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RESEARCH ARTICLE ScienceAsia 26 (2000) : 243-247
Theorems on a Modified Newton Method for
Solving Systems of Nonlinear Equations
Maitree Podisuk
Department of Mathematics and Computer Science Faculty of Science King Mongkut’s Institute
of Technology Chaokhuntaharn Ladkrabang Ladkrabang Bangkok 10520 Thailand.
Received 21 Apr 1999
Accepted 16 May 2000
ABSTRACT The modified Newton method for solving systems of nonlinear equations is one of the
Newton-like methods. In this paper, results that will ensure the existence and uniqueness solutions to
a system of nonlinear equations will be given. A second order convergence result is established.
KEYWORDS: modified Newton method, Newton-like methods, solving systems, nonlinear equations.
INTRODUCTION AND DEFINITIONS N(x,t) = {y : y∈x, ||x-y|| 0 are
Define r0(h) = ()11−−2h η k 0
h real numbers such that
r (h) = 1 ()11+−2h η.
1 h (8) a(t) + σKt
Then if N(x ,r (h)) ⊂ D , the sequence of iterates is isotonic on (0,r), and
0 0 0
defined by Newton Method exists, remains in (9) ||B(x)|| ≤ a(||x - x ||) +σK||x - x || - δ,
N(x ,r (h)) and converges to s in N(x ,r (h)) such 0 0
0 0 0 0
that f(s) = 0. If h < 1/2, s is the only root in N(x ,r (h))
0 1 for every x ∈ N(x ,r))
∩ D , and if h = 1/2, s is unique in N(x ,r (h)) ∩ D . 0
0 0 1 0
Furthermore, the sequence of the iterates satisfies -1 2
the error bounds then g(t) = t + (a(t)) (0.5 σK t - δt + a(0)||
-1
(A(x )) -f(x )||)
1 0 0
mm
||s-x || ≤ (/12)(1−−12h)2η.
m h -1
weakly majorizes G(x) = x - (A(x)) f(x) on N(x ,r).
0
The following theorem is given by Kantorovich Theorem 4.
and Akilov.3 This theorem together with Lemma 3
-1
Let f' ∈ LipK (N(x ,r)) and (A(x )) exist and be
and Definition 1 give the convergence of the 0 0
-1
sequence {x } in X when the sequence of {t } converges. bounded in the norm by (a(0)) .
k k
ScienceAsia 26 (2000) 245
If ||B(x )|| < a(0) and
0 11−−2h'
r' = ()1−βδ
00
1 βK
-1 2
(10) h' = K||(A(x )) f(x )||a(0)/(a(0) -||B(x )||) ≤
0 0 0 '
2 imply that f has a root r ∈ N(x , r ) which is unique
0 0
and in D ∩ N(x , r') where
0 1
' 1 ' 11−−2h'
(11) r =−(11−2ha’ )( (0)−||B(x)||)≤r
0 0 .
r = ()1−βδ
K 10
βK
' ' -1 '
Furthermore x = x - (A(x )) f(x )
Then if f has a unique zero s ∈ N(x ,r'), and m+1 m 0 m
0 0
converges to s from any x' ∈ D ∩ N(x , r' ).
'' −1 ' 0 0 1
xx=−(A(x)) f(x), m=01, ,...
mm+10m
If, in addition, β(δ + η ) < 1 ,
0 0
'
xN∈ (,xr)
converges to s from any such that
00 h= σβKα ≤ 1
2 2
1 ()1−−βη βδ
'' 00
|| xx−<|| r=(11−−2ha)( (0)−||B(x)||).
001 0
K δη+
11
where σ = max , and N(x , r ) ⊂ D,
(,1 K ) 0 0
If, in addition, σ, δ and a satisfy the conditions of
Theorem 3 and 11−−2h
r = ()1−−βη βδ
0 σβK 00
1 −1 1 -1
then x = x - (A(x )) f(x ) converges to s.
(12) and m+1 m m m
hK=≤σ ||(A(x) f(x)||a(0)
δ2 0 0 2
1 MAIN RESULTS
(13) r =−()11−2hrδ< then
0 σK
The following theorem of this section will ensure
(14) x = x - (A(x ))-1f(x ), m = 0, 1, ..., the convergence of the modified Newton method for
m+1 m m m solving a system of nonlinear equations which is a
converges to s. special kind of the Newton-like method.
In the following theorem, we impose one more Theorem 6.
condition on A(x) and one condition on B(x) instead Let D be an open convex subset of the space X
'
and f ∈ LipK (D). Assuming that f(x) and H(x)
of B(x0).
satisfy all the conditions of the previous theorems,
Theorem 5. then there exists a unique zero s in D so that for any
point x in D the sequence {x } where
Let f' ∈ Lip (D) where x ∈ D and D is an open 0 m
k 0
convex subset of X. Assume that x = x - (H(x ))-1 f(x )
m+1 m m m
(15) ||(A(x ))-1 f(x )|| ≤ α
0 0 converges to s.
(16) ||(A(x ))-1|| ≤ β
0 Proof
(17) ||(A(x)-A(x )|| ≤ η +η ||x - x ||, ∀ x ∈ D By the results of the previous theorems, the
0 0 1 0 existence and uniqueness of s in D is ensured and
(18) ||(B(x) ≤ δ +δ ||x - x ||, ∀ x ∈ D the sequence {xm} of points in D, where
0 1 0
βαK 1 x = x - (H(x ))-1 f(x )
' ' m+1 m m m
Then βδ <=1,h ≤ and N(x , r )
0 2 2 0 0
()1−βδ
0 for x in D, converges to this unique points.
⊂ D where 0
Theorem 7 is the last theorem of this section that
246 ScienceAsia 26 (2000)
shows that the order of the convergence of the 2
uv ++22u v−
modified Newton method for solving a system of f(u, v) = 2 .
uvuv+++1
nonlinear equations which is of second order.
which satisfies all the required conditions above.
Theorem 7. Note that u =1 and v =-1a is solution to this problem.
Let the conditions of Theorem 5 be satisfied and The iterative formulas are
δ = 0, that is
0
u = (2-v )/(v2+2)
(19) ||M(x)|| ≤ δ ||x-x ||, ∀ x ∈ D. m+1 m m
1 0
v = (-u -1)/(u2+1).
Then the order of the convergence of the method is m+1 m m
equal to 2. The iteration will be stopped when ||f(u , v )||
< 0.0000001. In this example we have m+1 m+1
Proof.
Let Q = sup (a(||x-x ||))-1, x ∈ N(x ,r )
0 0 0 2
x = R , D = (0,2)x(-2,0), D = [0, 2]x[-2, 0], r =
0 0
1 0.5, δ = 0, δ = 1, x = (u , v ),
P = Q ( K + δ ) 0 1 0 0 0
2 1
h
and e = ||s-x || then 2
k k ' vu++22v1
f(u, v) = 2 ,
21uv ++u 1
ek+1 = ||s - x ||
k+1 1 2
' -1 uu+−12v−1
f(u, v)) = 22 22 2 ,
−−21uv v +2
-1 uv+−34uv−uv+1
= ||s - x + (H(x )) f(x )||
k k k
-1 -1 1 0
= ||(H(x )) f(s) + (H(x )) f(x )+ s - x || 2
k k k k v 0 −1 v2 +2
Au(,v)= 2 ,(Au(,v)) =
01u + 1
0
-1 -1 u2 +1
= ||(H(x )) f(s) + (H(x )) f(x )+
k k k
-1
(H(x )) H(x )(s - x )||
k k k
and B(u, v) = 02uv .
20uv
-1
≤ ||(H(x )) ||⋅||f(s) - f(x ) + H(x )(s - x )||
k k k k
-1 ' Table 1. The following table 1 gives the results with
= ||(H(x )) ||⋅||f(s) - f(x ) + f(x )(s - x ) - different initial points.
k k k k
'
f(x )(s - x ) + H(x )(s - x )||
k k k k
u v r u v number of
0 0 0 m m
-1 ' iterations(m)
= ||(H(x )) ||⋅||f(s) - f(x ) + f(x )(s - x ) -
k k k k
'
(f(x ) - H(x )(s - x )||
k k k 0.75 -0.75 0.5 1 -1 19
-1 1 2 1.25 -0.75 0.5 1 -1 18
≤ (a(||x -x ||)) (Ke +−||f'()x H()x ||e )
0 k 2 k kkk0.75 -1.25 0.5 1 -1 18
1 2 1.25 -1.25 0.5 1 -1 19
-1 (|Ke + |M(x )||e )
= (a(||x -x ||)) k kk
0 k 2 1.00 -0.75 0.5 1 -1 18
1 1.00 -1.25 0.5 1 -1 18
2 2
-1 ()Ke +δe
≤ (a(||x -x ||)) k 1 k 0.75 -1.00 0.5 1 -1 18
0 k 2
1.25 -1.00 0.5 1 -1 18
1 2
≤ QK()+δ e
2 1 k
The solution by the above method always
≤ Pe 2 . converges to the point (1, -1) for all the initial points
k in the domain D.
EXAMPLE
We will use the above method to solve the
following problem
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