Geometrically constructed families of iterative methods

Transcription

1 Chapter 4 Geometrically constructed families of iterative methods 4.1 Introduction The aim of this CHAPTER 3 is to derive one-parameter families of Newton s method [7, 11 16], Chebyshev s method [7, 30 32, 59 61], Halley s method [7, 30 32, 60 73], super- Halley method [30 32, 61, 71 73], Chebyshev-Halley type methods [74], C-methods [30], Euler s method [7, 30, 64], osculating circle method [33] and ellipse method [31, 32, 61] for finding simple roots of nonlinear equations, permitting f (x n ) = 0 at some points in the vicinity of required root. Halley s method, Chebyshev s method, super-halley method, Chebyshev-Halley type methods, Euler s method, osculating circle method and as an exceptional case, Newton s method are seen as the special cases of these families. All methods of various families are cubically convergent to simple roots except Newton s or a family of Newton s method. Further, by making some semi-discrete modifications to above proposed cubically convergent families, a number of interesting new classes of third-order as well as fourth-order multipoint iterative methods can be derived which are free from second-order derivative. Furthermore, numerical examples show that all proposed one-point as well as multipoint iterative families are effective and comparable to the well-known existing classical methods. As discussed previously, one of the most basic problems in computational mathematics is to find solutions of nonlinear equations (1.1). To solve equation (1.1), one can use 3 The main contents of this CHAPTER have been published in [206]. 93

2 iterative methods such as Newton s method [7, 11 16] and its variants namely, Halley s method [7, 30 32, 60 73], Chebyshev s method [7, 30 32, 59 61] and super-halley method [30 32, 61, 71 73] respectively. Among all these iterative methods, Newton s method is probably the well-known and mostly used algorithm. Its geometric construction consists in considering a straight line y = ax + b, (4.1) and then determining unknowns a and b by imposing the tangency conditions : y(x n ) =, y (x n ) = f (x n ), (4.2) thereby obtaining a tangent line y(x) = f (x n )(x x n ), (4.3) to the graph of f (x) at x n, }. The point of intersection of this line with x axis, gives the celebrated Newton s method f (x n) f (x n ). (4.4) The tangent line in Newton s method can be seen as first degree polynomial of f (x) at x = x n. Though, Newton s method is simple and fast with quadratic convergence, but its major drawback is that, it may fail miserably if at any stage of computation, the derivative of the function f (x) is either zero or very small in the vicinity of the required root. Therefore, it has poor convergence and stability problems as it is very sensitive to initial guess. The well-known third-order methods which entail an evaluation of f (x) are close relatives of Newton s method and can be obtained by admitting geometric derivation from the quadratic curves, e.g. parabola, hyperbola, circle or ellipse. parabola In this category, the Euler s method [7, 30, 64] can be constructed by considering a and imposing the tangency conditions : x 2 + ax + by + c = 0, (4.5) y(x n ) =, y (x n ) = f (x n ) and y (x n ) = f (x n ), (4.6) 94

3 one gets y(x) = f (x n )(x x n ) + f (x n ) (x x n ) 2. (4.7) 2 The point of intersection of (4.7) with x axis gives an iteration scheme } 2 1 ± 1 2L f (x n ), (4.8) where L = f (x n) f (x n ) f (x n )} 2. (4.9) The well-known Halley s method [7, 30 32, 60 73] admits its geometric derivation from a hyperbola in the form axy + y + bx + c = 0. (4.10) After the imposition of conditions (4.6) on (4.10) and taking the intersection of (4.10) with x axis as next iterate, the Halley s iteration process reads as } 2 f (xn ) 2 L f (x n ). (4.11) The classical Chebyshev s method [7, 30 32, 59 61] admits its geometric derivation from parabola in a following form : ay 2 + y + bx + c = 0. (4.12) Similarly after imposing the tangency conditions (4.6) and taking the intersection of (4.12) with x axis as next iterate, Chebyshev s iteration process reads as 1 L } f (xn ) 2 f (x n ). (4.13) Amat et al. [30] studied another type of third-order methods by considering a hyperbola in a following form : a n y 2 + b n xy + y + c n x + d n = 0, (4.14) and by the similar conditions as before, one can get an iteration process reads as L f (x ( ) n ) b f (xn ) f (x n n ), (4.15) f (x n ) where b n is a parameter depending on n. For particular values of b n, some interesting particular cases of this family are 95

4 (i) For b n = 0, formula (4.15) corresponds to classical Chebyshev s method [7, 30 32, 59]. (ii) For b n = f (x n ) 2 f (x n ), formula (4.15) corresponds to famous Halley s method [60, 30 32]. (iii) For b n = f (x n ) f (x n ), formula (4.15) corresponds to super-halley method [71 73]. (iv) For b n ±, formula (4.15) corresponds to classical Newton s method [7, 11 16]. Osculating circle method [33] and ellipse method [31, 32, 61] are more cumbersome from computing point of view as compared to Halley s, Chebyshev s or super-halley methods. For more detailed survey of these most important techniques, some excellent text books [1, 7, 14, 21] are available in literature. The problem with existing methods is that these techniques may fail to converge in some cases if an initial guess is far from zero point or if the derivative of the function is small or even zero in the vicinity of required root. Recently, Kanwar and Tomar [33, 231, 232] using different approach derived new classes of iterative techniques having quadratic and cubic convergence respectively. These techniques provide an alternative to the failure situation of existing classical techniques. The purpose of this study is to eliminate the defects of existing classical methods by simple modifications of iteration processes. 4.2 Development of iteration schemes Based on the two theorems [5], namely Theorem and Theorem (discussed in Chapter 1), proposed methods can be derived by admitting geometrical derivation from the hyperbolic curve, cubic curve and general quadratic curve Derivation from a hyperbola Let us now, develop an iteration scheme by fitting the function f (x) of equation (1.1) in the following form of a hyperbola : 2 } a n ye α(x xn) } + bn (x x n ) ye α(x xn) } + ye α(x xn) + c n (x x n ) + d n = 0, (4.16) 96

5 where α R. The unknowns a n,c n and d n are now determined in terms of b n by using tangency conditions (4.6) on equation (4.16). This gives a n = f (x n ) + α 2 2α f (x n ) b n 2 f (x n ) α } 2 f (x n ) α,. (4.17) c n = f (x n ) α } and d n = 0. At a root and it follows from (4.16) that y(x n+1 ) = 0, (4.18) a n } 2 }. (4.19) c n b n If (4.16) is an exponentially fitted straight line, then a n = 0 = b n and from (4.19), one can have f (x n ) α. (4.20) This is a one-parameter family of Newton s method [205, 231] already discussed in Chapter 2 and do not fail even if f (x n ) = 0 or very small in the vicinity of required root, unlike Newton s method. Ignoring the term in α, one gets Newton s method [7, 11 16]. Further, making use of a n and c n in (4.19), one can obtain L f (x n) f (x } n ) 2 f (x 1 + b n ) f (x n n ) α, (4.21) where b n is a parameter and f (x n ) α } L f (x n) = f (x n) f (x n ) + α 2 2α f (x n ) } f (x n ) α } 2. (4.22) This is a modification over the formulas (4.15) of Amat et al. [30] and do not fail even if f (x n ) is very small or zero in the vicinity of required root. For particular values of b n, some interesting particular cases of (4.21) are as under : (i) For b n = 0, one can obtain } 2 L f (x n) f (x n ) α. (4.23) This is a modification over the well-known cubically convergent Chebyshev s method. 97

6 (ii) Putting b n = L f (x n), one gets 2u 2 2 L f (x n) f (x n ) α. (4.24) This is a modification over the well-known cubically convergent Halley s method. (iii) Substituting b n = L f (x n) u, one can get L f (x } n) 2 1 L f (x n) f (x n ) α. (4.25) This is a modification over the well-known cubically convergent super-halley method. (iv) Letting b n = α L f (x n) u,α R, one can have L f (x n) 2 1 α L f (x n) } f (x n ) α. (4.26) This is a modification over the well-known cubically convergent Chebyshev-Halley type methods [74]. (v) If b n ±, one obtains f (x n ) α. (4.27) This is a modification over the well-known quadratically convergent Newton s method. It can be verified that in the absence of the term in parameter α, method (4.23), method (4.24), method (4.25), method (4.26) and method (4.27) reduce to Chebyshev s method [7, 30 32, 59 61], Halley s method [60, 30 32, 61 73], super-halley method [30 32, 61, 71 73], Chebyshev-Halley type methods [74] and Newton s method [7, 11 16] respectively. The sign of parameter α in all these methods is chosen so that the denominator is largest in magnitude. Following theorem gives an order of convergence of the proposed iterative scheme (4.21) : 98

7 Theorem Let f : I R R be a sufficiently differentiable function defined on an open interval I, enclosing a simple zero of f (x) (say x = r I). Assume that initial guess x = x 0 is sufficiently close to r and f (x n ) α 0 in I. Then an iteration scheme defined by formula (4.21) is cubically convergent and satisfies the following error equation: e n+1 = 2C2 2 C 3 3αC } 2 α2 + b n (C 2 α) e 3 n + O ( e 4 n), (4.28) where e n = x n r is an error at n th iteration and C k = ( ) 1 f k (r) k! f (r),k = 2,3,... Proof. Using Taylor s series expansion for about x = r and taking into account that f (r) = 0 and f (r) 0, one can have = f (r) e n +C 2 e 2 n +C 3 e 3 n + O(e 4 n), (4.29) and Using equations (4.29)-(4.31), one gets and f (x n ) = f (r) 1 + 2C 2 e n + 3C 3 e 2 n + O(e 3 n), (4.30) f (x n ) = f (r) 2C 2 + 6C 3 e n + 12C 4 e 2 n + O(e 3 n). (4.31) f (x n ) α = e n (C 2 α)e 2 n ( 2C 3 2C αC 2 α 2) e 3 n + O ( e 4 ) n, (4.32) L f (x n) = f (x n) f (x n ) + α 2 2α f (x n ) } f (x n ) α } 2 =2(C 2 α)e n (6C 2 2 6C 3 6C 2 α + 3α 2 )e 2 n + O(e 3 n). (4.33) Using equations (4.32) and (4.33), one can obtain 1 L f (x n) b n f (x n ) α } } = (C 2 α)e n ( 3C 2 2 3C 3 3αC α2 +b n (C 2 α) ) e 2 n+o(e 3 n). Substituting equations (4.32) and (4.34) in formula (4.21), one obtains e n+1 = (4.34) 2C2 2 C 3 3αC } 2 α2 + b n (C 2 α) e 3 n + O ( e 4 n). (4.35) This completes proof of the theorem. 99

8 4.2.2 Derivation from a cubic curve Another iterative processes can also be derived by considering different exponentially fitted osculating curves. For instance, if one takes a cubic equation namely } 3 } 2+ } a n ye α(x xn) +bn ye α(x xn) ye α(x xn) +d n (x x n ) = 0, and then using tangency conditions (4.6) on (4.36), one gets At a root and it follows from (4.36) that a n =C f (x n ) + α 2 2α f (x n ) f (x n ) α } 2, b n = f (x n ) + α 2 2α f (x n ) 2 f (x n ) α } 2, d n = f (x n ) α } L f (x n) +C L f (x n) } } 2 (4.36). (4.37) y(x n+1 ) = 0, (4.38) f (x n ) α. (4.39) This is a modification over the Amat et al. C-methods [30, 71] and do not fail even if f (x n ) is very small or zero in the vicinity of required root. It can be verified that in the absence of the term in parameter α, method (4.39) reduces to Amat et al. C-methods [30, 71]. A mathematical proof for an order of convergence of the proposed iterative scheme (4.39) is given below : Theorem Let f : I R R be a sufficiently differentiable function defined on an open interval I, enclosing a simple zero of f (x) (say x = r I). Assume that initial guess x = x 0 is sufficiently close to r and f (x n ) α 0 in I. Then an iteration scheme defined by formula (4.39) is cubically convergent and satisfies the following error equation: e n+1 = (2 4C)C 2 2 C 3 + ( 3 + 8C) (C 2 α α2 2 )} e 3 n + O(e 4 n). (4.40) 100

9 Proof. Using equations (4.29)-(4.31), one gets and f (x n ) α = e n (C 2 α)e 2 n ( 2C 3 2C αC 2 α 2) e 3 n + O ( e 4 ) n, (4.41) L f (x n) = f (x n) f (x n ) + α 2 2α f (x n ) } f (x n ) α } 2 =2(C 2 α)e n (6C 2 2 6C 3 6C 2 α + 3α 2 )e 2 n + O(e 3 n). (4.42) Substituting equations (4.41) and (4.42) in formula (4.39), one can obtain e n+1 = (2 4C)C 2 2 C 3 + ( 3 + 8C) This completes proof of the theorem. (C 2 α α2 2 )} e 3 n + O(e 4 n). (4.43) Derivation from a general quadratic curve Let us consider a following general quadratic equation : ) 2 ( ) (x x n ) 2 + a n (ye α(x xn) + bn (x x n ) + c n ye α(x xn) + d n = 0, (4.44) and determine the unknowns b n, c n and d n in the terms of parameter a n by tangency conditions (4.6), one can have b n = 2[ 1 + a n f (x n ) α } 2] f (x n ) α } f (x n ) + α 2 2α f, (x n ) c n = 2[ 1 + a n f (x n ) α } 2] f (x n ) + α 2 2α f (x n ),. (4.45) d n =0. Taking next iterate x n+1 as a point of intersection of (4.44) with x axis, one gets another two-parameter family of iteration scheme as } f D 1, (4.46) (x n ) α D 2 101

10 where D 1 = 2 + a n f (x n ) α } 2 ( 2 + L f (x n) ), (4.47) and D 2 = 1 + a n f (x n ) α } 2 ± 1 + a n f (x n ) α } 2} 2 L f (x n ) D 1. (4.48) This is a modification over formula (20) of Sharma [32] and do not fail even if f (x n ) is very small or zero in the vicinity of required root. Iterative family (4.46) requires computation of square root at each step and sometimes is not convenient for practical usage. Therefore, it can be further reduced to two parameter rational iterative family ( similar to Chun [31] ), or by using the well-known approximation ( similar to Jiang and Han [61] ) and is given by Special cases: x 1 + x, x < 1, 2 2 [ 1 + a n f (x n ) α } 2] D 1 4 [ 1 + a n f (x n ) α } 2] 2 L f (x n )D 1 f (x n ) α. (4.49) For particular values of a n, some interesting particular cases of (4.46) are (i) For a n = 0, one can obtain 2 f (x n ) 1 ± 1 2L f (x n) f (x n ) α. (4.50) This is a modification over the well-known cubically convergent Euler s method [7, 30]. (ii) Taking a n = 1, one gets f (x n ) α D 1 D 2, (4.51) where D 1 = 2 + f (x n ) α } 2 ( 2 + L f (x n) ), (4.52) and D 2 = 1 + f (x n ) α } 2 ± 1 + f (x n ) α } 2} 2 L f (x n ) D 1. (4.53) This is a modification over the cubically convergent osculating circle method derived by Kanwar et al. [33]. 102

11 2 (iii) Letting a n = ), one can have f (x n )} (2 L 2 f (x n) 2 2 L f (x n) f (x n ) α. (4.54) This is a modification over the well-known cubically convergent Halley s method. (iv) Substituting a n = L f (x n) 4 f (x n )} 2 (1 L f (x n) ), one gets L f (x } n) 2 1 L f (x n) f (x n ) α. (4.55) This is a modification over the well-known cubically convergent super-halley method. (v) If a n ±, one gets } 2 L f (x n) f (x n ) α. (4.56) This is a modification over the well-known cubically convergent Chebyshev s method. (vi) For a n = 1/ f (x n )} 2, one can get f (x n ) α. (4.57) This is a modification over the well-known quadratically convergent Newton s method. It can be verified that in the absence of the term in parameter α, method (4.50), method (4.51), method (4.54), method (4.55), method (4.56) and method (4.57) reduce to Euler s method [7, 30, 64], osculating circle method [33], Halley s method [60, 30 32, 61 73], super-halley method [30 32, 61, 71 73], Chebyshev s method [7, 30 32, 59 61] and Newton s method [7, 11 16] respectively. The sign of parameter α in all these methods is chosen so that the denominator is largest in magnitude. Next, Theorem presents a mathematical proof for an order of convergence of the proposed iterative scheme (4.46). 103

12 Theorem Let f : I R R be a sufficiently differentiable function defined on an open interval I, enclosing a simple zero of f (x) (say x = r I). Assume that initial guess x = x 0 is sufficiently close to r and f (x n ) α 0 in I. Then an iteration scheme defined by formula (4.46) is cubically convergent for any finite a n except a n = 1/ f (x n )} 2 (in that case it has quadratic convergence) and satisfies the following error equation : ( 2C2 α ) α + a n f (r)} 2( 4C2 2 e n+1 = 6C 2α + 3α 2) } 2 ( 1 + a n f (r)} 2) C 3 e 3 n + O ( e 4 n). (4.58) Proof. Using equations (4.29)-(4.31), one can get and f (x n ) α = e n (C 2 α)e 2 n ( 2C 3 2C αC 2 α 2) e 3 n + O ( e 4 ) n, (4.59) L f (x n) = f (x n) f (x n ) + α 2 2α f (x n ) } f (x n ) α } 2 =2(C 2 α)e n ( 6C 2 2 6C 3 6C 2 α + 3α 2) e 2 n + O(e 3 n). (4.60) Using equations (4.29), (4.30) and (4.60), one can have D 1 = 2 ( 1 + a n f (r)} 2) + 2a n f (r)} 2( 5C 2 3α ) e n + O(e 2 n), (4.61) and D 2 = 2 ( 1 + a n f (r)} 2) + ( 4a n f (r)} 2( 2C 2 α ) 2(C 2 α) ) e n + O(e 2 n). (4.62) Substituting equations (4.59), (4.61) and (4.62) in formula (4.46), one gets ( 2C2 α ) α + a n f (r)} 2( 4C2 2 e n+1 = 6C 2α + 3α 2) } 2 ( 1 + a n f (r)} 2) C 3 e 3 n + O ( e 4 n). (4.63) This completes proof of the theorem. Alternatively, all these modified techniques ( (4.21), (4.39) and (4.46) ) can also be obtained by using the idea of Ben-Israel [211]. By applying the existing classical techniques to a modified function, say f (x) := e a(x)dx f (x), for some suitable function f (x), one can get the corresponding modified iterative schemes. 104

13 4.2.4 Worked examples Now, let us make an attempt to present some numerical results obtained by employing existing classical methods namely, Chebyshev s method (CM), Halley s method (HM), super- Halley method (SHM), Euler s method (EM), osculating circle method (OCM) and newly developed methods namely, modified Chebyshev s method (MCM) (4.23), modified Halley s method (MHM) (4.24), modified super-halley method (MSHM) (4.25), modified Euler s method (MEM) (4.50) and modified osculating circle method (MOCM) (4.51) respectively to solve nonlinear equations given in Table 4.1. Here, for simplicity, the formulae are tested for α = 1. The results are summarized in Table 4.2 (Number of iterations), Table 4.3 (Computational order of convergence) and Table 4.4 (Total number of function evaluations) respectively. Computations have been performed using MAT LAB R version 7.5 (R2007b) in double precision arithmetic. We use ε = as a tolerance error. The following stopping criteria are used for computer programs : (i) x n+1 x n < ε, (ii) f (x n+1 ) < ε. 105

14 Table 4.1 Test examples, interval (I) containing the root, initial guesses (x 0 ) and exact root (r) Example No. Examples I Initial guesses Exact root (r) exp(1 x) 1 = 0. [0,2] x exp( x) = 0. [1, 3] sin(x) = 0. [0, 2] exp( x) sin(x) = 0. [5, 7] arctan(x) = 0. [ 1, 2] x 3 + 4x 2 10 = 0. [0,2] x exp( x) 0.1 = 0. [ 2, 2] exp(x 2 + 7x 30) 1 = 0. [2,4] cos(x) x = 0. [ 2,2]

15 Table 4.2 Number of iterations (D and F below stand for divergent and failure respectively) Example No. CM MCM HM MHM SHM MSHM EM MEM OCM MOCM (4.23) (4.24) (4.25) (4.50) (4.51) D 1 D 1 D 1 D 1 D D D F 4 F 4 F 4 F 4 F D F 3 F 3 F 3 F 3 F D D D Converges to undesired root. 107

16 Table 4.3 Computational order of convergence (COC) Example No. CM MCM HM MHM SHM MSHM EM MEM OCM MOCM (4.23) (4.24) (4.25) (4.50) (4.51) == 2.90 == 2.96 == 2.86 == 2.96 == == 3.06 == 3.31 == 3.13 == 3.16 == Here, the symbols namely (i) represents COC for a case of undesired root. (ii)... represents COC for a case of divergence. (iii) == represents COC for a case of failure. (iv) represents COC for a case when number of iterations are not sufficient. 108

17 Table 4.4 Total number of function evaluations (TNOFE) Example CM MCM HM MHM SHM MSHM EM MEM OCM MOCM No. (4.23) (4.24) (4.25) (4.50) (4.51) = 12 = 12 = 12 = 12 = = 9 = 9 = 9 = 9 = Here, the symbols namely (i) represents TNOFE for a case of undesired root. (ii) represents TNOFE for a case of divergence. (iii) = represents TNOFE for a case of failure. 109

18 4.3 Generalized families of multipoint iterative methods without memory The practical difficulty associated with third-order methods given by formula (4.21) may be an evaluation of second-order derivative. In past and recent years, many new modifications of Newton s method free from second-order derivative have been proposed by researchers [7, 27, 34, 35, 82, 9, 47 55, 57, 58, 75 81, ] by discretization of second-order derivative or by considering different quadrature formulae for computation of an integral arising from Newton s theorem [8, 52 55] or by considering Adomian decomposition method [39 44]. All these modifications are targeted at increasing a local order of convergence with the view of increasing their efficiency index [13]. Some of these multipoint iterative methods are of third-order [7, 52 55, 57, 58, 75 81, 87 95], while others are of order four [7, 34, 35, 82, 9, 47 51, ]. Recently, fourth-order methods proposed by Nedzhibov et al. [101], Basu [102], Maheswari [103], Cordero et al. [104], Petković and Petković (a survey) [105], Ghanbari [106], Soleymani [47, 107], Cordero and Torregrosa [108], Chun et al. [109] etc. have optimal order of convergence [9]. The efficiency index [13] of these optimal multipoint iterative methods is equal to These multipoint methods calculate new approximations to a zero of f (x) by sampling f (x) and possibly its derivatives for a number of values of independent variable, per step. Nedzhibov et al. [101] have modified Chebyshev-Halley type methods [74] to derive several cubic and quartic order convergent multipoint iterative methods free from second-order derivative. Since all these techniques are variants of Newton s method and main practical difficulty associated with these multipoint iterative methods is that an iteration can be aborted due to the overflow or leads to divergence if the derivative of the function at any iterative point is singular or almost singular (namely f (x n ) 0). Here, it is intended to develop and unify the class of third-order and fourth-order multipoint iterative methods free from second-order derivative in which f (x n ) = 0 is permitted. The main idea of proposed methods lies in discretization of second-order derivative involved in family (4.21) (similar to Nedzhibov et al. [101]). Therefore, again this work can be viewed as a generalization over Nedzhibov et al. [101] families of multipoint iter- 110

19 ative methods. An additional feature of these processes is that they may possess a number of disposable parameters which can be used to ensure the convergence is to a certain order for simple roots. This section, deals with the derivation of three different families free from second-order derivative involved in the family (4.21) First Family Let us take u u(x n ) = f (x n ) α, and expanding the function f (x n u) about a point x = x n with 0 by Taylor s series expansion, one can have f (x n u) = u f (x n ) + u2 2 f (x n ) + O(u 3 ). (4.64) Inserting the value of u defined above in the coefficient of f (x n ), one can obtain f (x n u)( f (x n ) α ) + α } 2 f (x n ) α = u2 2 f (x n ) + O(u 3 ). (4.65) Further, L f (x n) from equation (4.22) may be written as follows : L f (x n) = f (x n ) f (x n ) α } 2 α f (x n ) α α f (x n) f (x n ) f (x n ) α } 2. (4.66) Taking the approximate value of f (x n ) from formula (4.65), one can simplify formula of L f (x n) given in (4.66) as L f (x n) 2 f (x n u) α 2 } 2 f. (4.67) (x n ) α If u is sufficiently small, then the second term in above equation can be neglected and one obtains L f (x n) = 2 f (x n u). (4.68) Substituting this approximate value of L f (x n) in (4.21), a following corresponding multipoint family is obtained 1 + f (x n u) ( [1 + b n f (x n ) α )] f (x n ) α. (4.69) 111

20 This is a modification over the formula (2.1) of Nedzhibov et al. [101] and do not fail even if f (x) is very small or zero in the vicinity of required root. For different specific values of b n, the following various families of multipoint iterative methods can be derived from (4.69), e.g. (i) For b n = 0, one gets formula f (x n ) α f (x n u) f (x n ) α. (4.70) This is a modification over the well-known cubically convergent Traub s method [7, pp. 168]. (ii) Letting b n = L f (x n) 2u, one can obtain } 2 f (x n ) α } f (x n u)}. (4.71) This is a modification over the well-known cubically convergent Newton-Secant method [7, pp. 180]. (iii) Substituting b n = L f (x n) u, one obtains [ ] f (xn ) f (x n u) f. (4.72) (x n ) α 2 f (x n u) This is a modification over the well-known quartically convergent Traub-Ostrowski s method [7, 82, 101]. It can be seen that in absence of the term in parameter α, method (4.70), method (4.71) and method (4.72) reduce to Traub s method [7, pp. 168], Newton-Secant method [7, pp. 180] and Traub-Ostrowski s method [7, 82, 101] respectively. The sign of parameter α in all these families is chosen so that the denominator is largest in magnitude. A mathematical proof for an order of convergence of the proposed iterative family (4.69) is given in following theorem : 112

21 Theorem Let f : I R R be a sufficiently differentiable function defined on an open interval I, enclosing a simple zero of f (x) (say x = r I). Assume that initial guess x = x 0 is sufficiently close to r and f (x n ) α 0 in I. Then an iteration scheme defined by formula (4.69), is cubically convergent for b n R and satisfies the following error equation : e n+1 = (C 2 α)(b n + 2C 2 α)e 3 n + O(e 4 n). (4.73) Proof. Using Taylor s series expansion for about x = r and taking into account that f (r) = 0 and f (r) 0, one can have = f (r) e n +C 2 e 2 n +C 3 e 3 n + O(e 4 n), (4.74) and From equations (4.74) and (4.75), one gets Further, one can get Using equations (4.74)-(4.77), one can have f (x n u) ( [1 + b n f (x n ) α f (x n ) = f (r) 1 + 2C 2 e n + 3C 3 e 2 n + O(e 3 n). (4.75) f (x n ) α = e n + ( C 2 + α)e 2 n + O(e 3 n). (4.76) f (x n u) = f (r) (C 2 α)e 2 n + O(e 3 n). (4.77) )] = (C 2 α)e n + ( (α C 2 )(b n + 3C 2 ) + 2C 3 α 2) e 2 n + O(e 3 n). Substituting equations (4.76) and (4.78) in formula (4.69), one gets (4.78) e n+1 = (C 2 α)(b n + 2C 2 α)e 3 n + O(e 4 n). (4.79) This completes proof of the theorem. 113

22 4.3.2 Second family Replacing second-order derivative in (4.22) by the finite difference between first-order derivatives as where u is as defined earlier. One obtains f (x n ) f (x n ) f (x n θu),θ R 0}, (4.80) θu L f (x n) f (x n ) f } (x n θu) θ f (x n ) α } + 2 α2 f 2α f (x n ) (x n ) α f (x n ) α } 2. (4.81) If u is sufficiently small, then the second term in above equation can be neglected and then one can get L f (x n) = f (x n ) f (x n θu) θ f (x n ) α } 2α f (x n ) f (x n ) α } 2. (4.82) Substituting this approximate value of L f (x n) in (4.21), a following multipoint family is obtained ( f (x n ) f (x n θu) ) } 2αθu f (x n ) θ ( f (x n ) + (b n α) ) f (x n ) α. (4.83) The formula (4.83) is a modification over formula (2.5) given by Nedzhibov et al. [101] and do not fail even if f (x n ) = 0 or very small in the vicinity of required root. For particular values of b n and θ, some particular cases of formula (4.83) are (i) For b n = L f (x n) 2u and θ = 1, one gets f (x n ) + f (x n 2 f (x n ) α ). (4.84) + 2α 2 f (x n ) f (x n ) α This is a modification over the well-known cubically convergent Weerakoon and Fernando method [8]. (ii) Putting b n = L f (x n) 2u family of multipoint iterative methods and θ = 1, one can get a following new cubically convergent 3 f (x n ) f (x n + 2 f (x n ) α ). (4.85) + 2α 2 f (x n ) f (x n ) α 114

23 (iii) Substituting b n = L f (x n) 2u and θ = 1 2, one can get an another new cubically convergent family of multipoint iterative methods 2 f (x n ) f (x n + 2 f (x n ) α } ). (4.86) + α 2 f (x n ) f (x n ) α (iv) Letting b n = L f (x n) 2u and θ = 1 2, one gets f (x n 2 f (x n ) α } ) + α 2 f (x n ) f (x n ) α. (4.87) This is a modification over the cubically convergent method derived by Traub [7, pp. 182]. (v) Taking b n = L f (x n) u and θ = 2 3, one can obtain f (x n ) α f (x n )} 2 f (x n ) α 3 4α f (x n) ( ). 2 f x n 2 3 u 2 f (x n ) f (x n )} 2 +α 2 } 2 f (x n ) α f ( x n 2 3 u ) f (x n ) (4.88) This is a modification over the well-known quartically convergent Jarratt s method [34]. It is easy to observe that in absence of the term in parameter α, method (4.84), method (4.85), method (4.86), method (4.87) and method (4.88) reduce to Weerakoon and Fernando method [8], third-order multipoint iterative method derived by Traub [7, pp. 182] and Jarratt s method [34, 35] respectively. Here, the parameter α should be chosen so that the denominator is largest in magnitude. A mathematical proof for an order of convergence of the proposed iterative family (4.83) is presented in the next theorem. 115

24 Theorem Let f : I R R be a sufficiently differentiable function defined on an open interval I, enclosing a simple zero of f (x) (say x = r I). Assume that initial guess x = x 0 is sufficiently close to r and f (x n ) 0 in I. Then an iteration scheme defined by formula (4.83) is cubically convergent and satisfies the following error equation : e n+1 = (2C C 3(2 3θ) + b n (C 2 α) 3C 2 α + 2α 2 ) e 3 n + O(e 4 n). (4.89) Proof. Using Taylor s series expansion for about x = r and taking into account that f (r) = 0 and f (r) 0, one can have = f (r) e n +C 2 e 2 n +C 3 e 3 n + O(e 4 n), (4.90) and From equations (4.90) and (4.91), one gets Further, one can get f (x n ) = f (r) 1 + 2C 2 e n + 3C 3 e 2 n + O(e 3 n). (4.91) f (x n ) α = e n + ( C 2 + α)e 2 n + O(e 3 n). (4.92) f (x n θu) = f (r) 1 + 2C 2 (1 θ)e n + ( 3C 3 (1 θ) 2 + 2C 2 (C 2 α)θ ) e 2 n + O(e 3 n). (4.93) Using equations (4.90)-(4.93), one can obtain (1 2αθu) f (x n ) f (x n θu) 2θ( f (x n ) + (b n α) ) ( = α + C ) 2 e n + 1 ( 3C3 + 2(αθ C 2 )(b n +C 2 2α) 4C 2 ) 2 e 2 θ 2θ n + O(e 3 n). (4.94) Substituting equations (4.92) and (4.93) in formula (4.83), one gets e n+1 = (2C C 3(2 3θ) + b n (C 2 α) 3C 2 α + 2α 2 ) e 3 n + O(e 4 n). (4.95) Proof of the theorem is now complete. 116

25 4.3.3 Third family Replacing second-order derivative in (4.22) by the finite difference as f (x n ) 5 f (x n ) 4 f ( x n u ) 2 f (x n u), (4.96) 3u where u is as defined earlier. One obtains L f (x n) 5 f (x n ) 4 f ( x n u ) 2 f } (x n u) 3 f +α 2 2 (x n ) α } f 2α f (x n ) (x n ) α f (x n ) α } 2. (4.97) If u is sufficiently small, then the second term in above equation can be neglected and one can obtain L f (x n) = 5 f (x n ) 4 f ( x n u ) 2 f (x n u) 3 f (x n ) α } 2α f (x n ) f (x n ) α } 2. (4.98) Substituting this approximate value of L f (x n) in (4.21), the following multipoint family is obtained ( 5 f (x n ) 4 f ( x n u ) 2 f (x n u) ) } 6αu f (x n ) ( f (x n ) + (b n α) ) Then for b n = L f (x n) 2u in (4.99), one gets f (x n ) α. (4.99) f (x n ) + 4 f (x n 2 f (x n ) α } 6 ) ( + f x n ). f (x n ) α + 6αu f (x n ) (4.100) This is a modification over the cubically convergent formula explored by Hasanov et al. [101, 230], which do not fail even if f (x n ) = 0 in the vicinity of required root. Here, the parameter α should be chosen such that the denominator is largest in magnitude. It can be observed that in absence of the term in parameter α, the family (4.100) reduces to Hasanov et al. method [101, 230]. Next, Theorem presents a mathematical proof for an order of convergence of the proposed iterative family (4.99). 117

26 Theorem Let f : I R R be a sufficiently differentiable function defined on an open interval I, enclosing a simple zero of f (x) (say x = r I). Assume that initial guess x = x 0 is sufficiently close to r and f (x n ) α in I. Then an iteration scheme defined by formula (4.99) is cubically convergent and satisfies the following error equation : e n+1 = ( 2C b n (C 2 α) 3C 2 α + 2α 2) e 3 n + O(e 4 n). (4.101) Proof. Using Taylor s series expansion for about x = r and taking into account that f (r) = 0 and f (r) 0, one can have and From equations (4.102) and (4.103), one gets Further, one can have and = f (r) e n +C 2 e 2 n +C 3 e 3 n + O(e 4 n), (4.102) f (x n ) = f (r) 1 + 2C 2 e n + 3C 3 e 2 n + O(e 3 n). (4.103) f (x n ) α = e n + ( C 2 + α)e 2 n + O(e 3 n). (4.104) f (x n u) = f (r) 1 + 2C 2 (C 2 α)e 2 n + O(e 3 n), (4.105) ( f x n u ) ( ) = f 3C3 (r) 1 +C 2 e n C 2(C 2 α) e 2 n + O(e 3 n). (4.106) Using equations (4.102)-(4.106), one can obtain (5 6αu) f (x n ) 4 f ( x n u 2) f (x n u) 6( f (x n ) + (b n α) ) =(C 2 α)e n + ( 2C 3 2α 2 (C 2 α)(b n + 3C 2 ) ) e 2 n + O(e 3 n). (4.107) Substituting equations (4.104) and (4.107) in the formula (4.99), one can get This completes proof of the theorem. e n+1 = ( 2C b n (C 2 α) 3C 2 α + 2α 2) e 3 n + O(e 4 n). (4.108) On the similar lines, one can further derive some more new cubically convergent families of multipoint iterative methods free from second-order derivative by discretizing the secondorder derivative involved in families (4.39) and (4.46). 118

27 4.3.4 Further exploration of family (4.69) One can also construct a different family of fourth-order convergent multipoint iterative methods by suitably choosing parameter b n. Again consider the family (4.69) f (x n u) 1 + ( )] [1 + b n where α R and u = f (x n ) α. f (x n ) α f (x n ) α, (4.109) Taking b n = 2C 2 + α f (x n ) f (x n ) + α, in an error equation (4.79), the family (4.109) has at least fourth-order convergence, i.e. after simplifications, one gets new fourth-order multipoint iterative family given by f (x n ) α f (x n u) f (x n ) f (x n )} 2 f (x n ). (4.110) Further, one can derive another fourth-order multipoint iterative family free from secondorder derivative by using (4.65) and is given by [ f 1 + (x n ) α f (x n u) f (x n ) u f (x n )} 2 2α } 2 2 f (x n u)( f (x n ) α ) (4.111) This is a modification over the well-known Traub-Ostrowski s method [7, 82] and is also an optimal fourth-order method. This family requires same evaluations of the function and its derivative as that of Traub-Ostrowski s method per iteration. Thus, an efficiency index [13] of this method is equal to 3 4 = 1.587, which is better than the one of Newton s method 2 2 = and method (4.72) (derived in this chapter) 3 3 = More importantly, this method will not fail even if the derivative of the function is very small or even zero in the vicinity of required root, unlike Traub-Ostrowski s method [1, 43]. Here, the parameter α should be chosen so that denominator is largest in magnitude. One can see in absence of the term in parameter α, the family (4.111) reduces to wellknown Traub-Ostrowski s method [7, 82]. A mathematical proof for an order of convergence of the proposed iterative family (4.111) is given by next theorem. ]. 119

28 Theorem Let f : I R R be a sufficiently differentiable function defined on an open interval I, enclosing a simple zero of f (x) (say x = r I). Assume that initial guess x = x 0 is sufficiently close to r and f (x n ) α in I. Then an iteration scheme defined by formula (4.111) has optimal order of convergence four and satisfies the following error equation : e n+1 = (C 2 α)(c2 2 C 3 αc 2 + 2α 4 )e 4 n + O(e 5 n). (4.112) Proof. Using Taylor s series expansion for about x = r and taking into account that f (r) = 0 and f (r) 0, one can have = f (r) e n +C 2 e 2 n +C 3 e 3 n + O(e 4 n), (4.113) and f (x n ) = f (r) 1 + 2C 2 e n + 3C 3 e 2 n + 4C 4 e 3 n + O(e 4 n). (4.114) From equations (4.113) and (4.114), one gets f (x n ) α = e n + ( C 2 + α)e 2 n + (2C 2 2 2C 3 2αC 2 + α 2 )e 3 n + O(e 4 n). (4.115) Further, one can obtain f (x n u) = f (r) (C 2 α)e 2 n + ( 2C C 3 + 2αC 2 α 2 )e 3 n + O(e 4 n). (4.116) Using equations (4.113), (4.114) and (4.116), one gets f (x n u) f (x n ) u f (x n )} 2 2α } 2 2 f (x n u)( f (x n ) α ) =(C 2 α)e n + ( C 2 2 +C 3 )e 2 n + ( 2C 2 C 3 + 3C 4 + 2C 2 2α C 3 α 3C 2 α 2 + 2α 3) e 3 n + O(e 4 n). (4.117) Substituting equations (4.115) and (4.117) in formula (4.111) and simplifying, one obtains e n+1 = (C 2 α)(c 2 2 C 3 αc 2 + 2α 4 )e 4 n + O(e 5 n). (4.118) This completes proof of the theorem. 120

29 4.3.5 Worked examples Here, an attempt has been made to present some numerical results obtained by employing existing classical methods namely, Steffensen s method (SM), Traub s method (T M), Newton-Secant method (NSM), Weerankoon and Fernando method (WFM), Hasanov et al. method (HAM), Traub-Ostrowski s method (T OM), Jarratt s method (JM) and newly developed methods namely, modified Traub s method (MT M) (4.70), modified Newton-Secant method (MNSM) (4.71), modified Weerankoon and Fernando method (MW FM) (4.84), modified Hasanov et al. method (MHAM) (4.100), modified Traub-Ostrowski s method (MT OM) (4.111) and modified Jarratt s method (MJM) (4.88) respectively to solve nonlinear equations given in Table (4.5). Here, for simplicity, the formulae are tested for α = 1. The results are summarized in Table 4.6 (Number of iterations), Table 4.7 (Computational order of convergence) and Table 4.8 (Total number of function evaluations) respectively. Computations have been performed using MAT LAB R version 7.5 (R2007b) in double precision arithmetic. We use ε = as a tolerance error. The following stopping criteria are used for computer programs : (i) x n+1 x n < ε, (ii) f (x n+1 ) < ε. 121

30 Table 4.5 Test examples, interval (I) containing the root, their initial guesses (x 0 ) and exact root (r) Example No. Examples I Initial guesses Exact root exp(1 x) 1 = 0. [0,2] sin(x) = 0. [0, 2] exp( x) sin(x) = 0. [5, 7] arctan(x) = 0. [ 1, 2] x 3 + 4x 2 10 = 0. [0,2] (x 1) 3 1 = 0. [0,3] exp(x 2 + 7x 30) 1 = 0. [2,4] cos(x) x = 0. [ 1,2] log(x) = 0. [.5, 3.5]

31 Table 4.6 Number of iterations (D and F below stands for divergent and failure respectively) Example SM TM MTM NSM MNSM WFM MWFM HAM MHAM TOM MTOM JM MJM No. (4.70) (4.71) (4.84) (4.100) (4.111) (4.88) D D D D D 5 D D D D 5 13 F 4 F 5 F 5 F 5 F 4 F F 1 F 1 F 5 F 1 F 1 F D D D D D 5 D 6 D 5 D 6 D Converges to undesired root 123

32 Table 4.7 Computational order of convergence (COC) Example SM TM MTM NSM MNSM WFM MWFM HAM MHAM TOM MTOM JM MJM No. (4.70) (4.71) (4.84) (4.100) (4.111) (4.88) == 2.88 == 2.98 == 2.98 == 2.98 == 3.90 == == == == 2.99 == == == Here, the symbols used, are already defined in this chapter under the Table

33 Table 4.8 Total number of function evaluations (TNOFE) Example SM TM MTM NSM MNSM WFM MWFM HAM MHAM TOM MTOM JM MJM No. (4.70) (4.71) (4.84) (4.100) (4.111) (4.88) = 12 = 15 = 15 = 20 = 12 = = 3 = 3 = 15 = 4 = 3 = Here, the symbols used, are already defined in this chapter under the Table

34 4.4 Discussion and conclusions This work proposes many new families of Chebyshev s method, Halley s method, super- Halley method, Chebyshev-Halley type methods, C-methods, Euler s method, osculating circle method and ellipse method for finding simple roots of nonlinear equations, permitting f (x n ) = 0, at some points in the vicinity of required root. Further, an attempt is made to propose some new families of third-order and fourth-order multipoint iterative methods free from second-order derivative by discretization. The numerical examples considered in Table 4.1 and Table 4.5 overwhelmingly support that proposed one-point as well as multipoint iterative methods can compete with any of the existing methods including classical Newton s method and Traub-Ostrowski s method. A reasonably close initial guess is necessary for these multipoint methods to converge. This condition, however, applies practically to all iterative methods for solving equations. Fourth-order multipoint iterative method of first family have optimal order with computational efficiency index [13] equal to 3 4 = and third-order multipoint iterative methods of first and second families have efficiency indices [13] equal to 3 3 = 1.442, which are better than the one of Newton s method 2 = In third family, a modified third-order multipoint family of Hasanov et al. method [101, 226] has been obtained, permitting f (x n ) = 0, at some points in the vicinity of required root. Further, one can also construct many new families of iterative methods free from first and second-order derivatives in computing process by the forward-difference approximations. Furthermore, the behaviours of existing classical iterative methods (one-point as well as multipoint) and the proposed modifications can be compared by their corresponding correction factors. The factor f (x n) f (x n ), which appears in the classical iterative schemes, is now f (x modified to n ) f (x n ) α, where α R. Here, the parameter α should be chosen such that the denominator is largest in magnitude. As is well-known that, if an initial guess x n is close enough to the solution r with f (r) 0, the classical iterates are well defined and converge to the required root. Now this modified factor has two major advantages over its predecessor. The first one is that it is always well defined, even if f (x n ) = 0. The second one is that an absolute value of denominator in the formula is greater than f (x n ) provided x n is not accepted as an approximation to the required root r. This means that the numerical 126

35 stability of iterative schemes having this factor is better than the classical iterates. Moreover, if x n is sufficiently close to r (simple root), then will be sufficiently close to zero, and in particular the considered denominator f (x n ) α will also be close to zero if f (x n ) 0. In these problems where f (x n ) 0, a small value of the parameter α may lead to divergence or failure. Such situations can be handled by choosing α in such a way such that the correction factor is as small as possible, i.e. denominator f (x n ) α is sufficiently large in magnitude. Finally, it is concluded that these techniques can be used as an alternative to existing techniques or in some cases where existing techniques are not successful, therefore, have definite practical utility. 127