On approximate bounds of zeros of polynomials within

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Special Issue: 10 | NCCOTII 2014 | Jun-2014, Available @ http://www.ijret.org 14
ON APPROXIMATE BOUNDS OF ZEROS OF POLYNOMIALS WITHIN
AND ON THE UNIT DISC
S.Sripriy12, Ajeet Singh2, P.K.Raina3
1Department of Mathematics, Lingayas University, Faridabad, Haryana
2Department of Mathematics, Lingayas University, Faridabad, Haryana
3Department of Mathematics, Institute of Education, J&K
Abstract
In this paper we obtain a result concerning the location of the zeros of a class of polynomials

  
n
j
n n j
j
j p z a z a where a and a s
0
( ) , 0, 1 ' are real and complex coefficient and numerically less than unity for
j  0,1,....., n. We obtain least upper bounds enclosing the zeros of polynomials as sharper bounds and compare these with the
ones given by Affane-Aji et al [1] and others [5, 6, 9].
-------------------------------------------------------------------***-------------------------------------------------------------------
1. INTRODUCTION
Current interest of researchers is to study the location of
zero of polynomials as such they have been writing
extensively on the works based on Gauss [4] and Cauchy
[2]. Numerous books and papers have been written in
modern areas of digital signal processing, Communication
theory, Control theory and Cryptography, to mention a few
and since then there is a greater need for improving the
bounds of the zeros of the polynomials. In this paper we
show that if all the coefficient of class of polynomials are
numerically less than unity. Then our method gives the
sharper bounds as compared to the ones given by Affane-Aji
et al [1]. Moreover, we show by way of examples that their
estimated bounds over estimate our bounds in all the cases
under present investigation.
The first result regarding the location of zeros of
polynomials is due to Gauss whose result was improved by
Cauchy [2] as given by the following theorem:
Theorem 1: Let 


 
1
0
( )
n
j
j
j
n p z z a z be the complex
polynomial. Then all the zeros of p(z) lie in the disc:
z : z   z : z 1 A; (1a)
Where j A  max a , 0  j  n 1, and  is a unique
positive root of the real coefficient polynomial
( ) . 1 0
2
2
1
1 Q x x a x a x a x a n
n
n
n
n        



(1b)
C-Affane-Aji et al [1] have given the following general
result of sharper bounds pertaining to the zeros of
polynomials that also include all the bounds obtained by
earlier authors [5, 6, 9].
Theorem 2.1: All the zeros of polynomials



 
1
0
( ) ,
n
j
j
j
n p z z a z
Lie in disc:
z z  z z  z z  z z  z z A k k               : : 1 : 1 ..... : 1 : 1 1 1     (2.1.1)
Where k  is the unique positive root of the characteristic equation:
( ) 0 1
2
1
1 1 1
  



    




    
Q x x   a x A k v
k
v
v
j
n j
k k v k j k v
k
k C C (2.1.2)

_______________________________________________________________________________________
For k  n or  n ,
Where A  max a , a  0, j  0 j j ,
0  j  n 1& nC  0, if n  r  0. r
We note here that
( )!( )!
!
n r r
n
Cr
n

 and may as will
be denoted by  


 


r
n
Result 1: For k = 2 from above theorem, the bound and
characteristic Eq. (2.1.2) turn out to be:
 : 1   : 1 , 2 z z    z z   A (2.1.3)
Where 2  is the unique positive root of the respective
characteristic equation
( ) (1 ) 0 1
2
2       Q x x a x A n (2.1.4)
The above equations coincide with the bounds given by
joyal et al [6].
Result 2: For k  3, equation (2.1.1) gives
z : z 1  z : z 1  z : z 1 A 3 2   ,
(2.1.5)
Where 3  is unique positive root of the characteristic
equation (2.1.2) is
( ) (2 ) (1 ) 0, 1 2
2
1
3
3            Q x x a x a a x A n n n
(2.1.6)
Which coincides with the bounds given by Sun [10]
Result 3 For k = 4, equation (2.1.1) turns out and to be the
disc
z : z   z : z 1  z : z 1  z : z 1  z : z 1 A 4 3 2    
, (2.1.7)
Where 4  is the unique positive root of characteristic
equation
( ) (3 ) (3 2 ) (1 ) 0 1 2 3
2
1 2
3
1
4
4                   Q x x a x a a x a a a x A n n n n n n
, (2.1.8)
Which coincides with the bounds given by Jain [5]
We, thus are able to show that this result provides a tool for
obtaining sharper bounds for the location of the zeros of a
class of polynomials with the real or complex co-efficient
each of which is numerically less than unity based on the
following remarks on the bounds of the zeros of polynomial:
2. BOUNDS ON ZEROS
The search for the real zeros of a polynomial function can be
reduced if bounds on the zeros are found. A number M is
bound on the root of a polynomial if every zero lies between
M and M inclusive. That is M is a bound to the zeros of a
polynomial f if M  any zerosof f  M .
Lemma 1:
Let f denote a polynomial function whose leading
coefficient is 1 as given by.
1 0
1
1 f (z) z a z ....... a z a n
n
n      

A bound M on the zero of f is the smaller of the two
numbers
max 1, , 1  , 0,1,...., 1
1
0
  









a Max a j n j
n
j
j
Where max means {} “Choose largest entry in  ”. The
bounds on the zeros of a polynomial provides good choice
for setting zero min and zero maximum (Z min and Z max)
of the viewing of an angular region.
We note here that we are interested in finding the unknown
radius  to determine the smaller circular bound containing
the zeros of the polynomials.
In this paper, using above Lemma 1, we replace 1+A where
A = Max j a 0  j  n 1 in above theorem [2.1] by B =
Max {1, D} where 



1
0
.
n
j
j D a
In order to show that our estimate of bounds are the least
upper bounds than obtained by C Affane Aji , we derive
the following theorem:
Theorem 3
All the zeros of polynomial



 
1
0
( ) ,
n
j
j
j
n p z z a z

_______________________________________________________________________________________
Lie in the disc:
 } { : } { : .... { : } { : }, 1 2            z z z z z z z z z k k (3.1.1)
Where k  is the unique positive root of the characteristic equation
( ) 0, 1
2
1
1
1
1
  



    




  
 
Q x x   C a x D k v
k x
j
n j
k v k j
k v k
k
k C

(3.1.2)
For k  n or  n,
Where 


 
1
0
max{1, } .
n
j
j B D subject to D a (3.1.3)
Here we note that for the class of polynomial under discussion, we show that for k  n or  n,
z : z } {z : z } ..... {z : z  {z : z 1 A}, k k           (3.1.4)
In view of the class of polynomials under the present discussion Here k  is as defined in the theorem [2.1] and k  lie within and
on the disc z   1.
To prove theorem 3, we first prove the following Lemma for k  n :
Lemma 2: Let 


 
1
0
( ) .
n
j
j
j
n p z z a z On simplifying equation (3.1.2)
We rewrite it as:
( ) ..... 1 .....  0, 1 2 1
2
1 2
1 2 2 1
1
1
1 1
       


    


      

 
 



Q x x C a x C C a a x a a a x D n n n k
k
n n
k k
k
n
k
k
k
(3.1.5)
For k > n, equation (3.1.5) can be written as:










  
  

 
  

  
 
 

  

 
1
1 0
1 1 2
1
( 1)
(1 .... ) (1 )
( 1) .....
2
( 1)
( )
( ) (1 )
n k
n
n n
n
n
n
k n
k
a a x D x
n a a
n n
x n a x
Q x x (3.1.6)
Now setting:
( ) ( ) .... (1 .... ) . 1 1 2 0
1
1 L x x n a x a a a x n n
n
n
n
n   

        (3.1.7)
We now give a simple proof for proving (3.16) based on mathematical induction:
Let P(k) denote the statement:
( ) (1 )  ( ) (1 ) , 1
1
( 1)  

      n k
n
k n
k Q x x L x D x (3.1.8)

_______________________________________________________________________________________
Where k  n  r, r  1,2,3,......... ..and n N, the set of natural numbers
Now for k  n 1, from Eq. (3.1.8), we have
R.H.S of Eq. (3.1.8) = (1 )  ( ) (1 )  ( ) , 1
( 1) 1
1
( 1 1) x L x D x L x D n
n n
n
n n      
  

   (3.1.9)
L.H.S of Eq. (3.1.8) = a x  a a x D
n
Q x x n
n
n
n
n      


 


  


 


   

 1 1 0
1
1 .... 1 ......
1
( ) (from(3.1.6)) (3.1.10)
= L x D n   ( ) 1 , using (3.1.7)
P(k) is true for n + 1.
Now let us assume P(k) is true for k = n + r, i.e
( ) (1 )  ( ) (1 ) , 1
1
1  


     r
n
r
n r Q x x L x D x , (3.1.11)
We now show that P(k) is true for (n + r) + 1, i.e
 r 
n
r
n r Q x x L x D x 
   ( )  (1 ) ( )  (1 ) ( ) 1 1 , (3.1.12)
We now eliminate ]jfrom the Eq. (3.1.10), (3.1.11) and (3.1.12) to give:
0,
( ) (1 ) 1
( ) (1 ) 1
( ) 1 1
1
1







r
n r
r
n r
n
Q x x
Q x x
Q x
which on simplifying gives:
 
(1 ) 1 ( ) (1 ) ( )
(1 ) 1
1
( ) 1
1
( ) 1 x Q x x x Q x
x
Q x n
r
n r
r
n r r 

      
 
 (3.1.13)
Since the statement that P(k) is proved true for n + 1 and
assumed to be true for k = n + r as explained above,
therefore by mathematical induction from Eq. (3.1.13), it is
evident, that P(k) is also true for k = (n + r) + 1. Thus
statement Lemma 2 is true for k = n + 1, (n + 1) + 1, (n + 1)
+ 2,
Here we make use of this Lemma to prove theorem 3.1
pertaining to the polynomials having the co-efficient of the
class of polynomials absolute less than unity. This process
helps us in obtaining the zeros of this class of polynomials
under discussion.
Now in order to obtain the bounds for the characteristic
equation Q(x) associated to the given polynomial P (z) n
we note that
( ) ( ) (1)
1 1
Lt Q x Lt Q x Q
x x
 
   
(3.2.14)
In view of Q(x) being a polynomial, is continuous
function. We therefore, express Q(x) in terms of Q (x) k to
study the two cases of k such as: k  n & k  n.
Case 1:- For k>n, from Eq. 1(b), we have:
  1 0
2
2
1
1 0
(1) ( ) (1 ) (1 ) (1 ) ..... 1
1
Q Lt Q x Lt a a a a k
n
n k
n
k
n
k
x n
k
         


  
   


_______________________________________________________________________________________
 


 


  






    


 

 
 






  


 



   


 


   



1 1 0
1 2
1
1
0
... 1 ....
1
1
1
1
...
1
a a a
n
n a
n
n n
Lt k k
n
k
n
k n k
n
k
n
k
k
     

 


 


  






  


 



   


 

 
   


 


  


 



   


 


   



1 1 0
1 2
1
1
0
1 ...
1
...
1
1
1
1
...
1
a a a
n
n n
n a
n
n n
Lt k k
n
k
n
k n k
n
k
n
k
k
     

   
...1 .....  1  1] (1 )  .
.... ( 1) ( 2) ...2
2
1
2
[
( 1) ( 1)
1 0
1 2 1
2
1
1
1
0
is positive
D
Here
D
a a
a n n a n a a a
n n
Lt n a
n k
k
k
n k
k
k
n
n n k
n
n k
n
n k
n
k
k
   

 





     
         


 


 


 

 
  


 


   




  

 
1 .....  (1 ) ]
1
2
...
3
2
2
1
1
...
1
1
2
1
[
( 1)
1 0
2
2 1
1 2
1
1 1
1
0
 

 

 


   
 




 





  


 



 


 




  


 




  


 



   


 


 


 

 
  


 


   
n k
n k k
k
n n
n
n k
n
n k
n
k
k
a a D
a a
a
n
n
a
n
n
n
n
a
n n
Lt n a
k
 

  
 
 1 
1
0
( ) (1 )
1  


   n k
n k
k
Lt L x D
k

 
(Using 3.17 of Lemma)





  k n
k
k k
k
Q
Q Lt
k
0 (1 )( 1)
1 ( )
(1)


 
(Using 3.18 of Lemma)
 (1)  0 sin ( )  0. n k Q ce Q  (Since k  as in (3.1.2) is positive root of Q (x)  0 k )
Therefore Q(1)  0  0 is 1 which gives0   1. k
Case 2:- Now if k  n, we have
(1) (1 ) (1 ) (1 ) ..... 1 , 1 0
1
1 0 1
Q Lt Lt a a a k
n
k
n
k k
x n
k
         
  
   

(by (1b))
 ( 1) 2 1 2 2 
( 1)
1
0
(1 ) (1 ) ..... (1 ) (1 ) ..... (1 )
1 k k k
n
k k
n k
n k k
n
k
n
k Lt a a D D D
n
k
       

             
 

 
 


 


 


 

  
      
 
 
 

 
k
n k
n k k
n k k
n
k
n
k Lt a a D
n
k 

  

(1 ) 1
(1 ) (1 ) ..... (1 )
1
( 1)
( 1)
1
0 1
Lt  a a a D k n k k
n
k n k
k
n k
k
k k
n k
k
k
k
        


 

 
  





 

      



( 1)
3
2
2
1
1
( 1)
0
(1 ) (1 ) (1 )
(1 )

_______________________________________________________________________________________
.. (1 ...... ]
( 2)
2
( 1)( 2)
[ ( 1
(1 )
1 ( 1)
2
1 2
1
1
( 1)
0
a a D
k a a
k k
Lt k a
n n k k
k
n n n
k
n k
k
n
k
n k
k
k
  




  
 
   


  

 


 


  



  k k
k
k Lt Q
k




(1 )
0



 ( )  0 k n Q 
In order to proof of above theorem we show that 1  k k   .
But before we give the proof, we first prove the following
the recurrence relation of Q (x) k as given below:
Recurrence relation of Q (x) k
Lemma 3 Q (x) k as defined above by Eq. (3.1.7), then
(1 ) ( ) ( ) 0 1      x Q x Q x Dx k k (4.1.1)
Proof:- From Eq. (3.1.8), we have
  1
1
1 1 ( ) ( ) (1 )  

      n k
k n
n k x Q x L x D x (4.1.2)
Replacing k by k-1 from eq. (3.1.8) we have
  2
1 1
2 1 ( ) ( ) (1 )  
 
      n k
k n
n k x Q x L x D x (4.1.3)
Now eliminating ( ) 1 L x n from above two equations, we
have
(1 )  ( ) (1 ) ( ) (1 ) (1 ) 1 1
1
1         

  x Q x x Q n D x x n k
k k
n k
This prove recurrence relation given by (4.1.1)
Proof of the Theorem:-
Since k1  is the root of ( ) 0 1 1  k k Q  (4.1.4)
Now put 1  k x  in the (4.1.1) is
1  1  1  0, 1 1 1 1 1       k k k k k k  Q  Q  
Where D & k1  are
Now
        1 1 1 1 1 1         k k R k k k k Q  Q   Q 
       , , 0 1. 1 D D a j n k j 
Hence   0, 1  R k Q  where as    0. R k Q  This implies
1  k k   this complete the proof of the theorem.
2.1 Some Examples based on the above Theorem:
Corresponding to the polynomial P z n we solve the so
called characteristic equation Q (x)  0. k The output of this
result is as given below in the following tables. In the Ist
column we have the degree from 1 to k of the algebraic
equation Q (x)  0 k in the second column we have the
corresponding bounds k 1 for the radius of the circle
that contains all the zeros of the polynomials and in the 3rd
column we tabulate the present bounds by the present class
of polynomials corresponding to the bounds max(1, ) k  .
On comparison we show that the present bounds obtained
are the infimum (lesser of lub) enclosing the respective
zeros of the polynomials as compared to the lub of
C Affane Aji. The output of these example is as given
below:
Example 1:
Let ( 1),
4
1
( ) 5 4 2 p z  z  z  z  true bound =
0.91119850
Values
of
K
Their Bound Our Bound Over estimate of
z 1 (of others) error k   z  max}1, } 1, k  3,: k 
1 1.25000000 {1, .75} For k = 10, the %
corresponding to the
bounds obtained by C.
2 2.00000000 {1,1.31872929=1.3872929
3 1.21972895 {1,.43861017}

_______________________________________________________________________________________
4 1.21372896 {1.38706535} Affane. Aji. et al [1]
over the present
estimate, errors by a
factor 2.6.
5 1.9237991 {1.33438766}
6 1.1802918 {1.29419863}
7 1.17815386 {1.27942836}
8 1.16181430 {1,.25511065}
9 1.156181430 {1,.23659091}
10 1.14753595 {1,.22058214}= 1.00000000
Example 2:
Let ,
4
1
3
1
5
1
3
1
( )
2
1
( ) 15 13 12 10 6 3 p z  z  z  z  z  z  z  z 
Values
of
K
Their Bound Our Bound Over estimate of error
z 1 (of others) k   z  max}1, } 1, k  3,: k 
1 1.50000000 {1,2.616}=2.6166666 Over estimation factor of
previous authors
corresponding to k=10, the
bounds obtained C. Affane.
Aji. et al [1] over estimate
the present bounds at least
by a factor 5.8 and in view
of error is 0.31393132:
0.053474
2 1.36602593 {1,2.19292693}=2.1929269
3 1.36602542 {1,.887149531}
4 1.36602542 {1,.71292043}
5 1.32471797 {1,.51521635}
6 1.30080360 {1,.51451182}
7 1.30080357 {1,.47310513}
8 1.29996768 {1,.46837437}
9 1.27138251 {1,.40011427}
10 1.26045701 {1,.37009931}
Example 3:
Let 0,
10
1
50
1
( )
10
1
( )
8
1
5
1
( ) 25 23 21 16 15 10 6 3 2 p z  z  z  z  z  z  z  z  z  z   true bound = 0.9225556
Values
of
K
z 1 (of others) k   z  max{1, }  k, k  2 k 
1 1.20000000 {1,0.995} Over estimation of previous
authors[1] , corresponding to
k=10 by < factor 2.4. In view
of the rel. errors of other
authors and present authors as:
0.18775156: 0.077444.
2 1.7082014 {1,1.657956}
3 1.708204 {1,.49147788}
4 1.15340459 {1,.44147179}
5 1.14820807 {1,.381310340}
6 1.14857797 {1,.33248603}
7 1.12732975 {1.29817784}
8 1.11938045 {1,.27032855}
9 1.11294632 {1,24874921}
10 1.11030725 {1.23107074}
Example 4:
Let
10 4 4), .89920688
6
35
(20 5 10 4 4 20
140
(5 10 4 4 )
20
1
( )
4 2
35 35 34 32 31 30 22 21 19 18 17 5
    
          
z z z true bound
z z z z z z
i
p z z z z z z z
Values
of
K
z 1 (of others) k   z  max{1, } 1, k  3 k 
1 1.50000000 {1,1.77}=1.7714205 Over estimate of error factors
corresponding to k = 10,the
bounds obtained C. Affane.
2 1.7539049 {1,1.75725359}=1.75725359
3 1.2670351 {1,.57764816}

_______________________________________________________________________________________
4 1.34667841 {1,.51961333} Aji. et al [1] overstate the
present bounds or least by a
factor 3. In view of negative
error of other authors and the
present as 0.32286685:
0.10079
5 1.31837386 {1,.51664084}
6 1.29831222 {1,39623964}
7 1.27480981 {1,.38846096}
8 1.25438958 {1,.36794695}
9 1.23839181 {1,.34069264}
10 1.22287373 {1,.3150619}
From the above table we note the upper bounds obtained by
C. Affane.Aji. et al [1] over estimate the corresponding
bounds due to the present investigation in all the cases at
least by a factor 2.5 and thereby is confirm that in all the
cases k  3, unit disc is the least upper bounds enclosing
the zeros of the class of polynomials under the present
discussion.
REFERENCES
[1] C. Affane-Aji, N.Agarwal & N.K. Govil,
Mathematical and Computer Modelling.
[2] A.L. Canchy, Exercises de mathmatiques, IV Annee
de Bure Freres, Paris, 1829.
[3] B. Datt, N.K. Govil, on the location of zeros of
polynomial, J. Approximation theory 24(1978), 78-
82.
[4] K. F. Gauss, Beitrage zur Theorie der algebraisshen
Gleichungen, Abh. Ges. Wiss. Gottingen 4(1850)
Ges. Werkr. Vol 3, pp73-102.
[5] V.K. Jain on Cauchy’s bounds for zeros of
polynomials Turkish.J. Math. 30(2006), 95-100.
[6] A. Joyal, G. Labelle, Q.I. Rehman on location of
zeros of polynomials Canada.math.bull. 10(1967),
53-63.
[7] M. Marden, Geometry of polynomials, Amer. Math
Society, Math surveys Vol. 3, Amer Math society,
providence, R.I, 1966.
[8] Z. Rabinstein, Some results in the location the zeros
of linear combinations of polynomials, Trans. Amer.
Math. Soc. 116(1965), 1-8.
[9] Y.J. Sun, J.G. Hsieh, Anote on circular bounds of
polynomials of zeros, IEEE Trans, Circuits system I,
Regul pap 43(1996), 476-478.
[10] M.S. Zilric, L.M. Roytman, P.L. Combettes, M.N.
Swamy, A bound for the zeros of polynomials, IEEE
Trans circuits systemI, 39(1992), 476-478.

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