Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 709832, 11 pages
doi:10.1155/2012/709832
Research Article
The Point Zoro Symmetric Single-Step
Procedure for Simultaneous Estimation of
Polynomial Zeros
Mansor Monsi,1 Nasruddin Hassan,2 and Syaida Fadhilah Rusli1
1
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia,
43400 UPM Serdang, Selangor, Malaysia
2
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
43600 Bangi, Selangor, Malaysia
Correspondence should be addressed to Nasruddin Hassan, nas@ukm.my
Received 15 November 2011; Revised 29 February 2012; Accepted 20 March 2012
Academic Editor: Martin Weiser
Copyright q 2012 Mansor Monsi et al. 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.
The point symmetric single step procedure PSS1 has R-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.
1. Introduction
The iterative procedures for estimating simultaneously the zeros of a polynomial of degree n
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
0
in Section 3. This procedure needs some preconditions for initial points xi 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
Journal of Applied Mathematics
2. Methods of Estimating polynomial zeros
Let p : C → C be a polynomial of degree n defined by
n
px ai xi ,
2.1
i0
ai ∈ C i 1, . . . , n and an /
0. Let x∗ x1∗ , x2∗ , . . . , xn∗ T be the distinct zeros of px 0,
expressed in the form:
px n
x − xi∗ 0,
2.2
i1
with an 1. Suppose that, for j 1, . . . , n, xj is an estimate of xj∗ , and let q : C → C be
defined by
qx n
x − xj .
2.3
j1
Then,
q xi n
xi − xj ,
i 1, . . . , n.
2.4
j/
i
By 2.2, if for i 1, . . . , n, xi / i, then
/ xj j 1, . . . , n; j pxi .
∗
x
−
x
i
j/
i
j
xi∗ xi − 2.5
Now, xj ≈ xj∗ j 1, . . . , n so by 2.5,
pxi j/
i xi − xj
xi∗ ≈ xi − i 1, . . . , n.
2.6
i 1, . . . , n k ≥ 0,
2.7
An iteration procedure PT1 of 2.6 is defined by
k1
xi
k
xi
k
p xi
− k
k
j/
i xi − xj
Journal of Applied Mathematics
3
which has been studied by Kerner 7. Furthermore, the following procedure PS1
k1
xi
k
xi
− i−1
k
xi
j1
k
p xi
k1
− xj
n
ji1
k
xi
k
− xj
2.8
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 PSS1Monsi
9. The procedure PSS1 is defined by
k,0
xi
k,1
xi
k
xi
− i−1
j1
k,2
xi
k
xi
− i−1
j1
k
xi
k
xi
i 1, . . . , n,
k
p xi
k,1 n
− xj
k
xi
ji1
k
p xi
k,1 n
− xj
k1
xi
k
xi
ji1
k,2
xi
k
xi
k,0
− xj
i 1, . . . , n,
2.9
k,2
− xj
i 1, . . . , n,
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 a p ≥ 1 such that for any null sequence {wk } generated from
{xk }, then the R-factor of the sequence {wk } is defined to be
Rp wk
⎧
1/k
⎪
⎨ lim sup wk ,
k→∞
k
⎪ lim sup wk 1/p ,
⎩
k→∞
p 1,
p > 1,
2.10
where Rp is independent of the norm · .
Definition 2.2. We next define the R-order of the procedure I in terms of the R-factor as
⎧
⎨∞ if Rp I, x∗ 0,
OR I, x∗ ⎩infp | p ∈ 1, ∞, R I, x∗ 1,
p
for p ≥ 1,
otherwise.
2.11
Suppose that Rp wk < 1, then it follows from Ortega and Rheinboldt 13 that the R-order
of I satisfies the inequality OR I, x∗ ≥ p.
4
Journal of Applied Mathematics
Theorem 2.3. Let I be an iterative procedure and let ΩI, x∗ be the set of all sequences
{xk } generated by I which converges to the limit x∗ . Suppose that there exists a p ≥ 1 and a constant
γ such that for any {xk } ∈ ΩI, x∗ ,
p
k1
− x∗ ≤ γ xk − x∗ ,
x
k ≥ k0 k0
xk .
2.12
Then, it follows that R-order of I satisfies the inequality OR I, x∗ ≥ p.
We will use this result in order to calculate the R-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
k,2
The value of xi
which is computed from 3.1c requires n−i multiplications, one division,
and n−i1 subtractions, increasing the lower bound on the R-order by unity compared with
k,3
which is computed from 3.1d requires
the R-order of PS1. Furthermore, the value of xi
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
k
the sequences {xi } i 1, . . . , n from
k,0
xi
k,1
xi
k
xi
− i−1
j1
k,2
xi
k
xi
− i−1
j1
k,3
xi
k
xi
− i−1
j1
k
xi
k
xi
k
xi
k1
xi
k
xi
k
p xi
k,1 n
k
k
p xi
k,1 n
− xj
k
k
p xi
k,3 n
− xj
k
− xj
− xj
− xj
− xj
k,3
xi
3.1a
i 1, . . . , n,
ji1
xi
ji1
xi
ji1
xi
k,0
k,2
k,2
i 1, . . . , n,
3.1b
i 1, . . . , n,
3.1c
i 1, . . . , n,
3.1d
i 1, . . . , n k ≥ 0.
3.1e
The procedure PZSS1 has the following attractive features.
From 3.1b, 3.1c, and 3.1d, it follows that for k ≥ 0,
k
i the values pxi i 1, . . . , n which are computed for use in 3.1b are reused in
3.1c and 3.1d.
k,2
ii xn
k,1
xn
k,3
and x1
k,2
x1
k,2
, so that xn
k,3
and x1
need not be computed.
Journal of Applied Mathematics
5
iii The product
i−1 j1
k
xi
k,1
− xj
i 2, . . . , n,
3.2
which are computed for use in 3.1b are reused in 3.1c.
iv The product
n k
xi
ji1
k,2
− xj
i n − 1, . . . , 1
3.3
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
pi x i−1
∗
x − xm
m1
n
∗
x − xm
i 1, . . . , n;
3.4
i 1, . . . , n,
3.5
mi1
iii qi : C → C is defined by
qi x i−1
x − xm m1
n
x − xm mi1
where xj /
m);
xm and xj /
xm (j, m 1, . . . , n; j /
iv φi : C → C is defined by
φi x qi x i−1 p x q x
n
pi xj qi x
i
j
i
x − xj
j x − xj
j1 qi xj
ji1 qi x
i 1, . . . , n,
3.6
then
φi x pi x
∀x ∈ C i 1, . . . , n.
3.7
0
Lemma 3.2. If i–iv of Lemma 3.1 are valid; v x̌i i 1, . . . , n are such that px̌i /
xm m 1, . . . , i − 1, x̌i /
xm m i 1, . . . , n, and
i 1, . . . , n, x̌i /
px̌i n
m mi1 x̌i − x
m1 x̌i − xm xi x̌i − i−1
i 1, . . . , n;
3.8
6
Journal of Applied Mathematics
i xi − xi∗ , and wi xi − xi∗ i 1, . . . , n, then
w̌i x̌i − xi∗ , w
⎧
⎫
i−1
n
⎨
⎬
wi w̌i
γ ij wj γij w
j
i 1, . . . , n,
⎩ j1
⎭
ji1
where
∗
− xm
γ ij − x̌i
∗
j − xm
m/
i,j x
γij qi xj xj − x̌i
m/
i,j xj
qi xj xj
3.9
j 1, . . . , i − 1 ,
j i 1, . . . , n .
3.10
Lemma 3.3. If i–v of Lemma 3.2 are valid; vi |x̌i − xi∗ | ≤ θd/2n − 1 and |xi − xi∗ | ≤
θd/2n − 1 i 1, . . . , n, where d min{|xi∗ − x∗j | | i, j 1, . . . n; j / i} and 0 < θ < 1, then
|wi | ≤ 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
0
|xi − xi∗ | ≤ θd/2n − 1 i 1, . . . , n, where 0 < θ < 1 and d min{|xi∗ −x∗j | | i, j 1, . . . ,
k
n : j/
i}, and the sequences {xi } i 1, . . . , n are generated from PZSS1 i.e., from 3.1a–
k
3.1e, then xi → xi∗ k → ∞ i 1, . . . , n and OR PZSS1, x∗ ≥ 4.
Proof. For i 1, . . . , n, let
q1,i x i−1 k,1
x − xm
m1
q2,i x i−1 k,1
x − xm
,
n k,2
x − xm
,
3.11
mi1
i−1 k,3
x − xm
m1
k,0
x − xm
mi1
m1
q3,i x n n k,2
x − xm
.
mi1
Then, by 3.5 and 3.6,
k,1
k,0
n
q1,i x
q1,i x
pi xj
pi xj
φ1,i x q1,i x ,
k,1
k,1
k,0
k,0
x − xj
x − xj
j1 q1,i xj
ji1 q1,i xj
k,1
k,2
i−1
n
q2,i x
q2,i x
pi xj
pi xj
φ2,i x q2,i x ,
k,1
k,1
k,2
k,2
x − xj
x − xj
j1 q2,i xj
ji1 q2,i xj
k,3
k,2
i−1
n
q3,i x
q3,i x
pi xj
pi xj
φ3,i x q3,i x ,
k,3
k,3
k,2
k,2
x − xj
x − xj
j1 q3,i xj
ji1 q3,i xj
i−1
where pi x is defined by 3.4.
3.12
Journal of Applied Mathematics
7
k
k,0
By Lemmas 3.1 and 3.2 with qi q1,i , x̌i xi , xi xi
n, it follows that, for i 1, . . . , n, k ≥ 0,
k,1
wi
k,1
, xi xi
, φi φ1,i i 1, . . . ,
⎧
⎫
i−1
n
⎨
⎬
k
k,1 k,1
k,0 k,0
,
wi
αij wj
αij wj
⎩ j1
⎭
ji1
3.13
where
k,s
k,s
xi
− xi∗ s 0, . . . , 3,
k,1
∗
x
−
x
m
m
i,j
j
/
j 1, . . . , i − 1 ,
k,1
k,1
k
xj
q1,i xj
− xi
wi
k,1
αij
k,0
∗
xj
− xm
k,0
k,0
k
xj
xj
q1,i
− xi
k,0
αij
j i 1, . . . , n .
m/
i,j
k
k,2
Similarly, by Lemmas 3.1 and 3.2, with qi q2,i , x̌i xi , xi xi
i 1, . . . , n, it follows that, for i 1, . . . , n, k ≥ 0,
k,2
wi
3.14
⎧
i−1
⎨
k
wi
⎩ j1
k,1
βij
k,1
wj
n
k,2
βij
ji1
k,2
wj
k,1
, xi xi
, φi φ2,i
⎫
⎬
3.15
,
⎭
where
k,1
βij
k,2
βij
k,1
∗
xj
− xm
k,1
k,1
k
xj
xj
q2,i
− xi
m/
i,j
k,2
m
/ i,j xj
k,2
xj
q2,i
−
k,2
xj
∗
xm
j 1, . . . , i − 1 ,
3.16
k
− xi
j i 1, . . . , n .
k
k,2
Similarly, by Lemma 3.1 and Lemma 3.2, with qi q3,i , x̌i xi , xi xi
i 1, . . . , n, it follows that, for i 1, . . . , n, k ≥ 0,
k,3
wi
⎧
⎫
i−1
n
⎨
⎬
k
k,3 k,3
k,2 k,2
,
wi
γij wj
γij wj
⎩ j1
⎭
ji1
k,3
, xi xi
, φi φ3,i
3.17
8
Journal of Applied Mathematics
where
k,3
k,3
∗
xj
− xm
k,3
k,3
k
xj
xj
q3,i
− xi
k,2
k,2
∗
x
−
x
m
m/
i,j
j
k,2
k,2
k
xj
q3,i xj
− xi
γij
γij
m/
i,j
j 1, . . . , i − 1 ,
3.18
0,1
j i 1, . . . , n .
0,0
It follows from 3.13-3.14 and Lemma 3.3 that |wi | ≤ θ|wi | i 1, . . . , n, and it follows
0,2
0,0
from 3.15-3.16 and Lemma 3.3 that |wi | ≤ θ2 |wi | i 1, . . . , n follows from 3.1e. It
follows from 3.17-3.18 and Lemma 3.3 that
0,0 0,3 wi ≤ θ3 wi 1,0
whence |wi
0,0
| ≤ θ3 |wi
k
3.19
| i 1, . . . , n. It then follows by induction on k that, for all k ≥ 0,
k
k,0 0,0 wi ≤ θ4 −1 wi whence xi
i 1, . . . , n,
i 1, . . . , n,
3.20
→ xi∗ k → ∞, i 1, . . . , n. Let
k,m
hi
2n − 1 k,m wi
i 1, . . . , n m 0, . . . , 3.
d
3.21
Then, by 3.13, 3.15, 3.17, and 3.21, for i 1, . . . , n, recall 3.1b, 3.1c, 3.1d
k,1
hi
⎧
⎫
i−1
n
⎨
⎬
1
k,0
k,1
k,0
,
≤
hj
hj
hi
⎩ j1
⎭
n − 1
ji1
3.22
for i n, . . . , 1,
k,2
hi
⎧
⎫
i−1
n
⎨
⎬
1
k,0
k,1
k,2
hi
≤
hj
hj
,
⎩ j1
⎭
n − 1
ji1
3.23
and for i 1, . . . , n,
k,3
hi
⎧
⎫
i−1
n
⎨
⎬
1
k,0
k,3
k,2
hi
.
≤
hj
hj
⎩ j1
⎭
n − 1
ji1
3.24
Journal of Applied Mathematics
9
Let
1,1
ui
1,2
ui
1,3
ui
⎧
⎨2,
i 1, . . . , n − 1,
⎩3,
i n,
⎧
⎨4,
i 1,
⎩3,
i 2, . . . , n,
⎧
⎨4,
i 1, . . . , n − 1,
⎩5,
i n.
3.25
For r 1, 2, 3, let
k1,r
ui
⎧
⎨4uk,r
,
i
i 1, . . . , n − 1,
⎩4uk,r 1,
i
i n.
3.26
Then, by 3.25–3.26, for all k ≥ 1,
⎧ ⎪
⎨2 4k−1 ,
k,1
ui
10 1
⎪
⎩
4k−1 − ,
3
3
⎧ ⎪
4 4k−1 ,
⎪
⎪
⎪
⎨ k−1
k,2
,
ui
3 4
⎪
⎪
⎪
⎪
⎩ 10 4k−1 − 1 ,
3
3
⎧ ⎪
⎨4 4k−1 ,
k,3
ui
16 1
⎪
⎩
4k−1 − ,
3
3
i 1, . . . , n − 1,
i n,
i 1,
3.27
i 2, . . . , n − 1
i n,
i 1, . . . , n − 1,
i n.
3.28
Suppose, without loss of generality, that
0,0
hi
≤h<1
i 1, . . . , n.
3.29
Then, by a lengthy inductive argument, it follows from 3.21–3.29 that for i 1, . . . , n, for
all k ≥ 1,
k1,1
k,1
≤ hui
k,2
≤ hui
k,3
≤ hui
hi
hi
hi
k1,2
k1,3
,
,
,
3.30
10
Journal of Applied Mathematics
whence, by 3.28 and 3.1e, for all k ≥ 1,
k
k
≤ h4
hi
i 1, . . . , n.
3.31
By 3.21 for m 3,
k,3 wi d
k,3
h
2n − 1 i
i 1, . . . , n,
3.32
k1 wi
d
k1
h
2n − 1 i
i 1, . . . , n.
3.33
then by 3.1e,
So,
k wi d
k
h
2n − 1 i
i 1, . . . , n k ≥ 0.
3.34
Let
k wk max wi ,
1≤i≤n
3.35
hk max hi k .
1≤i≤n
Then, by 3.22–3.35
wk ≤
d
k
h4
2n − 1
∀k ≥ 0.
3.36
So,
R4 w
k
lim sup
k→∞
w
k
1/4k ≤ lim sup
k→∞
d
2n − 1
1/4k h h < 1.
3.37
Therefore Ortega and Rheindboldt 13,
OR PZSS1, xi∗ ≥ 4 i 1, . . . , n.
3.38
4. Conclusion
The result above shows that the procedure PZSS1 has R-order of convergence at least 4 that
is higher than does PT1, PS1, and PSS1. The attractive features given in Section 3 of this
Journal of Applied Mathematics
11
procedure will give less computational time. Our experiences in the implementation of the
interval version of PZSS1, that is, the procedure IZSS1Rusli et al. 10 showed that this
procedure is more efficient for bounding the zeros simultaneously.
Acknowledgment
The authors are indebted to Universiti Kebangsaan Malaysia for funding this research under
the grant UKM-GUP-2011-159.
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International Journal of
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International Journal of
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Optimization
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