2,3Department of Mathematics. Sungkyunkwan University. Suwon, 440-746, KOREA. Abstract: In this paper, we derive a modified family of iteration functions.
International Journal of Pure and Applied Mathematics ————————————————————————– Volume 54 No. 1 2009, 19-26
A MODIFIED FAMILY OF LAGUERRE ITERATION FUNCTIONS OF QUARTICALLY CONVERGENCE Yoon Mee Ham1 , Sang-Gu Lee2 § , Duk-Sun Kim3 1 Department
of Mathematics Kyonggi University Suwon, 443-760, KOREA 2,3 Department of Mathematics Sungkyunkwan University Suwon, 440-746, KOREA Abstract: In this paper, we derive a modified family of iteration functions for finding simple zeros of analytic functions. The family includes, Traub’s quartic square root method and, as a limiting cases, the Kiss method, the Halley method and the Newton method. We also present a family of thirdderivative-free variants of this method. All the methods of the family are locally and quartically convergence for a simple zero. The asymptotic error constants for this methods of the family and numerical examples are given to show the performance of presented methods. AMS Subject Classification: 65-01, 65B99, 65H05 Key Words: Laguerre’s method, iterative methods, order of convergence
1. Introduction Let f (z) be an analytic function in a region. For any real number ν 6= 0, 1, define the Laguerre iteration function as (see [3, 6, 10]): νf (z) p . (1) zˆ = z − ′ f (z) + sgn(ν − 1) (ν − 1)2 f ′ (z)2 − ν(ν − 1)f (z) f ′′ (z) When f (z) is a polynomial of degree n, this is the Laguerre method for ν = n [1, 7]. It is well known that for every ν 6= 0, 1, iteration of (1) converges with order 3 in the neighborhood of a simple zero of f (see [6]). Received:
April 14, 2009
§ Correspondence
author
c 2009 Academic Publications
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Y.M. Ham, S.-G. Lee, D.-S. Kim
We derive a modified one parameter family of iteration functions for finding simple zeros of analytic functions and prove locally quartically convergence in Section 2. We give free from third-derivatives-free variants of this family of iteration functions prove that the method of the family has fourth-order convergence in Section 3. In the last section, we test the Laguerre method by a numerical experiment.
2. Derivation of the Modified Family Including Third-Derivative We shall use the method in [3, 9] toQderive a modified one parameter family of iteration functions. Let f (z) = ni=1 (z − ξi ), n ≥ 4. By computing the derivation of log |f (z)|, we have n f ′ (z) X 1 = , F1 (z) = f (z) z − ξi i=1 n f ′ (z) ′ X 1 = , F2 (z) = − (2) f (z) (z − ξi )2 i=1 n 1 1 f ′ (z) ′′ X = . F3 (z) = 2 f (z) (z − ξi )3 i=1
Let ξn be a zero to be determined and let z be an approximation to ξn . Define 1 Pn−1 1 1 α = z−ξn and β = n−1 i=1 z−ξi . Then (2) becomes Since β =
F1 (z)−α n−1 ,
Fk (z) = αk + (n − 1)β k , k = 1, 2, 3.
(3)
(n − 1)2 F3 (z) = (n − 1)2 α3 + (F1 (z) − α)3
which implies that
n(n − 2)α3 + 3F1 (z)α2 − 3F1 (z)2 α + F1 (z)3 − (n − 1)2 F3 (z) = 0. (4) Pn−1 1 3 If i=1 ( z−ξi − β) is sufficiently small when all other zeros are distinct from ξn , then by [8] − n(n + 1)α3 + 3(n + 1)F1 (z) α2 + 3[(n − 1)F2 (z) − 6F1 (z)2 ] α
+ (n − 1)2 F3 (z) − 3(n − 1)F1 (z) F2 (z) + 2F1 (z)3 = 0. (5)
Eliminating α3 in (4) and (5), it yields the quadratic equation (n + 1)F1 (z) α2 + (n − 2)F2 (z) − 3F1 (z)2 α
+ F1 (z)3 − (n − 2)F1 (z) F2 (z) − (n − 1)F3 (z) = 0. (6)
A MODIFIED FAMILY OF LAGUERRE ITERATION...
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Solving for ξn in (6), then we have ξn = z −
3F1
(z)2
where
2(n + 1)F1 (z) √ , − (n − 2)F2 (z) ± Rn
(7)
Rn = (3F1 (z)2 − (n − 2)F2 (z))2
− 4(n + 1)F1 (z)(F1 (z)3 − (n − 2)F1 (z) F2 (z) − (n − 1)F3 (z)).
We choose the sign in (7) to be maximize the absolute value of denominator of ξn as z → ξn . Since ξn is a simple zero of f (z), we have F2 (ξn ) F3 (ξn ) = 1, =1 F1 (ξn )2 F1 (ξn )3 and thus Rn → (5 − n)2 + 8(n + 1)(n − 2) = (3(n − 1))2 as z → ξn . Since |3 − (n − 2) − 3(n − 1)| ≤ |3 − (n − 2) + 3(n − 1)| = 2(n + 1), n ≥ 1
and the equality holds only for n = 1, (7) becomes 2(n + 1)F1 (z) √ ξn = z − 2 3F1 (z) − (n − 2)F2 (z) + Rn for n 6= ±1.
(8)
We apply this method to an analytic function f (z) with a simple zero at f (j) (z) ξ. Define u(z) = u = ff′(z) (z) and Aj (z) = Aj = j! f ′ (z) , j = 2, 3. Let ν be a real parameter with ν 6= ±1. We define a modified one parameter family of iteration functions by 2(ν + 1)u √ ψν (z) = z − , (9) 3 − (ν − 2)(1 − 2A2 u) + sgn(ν − 1) Rν where Rν = 9(ν −1)2 −12(ν −1)(2ν −1)A2 u+4(ν −2)2 (A2 u)2 +12(ν 2 −1)A3 u2 . (10)
Theorem 2.1. Suppose that ξ is a simple zero of f (z). Let f (z) = (z − ξ)g(z) be an analytic function with g(ξ) 6= 0. Then for ν 6= ±1, ψν (z) in (9) with (10) converges locally and quartically to ξ, i.e., 5ν − 7 g′ (ξ) g′′ (ξ) g(3) (ξ) 2(ν − 2) g′ (ξ) 3 ψν (z) − ξ − = + . lim z→ξ (z − ξ)4 3(ν − 1) g(ξ) 6(ν − 1) g(ξ)2 6 g(ξ) Proof. In view of an elementary evaluation of derivatives of ψν (z), we employ the symbolic computation of the Maple package to compute the Taylor expansion of ψν (z) around z = ξ (see [2] for details). We find after simplifying that
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Y.M. Ham, S.-G. Lee, D.-S. Kim
ψν (z) = ξ +
h 2(ν − 2) g′ (ξ) 3 3(ν − 1) g(ξ)
−
5ν − 7 g′ (ξ) g′′ (ξ) g(3) (ξ) i (z − ξ)4 + 6(ν − 1) g(ξ) g(ξ) 6 g(ξ)
+ O((z − ξ)5 ).
Example 2.1. When ν = 2 in (9), 2u 1 + 1 − 4A2 u + 4A3 u2 which is Traub’s quartic square root method [11]. ψ2 (z) = z −
√
Example 2.2. Rationalizing (9), we have √ u((5 − ν) + 2(ν − 2)A2 u − sgn(ν − 1) Rν . (11) ψν (z) = z − 2[(4 − 2ν) + (5ν − 7)A2 u − 3(ν − 1)A3 u2 ] From (10), R−1 = 36(1 − A2 u)2 and thus if Re(1 − A2 u) > 0, taking ν → −1 in (11), we obtain u(1 − A2 u) zˆ = z − 1 − 2A2 u + A3 u2 which is the Kiss method [4] of order 4. Example 2.3. For ν = 1 in (9), R1/2 = (2A2 u)2 . If Re(A2 u) > 0, letting ν → 1+0 in (11), we obtain zˆ = z − u
(12)
1−0
which is the Newton’s method. Taking a limit ν → in (11), we obtain ′ ff u = z − ′2 (13) zˆ = z − 1 − A2 u f − f f ′′ /2 which is the Halley’s method, see [5]. If Re(A2 u) < 0, we obtain the Halley method and Newton method by letting ν → 1+0 and ν → 1−0 , respectively.
3. Derivation of the Modified Family Free from Third-Derivative Let w = z −
f (z) f ′ (z) .
We consider Taylor expansion of f (w) about z
1 1 f (w) ≃ f (z) + f ′ (z)(w − z) + f ′′ (z)(w − z)2 + f ′′′ (z)(w − z)3 2 3! 1 f ′′′ (z)f (z)3 1 f ′′ (z)f (z)2 which implies f (w) ≃ 2 f ′ (z)2 − 3! f ′ (z)3 . Therefore, we have A3 u2 ≃ A2 u −
f (w) . f (z)
(14)
(15)
A MODIFIED FAMILY OF LAGUERRE ITERATION...
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Substituting (15) into (10), we obtain ψb ν (z) = z − where
2(ν + 1)u
q , bν 3 − (ν − 2)(1 − 2A2 u) + sgn(ν − 1) R
bν = R bν (z) = 3(ν − 1) − 2(ν − 2)A2 u R and w = z −
2
− 12(ν 2 − 1)
f (w) f (z)
(16)
(17)
f (z) f ′ (z) .
Theorem 3.1. Suppose that ξ is a simple zero of f (z). Let f (z) = (z − ξ)g(z) be an analytic function with g(ξ) 6= 0. Let f (z) = (z − ξ) g(z) with g(ξ) 6= 0. Then for ν 6= ±1, ψbν (z) in (16) with (17) converges locally and quartically to ξ, i.e., 5ν − 7 g′ (ξ) g′′ (ξ) 2(ν − 2) g′ (ξ) 3 ψbν (z) − ξ − = . lim z→ξ (z − ξ)4 3(ν − 1) g(ξ) 6(ν − 1) g(ξ)2 Proof. Let ξ be a simple zero of f . Consider the iteration function ψbν (z) in (16), where bν (z) = 9(ν − 1)2 − 12(ν − 1)(ν − 2)A2 u + 4(ν − 2)2 (A2 u)2 R
− 12(ν 2 − 1)
and w(z) = z −
f (w(z)) f (z)
(18)
f (z) f ′ (z) .
In view of an elementary evaluation of derivatives of ψbν (z), we employ the symbolic computation of the Maple package to compute the Taylor expansion of ψbν (z) around z = ξ (see [2] for details). We find after simplifying that h 2(ν − 2) g′ (ξ) 3 5ν − 7 g′ (ξ) g′′ (ξ) i b ψν (z) = ξ + (z − ξ)4 − 3(ν − 1) g(ξ) 6(ν − 1) g(ξ) g(ξ) + O((z − ξ)5 ).
4. Numerical Examples All computations were doing g++ compiler(gcc version 2.95.2) with MPPACK [12] with 60 significant digits. We accept an approximate solution rather than the exact root, depending on the precision (ǫ) of the computer. We use the following stopping criteria for computer programs: (i) |zn+1 − zn |
