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File: Solving Equations Pdf 181949 | V26 243 247
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 ...

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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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...Research article scienceasia theorems on a modified newton method for solving systems of nonlinear equations maitree podisuk department mathematics and computer science faculty king mongkut s institute technology chaokhuntaharn ladkrabang bangkok thailand received apr accepted may abstract the is one like methods in this paper results that will ensure existence uniqueness solutions to system be given second order convergence result established keywords introduction definitions n x t y...

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