In this article the problem of solving a system of singular nonlinear equations will be discussed. New the iterative 2-order method for this problem is presented. The article includes the numerical results for the method

M. Grzegórski, Stanisław

Łukasik, Edyta

English

University of Maria Curie-Skłodowska (UMCS): Scientific e-Journals / Uniwersytet Marii Curie-Skłodowskiej: e-czasopisma naukowe

Annales UMCS Informatica AI 2 (2004) 243-250 Annales UMCS Informatica Lublin-Polonia  Sectio AI http://www.annales.umcs.lublin.pl/  Efficiency of 2–order iterative methods  Stanisław M. Grzegórski, Edyta Łukasik∗ Department of Computer Science, Lublin University of Technology,  Nadbystrzycka 36b, 20-618 Lublin, Poland  Abstract In this article the problem of solving a system of singular nonlinear equations will be discussed. New the iterative 2-order method for this problem is presented. The article includes the numerical results for the method.   1. Problem Let : n nF D R R⊂ →  be a nonlinear operator, D is an open set. The point 0x D∈  is given.  The problem of solving a system of nonlinear equations consists in finding a solution *x D∈  such, that   0*)( =xF . (1) Such nonsingular problems were considered in [1,2]. In this paper we give some results for singular problems.  Definition 1. Operator F is regularly singular at the point x*, if exists c>0 such, that   * 2|| ( ) || || ||x DF x c x x∈∀ ≥ − . (2)  Note 1. If there exists i, 1 ≤ i ≤ n such, that matrix '' ( *)iF x  is nonsingular, then the operator F is regularly singular at the point x*.  Definition 2. The sequence {xk} is locally M-superquadratically convergent, if:  1 21|| ( ) ||lim 0|| ||kkk kF xx x+→∞ +=− . (3)                                                   ∗ Corresponding author: e-mail address: e.lukasik@pollub.pl Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCSStanisław M. Grzegórski, Edyta Łukasik 244Note 2. The condition (3) is equivalent to the following condition   21 1|| ( ) || || ||k k k kF x c x x+ +≤ ⋅ −     with    lim 0kk c→∞ = . (4) For the singular system of equations the Newton method is at most Q-linearly convergent. For the regularly singular system of equations condition (3) guarantees Q-superlinear convergence of the method [3].   2. New methods Definition 3. Method (I) Given nkx R∈ , we solve the equation  ' 1( ) ( ) 02Tk k k k k kF x F x s s B s+ + =  (5) and compute  kkk sxx +=+1 ,  k=0,1,… . The operator ''( )kF x  will be aproximated by the operator 1 2[ , ,..., ]nk k k kB B B B= , ( )i ik k TB B= , i n nkB R ×∈) for i=1,2,...,n.  Results: [3] If the operator Bk satisfies the secant equation  ' '1 1( ) (k k k kB s F x F x+ += −  (6) and  1lim || || 0k kk B B+→∞ − =  (7) then 1. If the system of equations is nonsingular, then method I is locally M-superquadratically convergent. 2. For the regularly singular system of equations, method I is superlinearly convergent to x*:  1|| * || || * ||, lim 0k k k kx x x x kα α+ →∞− ≤ − = . (8) The system of equations (5) is difficult to solve, because there are n quadratic equations with n variables. We propose the Newton method for solving this system.  Algorithm: Given: 0x D∈ , eps>0,  0i n nB R ×∈ ,    0 0( )i T iB B= ,   ''0 ( )i i 0B F x≅    for i=1,2,…,n.  maxiter – integer,  maxiter ≥ 0.  Denote:  1( ) ( ) '( ) ( , )2t k k t t kG s F x F x s s B s= + + t  (9) Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCS Efficiency of 2–order iterative methods 245 1. ;  t:=0; 0 : 0ks =2. while  ( ) ( )( ) maxtkG s eps t iter> ∧ ≤    do begin a) solve (for ks∆ ): ' ( ) ( )t tk k kG s s G s∆ = −  1t tk k ks s s+ = + ∆  b) t:=t+1; end  If maxiter=0, then the above algorithm is equivalent to the Newton method. The operator Bk is defined as  2)(1TikikikikEEBB ++=+   (10) where i Ti k kk Tk kr sEs s= , ,  for  i=1,2,...,n. 1( ) ( )i ik i k i k kr F x F x B s+= ∇ −∇ − Tk 3. Regularly singular system of equations Consider the operator 3 3:F R R→ :  2 2 21 1 2 3( ) ( 1) 1F x x x x= − + + −   2 2 22 1 2 3( ) ( 1) 1F x x x x= + + + −   3 1 2( ) 3F x x x x= + +  Calculate norm of the operator F: ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )2 2 2 21 2 32 2 22 2 2 2 2 21 2 3 1 2 3 1 2 32 2 22 2 2 2 2 21 1 2 3 1 1 2 3 1 2 3( ) ( ) ( ) ( )( 1) 1 ( 1) 12 2F x F x F x F xx x x x x x x x xx x x x x x x x x x x= + + == − + + − + + + + − + + + == − + + + + + + + + + = ( ) ( ) ( ) ( )( ) ( ) ( ) ( )2 2 22 2 2 2 2 2 2 2 2 2 2 21 2 3 1 1 2 3 1 2 3 1 1 2 32 2 22 22 2 2 2 2 21 2 3 1 2 3 1 2 3 1 2 324 42 22 || * || , * [0,0,0] .Tx x x x x x x x x x x x x xx x x x x x x x x x x x xx x where x= + + − + + + + + + + + ++ + + = + + + + + ≥ + + == − =22 =Hence  2( ) 2 *F x x x≥ − , so the operator F is regularly singular at the point x*. Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCSStanisław M. Grzegórski, Edyta Łukasik 246This problem can not be solved by the Newton method, because the matrix '( *)F x  is singular.   4. Experimental results We consider singular problems and nonsingular system of equations.  Problem1.  2 2 21 1 2 3( ) ( 1) 1F x x x x= − + + −   2 2 22 1 2 3( ) ( 1) 1F x x x x= + + + −   3 1 2( ) 3F x x x x= + +  Problem 2.  2 2 21 1 2 3( ) 1F x x x x= + + −   2 22 1 2( ) 2 4 3F x x x= + − x3  2 23 1 2( ) 3 4F x x x x= − +  Problem 3.   21 1 2( ) 8 16F x x x= + − 22 1( )xF x x e= −  Problem 4. Rosenbrock Function [4]  1 1( ) 1F x x= −   22 2( ) 10( )1F x x x= −  Problem 5. Powell Singular Function  1 1( ) 10 2F x x x= +   2 3( ) 5( )4F x x= − x   23 2 3( ) ( 2 )F x x x= −   24 1( ) 10( )2F x x= − x1 Problem 6. Wood Function  21 1 2 1( ) (200 ( )) (1 )F x x x x= − − − − x   22 2 1 2 4( ) 200( ) 20.2( 1) 19.8( 1)F x x x x x= − − − + −   23 3 4 3( ) (0.25 ( )) (1 )3F x x x x= − − − − x   24 4 3 4 2( ) 180( ) 20.2( 1) 19.8( 1)F x x x x x= − − − + −  Problem 7. Brown Almost-Linear Function  1( ) , 1,2,..., ( 1)ni i jjF x x x i n== + =∑ −   1( ) ( 1)nn jjF x x== −∏  Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCS Efficiency of 2–order iterative methods 247 Problem 8. Artificial Test Problem   21( ) 21 2( ) 3xF x e x= + − 2 1 2 1 2( ) sin(3 )F x x x x x= + + +  Problem 9. Freudenstein And Roth  3 21 1 2 2( ) 13 ( 5 2 )2F x x x x= − + + − + x2  3 22 1 2 2( ) 29 ( 14 )F x x x x= − + + + + x  Table 1 Problem number Start point X0Maxiter Number of iterations k Number of solving system equations Newton method 0 14 14 1 7 14 2 5 15 1 [1,2,1] 3 4 15 15 1 [1,1,1] 0 14 14 * 0 13 13 1 7 14 2 5 14 1 [0.5,0.5,1] 3 4 14 14 0 11 11 1 6 12 1 [0,1,0] 2 4 12 12 0 14 14 1 7 14 3 4 15 1 [2,3,1] 4 3 15 15 0 4 4 1 [0.001,0.002,0.001] 1 2 4 5 0 4 4 1 2 4 2 2 5 2 [0.5,0.5,0.5] 3 1 4 14 0 4 4 1 2 4 2 2 5 2 [1,1,0] 3 1 4 12 0 7 7 1 4 8 2 3 8 2 [0.1,0.1,0.1] 3 2 8 * Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCSStanisław M. Grzegórski, Edyta Łukasik 248Problem number Start point X0Maxiter Number of iterations k Number of solving system equations Newton method 0 10 10 1 5 10 2 4 11 2 [0.01,0,0.05] 3 3 11 28 0 5 5 3 [1,2] 1 4 8 6 0 5 5 3 [0.1,0.1] 1 4 8 6 0 5 5 1 4 8 3 [1,0] 2 3 9 6 0 4 4 1 3 6 3 [1,1] 2 2 5 5 0 10 10 1 5 10 4 [0.1,0.3] 2 4 11 11 0 3 3 1 2 4 4 [0.001,0.003] 3 1 3 3 0 11 11 1 6 12 2 4 12 5 [1,0.9,1,0.9] 3 3 12 12 0 12 12 1 6 12 2 4 12 5 [1,0,1,0] 3 3 12 13 6 [0,0,0,0] 0 14 14 15 6 [0.1,0.1,0.1,0.1] 0 10 10 11 0 12 12 1 12 24 3 6 21 6 [1,1,0,0] 4 4 16 13    Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCS Efficiency of 2–order iterative methods 249 Problem number Start point X0Maxiter Number of iterations k Number of solving system equations Newton method 0 6 6 1 6 12 7 [0,0,0] 2 4 11 * 0 8 8 1 7 14 2 6 17 7 [0.5,0.5,0.5] 3 5 17 * 8 [1,1] 0 7 7 9 0 ∞  1 5 10 8 [1,-1] 2 4 11 ∞ 0 ∞  1 ∞  2 ∞  3 21 81 4 4 15 8 [0.1,0.1] 5 4 15 ∞ 0 4 4 1 3 6 9 [0,0] 2 3 8 5 0 4 4 1 3 6 9 [1,2] 2 3 8 5 0 3 3 1 2 4 9 [20,1] 2 2 5 4 * – singularity ∞ – method is not convergent Number of iterations k is equal to the number of calculation operator Fand F’.  5. Results The proposed method for regularly singular system of equations is locally superlinearly convergent while the Newton method diverges or is at most linearly convergent. In our method the number of calculation operator F and 'F is generally less than in the Newton method even for nonsingular problems.   Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCSStanisław M. Grzegórski, Edyta Łukasik 250Acknowledgments We would like to thank Professor Marcin Paprzycki from Oklahoma State University for his helpful remarks and examples of the system of nonlinear equations.  References [1] Grzegórski S.M., Rescaled multilevel least-change almost secant methods, Matematyka Stosowana, XXXIII, (1991). [2] Grzegórski S.M., Orthogonal projections on convex sets for Newton-like methods, SIAM J.Numer. Anal., 22 (1985) 1208. [3] Grzegórski S.M., Łukasik E., Theory of convergence for 2-rank iterative methods, prepared for publication. [4] Dent D., Paprzycki M., Kucaba-Pietal A., Testing convergence of nonlinear system solvers, Proceeding of the First Southern Symposium on Computing, The University of Southern Mississippi, (1998). Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.plData: 04/08/2020 21:56:47UMCSPowered by TCPDF (www.tcpdf.org)

Efficiency of 2-order iterative methods

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Efficiency of 2-order iterative methods

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