General Mathematics Vol. 16, No. 1 (2008), 33-45
Nazir Ahmad Mir, Nusrat Yasmin, Naila Rafiq
In this paper, we present two-step quadrature based iterative
methods for solving non-linear equations. The convergence analysisof the methods is discussed. It is established that the new methodshave convergence order five and six Numerical tests show that thenew methods are comparable with the well known existing methodsand in many cases gives better results. Our results can be consideredas an improvement and refinement of the previously known resultsin the literature.
2000 Mathematics Subject Classification: 65F10
Key words: Iterative methods, Three-step methods, Quadrature rule,
Predictor-corrector methods, Nonlinear-equations.
Let us consider a single variable non-linear equation
Accepted for publication (in revised form) 15 February, 2007
Nazir Ahmad Mir, Nusrat Yasmin, Naila Rafiq
Finding zeros of a single variable nonlinear equation (1.1) efficiently, is
an interesting very old problem in numerical analysis and has many ap-plications in applied sciences. In recent years, researches have developedmany iterative methods for solving equation (1.1). These methods can beclassified as one-step, two-step and three-step methods, see [1-12]. Thesemethods have been proposed using Taylor series, decomposition techniques,error analysis and quadrature rules, etc. Abbasbandy [1], Chun [3] andGrau [7] have proposed many two-step and three-step methods.
In this paper, we present two-step quadrature based iterative methods
for solving non-linear equations. We prove that the new methods have orderof convergence five and six. The methods and their algorithms are describedin section 2. The convergence analysis of the methods is discussed in section3. Finally, in section 4, the methods are tested on numerical examples givenin the literature. It was noted that the new methods are comparable withthe well known existing methods and in many cases gives better results. Our results can be considered as an improvement and refinement to thepreviously known results in the literature.
Weerakoon and Fernando [12], Gyurhan Nedzhibov [11] and M. Frontiniand E. Sormani [5-6] have proposed various methods by the approximationof the indefinite integral
using Newton Cotes formulae of order zero and one. We approximate, herehowever the integral (2.1) by rectangular rule at a generic point λx+(1−λ)znwith the end-points x and zn. We thus have:
f ′(t)dt = (x − zn)f ′(λx + (1 − λ)zn),
Quadrature based two-step iterative methods for non-linear equations
−f (zn) = (x − zn)(f ′(zn) + λ(x − zn)2f ′′(zn)).
From (2.3), for λ = 0, we have Newton’s method and for λ = 1, we have
This formulation allows to suggest many one-step, two-step and three-
step methods. We suggest here, however the following two-step methods:
Algorithm 2.1. For a given initial guess x0, find the approximate solutionby the iterative scheme:
Algorithm 2.1 can further be modified by using an approximation for
f ′(yn) with the help of Taylor’s expansion.
Let yn be defined by (2.4). If we use Taylor expansion of f ′(yn):
f ′(yn) ≃ f ′(xn) + f ′′(xn)(yn − xn),
(where the higher derivatives are neglected) in combination with Taylorapproximation of f (yn):
f (yn) ≃ f (xn) + f ′(xn)(yn − xn) + f ′′(xn)(yn − xn)2,
we can remove the second derivative and approximate f ′(yn) as:
then Algorithm 2.1 can be written in the form of the following algorithm:
Nazir Ahmad Mir, Nusrat Yasmin, Naila Rafiq
Algorithm 2.2. For a given initial guess x0, find the approximate solutionby the iterative scheme:
Let us now discuss the convergence analysis of the above mentioned algo-rithms.
Theorem 3.1. Let α ∈ I be a simple zero of sufficiently differentiable
function f : I ⊆ R → R for an open interval I. If x0 is sufficiently close toα, then the algorithm 2.1 has sixth order convergence for λ = 1 .
Proof: Let α be a simple zero of f and xn = α + en.By Taylor’s expansionwe have:
(3.2) f ′(xn) = f ′(α)(1 + 2c2en + 3c3e2 + 4c
f ′′(xn) = f ′(α)(2c2 + 6c3en + 12c4e2 + 20c
, k = 2, 3, . . . and en = xn − α.
Quadrature based two-step iterative methods for non-linear equations
Using (3.1), (3.2) and (3.3) in (2.4), we have:
+(28λ2c32 + 4c32 + 38λc2c3 + 3c4 − 26λc32 − 8λ3c32 − 24λ2c2c3−
+42λc23 + 68λc2c4 − 132λ2c42 − 72λ3c22c3 + 20c3c22 − 16λ4c42+
+80λ3c42 − 36λ2c23 − 10c2c4 + 216λ2c22c3 − 20λc5 + 76λc42 − 6c23)e5 +
+(5c6 − 13c2c5 − 17c3c4 + 106λc2c5 + 144λc3c4 − 30λc6
+16c52 + 33c2c23 + 28c22c4 − 52c32c3 − 208c52λ−
−338λc2c23 − 280c22λc4 + 606c32λc3 − 144λ2c3c4+
+540λ2c23c2 + 404λ2c22c4 − 1236λ2c32c3 − 144λ3c22c4+
+516λ2c2c52 − 496λ3c52 + 208λ4c52 − 32λ5c52)e6 + O(e7 ).
f (yn) = f ′(α)(−c2(2λ − 1)e2 + (8λc2
+(32λ2c32 − 24λ2c2c3 − 8λ3c32 − 7c2c3 − 12λc4 + 38λc2c3−
−172λ2c42 − 6c23 + 68λc2c4 − 48λ2c2c4 − 20λc5 + 96λ3c42−
−12c42 + 240λ2c22c3 + 42λc23 + 100λc42 + 4c5 − 16λ4c42−
−36λ2c23 − 186c3λc22)e5 ) + O(e6 ).
f ′(yn) = f ′(α)(1 + 2c2(−2λc2 + c2)e2 + 2c
2(38λc2c3 − 24λ2c2c3 − 12λc4 − 7c2c3 + 3c4−
−26λc32 − 8λ3c32 + 28λ2c32 + 4c32) + 3c3(−2λc2 + c2)2)e4 +
Nazir Ahmad Mir, Nusrat Yasmin, Naila Rafiq
+(2(80λ3c42 − 72λ3c22c3 − 10c2c4 + 76λc42 − 36λ2c23 − 20λc5+
+68λc2c4 − 16λ4c42 + 4c5 + 216λ2c22c3 + 42λc32 − 132λ2c42+
+20c3c22 − 8c42 − 166c3λc22 − 48λ2c2c4 − 6c23)c2 + 6(−2λc2+
+c2)(8λc22 + 2c3 − 6λc3 − 4λ2c22 − 2c22)c3)e5 ) + O(e5 ),
Using (3.5), (3.6) and (3.7) in (2.5), we have:
xn+1 := α + (c32 + 4λ2c32 − 4λc32)e4 + (24λ2c2
+16λ3c42 − 20c3λc22 − 40λ2c42 + 24λc42)e5 + (−30λc
−13c2c5 − 17c3c4 + 106λc2c5 + 144λc3c4 − 192λ4c32c3−
−216λ3c2c23 + 38c52 + 41c2c23 + 40c22c4 − 93c32c3 + 972λ2c52−
−396λc52 − 864λ3c52 + 304λ4c52 − 32λ5c52 − 386λc2c23+
+500λ2c22c4 − 1824λ2c32c3 + 908λc32c3 − 144λ2c3c4+
+612λ2c23c2 − 144λ3c22c4 + 1120λ3c32c3 − 352λc22c4−
Thus, we observe that the algorithm 2.1 has sixth order convergence for
Theorem 3.2. Let α ∈ I be a simple zero of sufficiently differentiable
function f : I ⊆ R → R for an open interval I. If x0 is sufficiently close toα, then the algorithm 2.2 has fifth order convergence for λ = 1 .
Quadrature based two-step iterative methods for non-linear equations
Proof. Let α be a simple zero of f and xn = α + en. By Taylor’s expansion,we have:
(3.12) f ′(xn) = f ′(α)(1 + 2c2en + 3c3e2 + 4c
(3.13) f ′′(xn) = f ′(α)(2c2 + 6c3en + 12c4e2 + 20c
, k = 2, 3, . . . and en = xn − α.
Using (3.1), (3.2) and (3.3) in (2.7), we have:
+(28λ2c32 + 4c32 + 38λc2c3 + 3c4 − 26λc32 − 8λ3c32 − 24λ2c2c3−
+42λc23 + 68λc2c4 − 132λ2c42 − 72λ3c22c3 + 20c3c22 − 16λ4c42+
+80λ3c42 − 36λ2c23 − 10c2c4 + 216λ2c22c3 − 20λc5 + 76λc42 − 6c23)e5 + O(e6 ).
By Taylor’s series, we have:f (yn) = f ′(α)(−c2(2λ − 1)e2 + (8λc2
+(32λ2c32 − 24λ2c2c3 − 8λ3c32 − 7c2c3 − 12λc4 + 38λc2c3−
−172λ2c42 − 6c23 + 68λc2c4 − 48λ2c2c4 − 20λc5 + 96λ3c42−
Nazir Ahmad Mir, Nusrat Yasmin, Naila Rafiq
−12c42 + 240λ2c22c3 + 42λc32 + 100λc42 + 4c5 − 16λ4c42−
−36λ2c23 − 186c3λc22)e5 ) + O(e6 ).
By using (3.1), (3.2), (3.5) and (3.6) in (2.6), we have:
f ′(yn) = f ′(α)(1 + (−4λc22 − c3 + 2c22)e2 + (−4c3
16λc2c3 + 6c2c3 − 8λ2c32)e3 + (−16λ3c4
168λc2c5 − 60λc6 − 432λ3c2c23 − 384λ4c32c3 + 56c2c23+
46c22c4 − 106c32c3 + 1248λ2c52 − 464λc52 − 1216λ3c52+
604λc2c23 − 288λ2c3c4 + 1080λ2c23c2 − 288λ3c22c4+
Using (3.5), (3.6) and (3.7) in (2.8) , we have:
xn+1 = α + (2λc2c3 − c2c3 + 4λ2c32 − 4λc32 + c32)e4 + (−2c
+16λ3c42 − 40λ2c42 + 6λc23 + 4λc2c4)e5 + O(e6 ),
or en+1 = (−c3c22 + c23)e5 + O(e6 ).
Thus, we observe that the algorithm 2.2 has fifth order convergence for
Quadrature based two-step iterative methods for non-linear equations
We consider here some numerical examples to demonstrate the performanceof the new developed two-step iterative methods, namely algorithm 2.1 and2.2.We compare the classical Newton’s method (N M ), the method of Grau(GM )[7] and the new developed two-step method algorithm 2.1 (M N 6) andalgorithm 2.2 (M N 5), in this paper. All the computations are performedusing Maple 10.0. We take ∈= 10−15 as tolerance. All the values arecomputed using 128 significant digits.
The following criteria is used for estimating the zero:
The following examples are used for numerical testing:
For convergence criteria, it was required that δ, the distance between
two consecutive iterates was less than 10−15, n represents the number ofiterations and f (xn), the absolute value of the function. The numericalcomparison is given in Table 4.1.
Nazir Ahmad Mir, Nusrat Yasmin, Naila Rafiq
Quadrature based two-step iterative methods for non-linear equations
From Table 4.1, we observe that our two-step iterative methods of conver-gence orders five and six are comparable with the sixth order method of M. Grau, J.L. Diaz-Barrero[7] and in many cases gives better results in terms ofthe function evaluation f (xn).The computational efficiency of the methodsdescribed in this paper is better than the efficiency of most of the othermethods defined in the literature. For algorithm 2.1 with convergence order
6, the computational efficiency is 65 = 1.430969 and for algorithm 2.2 with
convergence order 5, the computational efficiency is 54 = 1.495349.
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tions by modified Adomian decomposition method, Appl. Math. Com-put. 145(2003)887-893.
Nazir Ahmad Mir, Nusrat Yasmin, Naila Rafiq
[2] R. L. Burden, J. D. Faires, Numerical Analysis, PWS publishing com-
[3] C. Chun,Iterative methods improving Newton’s method by the decom-
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[6] M. Frontini, E. Sormani, Third order methods for quadrature formulae
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[7] M. Grau, J.L. Diaz-Barrero, An improvement to Ostrowski root-finding
method, Appl. Math. Comput. 173(2006) 450-456.
[8] Jisheng Kou, Yitian Li and Xiuhua Wang,Third order modification of
Newton’s method, Appl. Math. and Comput.,(2006, in press).
[9] M. V. Kanwar, V. K. Kukreja, S. Singh,On a class of quadratically
convergent iteration formulae, Appl. Math. Comput. 166 (3) (2005),633-637.
[10] Mamta, V. Kanwar, V. K. Kukreja, S. Singh,On some third order iter-
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[11] Gyurhan Nedzhibov, On a few iterative methods for solving nonlinear
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Quadrature based two-step iterative methods for non-linear equations
[12] S. Weerakoon, T. G. I. Fernando,A variant of Newton’s method with
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Nazir Ahmad MirDepartment of Mathematics,COMSATS Institute of Information Technology,Plot No.30,Sector H-8/1,Islamabad 44000PakistanE-mail: namir@comsats.edu.pk
Nusrat Yasmin, Naila RafiqCentre for Advanced Studies in Pure and Applied MathematicsBahauddin Zakariya University,Multan,PakistanE-mail: nusyasmin@yahoo.comE-mail: rafiqnaila@yahoo.com

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