New simultaneous methods for finding all zeros of a polynomial

0.14 MB·English

by  Jun-Seop Song

NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A
POLYNOMIAL
JUN-SEOP SONGA,∗
ACOLLEGE OF MEDICINE, YONSEI UNIVERSITY, SEOUL, REPUBLIC OF KOREA
5 Abstract. The purpose of this paper is to present three new methods for finding all
1
0 simple zeros of polynomials simultaneously. First, we give a new method for finding
2
n simultaneously all simple zeros of polynomials constructed by applying the Weierstrass
a
J method to the zero in the trapezoidal Newton’s method, and prove the convergence

1
of the method. We also present two modified Newton’s methods combined with the
2

derivative-free method, which are constructed by applying the derivative-free method
]
A
to the zero in the trapezoidal Newton’s method and the midpoint Newton’s method,
N
. respectively. Finally, we give a numerical comparison between various simultaneous
h
t
a methods for finding zeros of a polynomial.
m
[

1
v
3 I. Introduction
3
0
5 Witha typical iterationmethodsuch asNewton’s method, aninitial approximation ofa
0
.
1 zero converges to a specific zero, but the Weierstrass method (or Durand-Kerner method)
0
5 approximates all simple (real or complex) zeros of polynomial simultaneously (see [2, 4]).
1
:
v Let P(z) = zn +a zn−1 +···+a z +a be a polynomial of degree n having simple
1 n−1 n
i
X
zeros with constants a ,a ,...,a . Let r ,...,r be the distinct zeros of P(z) and let
r 1 2 n 1 n
a
distinct complex numbers z ,...,z be their approximations. The Weierstrass method
1 n
(Durand-Kerner method) is defined as
P(zm)
zm+1 = zm − i , (1)
i i (zm −zm)
j6=i i j
Q
2010 Mathematics Subject Classification. 65H04, 65H05.
Key words and phrases. Polynomial zeros, Simultaneous methods, Weierstrass method, Convergence,
Newton’s method.
*Corresponding author.
1
2 JUN-SEOPSONG
for m ≥ 0, and this method is one of the most frequently used iterative methods which
give simultaneous computation of all zeros of P. If a function W (z) is defined by
i
P(z)
W (z) = ,
i
(z −z )
j6=i j
Q
then W (z) has the same zeros as the polynomial P, and so the problem of finding the
i
zeros of P reduces to that of zeros of the function W (z). If we denote W = W (z ) for
i i i i
i = 1,2,...,n in the case of z = z , (1) can be written as
i
zˆ = z −W , (2)
i i i
where z is a current approximation and zˆ is a new approximation to a zero of polyno-
i i
mial P(z). The method constructed by (2) is called the Weierstrass-like method (briefly,
WLM).
The aim of this paper is to present three new methods for finding all simple zeros
of polynomials simultaneously. These new methods are based on the Frontini-Sormani’s
midpoint Newton’s method ([7]) and the Weerakoon’s trapezoidal Newton’s method ([8])
which were modifications of the Newton’s method through iterative approximations.
ItiswellknownthatNewton’smethodisdefinedbyx∗ = x−f(x) withanapproximation
f′(x)
x and a new approximation x∗ of a zero, and is efficient to find a zero of an equation
f(x) = 0 for a differentiable function f with proper conditions and a sufficiently close
initial value (see [8]).
In [8], Weerakoon proposed the trapezoidal Newton’s method defined by
2f(x)
xˆ = x− . (3)
f′(x)+f′(x∗)
He applied Newton’s method to the x∗ of the denominator.
Along with (3), the midpoint Newton’s method that Frontini-Sormani proposed in [7] is
constructed as
f(x)
xˆ = x− . (4)
f′ x+ 1(x∗ −x)
2
TheyalsoappliedNewton’s methodtothe(cid:0)x∗ ofthedenom(cid:1)inator, andsoset x∗ = x− f(x).
f′(x)
NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL 3
Both the trapezoidal Newton’s method and the midpoint Newton’s method are of cubic
order, while the original Newton’s method was of quadratic order. A variety of methods
can be applied to the x∗ in addition to Newton’s method. Petkovi´c [5] derived the fol-
lowing simultaneous method for finding all simple zeros of polynomials by applying the
Weierstrass method to the x∗ in the midpoint Newton’s method:
P(z )
i
zˆ = z − , (5)
i i P′ z − 1W
i 2 i
(cid:0) (cid:1)
which is called Newton-Weierstrss method (or NWM). Also, Petkovi´c [6] found the fol-
lowing derivative-free method (or DFM) defined as
W
i
zˆ = z − , (6)
i i
1−P(z −W )/P(z )
i i i
which has a similar form with the one above and this method is of cubic order.
In this paper, we present three new methods for the simultaneous approximation of all
simple zeros of polynomials by applying the Weierstrass-like method and the derivative-
freemethodtox∗ inthetrapezoidalNewton’smethodandthemidpointNewton’smethod.
Throughout this paper, the convergence of zeros will be discussed and the order will
be calculated for new constructed methods. We will use the notation a = O (b) for two
M
complex numbers a and b whose moduli are of the same order, that is, |a| = O(|b|). In
addition, the error is defined as |e| = max {|e |} with e = z −r for i = 1,...,n.
i i i i
i=1,...,n
In all discussions, the order related to e , which is an error of the previously approx-
i
imated zeros z , is presumed to be the same. After that, we will show that the order
i
related to e , which is an error of the approximated zeros concerning each method, is
i
identical. For the same being, the order related to the already approximated zeros eˆ is
i
hypothesized to be identical as follows:
e = O (e) for all i.
i M
In Section 2, we give a new method for finding simultaneously all simple zeros of poly-
nomials constructed by applying the Weierstrass method to the x∗ in the trapezoidal
Newton’s method, and prove the convergence of the method. In Section 3, we present
4 JUN-SEOPSONG
two modified Newton’s methods combined with the derivative-free method. They are
constructed by applying the derivative-free method to the x∗ in the trapezoidal Newton’s
method and the midpoint Newton’s method, respectively. In Section 4, we give a numer-
ical comparison between various simultaneous methods for finding zeros of a polynomial.
Finally, we conclude that the convergence of all new constructed methods in this paper
are similar or superior than other iterative methods of cubic order.
II. Weierstrass-like Trapezoidal Newton’s method
In this section, we construct a new method for finding simultaneously all simple zeros
of polynomials of cubic order. By applying the Weierstrass method (5) to the x∗ in the
trapezoidal Newton’s method (3), we derive a new method constructed as follows:
2P(z )
i
zˆ = z − . (7)
i i P′(z )+P′(z −W )
i i i
We call (7) the Weierstrass-like trapezoidal Newton’s method, and from this, simply, call
it Method 1.
The calculation and discussion of the order of Method 1 are similar to those of the
Newton-Weierstrass method, which is an alteration of Petkovi´c’s midpoint Newton’s
method (see [5]). From (7), we have the following theorem.
Lemma 1. For a polynomial P(z), we have
P(z ) P′′(z )P(z )
i i i O (e)+O (e2) = O (e3).
P′(z ) 2P′(z )2 M M M
i (cid:18) i (cid:19)
Proof. By the Taylor’s expansion around r , we have that
i
P′′(r )
P(z ) = P′(r ) e + i e2 +O (e3) ,
i i i 2P′(r ) i M
(cid:18) i (cid:19)
P′′(r ) P′′′(r )
P′(z ) = P′(r ) 1+ i e + i e2 +O (e3) , (8)
i i P′(r ) i 2P′(r ) i M
(cid:18) i i (cid:19)
P′′(r ) P′′′(r ) P′′′′(r )
P′′(z ) = P′(r ) i + i e + i e2 +O (e3) .
i i P′(r ) P′(r ) i 2P′(r ) i M
(cid:18) i i i (cid:19)
NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL 5
From (8), we obtain
P(z ) P′′(z )P(z )
i i i O (e)+O (e2)
P′(z ) 2P′(z )2 M M
i (cid:18) i (cid:19)
1 P′′(r ) 2 P′′(r ) P′′′(r ) 3
= e + i e2 +O (e3) 1− i e + i e2 +O (e3)
2 i 2P′(r ) i M P′(r ) i 2P′(r ) i M
(cid:18) i (cid:19) (cid:18) i i (cid:19)
P′′(r ) P′′′(r ) P′′′′(r )
× i + i e + i e2 +O (e3) O (e)+O (e3)
P′(r ) P′(r ) i 2P′(r ) i M M M
(cid:18) i i i (cid:19)
1 P′′(r )
= (O (e))2(1+O (e))3 i +O (e) O (e)+O (e3) = O (e3).
2 M M P′(r ) M M M M
(cid:18) i (cid:19)
(cid:3)
From Lemma 1, we have the following theorem.
Theorem 1. If the approximate zero x grounded from Method 1 is close enough to r
i i
and the order of e is the same, then the order of eˆ is identical, and |eˆ| = O (|e|3) is
i i i M
formed.
Proof. We easily see that the following equation is satisfied.
n
P(z ) = (z −r ) = (z −r ) (z −r ) = e (z −r ) = O (e).
i i j i i i j i i j M
j=1 j6=i j6=i
Y Y Y
That is,
W = W (z ) = O (P(z )) = O (e). (9)
i i i M i M
If Q(z) = P(z)− n (z −z ), then Q(z) is a polynomial of order n−1, and Q(z ) =
j=1 j i
P(z ) for all i. TherQefore, Q(z) is the Lagrange interpolation of points z ,z ,...,z , and
i 1 2 n
so we have
n n
z −z z −z
k k
Q(z) = P(z ) = W (z −z )
j j j k
z −z z −z
j k! ! j k!
j=1 k6=j j=1 k6=j k6=j
X Y X Y Y
n n n
W
j
= W (z −z ) = (z −z ).
j k k
z −z
! j!
j=1 k6=j j=1 k=1
X Y X Y
6 JUN-SEOPSONG
Therefore, we obtain
n n
W
j
P(z) = 1+ (z −z ) (10)
j
z −z
j!
j=1 j=1
X Y
n
W
j
= W (z −z )+ 1+ (z −z ).
i j j
z −z
j!
j6=i j6=i j=1
Y X Y
From (10), it follows that
P′(z) = 1 + 1+ j6=i zW−zjj −(z −zi) j6=i (z−Wzjj)2. (11)
P(z) z −zj! WP+(z −z ) 1+ P Wj
Xj6=i i i j6=i z−zj
(cid:16) (cid:17)
P
Substituting z = z in (11), we obtain
i
P′(z ) 1 1+ Wj
i = + j6=i zi−zj .
P(z ) z −z W
i i j! P i
j6=i
X
Therefore, we have
P(z ) 1+ Wj P(z )
W = i · j6=i zi−zj = i (1+O (e)). (12)
i P′(zi) 1− PP′((zPzii)) j6=i zi−1zj P′(zi) M
P
Now we will find the order of Method 1. If the Taylor’s expansion is applied to P′(z −
i
W ), then we have
i
2P(z )
i
zˆ = z −
i i P′(z )+P′(z −W )
i i i
2P(z )
i
= z −
i P′(z )+ P′(z )−P′′(z )W + 1P′′′(z )W2 +···
i i i i 2 i i
(cid:0) P(zi) (cid:1)
= z − .
i P′(z )− 1P′′(z )W + 1P′′′(z )W2 +···
i 2 i i 4 i i
NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL 7
By (9), (12) and Lemma 1, we have that
P(z )
i
zˆ = z −
i i P′(z ) 1− P′′(zi)W +O (e2)
i 2P′(zi) i M
(cid:16) (cid:17)
P(z ) P′′(z )
= z − i · 1+ i W +O (e2)
i P′(z ) 2P′(z ) i M
i (cid:18) i (cid:19)
P(z ) P′′(z )P(z )
= z − i · 1+ i i (1+O (e))+O (e2)
i P′(z ) 2P′(z )2 M M
i (cid:18) i (cid:19)
P(z ) P′′(z )P(z )2 P(z ) P′′(z )P(z )
= z − i − i i − i i i O (e)+O (e2)
i P′(z ) 2P′(z )3 P′(z ) 2P′(z )2 M M
i i i (cid:18) i (cid:19)
P(z ) P′′(z )P(z )2
= z − i − i i −O (e3).
i P′(z ) 2P′(z )3 M
i i
The Chebyshev’s method is defined by
f′′(x)f(x) f(x)
xˆ = x− 1+ ,
2f′(x)2 f′(x)
(cid:18) (cid:19)
and of cubic order (see [7, Section 5.2]). According to the Chebyshev’s method, we see
that
P(z ) P′′(z )P(z )2
z − i − i i −r = O (e3).
i P′(z ) 2P′(z )3 i M
i i
Therefore, the order of eˆ is calculated as follows:
i
P(z ) P′′(z )P(z )2
|eˆ| = |zˆ −r | = z − i − i i −r +O (e3) = O |e|3 .
i i i i P′(z ) 2P′(z )3 i M M
(cid:12) i i (cid:12)
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
(cid:3)
(cid:12) (cid:12)
III. Modified Newton’s methods combined with Derivative-free method
Inthissection, wepresenttwomodifiedNewton’smethodscombinedwiththederivative-
free method (6) for finding all simple zeros of a polynomials simultaneously. The one is
a form that the derivative-free method is applied to the x∗ in the trapezoidal Newton’s
method (3) as follows:
2P(z )
i
zˆ = z − , (13)
i i
P′(z )+P′ z − Wi
i i 1−P(zi−Wi)/P(zi)
(cid:16) (cid:17)
which is called the Derivative-free trapezoidal Newton’s method, or simply, Method 2.
8 JUN-SEOPSONG
From (13), we have the following theorem.
Theorem 2. If the approximate zero x grounded from Method 2 is close enough to r
i i
and the order of e is the same, then the order of eˆ is identical, and |eˆ| = O (|e|3) is
i i i M
formed.
Proof. Since Petkovi´c’s derivative-free method (6) is of cubic order
W
z − i −r = O (e3), (14)
i i M
1−P(z −W )/P(z )
i i i
(see [6]). Using (14) and the Taylor’s expansion, zˆ is calculated as follows. (In this case,
i
C = 1 · P(j)(ri)).
j j! P′(ri)
2P(z )
i
zˆ = z −
i i
P′(z )+P′ z − Wi
i i 1−P(zi−Wi)/P(zi)
(cid:16) (cid:17)
2P(r +e )
i i
= z −
i P′(r +e )+P′(r +O (e3))
i i i M
2 P(r )+P′(r )e + 1P′′(r )e2 + 1P′′′(r )e3 +···
= z − i i i 2 i i 6 i i
i P′(r )+P′′(r )e + 1P′′′(r )e2 +··· +(P′(r )+P′′(r )O (e3)+···)
i (cid:0) i i 2 i i i i M(cid:1)
= z − (cid:0)2P′(ri)(ei +C2e2i +C3e3i +OM(e4))(cid:1)
i 2P′(r ) 1+C e + 3C e2 +O (e3)
i 2 i 2 3 i M
(cid:0) (cid:1) 3
= z −e 1+C e +C e2 +O (e3) 1−C e + C e2 +O (e3)
i i 2 i 3 i M 2 i 2 3 i M
(cid:18) (cid:19)
(cid:0) (cid:1)
= z −e +O (e3).
i i M
Therefore, the order of eˆ is calculated as follows.
i
|eˆ| = |zˆ −r | = z −e +O (e3)−r = O |e|3 .
i i i i i M i M
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
(cid:3)
Now we apply the derivative-free method to the x∗ in the midpoint Newton’s method
(4) and construct the iteration as follows:
P(z )
i
zˆ = z − , (15)
i i
P′ z − 1 · Wi
i 2 1−P(zi−Wi)/P(zi)
(cid:16) (cid:17)
NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL 9
which is called the Derivative-free midpoint Newton’s method. From this, we call it
Method 3 simply. From (15), we have the following theorem.
Theorem 3. If the approximate zero x grounded from Method 3 is close enough to r
i i
and the order of e is the same, then the order of eˆ is identical, and |eˆ| = O (|e|3) is
i i i M
formed.
Proof. By using (14) and Taylor’s expansion, zˆ is calculated as follows: (In this case,
i
C = 1 · P(j)(ri)).
j j! P′(ri)
P(z )
i
zˆ = z −
i i
P′ z − 1 · Wi
i 2 1−P(zi−Wi)/P(zi)
(cid:16) (cid:17)
P(z )
i
= z −
i P′ 1z + 1r +O (e3)
2 i 2 i M
(cid:0) P(ri +ei) (cid:1)
= z −
i P′ r + 1e +O (e3)
i 2 i M
= z − (cid:0) P(ri)+P′(ri(cid:1))ei + 21P′′(ri)e2i + 61P′′′(ri)e3i +···
i P′(r )+P′′(r ) 1e +O (e3) + 1P′′′(r ) 1e +O (e3) 2 +···
i i 2 i M 2 i 2 i M
= z − P′(ri)(ei +C2e(cid:0)2i +C3e3i +OM(cid:1)(e4)) (cid:0) (cid:1)
i P′(r ) 1+C e + 3C e2 +O (e3)
i 2 i 4 3 i M
(cid:0) (cid:1) 3
= z −e 1+C e +C e2 +O (e3) 1−C e + C e2 +O (e3)
i i 2 i 3 i M 2 i 4 3 i M
(cid:18) (cid:19)
(cid:0) (cid:1)
= z −e +O (e3).
i i M
Therefore, the order of eˆ can be calculated as follows.
i
|eˆ| = |zˆ −r | = z −e +O (e3)−r = O |e|3 .
i i i i i M i M
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
(cid:3)
IV. Numerical comparison
In this section, we give numerical experiments and comparisons between various simul-
taneous methods for finding zeros of a polynomial. These methods are all of cubic order.
They include Method 1, Method 2, Method 3, the derivative-free method (DFM), the
Petkovi´c’s Newton-Weierstrass method(NWM),andtheWeierstrass-like method(WLM).
10 JUN-SEOPSONG
For apolynomial P(z) = zn+a zn−1+···+a z+a , we choose initial approximations
1 n−1 n
as Aberth’s approach (see [1]):
a iπ 3
(0) 1
z = − +Rexp 2k − .
k n n 2
(cid:18) (cid:18) (cid:19)(cid:19)
In this case, R is a radius of a circle, where the initial zeros by Aberth’s approach are
located in complex number plane. We use the following Henrici’s fomula to select R (see
[3]):
R = 2 max |a |1/k.
k
1≤k≤n
According to Henrici’s formula, a disk {z : |z| < R} centered at the origin contains all
zeros of polynomial P(z).
The polynomials that we used on numerical comparison are as follows:
P = (x−1)(x−2)(x−3)(x−4)
1
P = (x−1)(x−2)(x−3)(x−4)(x−5) (16)
2
P = (x−1)(x−2)(x−3)(x−4)(x−5)(x−6)
3
P = x8 +5x7 +3x6 +7x5 +6x4 +8x3 +x2 +3x+7
4
Here P (x),P (x) and P (x) are Wilkinson’s polynomials when n = 4,5,6, respectively.
1 2 3
We approximated the zeros until it satisfy the following condition:
max P(z(m)) < 10−10. (17)
k
1≤k≤n
(cid:12) (cid:12)
(cid:12) (cid:12)
In Table I, we give a numerical comparison between several methods to find all zeros of
(cid:12) (cid:12)
(m)
those polynomials (16). It contains the iteration number m and the value max P(z )
k
1≤k≤n
(cid:12) (cid:12)
of iterative methods, after we approximated (17) to a satisfying label. The sm(cid:12)aller the(cid:12)
(cid:12) (cid:12)
m, the faster approximated on the zeros, When m is the same, it can be interpreted that
(m)
a smaller max P(z ) leads to a higher accuracy of approximation. All computations
k
1≤k≤n
(cid:12) (cid:12)
have been done(cid:12)using M(cid:12)ATLAB.
(cid:12) (cid:12)