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A note on “A family of methods for solving nonlinear equations”
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Applied Mathematics and Computation 195 (2008) 819–821 www.elsevier.com/locate/amc
A note on ‘‘A family of methods for solving nonlinear equations’’ Arif Rafiq Department of Mathematics, COMSATS Institute of Information Technology, Plot No. 30, Sector H-8/1, Islamabad 44000, Pakistan
Abstract Ujevic´ et al. [Nenad Ujevic´, Goran Erceg, Ivan Lekic´, A family of methods for solving nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.03.007] introduced a family of methods for solving nonlinear equations. For certain choices of parameters, firstly they showed that the classical Newton’s method is a member of this family and then they introduced a particular method. However the main Algorithm 1 put forward by Ujevic´ et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujevic´ et al. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Iterative methods; Convergence order; Nonlinear equations
1. Introduction and main results In [2,3], Ujevic´ introduced the following algorithms for solving the nonlinear equation: f ðxÞ ¼ 0: Algorithm 1 [2, Algorithm 1, p. 1420]. Step 1. Choose x0, a 2 (0, 1), e > 0 and a positive integer m. Set k = 0. Step 2. Calculate f ðxk Þ ; zk ¼ x k a 0 f ðxk Þ f ðxk Þ xkþ1 ¼ xk þ 4ðzk xk Þ : 3f ðxk Þ 2f ðzk Þ Step 3. If jxk+1 x kj < e or k > m then stop. Step 4. Set k ! k + 1 and go to step 2. E-mail address: arafi[email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.116
820
A. Rafiq / Applied Mathematics and Computation 195 (2008) 819–821
Algorithm 2 [3, Algorithm 1, p. 211]. Step 1. Choose x0, a 2 (0, 1), e > 0 and a positive integer m. Set k = 0. Step 2. Calculate zk ¼ x k a
f ðxk Þ ; f 0 ðxk Þ
xkþ1 ¼ xk þ 3ðzk xk Þ
f ðxk Þ : 2f ðxk Þ f ðzk Þ
Step 3. If jxk+1 x kj < e or k > m then stop. Step 4. Set k ! k + 1 and go to step 2.
Remark 1 1. Using Maple 10, it can be easily seen that Algorithms 1 and 2 have the same efficiency index 1.2599 for a ¼ 12 : 2. In comparison to classical Newton’s method, Algorithms 1 and 2 are less efficient. 3. For further discussion on Algorithms 1 and 2, see for example [1]. Recently Ujevic´ et al. [4] introduced a family of methods for solving nonlinear equations: zk ¼ xk a
f ðxk Þ ; f 0 ðxk Þ
xkþ1 ¼ xk þ ðzk xk Þ
0 6 a 6 1; f ðxk Þ ; bf ðxk Þ þ cf ðzk Þ
ð1:1Þ b; c 2 R:
We first discuss the convergence analysis of the method (1.1). Theorem 2. Assume that the function f : D R ! R for an open interval D has a simple root w 2 D. Let f(x) be sufficiently smooth in the neighborhood of the root w, then the order of convergence of the method defined by (1.1) a is at least two for bþcð1aÞ ¼ 1. Proof. The iterative scheme is given by zn ¼ x n a
f ðxn Þ ; f 0 ðxn Þ
xnþ1 ¼ xn þ ðzn xn Þ
f 0 ðxn Þ 6¼ 0;
ð1:2Þ
f ðxn Þ : bf ðxn Þ þ cf ðzn Þ
ð1:3Þ
Let w be a simple zero of f . By Taylor’s expansion, we have f ðxn Þ ¼ f 0 ðwÞ½en þ c2 e2n þ Oðe3n Þ; 0
0
f ðxn Þ ¼ f ðwÞ½1 þ 2c2 en þ
3c3 e2n
þ
ð1:4Þ Oðe3n Þ;
ð1:5Þ
where ðkÞ 1 f ðwÞ ; ck ¼ k! f 0 ðwÞ
k ¼ 2; 3; . . . ;
and en ¼ xn w:
From (1.4) and (1.5), we have f ðxn Þ ¼ en c2 e2n þ Oðe3n Þ: f 0 ðxn Þ
ð1:6Þ
A. Rafiq / Applied Mathematics and Computation 195 (2008) 819–821
821
Using (1.2) and (1.6), we have zn ¼ w þ ð1 aÞen þ ac2 e2n þ Oðe3n Þ:
ð1:7Þ
By Taylor’s series, we have f ðzn Þ ¼ f 0 ðwÞ½ð1 aÞen þ ð1 a þ a2 Þc2 e2n þ Oðe3n Þ:
ð1:8Þ
Using (1.3), (1.4) and (1.8), we get a b cð1 aÞ þ ca2 2 xnþ1 ¼ w þ 1 en þ Oðe3n Þ; en þ ac2 b þ cð1 aÞ ðb þ cð1 aÞÞ2 implies enþ1 ¼
1
a b cð1 aÞ þ ca2 2 en þ Oðe3n Þ: en þ ac2 2 b þ cð1 aÞ ðb þ cð1 aÞÞ
From the last error equation, we can easily deduce the desired results.
h
Remark 3 a 1. In [4], Ujevic´ et al. proved that for c ¼ 0; bþcð1aÞ ¼ 1 reduces to a = b and the family of methods (1.1) reduce to the classical Newton’s method. a 2. It is interesting to note that bþcð1aÞ ¼ 1 is equivalent to c ¼ ab , for which a 5 1. Thus Algorithm 1 put 1a forward by Ujevic´ et al. [4, p. 7] does not exist and is wrong. a 3. Also for a ¼ 12 ; bþcð1aÞ ¼ 1 reduces to b ¼ 12 ð1 cÞ. Now it can easily be seen that Algorithms 1 and 2 are obtained from (1.1) by taking values 12 and 13 respectively for c.
2. Conclusion The main Algorithm 1 put forward by Ujevic´ et al. [4, p. 7] is wrong. References [1] L.D. Petkovic, M.S. Petkovic, A note on some recent methods for solving nonlinear equations, Appl. Math. Comput. 185 (2007) 368– 374. [2] N. Ujevic´, A method for solving nonlinear equations, Appl. Math. Comput. 174 (2) (2006) 1416–1426. [3] N. Ujevic´, An iterative method for solving nonlinear equations, J. Comput. Appl. Math. 201 (1) (2007) 208–216. [4] Nenad Ujevic´, Goran Erceg, Ivan Lekic´, A family of methods for solving nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.03.007.
Applied Mathematics and Computation 195 (2008) 819–821 www.elsevier.com/locate/amc
A note on ‘‘A family of methods for solving nonlinear equations’’ Arif Rafiq Department of Mathematics, COMSATS Institute of Information Technology, Plot No. 30, Sector H-8/1, Islamabad 44000, Pakistan
Abstract Ujevic´ et al. [Nenad Ujevic´, Goran Erceg, Ivan Lekic´, A family of methods for solving nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.03.007] introduced a family of methods for solving nonlinear equations. For certain choices of parameters, firstly they showed that the classical Newton’s method is a member of this family and then they introduced a particular method. However the main Algorithm 1 put forward by Ujevic´ et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujevic´ et al. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Iterative methods; Convergence order; Nonlinear equations
1. Introduction and main results In [2,3], Ujevic´ introduced the following algorithms for solving the nonlinear equation: f ðxÞ ¼ 0: Algorithm 1 [2, Algorithm 1, p. 1420]. Step 1. Choose x0, a 2 (0, 1), e > 0 and a positive integer m. Set k = 0. Step 2. Calculate f ðxk Þ ; zk ¼ x k a 0 f ðxk Þ f ðxk Þ xkþ1 ¼ xk þ 4ðzk xk Þ : 3f ðxk Þ 2f ðzk Þ Step 3. If jxk+1 x kj < e or k > m then stop. Step 4. Set k ! k + 1 and go to step 2. E-mail address: arafi[email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.116
820
A. Rafiq / Applied Mathematics and Computation 195 (2008) 819–821
Algorithm 2 [3, Algorithm 1, p. 211]. Step 1. Choose x0, a 2 (0, 1), e > 0 and a positive integer m. Set k = 0. Step 2. Calculate zk ¼ x k a
f ðxk Þ ; f 0 ðxk Þ
xkþ1 ¼ xk þ 3ðzk xk Þ
f ðxk Þ : 2f ðxk Þ f ðzk Þ
Step 3. If jxk+1 x kj < e or k > m then stop. Step 4. Set k ! k + 1 and go to step 2.
Remark 1 1. Using Maple 10, it can be easily seen that Algorithms 1 and 2 have the same efficiency index 1.2599 for a ¼ 12 : 2. In comparison to classical Newton’s method, Algorithms 1 and 2 are less efficient. 3. For further discussion on Algorithms 1 and 2, see for example [1]. Recently Ujevic´ et al. [4] introduced a family of methods for solving nonlinear equations: zk ¼ xk a
f ðxk Þ ; f 0 ðxk Þ
xkþ1 ¼ xk þ ðzk xk Þ
0 6 a 6 1; f ðxk Þ ; bf ðxk Þ þ cf ðzk Þ
ð1:1Þ b; c 2 R:
We first discuss the convergence analysis of the method (1.1). Theorem 2. Assume that the function f : D R ! R for an open interval D has a simple root w 2 D. Let f(x) be sufficiently smooth in the neighborhood of the root w, then the order of convergence of the method defined by (1.1) a is at least two for bþcð1aÞ ¼ 1. Proof. The iterative scheme is given by zn ¼ x n a
f ðxn Þ ; f 0 ðxn Þ
xnþ1 ¼ xn þ ðzn xn Þ
f 0 ðxn Þ 6¼ 0;
ð1:2Þ
f ðxn Þ : bf ðxn Þ þ cf ðzn Þ
ð1:3Þ
Let w be a simple zero of f . By Taylor’s expansion, we have f ðxn Þ ¼ f 0 ðwÞ½en þ c2 e2n þ Oðe3n Þ; 0
0
f ðxn Þ ¼ f ðwÞ½1 þ 2c2 en þ
3c3 e2n
þ
ð1:4Þ Oðe3n Þ;
ð1:5Þ
where ðkÞ 1 f ðwÞ ; ck ¼ k! f 0 ðwÞ
k ¼ 2; 3; . . . ;
and en ¼ xn w:
From (1.4) and (1.5), we have f ðxn Þ ¼ en c2 e2n þ Oðe3n Þ: f 0 ðxn Þ
ð1:6Þ
A. Rafiq / Applied Mathematics and Computation 195 (2008) 819–821
821
Using (1.2) and (1.6), we have zn ¼ w þ ð1 aÞen þ ac2 e2n þ Oðe3n Þ:
ð1:7Þ
By Taylor’s series, we have f ðzn Þ ¼ f 0 ðwÞ½ð1 aÞen þ ð1 a þ a2 Þc2 e2n þ Oðe3n Þ:
ð1:8Þ
Using (1.3), (1.4) and (1.8), we get a b cð1 aÞ þ ca2 2 xnþ1 ¼ w þ 1 en þ Oðe3n Þ; en þ ac2 b þ cð1 aÞ ðb þ cð1 aÞÞ2 implies enþ1 ¼
1
a b cð1 aÞ þ ca2 2 en þ Oðe3n Þ: en þ ac2 2 b þ cð1 aÞ ðb þ cð1 aÞÞ
From the last error equation, we can easily deduce the desired results.
h
Remark 3 a 1. In [4], Ujevic´ et al. proved that for c ¼ 0; bþcð1aÞ ¼ 1 reduces to a = b and the family of methods (1.1) reduce to the classical Newton’s method. a 2. It is interesting to note that bþcð1aÞ ¼ 1 is equivalent to c ¼ ab , for which a 5 1. Thus Algorithm 1 put 1a forward by Ujevic´ et al. [4, p. 7] does not exist and is wrong. a 3. Also for a ¼ 12 ; bþcð1aÞ ¼ 1 reduces to b ¼ 12 ð1 cÞ. Now it can easily be seen that Algorithms 1 and 2 are obtained from (1.1) by taking values 12 and 13 respectively for c.
2. Conclusion The main Algorithm 1 put forward by Ujevic´ et al. [4, p. 7] is wrong. References [1] L.D. Petkovic, M.S. Petkovic, A note on some recent methods for solving nonlinear equations, Appl. Math. Comput. 185 (2007) 368– 374. [2] N. Ujevic´, A method for solving nonlinear equations, Appl. Math. Comput. 174 (2) (2006) 1416–1426. [3] N. Ujevic´, An iterative method for solving nonlinear equations, J. Comput. Appl. Math. 201 (1) (2007) 208–216. [4] Nenad Ujevic´, Goran Erceg, Ivan Lekic´, A family of methods for solving nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.03.007.