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### Bp31457463

1. 1. Dr.V.Kusuma Kumari, / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.457-463 Iterative Method For The Solution Of Simple And Multiple Roots Of Non Linear System Of Equations Author: Dr.V.Kusuma Kumari, Professor, Humanities and Basic Sciences Department, Godavari Institute of Engineering and Technology, Rajahmundry, Andhra Pradesh.Abstract In this paper a method called x1( k 1)  f1 ( x1( k ) , x2k ) , x3k ) ,..., xnk ) ) ( ( (Parametric Method of Iteration is developed forsolving the non linear system of equations andshowed the efficiency of this method over x2k 1)  f 2 ( x1( k ) , x2k ) , x3(k ) ,..., xn(k ) ) ( (iteration method and Newton Raphson methodby considering some examples. x3k 1)  f 3 ( x1( k ) , x2k ) , x3k ) ,..., xnk ) ) ( ( ( ( xnk 1)  f n ( x1( k ) , x2k ) , x3k ) ,..., xnk ) ) ,(k=0,1,2,…) (1.4) ( ( ( (Key words: N-R Method, Iterative Method As is well known that the iterations (1.4)I. INTRODUCTION are to be repeated successively until the It is well known that the solution of convergence is achieved upto the desired accuracysystems of non-linear equations play an important and this method convergences under the condition.role in all the fields of science and engineeringapplications and there exist a great variety ofproblems for the solution of such problems. Many n fi ( X )applications in Science and Engineering are j 1 x j 1 (1.5)described by a set of n-coupled, non-linear algebraic x x (k )equations in n-variables x1 , x2 ,..., xn of the form (i=1,2,….,n), for each k.F1 ( x1 , x2 ,..., xn )  0 whereas in the multivariable Newton-F2 ( x1 , x2 ,..., xn )  0 (1.1) Raphson method, the system (1.2) is solved by iterating the schemeFn ( x1 , x2 ,..., xn )  0 1The equations (1.1) can be expressed as a system x ( k 1)  x( k )   J ( k )  F ( x ( k ) )   (1.6)Fi ( X )  0 (i  1, 2,..., n) (k=0,1,2,..)(1.2) where X is an n-dimensional vector. where the Jacobian J ( k ) is  F1 F1 F1   x . . . xn  Though we have various techniques like x2the method of steepest discent, Riks/Wempner  1 method, Von-Mises method, Power method,  F2 F2 F2   x . . . xn Gradient method and Graphical method [1,2] for the x2solution of (1.2), the most generally used methods  1 for its solution are the successive approximation  . . . . . . method also known as method of iteration, and the  multivariable Newton-Raphson method.  . . . . . .  In the Successive approximation mehod the  . . . . . . system (1.2) is written as  xi  fi ( x1 , x2 ,..., xn )  Fn Fn . . . Fn   x1 x2 xn  (i  1, 2,..., n) (1.3)   The multivariable Newton-Raphson (0) (0) (0) If x , x2 ,..., xn 1 be the initial method converges if the functions Fi ( X ) , all haveapproximation to the solution of (1.2), then in this continuous first order partial derivatives near a rootiterative method the improved values are found asindicated below X * , and if the Jacobian is non-singular in the 457 | P a g e
2. 2. Dr.V.Kusuma Kumari, / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.457-463 (k ) neighbourhood of this root and if X is taken * sufficiently close to the solution X . f1 f f 1  (1   2 )   2 2  ...   n n  1 (2.4) In this paper we develop a method for the solution x2 x2 x2 of the simultaneous non-linear equations (1.1) which will be presented in the next section. This method is also compared with the other methods by f1 f f considering some numerical examples. 1   2 2  ...  (1   n )   n n  1 x1 xn xn 2. PARAMETRIC METHOD OF ITERATION hold true for all x  x ( k ) and    ( k ) for each To solve the system (1.1), we rewrite each k. equation of (1.1) by introducing a set of parameters If we choose 1 ,  2 ,...,  n all of them are positive, in the form  f  1  (k )  1  i (k )  , (i=1,2,…,n), for each k (2.5)  xi i x x x1  (1  1 ) x1  1 f1 ( x1 , x2 ,..., xn )  x2  (1   2 ) x2   2 f 2 ( x1 , x2 ,..., xn ) (2.1) Then the iteration process (2.3) takes the form xn  (1   n ) xn   n f n ( x1 , x2 ,..., xn ) For the solution of the system (1.1), the method is  f   f1  xi( k 1)   f1( k )  x1( k ) 1  / 1   x1 x x   x1 x x (k ) (k ) defined as  x1( k 1)  (1  1(k ) ) x1(k )  1(k ) f1 ( x1(k ) , x2(k ) , x3(k ),..., xn(k ) )  f   f 2  x2k 1)   f 2( k )  x2k ) 2 ( (  / 1   (2.6)x( k 1)  (1   (k ) )x(k )  (k ) (k ) (k ) f 2 ( x , x , x ,..., x ) (k) (k)  x2 x  x( k )   x2 x  x( k )  2 2 2 2 1 2 3 n .x( k 1)  (1   (k ) )x(k )  (k ) f ( x1( k ) , x2k ) , x3k ) ,..., xnk ) ) (2.2) ( ( ( 3 3 3 3 3  f   f n  xnk 1)   f n(k )  xnk ) n ( (  / 1   xn x x   xn x x (k ) (k ) xnk 1)  (1   nk ) ) xnk )   nk ) f n ( x1( k ) , x2k ) , x3k ) ,..., xnk ) ) ( ( ( ( ( ( (   (k = 0,1,2,…) The equations (2.2) are expressed as For the choices of  i given (2.5) , the conditions for convergence of the method (2.6) can be obtained x1( k 1)  (1  1( k ) ) x1( k )  1( k ) f1( k ) from (2.4) as f 2 f f x2k 1)  (1   2k ) ) x2k )   2k ) f 2( k ) ( ( ( ( 2   3 3  ...   n n  1 x1 x1 x1 x3k 1)  (1   3 k ) ) x3k )   3 k ) f3( k ) ( ( ( ( (2.3) f1 f f x ( k 1)  (1   (k ) )x (k )  (k ) f (k ) 1   3 3  ...   n n  1 (2.7) n n n n n x2 x2 x2 It can directly be seen that the method (2.3) f1 f f  i( k )  1 1   2 2  ...   n1 n1  1 coincides with ( 1.4) when for each i and xn xn xn k. As the parameters  i (k ) in (2.3) accelerates or improves the convergence of (1.4), the method (2.3) For all x  x ( k ) and    ( k ) for each k. may be called as parametric method of iteration. For solving non-linear equations in single variable As given in [3] and [1], one can easily of the form show that the method (2.3) converges when the F(x)=0 (2.8) conditions The Parametric method of iteration can be defined from (2.3) as f1 f f (1  1 )  1   2 2  ...   n n  1 x ( k 1)  (1   ( k ) ) x ( k )   ( k ) f ( x ( k ) ) (2.9) x1 x1 x1 Where  is a parameter, whose choice using (2.5) obtained as 458 | P a g e
3. 3. Dr.V.Kusuma Kumari, / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.457-463 1 The parametric method of iteration (2.6 ) for the  (k )  for each k (2.10) solution of (3.1) can be written as 1  f ( x)   x  x( k )  With this choice of (2.9) the method (2.8) reduce to  f   f  x1( k 1)  k  f1( k )  x1( k ) 1 x  xk  / 1  1 x  xk   x1   x1 x( k 1)   f ( x ( k ) )  x ( k ) f ( x)  x  x( k )  / 1  f ( x)   x  x( k )  2.11)   ( f   f  x2k 1)  k  f 2( k )  x2k ) 2 x  xk  / 1  2 x  xk  ( As it is well known that the Newton-Raphson  x2   x2  method for the solution of (2.8) is given by F ( x( k ) )  ( f   f  x ( k 1) x (k )  (2.12) x3( k 1)  k  f3( k )  x3k ) 3 x  xk  / 1  3 x  xk  F ( x)  x3   x3  x  x( k ) It is interesting to note that F(x) of equation (2.8) is (3.3) expressed in the form The iteration method for solving (3.1) can be taken by writing (3.1) into the form (3.2) as F ( x)  x  f ( x) (2.13) Then the Newton-Raphson iteration (2.12) exactly x1( k 1)  (cos x2k ) x3k )  0.5  x1( k ) ) / 4, ( ( * ( k 1) takes the form (2.11). x2   x1( k 1) / 25, (3.4) x3k 1)  ( x3k )  9  e x1 (k ) (k ) 3. NUMERICAL EXAMPLES: ( ( x2 ) / 21 We consider the non-linear system of three (*Here + or – sign should be taken in accordance equations in three unknowns: with the initial guess) F1 ( x1 , x2 , x3 )  3 x1  cos x2 x3  0.5  0 And, the Newton-Raphson iteration for the solution F2 ( x1 , x2 , x3 )  x12  625 x2  0 2 of (3.1) is F3 ( x1 , x2 , x3 )  e  x1x2  20 x3  9  0 1 x1( k 1)  x( k )   J ( x( k )  F ( x( k ) )   (3.5) (3.1) Where X  ( x1 , x2 , x3 ) T As noted by William W.Hager [4], this system has and the Jacobian J is a more than one solution. Expressing the system (3.1) 3x3 matrix: as x1  f1 ( x1 , x2 , x3 )  F1 ( X ) F1 ( X ) F1 ( X )    x2  f 2 ( x1 , x2 , x3 ) (3.2)  x1 x2 x3   F ( X ) F2 ( X ) F2 ( X )  (3.6) x3  f3 ( x1 , x2 , x3 ) J ( x)   2  f1  (cos x2 x3  0.5  x1 ) / 4,  x1 x2 x3   F3 ( X ) F3 ( X ) F3 ( X )    where * f 2   x1 / 25,  x1 x2 x3  f 3  ( x3  9  e  x1x2 ) ) / 21 to compare the methods (3.3), (3.4) and (3.5) for the solution of (3.1), we tabulate the following results (*Here + or – sign should be taken in accordance for various initial approximation to the unknowns with the initial guess) 459 | P a g e
4. 4. Dr.V.Kusuma Kumari, / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.457-463 TABLE 3.1 For x1(0) = 0, x2(0) = 0, x3(0) = 0Iteration Parametric Method of Variable Iteration Method Newton- Raphson MethodNumber Iteration X1 0.375 0.5 Since J-1 does not exist, Newton-1 X2 0.015 0.02 Raphson Method fails to converge X3 -0.475923371 -0.4995024917 X1 0.4687436296 0.49998336662 X2 0.01874974519 --- 0.01999933467 X3 -0.4984368122 -0.4995025246 X1 0.49217499 0.49998336773 X2 0.0196869996 --- 0.01999933471 X3 -0.4994663883 -0.4995025246 X1 0.49803166164 X2 0.01992126647 --- --- X3 -0.4995044772 X1 0.49949553835 X2 0.01997982153 --- --- X3 -0.4995035371 X1 0.49986143476 X2 0.01999445739 --- --- X3 -0.4995028027 X1 0.49995289057 X2 0.01999811562 --- --- X3 -0.4995025953 X1 0.49997574998 X2 0.01999903 --- --- X3 -0.4995025424 X1 0.49998146379 X2 0.01999925855 --- --- X3 -0.4995025291 X1 0.499982891810 X2 0.01999931567 --- --- X3 -0.4995025257 X1 0.499983248811 X2 0.01999932995 --- --- X3 -0.4995025249 X1 0.49998333812 X2 0.01999933352 --- --- X3 -0.4995025247 X1 0.499983360313 X2 0.01999933441 --- --- X3 -0.4995025246 X1 0.499983365914 X2 0.01999933463 --- --- X3 -0.4995025246 X1 0.499983367315 X2 0.0199933469 --- --- X3 -0.4995025246 X1 0.499983367616 X2 0.0199993347 --- --- X3 -0.4995025246 X1 0.499983367617 X2 0.01999933471 --- --- X3 -0.4995025246 CONVERGED IN CONVERGED IN 17CONCLUSION NO CONVERGENCE 3 ITERATIONS ITERATIONS 460 | P a g e
5. 5. Dr.V.Kusuma Kumari, / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.457-463 TABLE 3.2 For x1(0) = 1, x2(0) = 1, x3(0) = 0Iteration Parametric Method of Variable Iteration Method Newton- Raphson MethodNumber Iteration X1 0.625 0.5 0.51 X2 0.025 0.5 0.02 X3 -0.475452213 -0.4867879441 -0.4995024917 X1 0.5312323397 0.4996539197 0.49998336662 X2 0.02124929359 0.2503994463 0.01999933467 X3 -0.4982965416 -0.493806505 -0.4995025246 X1 0.5077940706 0.4999491556 0.49998336773 X2 0.02031176283 0.1259982842 0.01999933471 X3 -0.4994302551 -0.4968557293 -0.4995025246 X1 0.5019356544 0.49997931444 X2 0.02007742618 0.06458633398 --- X3 -0.4994953947 -0.4983882295 X1 0.5004713422 0.49998292425 X2 0.02001885369 0.0353895858 --- X3 -0.4995012645 -0.4991178073 X1 0.500105337 0.499983336 X2 0.02000421348 0.02334579615 --- X3 -0.4995022346 -0.4994188656 X1 0.500013854 0.49998336627 X2 0.02000055416 0.02023918093 --- X3 -0.4995024533 -0.4994965282 X1 0.4999909878 0.49998336778 X2 0.01999963951 0.02000075587 --- X3 -0.4995025069 -0.4995024891 X1 0.4999852724 0.49998336779 X2 0.01999941089 0.01999933476 --- X3 -0.4995025202 -0.4995025246 X1 0.4999838438 0.499983367710 X2 0.01999935375 0.01999933471 --- X3 -0.4995025235 -0.4995025246 X1 0.499983486711 X2 0.01999933947 --- --- X3 -0.4995025243 X1 0.499983397512 X2 0.0199993359 --- --- X3 -0.4995025245 X1 0.499983375213 X2 0.01999933501 --- --- X3 -0.4995025246 X1 0.4999833696 ---14 X2 0.01999933478 X3 -0.4995025246 --- X1 0.499983368215 X2 0.01999933473 --- --- X3 -0.4995025246 X1 0.499983367816 X2 0.01999933471 --- --- X3 -0.4995025246 461 | P a g e
6. 6. Dr.V.Kusuma Kumari, / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.457-463 CONVERGED CONVERGED IN CONVERGED INCONCLUSION IN 16 10 3 ITERATIONS ITERATIONS ITERATIONS TABLE 3.3 For x1(0) = 5, x2(0) = 5, x3(0) = 5Iteration Parametric Method of Variable Iteration Method Newton- Raphson MethodNumber Iteration X1 1.622800703 -1.257919398 0.4970676041 X2 0.06491202812 2.493987329 0.01988270416 X3 -0.2333342432 -0.4500000001 -0.4995082814 X1 0.7806715004 -0.1892295183 0.49998356082 X2 0.03122686002 1.246638798 0.01999934243 X3 -0.4861548126 -1.343104476 -0.4995025242 X1 0.5701390675 0.751266281 0.49998336773 X2 0.0228055627 0.6231139626 0.01999933471 X3 -0.4987255541 -0.5170993363 -0.4995025246 X1 0.5175185969 0.5013238454 X2 0.02070074387 0.3117994466 --- X3 -0.4994318911 -0.494510231 X1 0.5043662885 0.4999501915 X2 0.02017465154 0.1565410286 --- X3 -0.4994908605 -0.4961226146 X1 0.5010788789 0.49997544176 X2 0.02004315515 0.07954800913 --- X3 -0.4994999011 -0.4980152391 X1 0.5002571909 0.49998245987 X2 0.0200128764 0.04228803314 --- X3 -0.4995018832 -0.49894539 X1 0.5000518099 0.49998328118 X2 0.0200020724 0.02587317067 --- X3 -0.499502365 -0.4993556857 X1 0.5000004749 0.49998336299 X2 0.02000001899 0.02066608603 --- X3 -0.4995024848 -0.4994848556 X1 0.4999876437 0.499983367710 X2 0.01999950575 0.02001009043 --- X3 -0.4995025147 -0.4995022556 X1 0.4999844365 0.499983367711 X2 0.01999937746 0.0199993376 --- X3 -0.4995025221 -0.4995025245 X1 0.4999836349 0.499983367712 X2 0.0199993454 0.01999933471 --- X3 -0.499502524 -0.4995025246 X1 0.499983434513 X2 0.01999933738 --- --- X3 -0.4995025245 X1 0.499983384414 X2 0.01999933538 --- --- X3 -0.4995025246 X1 0.499983371915 X2 0.01999933488 --- --- X3 -0.4995025246 X1 0.499983368816 X2 0.01999933475 --- --- X3 -0.4995025246 462 | P a g e
7. 7. Dr.V.Kusuma Kumari, / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.457-463 X1 0.49998336817 X2 0.01999933472 --- --- X3 -0.4995025246 X1 0.499983367818 X2 0.01999933471 --- --- X3 -0.4995025246 CONVERGED CONVERGED IN CONVERGED INCONCLUSION IN 18 12 3 ITERATIONS ITERATIONS ITERATIONS Conclusion: From the above tabulated results it can be observed that the Parametric method of iteration looks more efficient than the Newton-Raphson method and Iteration method for the problems considered. It can also observed that Parametric method of Iteration converges even though Newton-Raphson method fails to converge from example (3.1). REFERENCES [1] GOURDIN, A., BOUMAHRAT, M., Applied Numerical Methods, Prentice- Hall of India Pvt Ltd, (1989). [2] RAJASEKARAN, S., (1986) ‘Numerical Methods in Science and Engineering’, A.H.wheeler & Co., Pvt Ltd.,1st edition. [3] SCARBOUROUGH, J. B. (1966), ‘ Numerical Mathematical Analysis’; Oxford & IBH Publishing Co., Pvt.Ltd. [4] WILLIAM W.HAGER, Applied Numerical Linear Algebra, Prentice Hall, Englrwood Cliffs, New Jersey 07637, (1988). 463 | P a g e