If(i)
p:C→C is defined by (2.1);(ii)
pi:C→C is defined by
(3.4)pi(x)=∏m=1i-1(x-xm*)∏m=i+1n(x-xm*)(i=1,…,n);(iii)
qi:C→C is defined by
(3.5)qi(x)=∏m=1i-1(x-x̅m)∏m=i+1n(x-x̂m)(i=1,…,n),
where x̅j≠x̅m and x̂j≠x̂m (j,m=1,…,n;j≠m);(iv)
ϕi:C→C  is  defined by
(3.6)ϕi(x)=qi(x)+∑j=1i-1pi(x̅j)qi(x)qi′(x̅j)(x-x̅j)+∑j=i+1npi(x̂j)qi(x)qi′(x̂j)(x-x̂j)(i=1,…,n),
then
(3.7)ϕi(x)=pi(x)(∀x∈C)(i=1,…,n).Lemma 3.2.
If (i)–(iv)  of Lemma3.1 are valid; (v) x̌i  (i=1,…,n) are such that  p(x̌i)≠0 (i=1,…,n), x̌i≠x̅m (m=1,…,i-1), x̌i≠x̂m (m=i+1,…,n), and
(3.8)x̅i=x̌i-p(x̌i)∏m=1i-1(x̌i-x̅m)∏m=i+1n(x̌i-x̂m)(i=1,…,n);
w̌i=x̌i-xi*, ŵi=x̂i-xi*, and  w̅i=x̅i-xi* (i=1,…,n), then
(3.9)w̅i=w̌i{∑j=1i-1γ̅ijw̅j+∑j=i+1nγ̂ijŵj}(i=1,…,n),
where
(3.10)γ̅ij=∏m≠i,j(x̅j-xm*)qi′(x̅j)(x̅j-x̌i)(j=1,…,i-1),γ̂ij=∏m≠i,j(x̂j-xm*)qi′(x̂j)(x̂j-x̌i)(j=i+1,…,n).Lemma 3.3.
If (i)–(v) of Lemma3.2 are valid; (vi)  |x̌i-xi*|≤θd/(2n-1) and |x̂i-xi*|≤θd/(2n-1) (i=1,…,n),  where  d=min{|xi*-xj*|∣i,j=1,…n;j≠i} and 0<θ<1, then  |w̅i|≤θ|w̌i| (i=1,…,n).Theorem 3.4.
If (i)p:C→C defined by  (2.1) has n distinct zeros  xi* (i=1,…,n); (ii) |xi(0)-xi*|≤θd/(2n-1) (i=1,…,n), where 0<θ<1 and d=min{|xi*-xj*|∣i,j=1,…,n:j≠i},  and the sequences {xi(k)} (i=1,…,n) are generated from PZSS1 (i.e., from (3.1a)–(3.1e)), then xi(k)→xi* (k→∞) (i=1,…,n) and  OR(PZSS1,x*)≥4.Proof.
Fori=1,…,n, let
(3.11)q1,i(x)=∏m=1i-1(x-xm(k,1))∏m=i+1n(x-xm(k,0)),q2,i(x)=∏m=1i-1(x-xm(k,1))∏m=i+1n(x-xm(k,2)),q3,i(x)=∏m=1i-1(x-xm(k,3))∏m=i+1n(x-xm(k,2)).
Then, by (3.5) and (3.6),
(3.12)ϕ1,i(x)=q1,i(x)+∑j=1i-1pi(xj(k,1))q1,i(x)q1,i′(xj(k,1))(x-xj(k,1))+∑j=i+1npi(xj(k,0))q1,i(x)q1,i′(xj(k,0))(x-xj(k,0)),ϕ2,i(x)=q2,i(x)+∑j=1i-1pi(xj(k,1))q2,i(x)q2,i′(xj(k,1))(x-xj(k,1))+∑j=i+1npi(xj(k,2))q2,i(x)q2,i′(xj(k,2))(x-xj(k,2)),ϕ3,i(x)=q3,i(x)+∑j=1i-1pi(xj(k,3))q3,i(x)q3,i′(xj(k,3))(x-xj(k,3))+∑j=i+1npi(xj(k,2))q3,i(x)q3,i′(xj(k,2))(x-xj(k,2)),
where pi(x) is defined by (3.4).
By Lemmas3.1 and 3.2 with qi=q1,i, x̌i=xi(k), x̂i=xi(k,0), x̅i=xi(k,1), ϕi=ϕ1,i (i=1,…,n), it follows that, for i=1,…,n, k≥0,
(3.13)wi(k,1)=wi(k){∑j=1i-1αij(k,1)wj(k,1)+∑j=i+1nαij(k,0)wj(k,0)},
where
(3.14)wi(k,s)=xi(k,s)-xi*(s=0,…,3),αij(k,1)=∏m≠i,j(xj(k,1)-xm*)q1,i′(xj(k,1))(xj(k,1)-xi(k))(j=1,…,i-1),αij(k,0)=∏m≠i,j(xj(k,0)-xm*)q1,i′(xj(k,0))(xj(k,0)-xi(k))(j=i+1,…,n).
Similarly, by Lemmas 3.1 and 3.2, with  qi=q2,i, x̌i=xi(k), x̂i=xi(k,2), x̅i=xi(k,1), ϕi=ϕ2,i (i=1,…,n), it follows that, for i=1,…,n, k≥0,
(3.15)wi(k,2)=wi(k){∑j=1i-1βij(k,1)wj(k,1)+∑j=i+1nβij(k,2)wj(k,2)},
where
(3.16)βij(k,1)=∏m≠i,j(xj(k,1)-xm*)q2,i′(xj(k,1))(xj(k,1)-xi(k))(j=1,…,i-1),βij(k,2)=∏m≠i,j(xj(k,2)-xm*)q2,i′(xj(k,2))(xj(k,2)-xi(k))(j=i+1,…,n).
Similarly, by Lemma 3.1 and Lemma 3.2, with  qi=q3,i, x̌i=xi(k), x̂i=xi(k,2), x̅i=xi(k,3), ϕi=ϕ3,i (i=1,…,n), it follows that, for i=1,…,n, k≥0,
(3.17)wi(k,3)=wi(k){∑j=1i-1γij(k,3)wj(k,3)+∑j=i+1nγij(k,2)wj(k,2)},
where
(3.18)γij(k,3)=∏m≠i,j(xj(k,3)-xm*)q3,i′(xj(k,3))(xj(k,3)-xi(k))(j=1,…,i-1),γij(k,2)=∏m≠i,j(xj(k,2)-xm*)q3,i′(xj(k,2))(xj(k,2)-xi(k))(j=i+1,…,n).
It follows from  (3.13)-(3.14) and Lemma 3.3 that |wi(0,1)|≤θ|wi(0,0)| (i=1,…,n), and it follows from (3.15)-(3.16) and Lemma 3.3 that  |wi(0,2)|≤θ2|wi(0,0)|  (i=1,…,n) follows from (3.1e). It follows from (3.17)-(3.18) and Lemma 3.3 that
(3.19)|wi(0,3)|≤θ3|wi(0,0)|(i=1,…,n),
whence  |wi(1,0)|≤θ3|wi(0,0)|  (i=1,…,n). It then follows by induction on k that, for all k≥0,
(3.20)|wi(k,0)|≤θ4k-1|wi(0,0)|(i=1,…,n),
whence xi(k)→xi*(k→∞), (i=1,…,n).  Let
(3.21)hi(k,m)=(2n-1)d|wi(k,m)|(i=1,…,n)(m=0,…,3).
Then, by (3.13), (3.15), (3.17), and (3.21), for i=1,…,n, (recall (3.1b), (3.1c), (3.1d))
(3.22)hi(k,1)≤1(n-1)hi(k,0){∑j=1i-1hj(k,1)+∑j=i+1nhj(k,0)},
for i=n,…,1,
(3.23)hi(k,2)≤1(n-1)hi(k,0){∑j=1i-1hj(k,1)+∑j=i+1nhj(k,2)},
and for i=1,…,n,
(3.24)hi(k,3)≤1(n-1)hi(k,0){∑j=1i-1hj(k,3)+∑j=i+1nhj(k,2)}.
Let
(3.25)ui(1,1)={2,(i=1,…,n-1),3,(i=n),ui(1,2)={4,(i=1),3,(i=2,…,n),ui(1,3)={4,(i=1,…,n-1),5,(i=n).
For r=1,2,3, let
(3.26)ui(k+1,r)={4ui(k,r),(i=1,…,n-1),4ui(k,r)+1,(i=n).
Then, by (3.25)–(3.26), for allk≥1,
(3.27)ui(k,1)={2(4(k-1)),(i=1,…,n-1),103(4(k-1))-13,(i=n),ui(k,2)={4(4(k-1)),(i=1),3(4(k-1)),(i=2,…,n-1)103(4(k-1))-13,(i=n),(3.28)ui(k,3)={4(4(k-1)),(i=1,…,n-1),163(4(k-1))-13,(i=n).
Suppose, without loss of generality, that
(3.29)hi(0,0)≤h<1(i=1,…,n).
Then, by a lengthy inductive argument, it follows from (3.21)–(3.29) that for  i=1,…,n, for all k≥1,
(3.30)hi(k,1)≤hui(k+1,1),hi(k,2)≤hui(k+1,2),hi(k,3)≤hui(k+1,3),
whence, by (3.28)  and (3.1e), for all k≥1,
(3.31)hi(k)≤h4k(i=1,…,n).
By (3.21) for m=3,
(3.32)|wi(k,3)|=d(2n-1)hi(k,3)(i=1,…,n),
then by (3.1e),
(3.33)|wi(k+1)|=d(2n-1)hi(k+1)(i=1,…,n).
So,
(3.34)|wi(k)|=d(2n-1)hi(k)(i=1,…,n)(k≥0).
Let
(3.35)w(k)=max⁡1≤i≤n⁡{|wi(k)|},h(k)=max⁡1≤i≤n⁡{hi(k)}.
Then, by (3.22)–(3.35)
(3.36)w(k)≤d(2n-1)h4k(∀k≥0).
So,
(3.37)R4(w(k))=lim⁡k→∞⁡sup⁡⁡{(w(k))1/4k}≤lim⁡k→∞⁡sup⁡⁡{(d2n-1)1/4kh}=h<1.
Therefore (Ortega and Rheindboldt [13]),
(3.38)OR(PZSS1,xi*)≥4(i=1,…,n).


## 4. Conclusion

The result above shows that the procedure PZSS1 hasR-order of convergence at least 4 that is higher than does PT1, PS1, and PSS1. The attractive features given in Section 3 of this procedure will give less computational time. Our experiences in the implementation of the interval version of PZSS1, that is, the procedure IZSS1(Rusli et al. [10]) showed that this procedure is more efficient for bounding the zeros simultaneously.



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