Generalization of Ehrlich-Kjurkchiev method for multiple roots of
algebraic equations
ANTON I. ILIEV
In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev
method is developed. The method allows to find simultaneously all roots of the algebraic
equation in the case when the roots are supposed to be multiple with known
multiplicities. The offered generalization does not demand calculation of derivatives of
order higher than first simultaneously keeping quaternary rate of convergence which
makes this method suitable for application from the practical point of view.
Introduction. The problem of simultaneous finding all roots (SFAR) of
polynomial equation is connected with the works of Weierstrass [1] and
Dochev [2]. Their formula represents the modification of the Newton
method for individual determination of the roots of the algebraic equation.
The method suggested by Weierstrass-Dochev has quadratic rate of
convergence. Using divided differences with multiple knots Semerdzhiev
[3] generalizes the method of Weierstrass-Dochev, when the roots are
multiple. Another approach to SFAR is proposed by Ehrlich [4]. He shows
that for his method the modification of Gauss-Seidel is applicable. The
Ehrlich method is generalized for the case of multiple roots of algebraic,
trigonometric and exponential equations in [5]. The presented there method
is of the same complexity as the method of Ehrlich for simple roots. The
same method is also generalized [6] for the case of polynomial equations on
arbitrary Chebyshev system having multiple roots with known
multiplicities. The rate of convergence is also cubic, but at each step
determinants should be calculated.
In this paper a new method representing a generalization of the method
of Ehrlich-Kjurkchiev [7] for algebraic equations is obtained. Our method
has also quaternary rate of convergence and is effective from a
computational point of view and it can be used for SFAR, when they have
known multiplicities. This new method is of the same complexity as the
method of Ehrlich-Kjurkchiev for simple roots.
Let the algebraic polynomial
(1)
An ( x ) = x n + a1 x n −1 +...+ a n
be given and x1 , x 2 ,..., x m be its roots with known multiplicities
α 1 , α 2 ,..., α m (α 1 + α 2 +...+α m = n) . When the roots are simple (α 1 = α 2 =...α m = 1)
Kjurkchiev [7] proposed the following iteration formula for SFAR
2
[
xi[ k +1] = xi[ k ] − An ( xi[ k ] ) / An′ ( xi[ k ] ) − An ( xi[ k ] )Wi′[ k ] ( xi[ k ] ) / Wi [ k ] ( xi[ k ] )
+ An ( xi[ k ] )
(2)
n
∑
j =1 , j ≠ i
( )(
An x [jk ] xi[ k ] − x [jk ]
)
−2

/ W j[ k ] x [jk ] 

( )
i = 1, n , k = 0,1,2,...
where Wp[ k ] ( x [pk ] ) =
∏ (x
n
l =1 , l ≠ p
[k]
p
)
− xl[ k ] , p = 1, n .
Our principal aim is to generalize (2) for the case when the roots are not
simple, i.e. when α 1 ,α 2 ,...,α m are arbitrary.
So, we define
(3)
x
[ k +1]
i
=x
[k ]
i

− α i  S i[ k ] ( xi[ k ] ) +

m
∑
j =1 , j ≠ i
(
αj x
[k ]
j
−x
[k ]
i
)
−2
( )[ S ( x ) / α ]
An x
[k ]
j
[k ]
j
[k ]
j
α j −1
/Q
j
[k ]
j
(x )
[k ]
j



−1
i = 1, m , k = 0,1,2,...
where S [pk ] ( x [pk ] ) = An′ ( x [pk ] ) / An ( x [pk ] ) − Q ′p[ k ] ( x [pk ] ) / Q [pk ] ( x [pk ] ) , p = 1, m
and Q [pk ] ( x [pk ] ) =
∏ (x
m
l =1 , l ≠ p
[k ]
p
− x [l k ]
)
αl
, p = 1, m .
The following theorem shows, that the iterative formula (3) is
convergent to the roots with quaternary rate of convergence.
Theorem. Let c, q and d
def
min xi − x j be real positive constants, so
i≠ j
that the following inequalities be fulfilled
q<1, d−2c>0, 2c 2 n( d − 2c)
where M
def
−2
[c( d − 2c)
[[1 + c / ( d − 2c)] − 1]
−1
+ (1 + c( d − 2c)
and N
n
−1
[
)[ N + MN + M ]] < α
def 
(
)2
 1 + n( c / d − 2c )
]
n −1
i
, i = 1, m

− 1. It is
assumed that the initial approximations x1[ 0] , x 2[ 0] ,..., x m[ 0] to the roots of (1)
x1 , x 2 ,..., x m are chosen so that the inequalities xi[ 0] − xi < cq , i = 1, m hold
true. Then for every natural number k the inequalities
(4) xi[ k ] − xi < cq 4 , i = 1, m
are satisfied.
Proof. Let us assume that the inequalities (4) be fulfilled for some
natural k. It will be proved, that xi[ k +1] − xi < cq 4 , i = 1, m .
Easily one can find that
k
k +1
(5) Qi′[ k ] ( xi[ k ] ) / Qi[ k ] ( xi[ k ] ) =
and
m
∑
j =1 , j ≠i
(
)
α j / xi[ k ] − x [jk ] , i = 1, m
3
(6) S
[k]
i
(x ) = α / (x
[k ]
i
i
[k ]
i
− xi ) +
m
∑
j =1 , j ≠ i
(
α j x j − x [jk ]
)[( x
[k ]
i
)(
− x [jk ] xi[ k ] − x j
)]
−1
, i = 1, m .
If we subtract xi from both sides of the iterative formula (3), then using
(5),(6) and reducing under common denominator the right side of (3) we
receive
m

xi[ k +1] − xi = α i + ( xi[ k ] − xi ) ∑ α j x j − x [jk ]

j =1 , j ≠ i
(7)
(
][
)[( x
[k ]
i
)(
− x [jk ] xi[ k ] − x j
+ ( xi[ k ] − xi ) Pi [ k ] ( xi[ k ] ) − α i Si[ k ] ( xi[ k ] ) + Pi [ k ] ( xi[ k ] )
where Pi [ k ] ( xi[ k ] ) =
m
∑
j =1 , j ≠ i
( )(
α j An x [jk ] x [jk ] − xi[ k ]
]
−1
[k ]
j
−1
, i = 1, m
) [S ( x ) / α ]
−2
)]
[k ]
j
α j −1
j
( )
/ Q [j k ] x [jk ] .
Further, if from the numerator and the denominator of (7) factors ( xi[ k ] − xi )
−1
and ( xi[ k ] − xi ) respectively are separated, then (7) can be written in the form
x
− xi = ( x
[ k +1]
i
[k]
i
− xi )
2
 m
[k ]
 ∑ α j xj − xj
 j =1 , j ≠i
(
m

(8) × α i + ( xi[ k ] − xi ) ∑ α j x j − x [jk ]

j =1 , j ≠ i
(
)[( x
[k ]
i
)[( x
−x
[k ]
i
[k]
j
)(
− x [jk ] xi[ k ] − x j
)( x
[k ]
i
− xj
)]
−1
)] + ( x
−1
[k ]
i

+ Pi [ k ] ( xi[ k ] ) 


− xi ) Pi ( x ) 

[k ]
−1
[k]
i
i = 1, m .
Now we transform Pi [ k ] ( xi[ k ] ) , i = 1, m by the following way
Pi
(x ) = ∑ (
=
∑ α (x
m
[k]
×
[k]
i
j =1, j ≠i
m
j =1, j ≠i
j
∏ (x
m
s=1, s≠ j
[k]
j
[k ]
j
α j x[jk ] − xi[k ]
)(
) [ S ( x ) ] ∏( x
[k]
j
)
− xs
)
− xj x − x
− xs[ k ]
[k]
i
) (x
−α s
[k]
j
−2
−2
αs
[k]
j
[k]
j
/αj
α j −1 m
l =1

[k]
1 + x j − x j / α j

((
[k]
j
− xl
) ∏ (x
αl
m
s=1, s≠ j
) ) ∑α ( x − x )[( x
[k]
j
m
l =1, l ≠ j
l
l
[k]
l
[k ]
j
− x[sk ]
)(
)
−α s
− xl x − x
[k ]
j
[k ]
l
)]
−1
α j −1



, i = 1, m.
If we multiply the numerator and the denominator of the previous equation
with ( xi[ k ] − x j ) and put the obtained expression for Pi [ k ] ( xi[ k ] ) , i = 1, m into (8)
we receive
4
− xi = ( x − xi )
[ k +1]
i
[k ]
i
x
∑ α (x
)(
(9) 
j
j =1 , j ≠i

×1− xi[ k ] − x j xi[ k ] − x[jk ]

(
)[(
m
2
)
−1
)(
− x[jk ] xi[ k ] − x[jk ] xi[ k ] − x j
j
( ) ∏ (x
m
R[jk ] x[jk ]
s=1 , s≠ j
) [(
(
− xs[ k ]
[k ]
j
) (x
−α s
)(
×α i + ( xi[ k ] − xi ) ∑ α j x j − x[jk ] xi[ k ] − x[jk ] xi[ k ] − x j

j =1 , j ≠i
×
m
∏ (x
m
s=1 , s≠ j
where R
Denote
(10)
[k]
ij
Y
[k ]
j
(x
[k ]
j
− xs[ k ]
(x )
[k ]
j
[k]
i
,x
[k]
j
) (x
−α s
[k ]
j
− xs
)
αs
)]
− xs
[k ]
j
−1
−x
(
)(
) ( )
− x j xi[ k ] − x[jk ] R[jk ] x[jk ]
−1
)
1− x




 , i = 1, m

)
[k ]
i
αs
[k ]
i
) ∑ α ( x − x )[( x
def
)
)] [1− ( x

= 1 + x [jk ] − x j / α j

((
−1
)(
− xj x
)(
m
l
l =1 , l ≠ j
[k]
i
[k ]
j
)
−1
[k ]
l
l
− xl x
[k ]
j
(x ) ∏ (x
m
R
[k ]
j
[k]
j
s =1 , s ≠ j
[k]
j
−x
[k]
j
− x s[ k ]
[k ]
l
)]
) (x
−α s
−1
[k ]
j
α j −1



− xs
i , j = 1, m , i ≠ j.
The expression for R [j k ] ( x [jk ] ) , j = 1, m can be transformed in the form
α j −1
R
(11)
[k ]
j
( x ) = 1 + ∑ (α
[k ]
j
r =1
j
)[ (

×  x [jk ] − x j / α j

((
)
)]
− 1 ! r! α j − 1− r !
−1
) ∑ α ( x − x )[( x
m
l =1 , l ≠ j
l
[k ]
l
l
[k ]
j
)(
− xl x
[k ]
j
−x
[k ]
l
)]
−1

 , j = 1, m .

r
Using the following inequalities
x [jk ] − x l ≥ x j − x l − x j − x [jk ] ≥ d − cq 4 > d − c > d − 2c
k
(12)
x [jk ] − x l[ k ] ≥ x [jk ] − x l − x l − x l[ k ] ≥ d − 2cq 4 > d − 2c , l , j = 1, m , l ≠ j
k
the estimate for R [j k ] ( x [jk ] ) , j = 1, m , can be found.
Namely, from (11), (12) and N
(13) R [j k ] ( x [jk ] ) ≤ 1 + ( q 4
k
)
2
[
def 
(
)2
 1 + n( c / d − 2c )
N , j = 1, m .
Further, we make the following transformation
]
n −1

− 1 it follows that
.
)
αs
5
Z
[k ]
j
(x )
[k ]
j
∏ (x
def
m
s=1 , s≠ j
 m
=  ∏ x [jk ] − x [s k ]
 s=1 , s≠ j
(
[k ]
j
− x [s k ]
) (x
−α s
 m
+...+ ∏ x [jk ] − x [s k ]
 s= m−1 , s≠ j
(
 m
+  ∏ x [jk ] − x s[ k ]
 s= m , s≠ j
(
) (x
−α s
− xs
[k ]
j
) (x
−α s
) (x
−α s
[k ]
j
)
[k ]
j
− xs
αs
−
− xs
)
− xs
[k ]
j
)
αs
∏ (x
m
s= 2 , s≠ j
)
αs
−
[k ]
j
− x [s k ]
∏ (x
m
s= m , s≠ j
[k ]
j
) (x
−α s
− x [s k ]
[k ]
j
− xs
) (x
−α s
)
[k ]
j
αs



− xs
)



αs

− 1 + 1 , j = 1, m .

αs
Obviously, the expression Z [j k ] ( x [jk ] ) , j = 1, m can be presented in the form
( x ) = 1+ ∑ ( x
m
Z
(14)
[k ]
j
[k ]
j
αl
l =1 , l ≠ j
(
× ∑ x [jk ] − x l
p =1
) (x
αl − p
[k ]
l
[k ]
j
(
− xl ) x
− x l[ k ]
)
p −1
On the other hand we have
(15) Z [j k ] ( x [jk ] ) =
m
∏
s =1 , s ≠ j
[1 + ( x
[k ]
s
[k ]
j
−x
[k ]
l
)
∏ (x
m
−α l
s = l +1 , s ≠ j
[k ]
j
− x s[ k ]
) (x
−α s
[k ]
j
− xs
)
αs
, j = 1, m .
(
− x s ) / x [jk ] − x s[ k ]
Now, using (12), (15) and M
)]
αs
, j = 1, m .
[[1 + c / ( d − 2c)] − 1]
def
n
one can find the
following estimate
(16) Z [j k ] ( x [jk ] ) ≤ 1 + q 4 M , j = 1, m .
k
In order to receive an estimate for Yi [jk ] ( xi[ k ] , x [jk ] ) , i , j = 1, m , i ≠ j we transform
this expression by the use of (11) and (14) as follows
(
)
(
)(
Yi[jk ] xi[ k ] , x[jk ] = 1− xi[ k ] − x j xi[ k ] − x[jk ]
(17) α −1
) −( x
−1
[k ]
i

× ∑ α j − 1 ! r ! α j − 1− r !  x[jk ] − x j / α j

 r =1
j
+
∑ (x
m
l =1, l ≠ j
[k ]
l
) ] ((
)[ (
(
(
− xl ) x[jk ] − xl[ k ]
−1
) ∏(
−α l
m
s=l +1, s≠ j
)[ (
) ] ((
)
−1
) ) ∑ α ( x − x )[( x
x[jk ] − xs[ k ]
) (
−α s
 α j −1
−1
× 1 + ∑ α j − 1 ! r ! α j − 1 − r !  x[jk ] − x j / α j

 r=1
(
)(
− x j xi[ k ] − x [jk ]
m
l
l =1 , l ≠ j
x[jk ] − xs
[k ]
l
l
αl
) ∑( x
αs
p=1
[k ]
j
) ) ∑ α ( x − x )[( x
− xl
m
l =1, l ≠ j
l
l
[k ]
l
[k ]
j
[k ]
j
)(
− xl x − x
) (x
αl − p
)(
− xl x
i, j = 1, m , i ≠ j.
Using (12), (13), (16) and (17) it is easy to receive the estimate
[k ]
j
[k ]
j
[k]
j
[k]
l
− xl[ k ]
−x
[k ]
l
)
)]
)]
−1



r
p−1
−1



r



6
[
k 
k
k
k
−1
−1 
Yi[jk ] xi[ k ] , x [jk ] ≤ q 4 c( d − 2c) + 1 + cq 4 ( d − 2c) q 4 N + M 1 + q 4

(18)
(
)
(
)
( )
2
]

N 

i , j = 1, m , i ≠ j.
Now, from (9) we obtain
x
[ k +1]
i
(19)
− xi ≤ ( x
[
× αi − x
[k ]
i
− xi )
[k ]
i
− xi
m
∑
j =1 , j ≠ i
m
2
∑
j =1 , j ≠ i
[
α j x j − x [jk ] xi[ k ] − x [jk ] xi[ k ] − x j
[
α j x j − x [jk ] xi[ k ] − x [jk ] xi[ k ] − x j
]
−1
]
−1
(
Yi[jk ] xi[ k ] , x [jk ]
)]
(
Yi[jk ] xi[ k ] , x [jk ]
−1
)
, i = 1, m .
Finally, with the help of inequalities (12) the estimate (18) and using the
conditions of the theorem from (19) we receive
( ) ncq
xi[ k +1] − xi < c2 q 4
k
2
4k


[
( d − 2c) −2 q 4 c / ( d − 2c) + (1 + cq 4 ( d − 2c) −1 )q 4 N + M 1 + ( q 4

k
k
k

k 2
k
k
k
−2
−1
−1  k
× αi − c2 q4 n( d − 2c) q 4 c( d − 2c) + 1 + cq 4 ( d − 2c) q4 N + M 1 + q 4


( )
< cq 4 c2 n( d − 2c)
k +1
(
−2

−2
× α i − c2 n( d − 2c)

[
)
[c / ( d − 2c) + (1+ c( d − 2c) )[ N + MN + M]]
[c( d − 2c) + (1+ c(d − 2c) )[ N + MN + M]]]
( )
2
]
k
)
2
]

N 

−1

N 

−1
−1
−1
−1
k +1
< cq 4 , i = 1, m.
Thus the theorem is completely proved.
Remark. In the case when α 1 = α 2 =... = α m = 1 the method (3) coincides with
the method (2).
Numerical example. For the equation A6 ( x ) = ( x + 2) 2 ( x − 1)( x − 3) 3 = 0 using
. , x 3[ 0 ] = 4 with the help of the method
initial approximation x1[ 0] = −3, x 2[ 0] = 01
(3) the roots with accuracy of 18 decimal digits have been found only after 3
iterations.
k
0
1
2
3
x1[k ]
-3.00000000000000000
-1.98938060918119354
-1.99999999967737963
-2.00000000000000000
x 2[k ]
0.100000000000000000
0.995064651338749428
0.999999994237752166
1.000000000000000000
x 3[k ]
4.00000000000000000
3.02604710332169412
3.00000000683325288
3.00000000000000000
Acknowledgements. The author is deeply grateful to Dr. Khristo
Semerdzhiev for many useful discussions concerning this paper.
REFERENCES
[1] K. WEIERSTRASS. Neuer Beweis des Satzes, dass jede Ganze Rationale Function einer
Veränderlichen dargestellt werden kann als ein Produkt aus Linearen Funktionen
derselben Veränderlichen, Ges. Werke 3 (1903), 251-269.
7
[2] K. DOCHEV. An alternative method of Newton for simultaneous calculation of all the
roots of a given algebraic equation. Physico-Mathematical Journal, 5, 2 (1962), 136-139
(in Bulgarian).
[3] KHR. SEMERDZHIEV. A method for simultaneous finding all roots of an algebraic
equation if their multiplicities are given. C. R. Acad. Bulg. of Sci. 35, 8 (1982), 10571060 (in Russian).
[4] L. EHRLICH. A modified Newton’s method for polynomials, Comm. ACM 10, 2
(1967), 107-108.
[5] A. ILIEV. A generalization of Obreshkoff-Ehrlich method for multiple roots of
polynomial equations. C. R. Acad. Bulg. of Sci. 49, 5 (1996), 23-26.
[6] I. MAKRELOV, KHR. SEMERDZHIEV, S. TAMBUROV. A method for simultaneous
finding all zeros of a given generalized polynomial over a Chebyshev system. Serdica
Bulg. math. publ. 12, 4 (1986), 351-357 (in Russian).
[7] N. KJURKCHIEV. On some modifications of Ehrlich’s method for simultaneous solving
of algebraic equations. Pliska Stud. Math. Bulg. 5 (1983), 43-50 (in Russian).
University of Plovdiv
http://www.pu.acad.bg
Faculty of Mathematics and Informatics
http://www.fmi.pu.acad.bg
Department of Numerical Methods
24 Tzar Assen str.
4000 Plovdiv, BULGARIA
e-mail: aii@pu.acad.bg
URL: http://anton.iliev.tripod.com
1991 Mathematics Subject Classification: 65H05 Single equations.
Keywords: Simultaneous roots finding; quaternary convergence.