Neural network algorithm for the simultaneous extraction of all roots of algebraic polynomial

Zhang Ying Railway Traffic Department

Hunan Railway Professional Technology College Zhuzhou 412001, China

zhing1230@163.com

Zeng Zhe-zhao College of Electrical and Information Engineering Changsha University of Science and Technology

Changsha 410004，China hncs6699@yahoo.com.cn

Abstract— In this paper, we construct a neural network iteration method for simultaneous extraction of all roots of algebraic polynomial with variable learning rate. Its convergence was researched. The specific examples showed that the proposed method can simultaneously find all roots of algebraic polynomial at a very rapid convergence and very high accuracy with less computation.

Keywords- neural network; roots of algebraic polynomial; algorithm; convergence

I. INTRODUCTION (HEADING 1) In this paper we will consider a neural-network iteration

method for the simultaneous approximation of all zeros of a polynomial of degree n ,

0,)( 01 ≠+++= nn

n aaxaxaxf , (1)

with real coefficients. Let nξξξ ,,, 21 denote all the simple roots of the polynomial (1). Some authors have studied the parallel iterations for simultaneously finding all zeros of )(xf [1-5]. Xin Zhang, Hong Peng, and Guiwu Hu construct a high order iteration formula for the simultaneous inclusion of polynomial zeros using Lagrange interpolation with special nodes [6]. Gyurhan H. Nedzhibov and Milko G. Petkov presented a family of multi-point iterative methods for simultaneous determination of all roots of polynomial equation [7]. M.S. Petković and D.Milośević proposed improved Halley-like methods for the inclusion of polynomial zeros [8]. In this paper, we proposed a neural-network iteration method for the simultaneous extraction of polynomial zeros with variable parameters.

In section 2 and 3 we constructed a neural-network iteration method and analyzed the convergence, respectively. In section 4 we proposed the improved algorithm and its steps. The efficiency of the considered method was illustrated by numerical examples in section 5.

II. CONSTRUCTION OF THE NEURAL-NETWORK ITERATION METHOD

Let us suppose that 002

01 ,,, nxxx are distinct, reasonably

close initial approximations of the roots:

nξξξ ,,, 21 respectively. According to the formula (2) [7], we have

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−+−=+

)(2)()(11

kkk

kkkkk

uxfxfuxfuxx

λ (2)

where )()(

k

kk

xfxfu

′= , and λ is an arbitrary real parameter.

This family was proposed by Nedzhibov et al. [7]. These are order three iterative methods for ±∞≠ ,1λ and order four when 1=λ . Let

)()(

k

kkkkk

xfxfxuxy

′−=−= (3)

then the formula (2) has the following form

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−=+

)(2)()(11

kk

kkkk

yfxfyfuxx

λ (4)

Since kkk xxx Δ+=+1 (5)

hence we can obtain from the formula (4)and (5)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−=Δ)(2)(

)(1 kk

kkk

yfxfyfux

λ (6)

III. ANALYSIS OF CONVERGENCE In this section we analyzed the convergence property of the

formula (4). The following theorem deals with the choice of adaptive variable parameters λ . Theorem 1. Let 0)( =xf be an algebraic polynomial, with

n number of simple roots nξξξ ,,, 21 . If 00

201 ,,, nxxx are the initial approximations of the roots

respectively, then for sufficiently close initial approximations, only when the parameters satisfy

2009 International Conference on Computational Intelligence and Security

978-0-7695-3931-7/09 $26.00 © 2009 IEEE

DOI 10.1109/CIS.2009.78

161

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−<<⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛− 11

2111

21

kk TTλ (7)

The formula (4) is optimally convergent, where )(/)( kkk xfyfT = .

Proof. Define an error function and a Lyapunov function respectively as follows

)()(0)( kk xfxfke −=−= (8)

)(21)( 2 kekV = (9)

Then we have

)(21)1(

21)( 22 kekekV −+=Δ (10)

Since k

k xdx

kdekekekeke Δ+=Δ+=+ )()()()()1( (11)

and kkk

k xxfxdx

kdeke Δ′−=Δ=Δ )()()( (12)

According to formula (10) ,(11), and (12), we have

2

2

])([21)()(

)]([21)()()(

kkkkk xxfxxfxf

kekekekV

Δ′+Δ′=

Δ+Δ=Δ (13)

Known from the (6) that

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−=Δ)(2)(

)(1 kk

kkk

yfxfyfux

λ (14)

Let )(/)( kkk xfyfT = , then formula (14) has the following form

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−=Δ)21

1 k

kkk

TTuxλ

(15)

Substituting formula (15) into the formula (13) gives

2

211)(

21

211)()()(

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

+′+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+′−=Δ

k

kkk

k

kkkk

TTuxf

TTuxfxfkV

λ

λ (16)

If the formula (2) or formula (4) is convergent, i.e. 0)( <Δ kV , then it is easy to see from (16) that

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+′<

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

+′<

k

kkkk

k

kkk

TTuxfxf

TTuxf

λ

λ

211)()(

211)(

210

2

(17)

Hence we can obtain from the formula (17)

kk

k

k

k

uxfxf

TT

)()(2

211

′<

−+

λ (18)

Since 1)()( =

′ kk

k

uxfxf

, hence we can get the following form

from the formula (18)

221

1 <−

+ k

k

TT

λ (19)

Since

k

k

k

k

TT

TT

λλ 211

211

−+≤

−+ (20)

According to formula (19) and (20), we have

121

<− k

k

TT

λ (21)

Finally we can get from formula (21)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−<<⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛− 11

2111

21

kk TTλ (22)

Usually we chose the optimal parameters λ as follows

)/11)(3~2( kopt T−=λ (23)

which proved the theorem.

IV. IMPROVED ALGORITHM AND ITS STEPS

A. Improved algorithm

Let2

)(/1 kk xf ′=η , and )()( kxfke −= , then

)()()()( kk

k

kk xfke

xfxfu ′−=

′= η . Set )()(ˆ kyfke −= .

We can obtain from formula (4)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−′+=+

)()(ˆ2)(ˆ1)()(1

kekekexfkexx kkkk

λη (24)

where )()( kkkkkk xfkexuxy ′+=−= η (25)

Usually the optimal choice of kη is as follows

162

⎪⎩

⎪⎨⎧

>′′

≤′=

1)(,)(/1

1)(,5.02 kk

kkopt

xfxf

xfη

B. Algorithm steps To find simultaneously zeros of function 0)( =xf

Given one approximation Tnxxx ],,,[ 00

201

0 =x :

INPUT: 0x (real or complex vector); tolerance Tol ; maximum number of iterations N ; let 0=k ; OUTPUT: approximate solution kx or message of failure. Step 1: While Nk ≤ do Steps 2-5 Step 2: )()( kfk xe −= , )( kf x′ , and

⎪⎩

⎪⎨⎧

>′′

≤′=

1xx

1xη

)(,)(/.1

)(,5.02 kk

kkopt

ff

f;

Step 3: )(*).(*. kkopt

kk fk xeηxy ′+= ,

)()(ˆ kfk ye −= , and ))(/).(1(2 kk ff yxλ −= ;

Update weight vector:

)]}()(ˆ*.2/[).(ˆ1{

*).(*).(*.1

kkk

fkk

kkopt

kk

eeλe

xeηxx

−−

′+=+

Step 4: If Tolf k ≤+ ))(max( 1x then

OUTPUT ( 1+kx ); (The procedure was successful.) STOP Step 5: Set 1+= kk Go back to step 2 Step 6: OUTPUT (‘the method failed after N iterations,

=n ’ k ); (The procedure was unsuccessful.)

STOP

V. NUMERICAL EXAMPLES We give three examples in this section. The results of

three examples verify that the method proposed is high order convergent.

Example 1. Let 1010)( 3457 +−−−+= zzzzzzf

with zeros 21 =ξ , 12 =ξ , 13 −=ξ , i=4ξ , i−=5ξ ,

i216 +−=ξ , i217 −−=ξ . As initial approximations, the following complex numbers have been chosen [6]:

iz 23.066.101 += , iz 31.036.10

2 −= ,

iz 18.076.003 +−= , iz 17.135.00

4 +−= ,

iz 37.129.005 −= , iz 36.275.00

6 +−= ,

iz 62.127.107 −−= .

The Table 1 displayed the absolute errors:

iki

ki ze ξ−= ( 7,6,5,4,3,2,1=i ) using the method

proposed and the formula (11) in [6]. Where, the formula (11) in [6] is as follows

∑ ≠

+

++++−=

ijkij

ki

ki

ki

kik

iki

pWss

Wxx4)1(1

22

1 ,

∑≠ −

=ij

kj

ki

kjk

i xxW

s)(

,

))(( kj

ki

ki

kj

ki

kjk

ij xWxxxu

p−−−

= .

TABLE I. THE ABSOLUTE ERRORS FOR EXAMPLE 1

Method proposed Iteration formula (11) [6] kie k=8 k

ie k=3

ke1 0.000000 e+00 ke1 0.296524e-13

ke2 0.000000e+00 ke2 0.292952e-13

ke3 0.000000e+00 ke3 0.582335e-20

ke4 0.988305 e-18 ke4 0.750127e-18

ke5 0.289402e-18 ke5 0.335764e-17

ke6 0.000000e+00 ke6 0.111022e-15

ke7 0.000000e+00 ke7 0.000000e+00

* iki

ki ze ξ−=

Example 2. Let 1)( 4 −= zzf

with zeros 11 =ξ , 12 −=ξ , i=3ξ , i−=4ξ . As initial approximations, the following complex numbers have been chosen [6]:

iz 5.05.001 += , iz 42.036.10

2 +−= ,

iz 28.125.003 +−= , iz 37.146.00

4 −= The Table 2 displayed the absolute errors

1) iki

ki ze ξ−= ( 4,3,2,1=i ) using the adaptive

method proposed and the formula (11) in [6].

Example 3. Consider the algebraic polynomial

1)2045cos(2045)(

23

23

−−−++−−+=

xxxxxxxf

163

With zeros 51 −=ξ , 22 −=ξ , 23 =ξ . As initial approximations, the following real numbers have been chosen [7]: T]1.2,9.1,1.5[0 −−=x , and

T]4.2,7.1,3.5[0 −−=x . Table 3 showed the absolute

errors iki

ki xe ξ−= ( 3,2,1=i ) using the adaptive method

proposed and the formula (10) in [7]. Where, the formula (10) in [7] is as follows

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−=+

)(2)()(11

ki

ki

kik

iki

ki yfxf

yfWxx ,

∏≠

−= n

ij

kj

ki

kik

i

xx

xfW)(

)(, and k

iki

ki Wxy −=

TABLE II. THE ABSOLUTE ERRORS FOR EXAMPLE 2

Method proposed* Iteration formula (11) [6] kie k=8 k

ie k=3

ke1 0.0000e+00 ke1 0.6204 e-24

ke2 0.0000e+00 ke2 0.0000e+00

ke3 0.000e+00 ke3 0.2068 e-24

ke4 0.000e+00 ke4 0.0000e+00

* iki

ki ze ξ−=

TABLE III. THE ABSOLUTE ERRORS FOR EXAMPLE 3

Initial approximation

Method proposed

Iteration formula (10) [7]

k err k err [-5.1, -1.9, 2.1] 7 0.00e+0 5 0.84e-1 [-5.3,-1.7,2.4] 8 0.00e+0 3 0.49e-3

* iki

ki ze ξ−=

VI. CONCLUSIONS We can know from the table 1 to table 3 that the method

proposed can simultaneously find the real or complex roots of polynomials with fast convergence and high accuracy. The results from table 1 to table 3 show that the neural network iteration method proposed can simultaneously obtain exact roots of polynomial. Although the iterative numbers using the method proposed are almost three times as same as the iterative formula (11) in [6] , see Table 1 and Table 2, the iterative formula (11) in [6] has more computation than the method proposed at each iterative. The results in example 3 show that the method proposed has much higher accuracy than the iterative formula (10) in [7].

REFERENCES

[1] M.S. Petkoić, Iteration Methods for the Simultaneous Inclusion of Polynomial Zeros, Springer, Berlin, 1989.

[2] M.S. Petkoić, On initial conditions for the convergence of simultaneous root finding methods, Computing 57(1996)163-177.

[3] T. Sakural, M.S. Petkoić, On some simultaneous methods based on Weiesstrass’ correction, J. Comput. Appl. Math. 72(1996)275-291.

[4] S. Zheng, F. Sun, some simultaneous iterations for finding all zeros of a polynomial with high order convergence, Appl. Math. Comput. 99(1999)233-240.

[5] C. Carstensen, M.S. Petkoić, On iteration methods without derivatives for the simultaneous determination of polynomial zeros, J. Comput. Appl. Math. 45(1993)251-267.

[6] Xin Zhang, Hong Peng, Guiwu Hu, A high order iteration formula for the simultaneous inclusion of polynomial zeros, Appl. Math. Comput. 179 (2006)545 -552.

[7] Gyurhan H. Nedzhibov, Milko G. Petkov, On a family of iterative methods for simultaneous extraction of all roots of algebraic polynomial, Appl. Math. Comput. 162(2005)427-433

[8] M.S. Petkoić, D. Milośević, Improved Halley-like methods for the inclusion of polynomial zeros, Appl. Math. Comput. 169(2005)417-436.

164