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A remark on He’s variational iteration technique for solving nonlinear equations
Journal of Computational and Applied Mathematics 233 (2009) 1187–1189
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Letter to the editor
A remark on He’s variational iteration technique for solving nonlinear equations M.S. Petković Faculty of Electronic Engineering, Department of Mathematics, University of Niš, 18000 Niš, Serbia
article
abstract
info
Article history: Received 2 August 2009
The variational iteration method (VIM) can be usefully applied for solving many linear and nonlinear scientific and engineering problems. In this note we show that He’s approach for solving nonlinear equations, arising from the VIM, is, actually, Schröder’s method presented in his classical work from 1870. © 2009 Elsevier B.V. All rights reserved.
MSC: 65H05 35A15 Keywords: Variational iteration method Nonlinear equations Schröder basic sequence
In 2007 two special issues of Journal of Computational and Applied Mathematics (Vol. 207) and Computers and Mathematics with Applications (Vol. 54) were dedicated to variational iteration methods (VIM) and their applications. A lot of results concerned with VIM can be found in the papers cited in [1]. One of the applications of VIM was in solving nonlinear equations, presented in this journal by He [2]. In this short note we restrict our attention to this particular application and show that the root-solver arising from VIM presented by He is, actually, Schröder’s method developed in his classical work [3] from 1870. Let f : [a, b ] → R be a real function with a simple real zero α isolated on an interval D ⊂ [a, b]. Very recently, following He’s works [2,4], some authors have applied He’s variational iteration method for constructing root-finding methods. He’s approach is based on the use of a Lagrange multiplier λ and the optimality principle. Starting from an iteration function of the form S2 (x) = x + λf (x),
(1)
and solving the equation = 0 in λ, one obtains λ = −1/f (x). In this way, the following iteration function, actually Newton’s iteration, is obtained: dS2 dx
0
f (x) S 2 ( x) = x − 0 . f (x)
(2)
Furthermore, employing the iteration function in the form f (x) S3 (x) = S2 (x) + λf (x)2 = x − 0 + λf (x)2 f (x) and finding λ = −f 00 (x)/(2f 0 (x)3 ) from the equation method)
dS3 dx
= 0, the iteration method of the third order (known as Chebyshev’s
E-mail address: [email protected]. 0377-0427/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.08.107
1188
M.S. Petković / Journal of Computational and Applied Mathematics 233 (2009) 1187–1189
f (x)2 f 00 (x) f (x) − S3 (x) = x − 0 f (x) 2f 0 (x)3
(3)
is obtained, and so on; see [2, p. 5] and [4]. However, the origin of the construction presented for generalized iteration functions dates back to 1870 when Schröder [3] published two general classes of root-finding methods. Schröder used a function multiplier ϕ(x) instead of a constant multiplier λ and the following convergence theorem. Theorem 1. Let α be a simple zero of f . Let ψ be an iteration function such that ψ (p) is continuous in a neighborhood of α . Then ψ is of order p if
ψ (j) (α) = 0
ψ(α) = α,
(j = 1, 2, . . . , p − 1),
ψ (p) (α) 6= 0.
(4)
Starting from the relation
ψ2 (x) = x − f (x)ϕ1 (x),
(5)
similar to (1), in order to obtain a quadratically convergent iteration function Schröder used the condition ψ2 (x) = 0 (see (4)) and found 0
0 = 1 − (f 0 (x)ϕ1 (x) + f (x)ϕ10 (x)), and therefrom
ϕ1 (x) =
1
when f (x) = 0.
f 0 (x)
(6)
f (x)
Hence, ψ2 (x) = x − f 0 (x) , which is equivalent to (2). Remark 1. Ignoring the restriction x = α (see (6)) and taking ϕ1 (x) = λg (x) in (5), where g is an auxiliary function, one obtains the following ‘‘general iteration’’ (a term used in [2]):
φ(x) = x −
g (x)f (x) g (x)f 0 (x) + g 0 (x)f (x)
,
derived by He [2] and exploited by other authors in the papers that followed [2]. Schröder proceeded with more general approach; he noted that the quadratic convergence will be preserved for arbitrary x, setting
ϕ1 (x) =
1 f 0 (x)
+ f (x)ϕ2 (x),
in which ϕ2 remains free. Therefore, the iteration function
ψ3 (x) = x −
f (x) f 0 ( x)
− f (x)2 ϕ2 (x)
(7)
includes all iteration methods of the second order and possibly some methods of higher order. In particular, for ϕ2 (x) ≡ 0, from (7) one obtains Newton’s method (2). From the condition ψ30 (x) = 0 it follows that ϕ2 (x) = f 00 (x)/(2f 0 (x)3 ), which gives the iteration function (3). By the same reasoning, general third-order methods can be stated by using the condition ψ30 (x) = 0 but without the restriction x = α . It is sufficient to set
ϕ2 (x) =
f 00 (x) 2f 0 (x)3
+ f (x)ϕ3 (x).
Then the general method with cubic convergence has the form
ψ3 (x) = x −
f (x) f 0 ( x)
−
f (x)2 f 00 (x) 2f 0 (x)3
− f (x)3 ϕ3 (x).
In particular, for ϕ3 (x) ≡ 0, we obtain the third-order Chebyshev method (3). Schröder continued with this procedure and constructed the so-called basic sequence, or Schröder’s method of the first kind (see [3]) Sn (x) = x +
n−1 X f (x)r −1 (r ) (−1)r (f ) (f (x)), r! r =1
(8)
M.S. Petković / Journal of Computational and Applied Mathematics 233 (2009) 1187–1189
1189
where f −1 is the inverse of f . The order of convergence of the method (8) is n (n ≥ 2). Note that Schröder’s sequence {Sn } can be simply generated using Traub’s difference–differential equation (see Lemma 5-3 in [5]) Sk+1 (x) = Sk (x) −
u(x) 0 Sk (x), k
S2 (x) = x − u(x),
f (x) u(x) = 0 , (k ≥ 2). f (x)
We observe that He’s variational iteration technique for solving nonlinear equations is, actually, a special case of dψk (x) Schröder’s method. In essence, both methods are generated according to the optimality principle dx = 0, which is closely connected to the conditions (4) of Theorem 1. Both procedures, He’s method which uses a constant multiplier and Schröder’s method which handles a function multiplier, can be easily programmed in the programming package Mathematica to give the sequence of Schröder’s iteration methods of the first kind, as follows:
kg=6; g=f[x]; g1=D[g,x]; S=x; For[k=0, k
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Letter to the editor
A remark on He’s variational iteration technique for solving nonlinear equations M.S. Petković Faculty of Electronic Engineering, Department of Mathematics, University of Niš, 18000 Niš, Serbia
article
abstract
info
Article history: Received 2 August 2009
The variational iteration method (VIM) can be usefully applied for solving many linear and nonlinear scientific and engineering problems. In this note we show that He’s approach for solving nonlinear equations, arising from the VIM, is, actually, Schröder’s method presented in his classical work from 1870. © 2009 Elsevier B.V. All rights reserved.
MSC: 65H05 35A15 Keywords: Variational iteration method Nonlinear equations Schröder basic sequence
In 2007 two special issues of Journal of Computational and Applied Mathematics (Vol. 207) and Computers and Mathematics with Applications (Vol. 54) were dedicated to variational iteration methods (VIM) and their applications. A lot of results concerned with VIM can be found in the papers cited in [1]. One of the applications of VIM was in solving nonlinear equations, presented in this journal by He [2]. In this short note we restrict our attention to this particular application and show that the root-solver arising from VIM presented by He is, actually, Schröder’s method developed in his classical work [3] from 1870. Let f : [a, b ] → R be a real function with a simple real zero α isolated on an interval D ⊂ [a, b]. Very recently, following He’s works [2,4], some authors have applied He’s variational iteration method for constructing root-finding methods. He’s approach is based on the use of a Lagrange multiplier λ and the optimality principle. Starting from an iteration function of the form S2 (x) = x + λf (x),
(1)
and solving the equation = 0 in λ, one obtains λ = −1/f (x). In this way, the following iteration function, actually Newton’s iteration, is obtained: dS2 dx
0
f (x) S 2 ( x) = x − 0 . f (x)
(2)
Furthermore, employing the iteration function in the form f (x) S3 (x) = S2 (x) + λf (x)2 = x − 0 + λf (x)2 f (x) and finding λ = −f 00 (x)/(2f 0 (x)3 ) from the equation method)
dS3 dx
= 0, the iteration method of the third order (known as Chebyshev’s
E-mail address: [email protected]. 0377-0427/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.08.107
1188
M.S. Petković / Journal of Computational and Applied Mathematics 233 (2009) 1187–1189
f (x)2 f 00 (x) f (x) − S3 (x) = x − 0 f (x) 2f 0 (x)3
(3)
is obtained, and so on; see [2, p. 5] and [4]. However, the origin of the construction presented for generalized iteration functions dates back to 1870 when Schröder [3] published two general classes of root-finding methods. Schröder used a function multiplier ϕ(x) instead of a constant multiplier λ and the following convergence theorem. Theorem 1. Let α be a simple zero of f . Let ψ be an iteration function such that ψ (p) is continuous in a neighborhood of α . Then ψ is of order p if
ψ (j) (α) = 0
ψ(α) = α,
(j = 1, 2, . . . , p − 1),
ψ (p) (α) 6= 0.
(4)
Starting from the relation
ψ2 (x) = x − f (x)ϕ1 (x),
(5)
similar to (1), in order to obtain a quadratically convergent iteration function Schröder used the condition ψ2 (x) = 0 (see (4)) and found 0
0 = 1 − (f 0 (x)ϕ1 (x) + f (x)ϕ10 (x)), and therefrom
ϕ1 (x) =
1
when f (x) = 0.
f 0 (x)
(6)
f (x)
Hence, ψ2 (x) = x − f 0 (x) , which is equivalent to (2). Remark 1. Ignoring the restriction x = α (see (6)) and taking ϕ1 (x) = λg (x) in (5), where g is an auxiliary function, one obtains the following ‘‘general iteration’’ (a term used in [2]):
φ(x) = x −
g (x)f (x) g (x)f 0 (x) + g 0 (x)f (x)
,
derived by He [2] and exploited by other authors in the papers that followed [2]. Schröder proceeded with more general approach; he noted that the quadratic convergence will be preserved for arbitrary x, setting
ϕ1 (x) =
1 f 0 (x)
+ f (x)ϕ2 (x),
in which ϕ2 remains free. Therefore, the iteration function
ψ3 (x) = x −
f (x) f 0 ( x)
− f (x)2 ϕ2 (x)
(7)
includes all iteration methods of the second order and possibly some methods of higher order. In particular, for ϕ2 (x) ≡ 0, from (7) one obtains Newton’s method (2). From the condition ψ30 (x) = 0 it follows that ϕ2 (x) = f 00 (x)/(2f 0 (x)3 ), which gives the iteration function (3). By the same reasoning, general third-order methods can be stated by using the condition ψ30 (x) = 0 but without the restriction x = α . It is sufficient to set
ϕ2 (x) =
f 00 (x) 2f 0 (x)3
+ f (x)ϕ3 (x).
Then the general method with cubic convergence has the form
ψ3 (x) = x −
f (x) f 0 ( x)
−
f (x)2 f 00 (x) 2f 0 (x)3
− f (x)3 ϕ3 (x).
In particular, for ϕ3 (x) ≡ 0, we obtain the third-order Chebyshev method (3). Schröder continued with this procedure and constructed the so-called basic sequence, or Schröder’s method of the first kind (see [3]) Sn (x) = x +
n−1 X f (x)r −1 (r ) (−1)r (f ) (f (x)), r! r =1
(8)
M.S. Petković / Journal of Computational and Applied Mathematics 233 (2009) 1187–1189
1189
where f −1 is the inverse of f . The order of convergence of the method (8) is n (n ≥ 2). Note that Schröder’s sequence {Sn } can be simply generated using Traub’s difference–differential equation (see Lemma 5-3 in [5]) Sk+1 (x) = Sk (x) −
u(x) 0 Sk (x), k
S2 (x) = x − u(x),
f (x) u(x) = 0 , (k ≥ 2). f (x)
We observe that He’s variational iteration technique for solving nonlinear equations is, actually, a special case of dψk (x) Schröder’s method. In essence, both methods are generated according to the optimality principle dx = 0, which is closely connected to the conditions (4) of Theorem 1. Both procedures, He’s method which uses a constant multiplier and Schröder’s method which handles a function multiplier, can be easily programmed in the programming package Mathematica to give the sequence of Schröder’s iteration methods of the first kind, as follows:
kg=6; g=f[x]; g1=D[g,x]; S=x; For[k=0, k