Starting from q-Taylor formula, we suggest a new q-variant of Stef-fensen's method of fourth-order convergence for solving non-linear equations

Ashour, Ola Abdelnaby

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ISSN 2347-1921                                                           3443 | P a g e                             A p r i l  2 3 ,  2 0 1 5                                                           A q -VARIANT OF STEFFENSEN’S METHOD OF FOURTH-ORDER CONVERGENCE Ola A. Ashour Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt oaashour@hotmail.com  Abstract: Starting from q -Taylor formula, we suggest a new q -variant of Steffensen’s method of fourth-order convergence for solving non-linear equations. Keywords: q -Taylor series; Jackson q -difference operator; Steffensen’s method; Nonlinear equations. AMS subject classification 2010: Primary 65D25; Secondary 65E99.                       Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol .10, No.4 www.cirjam.com , editorjam@gmail.com ISSN 2347-1921                                                           3444 | P a g e                             A p r i l  2 3 ,  2 0 1 5                                                           1 INTRODUCTION Finding the zeros of a nonlinear equation, 0=)(xf , is a classical problem of numerical analysis. Analytic methods for solving such equations rarely exit, and therefore, one can hope to obtain only approximate solutions by relying on iteration methods. For a survey of the most important algorithms, some excellent textbooks are available (see, [4, 8, 10] ). The classical Newton’s method:   .0,1,2,=,)()(=1 nxfxfxxnnnn  (1) Being quadratically convergent, Newton’s method is probably the best known and most widely used algorithm. Time to time the method has been derived and modified in a variety of ways. One such method derived from Newton’s method by approximating the derivative with non-derivative term of difference quotient is Steffensen’s method [9, 11]. The method requires two evaluations of function and is quadratically convergent. The interesting iterative scheme is Steffensen’s method that has the following form:   .0,1,2,=,))())((()(=21 nxfxfxfxfxxnnnnnn  (2) In order to control the approximation of the derivative and the stability of the iteration, a Steffensen’s type method has been proposed in [2], this approach is based on a better approximation to the derivative )( nxf   in each iteration. It has the following form:   .)(|)(|))/())(|)(|(()(=1nnnnnnnnnnnxfxfxfxfxfxfxfxx   (3) After that, the paper [1] has extended the above result on Banach spaces, obtained its local and semi-local convergence theorems, and made its applications on boundary-value problems by multiple shooting methods. A family of fourth order methods free from any derivative, satisfying the highest convergence order were established in [12, 13]. 2  q -Calculus In the following, q  is a positive number, 1<<0 q . For }{0,1,= Nn , }{1,2,= Zk  and Ckaaa ,,, 1  , the q -shifted factorial, the multiple q -shifted factorial and the q binomial coefficients are defined by   ,);(:=);,,,(),(1:=);(1,:=);(1=2110=0 njkjnkjnjn qaqaaaaqqaqa    (4)  and   ,);()(1))(1(1:=1,:=011nnaaaqqqqqqqnaanda  (5)  respectively. The limit, nn qa );(lim  , is denoted by );( qa . Moreover nqa );(  has the representation, cf. [5],   .1)(=);( 1)/2(0=kkkqknkn aqknqa   (6)  The q Gamma function, [5, 6], is defined by   1,|<|,,)(1);();(:=)( 1 qzqqqqqz zzqC  (7)  where we take the principal values of zq  and zq  1)(1 . In particular  ISSN 2347-1921                                                           3445 | P a g e                             A p r i l  2 3 ,  2 0 1 5                                                            .,)(1);(=1)( N nqqqnnnq  Let C  be fixed. A set CA  is called a  geometric set if for Ax , Ax . Let f  be a function defined on a q -geometric set CA . The q -difference operator is defined by the formula    .0,)()(:=)( AxqxxqxfxfxfDq  (8)  If A0 , we say that f  has q -derivative at zero if the limit   Axxqfxqfnnn,(0))(lim  (9) exists and does not depend on x . We then denote this limit by (0)fDq . The q -integration of F. H. Jackson [7] is defined for a function f  defined on a q -geometric set A  to be   ,,,)()(:=)(00Abatdtftdtftdtf qaqbqba   (10)  where   ,),()(1:=)(0=0Axxqfqxqtdtf nnnqx (11)  provided that the series converges. A function f  which is defined on a q -geometric set A , A0 , is said to be q-regular at zero if   .(0),=)(lim Axeveryforfxqfnn The rule of q integration by parts is   .)()())((lim))((=)()(00xdqxfxgDaqfgafgxdxfDxg qqannqqa  (12)  If gf ,  are q regular at zero, the ))((limnn aqfg  on the right hand side of (12) will be replaced by )(0)( fg . The two variable polynomial ),( axn , Cax, , are defined to be    0.=,1)(0,,;/:=),(1,:=),(21)(0xaqxqxaxaxaxnnnnnnn  (13)  In [3], Annaby and Mansour gave q -Taylor series in the following forms   .)(),()(1),(1)()(=)( 110=tdtfDqtxnaxkafDxf qnqnxaqkqkqnk   (14)    ,)(),()(13),(1)()(1)(=)(1121)(10=tdtfDqtxncmxakaqfDqxfqnqnxnaqqkqkkqkkknk (15) ISSN 2347-1921                                                           3446 | P a g e                             A p r i l  2 3 ,  2 0 1 5                                                           3  A q -Steffensen-secant method In the following we set axe nn = , aye nn =*, )(= nnn xqfxz  , ],[)/(= nnnnn zxfxfxy  , where babfafbaf )()(=],[ ,  ,(4))()(1)(1(3))()(1(2))(=3222qqqqqq afDqqaafDqaafDA (16)    ,(4))())(2(1(3))(=32qqqq afDqqaafDB (17)  and  .(4))(=3qq afDC (18)   Now, we state and prove our q-Steffensen-secant Theorem with fourth order convergence. Theorem 3.1 Let RD:f  be a real-valued function with a root Da , RD , and let 0x  be closed enough to a . If )(xDkq , 1,2,3=k  exist, and 0)( aDq , then     ,),(],[],[],[],[=21N nyfyxfxzfyzfyxfyx nnnnnnnnnnn  (19)  is fourth-order convergent, and satisfies the following error equation   .),()]22(3)(1)[(1=54222111 N neOeAqqABAqACAqABAe nnn  (20)  Proof: Using the Taylor expansion in (14), we have    .)(),((4)1))()(((4))())(((3))()((2))(=)(43232tdtfDqtaaqxqaxaxafDqaxaxafDaxafDxfqqnxaqnnnqqnnqqnqqn (21)    ISSN 2347-1921                                                           3447 | P a g e                             A p r i l  2 3 ,  2 0 1 5                                                             Rearranging the above equation again gives:    ),(=)( 432 nnnnn eOCeBeAexf   (22)  that is ).())(())(())((=))(1)())((1)())((((4))())(1)())((((3))())(((2))()(=))())(())((((4))())())((((3))())(((2))()(),((4)1=))((=)(432232423243)(nnnnnnnnnnnnnqqnnnnqqnnqqnnnnnnnqqnnnnqqnnqqqqnxqfnxaqnnneOxqfeCxqfeBxqfeAqaxqfeqaxqfexqfeafDqaxqfexqfeafDxqfeafDeOxqfaqxxqfqaxxqfaxafDxqfqaxxqfaxafDxqfaxafDtdtfDqtaxqfxfzf   (23)  Thus,    ).(]][12]34[1[]3[1][1=)(4323322222nnnnneOeqAqBAqAqqACeAqqABeqAAzf (24)  Moreover,     ).(]]3[3[][2=)()())((=],[32222nnnnnnnnneOeqBAqqACeqABAxqfxfxqfxfxzf (25)  Therefore,   ISSN 2347-1921                                                           3448 | P a g e                             A p r i l  2 3 ,  2 0 1 5                                                            .]]3[2]][2[1[][1)(=],[)(:=)(32212122214nnnnnnnneAqqACABqAqAqABAeqABAeeOxzfxfxg (26)  Consequently,   ).())(())(())((=))(1)())((1)())((((4))())(1)())((((3))())(((2))()(=)(),((4)1))())(())((((4))())())((((3))())(((2))(=))((=)(432232443)(232nnnnnnnnnnnnnqqnnnnqqnnqqnqqnxgnxaqnnnnnnqqnnnnqqnnqqnnneOxgeCxgeBxgeAqaxgeqaxgexgeafDqaxqfexgeafDxgeafDeOtdtfDqtaxgaqxxgqaxxgaxafDxgqaxxgaxafDxgaxafDxgxfyf (27)  This means   ,]]3[2]][2[1[][1)(=)(32222124nnnneAqqACqBqAqABAeqABeOyf (28)  and   .]]3[2]][2[1[][1)(=32212122214*nnnneAqqACABqAqAqABAeqABAeOe (29)  On the other hand  ).(]][1[=)()()(=],[3221nnnnnnnneOeqABACBeAxgyfxfyxf (30)  Hence   .]][12[2]]2[3[22)(=],[33122242nnnnnneqABABCeqABACABeAeOyxf (31)  But   ).()]24(1)(1[)(1=)()()()(=],[322212nnnnnnnnneOeqAqABAqACeqABAxgxqfyfzfyzf (32) ISSN 2347-1921                                                           3449 | P a g e                             A p r i l  2 3 ,  2 0 1 5                                                            So that   ).()]43(5)(22[)]22(3)(1[=],[],[],[],[=)(43222432223212nnnnnnnnnnnneOeAqqABAqABCAeAqqABAqACAAxyfxzfyzfxyfxH (33)  If we multiply )( nxH  by )( nyf  we get   .)]()]43(5)(22[)]22(3)(1[[1=],[)(=)()(*432223222221*nnnnnnnnneeOeAqqABAqABCAeAqqABAqACAeayfxHyfxH (34)  Taking in consideration that 1nx  is nothing but )()( nnn yfxHy   we get   .)]()]43(5)(22[)]22(3)(1[[1=)()(=*4322232222211nnnnnnnnneeOeAqqABAqABCAeAqqABAqACAxyfxHyx (35)  Thus   ).()]22(3)(1][[1=)]()22(3)(1[=5422211*222211nnnnnneOeAqqABAqACAqABAeeeOAqqABAqACAe (36) This completes the proof.    References [1]  V. Alarcón, S. Amat, S. Busquier, and D.J López.  On a Steffensen’s type method in Banach spaces with applications on boundary-value problems,   Appl. Math. Comput, 216:234-–250, 2008. [2]  S. Amat, S. Busquier.  On a Steffensen’s type method and its behavior for semismooth equations,   Appl. Math. Comput, 177:819–-823, 2006. [3]  M.H. Annaby and Z.S. Mansour.  q -Taylor and interpolation series for Jackson q -difference operators.   J. Math. Anal. Appl, 344:472–483, 2008. [4]  J.E. Dennis, R.B. Schnable.  Numerical Methods for Unconstrained Optimization and Nonlinear Equations.  Prentice-Hall. Englewood Cliffs, NJ, 1983. [5]  G. Gasper and M. Rahman. Basic Hypergeometric Series, Cambridge university Press, second edition,2004. [6]  F.H. Jackson, A generalization of the function )(n  and nx ,  Proc. Roy. Soc. London ,74:64–72, 1904. [7]  F.H. Jackson,   On q -definite integrals,   Quart. J. Pure and Appl. Math.41:193–203, 1910. [8]  P. Jarratt,  A Review of Methods for Solving Nonlinear Algebraic Equations,   Gordon and Breach, Science Publishers, London, 1970. [9]  L.W. Johnson, R.D. Riess,  Numerical Analysis, Addison-Wesley,  Reading, MA, 1977. [10]  A.M. Ostrowski,  Solution of Equations in Euclidean and Banach Space,   Academic Press, New York, third edition, 1973. [11]  I.F. Steffensen,  Remarks on iteration,   Skand. Aktuarietidskr, 16:64–-72, 1933. [12]  Q. Zheng, J. Wanchao, P. Zhao, and L. Zhang   A Steffensen-like method and higher-order variants,   Appl. Math. Comput,209:206–210, 2009. [13]  Q. Zheng, J. Wanchao, P. Zhao, and L. Zhang   variants of Steffensen-secant method and applications ,   Quart. J. Pure and Appl. Math.,216:3486–3496, 2010. 

A q-VARIANT OF STEFFENSEN'S METHOD OF FOURTH-ORDER CONVERGENCE

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A q-VARIANT OF STEFFENSEN'S METHOD OF FOURTH-ORDER CONVERGENCE

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