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Keywords: Iterative method, Newton's method, order of convergence, discrete modification. 1 Introduction. In [1] and ... In Section 2 by using a new quadrature approach several iterative formulas are received. In Section ..... Cliffs, New Jersey. 8.
On a few Iterative Methods for Solving Nonlinear Equations Gyurhan Nedzhibov Laboratory of Mathematical Modelling, Shumen University, Shumen 9712, Bulgaria e-mail: [email protected]

Abstract In this study an unpopular method of quadrature formulas for receiving iterative methods for solving nonlinear equations is applied. It is proved for the presented iterative methods that the order of convergence is equal to two or three. The executed comparative numerical experiments show the efficiency of the presented methods. Keywords: Iterative method, Newton’s method, order of convergence, discrete modification.

1

Introduction

In [1] and [2] a method of quadrature formulas for solution of nonlinear equations is proposed. Here we continue this approach to receive new classes of iterative processes on the same problem. In Section 2 by using a new quadrature approach several iterative formulas are received. In Section 3 the discrete modifications of some of these formulas are given. Analysis of convergence is made in Section 4. Geometrical interpretation is included in Section 5. The study finishes with some numerical experiments (Section 6) and conclusion (Section 7).

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Several classes of iterative formulas obtained by using quadrature formulas

We will bring out some classes of iterative processes for solving the equation: f (x) = 0. Let consider the identity:

Z

xk+1

(1) 0

f (η)dη.

f (xk+1 ) = f (xk ) +

(2)

xk

In this identity we put the left side f (xk+1 ) = 0 (i.e. suppose that xk+1 is root of equation (1)) and approximate the integral with different quadrature formulae, which correspond to different iterative functions. 2.1. First family of iterative functions Let approximate and substitute the integral in (2) with the quadrature formula: Z xk+1 m−1 0 xk+1 − xk X 0 (xk+1 − xk ) f (η)dη = . f (δi ), where δi = xk + i m m xk i=0 Further on we replace xk+1 − xk = −uk which participate in δi , where uk = equation (2) we get: 0 = f (xk ) +

m−1 xk+1 − xk X 0 f (δi ), where m i=0

1

δi = xk − i

f (xk ) . f 0 (xk )

uk . m

Then from the

Finally we express xk+1 and obtain the iterative formula: mf (xk ) f (x ) ³ ´ , uk = 0 k . (i−1) f (xk ) xk − m uk i=1 f

xk+1 = xk − P m

(3)

0

This is a class of iterative processes(depends from m ∈ N ).It is easy to verify that when m = 1 from (3) Newton’s iterative algorithm is obtained. When m = 2 we obtain the formula: xk+1 = xk −

2f (xk ) ¡ ¢. f (xk ) + f 0 xk − 12 uk

(3.1)

0

2.2.Second family of iterative functions Now we shall use the quadrature formula: Z

xk+1

xk

µ ¶ m xk+1 − xk X 0 δi−1 + δi i f , where δi = xk + uk . f (η)dη = m 2 m i=1 0

By analogy such as in 2.1. we obtain the following class of iterative formulae: f (x ) mf (xk ) ³ ´ , uk = 0 k (2i−1) f (xk ) xk − 2m uk i=1 f

xk+1 = xk − P m In particular we have:

xk+1 = xk − and xk+1 = xk −

(4)

0

f

0

¡

f (xk ) ¢ , f orm = 1; xk − 12 uk

(4.1)

2f (xk ) ¡ ¢ ¡ ¢ , f orm = 2 1 f xk − 4 uk + f 0 xk − 34 uk

(4.2)

0

2.3.Third family of iterative functions Let substitute the integral in (2) with the quadrature formula of the trapeziums: Ã ! Z xk+1 m−1 X 0 0 0 0 xk+1 − xk (xk+1 − xk ) f (η)dη = f (xk ) + 2 f (δi ) + f (xk+1 ) , δi = xk + i . 2m m xk i=1 In this case we obtain the class of iterative processes: xk+1 = xk −

0

f (xk ) + 2

Pm−1

For m = 1 we have: xk+1 = xk −

i=1

2mf (xk ) . ¡ ¢ f 0 xk − mi uk + f 0 (xk − uk )

2f (xk ) . f 0 (xk ) + f 0 (xk − uk )

(5)

(5.1)

This formula by is proposed by S.Weerakoon and T.G.I.Fernando in [1] . When m = 2 the iterative algorithm is: xk+1 = xk −

af (xk ) ¡ ¢ , where f 0 (xk ) + bf 0 xk − 12 uk + f 0 (xk − uk )

a=4

and b = 2.

(5.2)

A similar iterative formula by V.I.Hasanov, I.G.Ivanov and G.Nedzhibov for a = 6 and b = 4 is observed in [2]. This one can be obtained from equation (2) if one use Simpson’s quadrature formula. In [2] the third order of convergence for this case is proved. The iterative processes (3.2) and (4.1) also are investigated by J.F.T raub in [4].

2

3

Discrete modifications

To reduce the number of the computations of derivatives one may use some corresponding discrete modifications. Here we will get the discrete modifications of the formulas (3.1), (4.2), (5.1). 3.1.Discrete modifications of the formula (3.1) After the substitution of the denominator µ ¶ µ ¶¶ µ 0 0 1 1 1 f (xk ) + f xk − uk with f (xk ) + f xk − uk 2 uk 2 we receive the following discrete modification: xk+1 = xk −

uk f (xk ) ¢ , where ¡ f (xk ) − f xk − 12 uk

uk = u(xk ) =

f (xk ) f 0 (xk )

(3.1.1)

3.2.Discrete modifications of the formula (4.2) By analogical way as in 3.2. we obtain the discrete modification of iterative formula (4.2): xk+1 = xk −

¡

2 f (xk −

uk f (xk ) ¡ 1 4 uk ) − f xk



3 4 uk

¢¢ , where

uk = u(xk ) =

f (xk ) f 0 (xk )

(4.2.1)

3.3.Discrete modifications of the formulas (5.1) and (5.2) The discrete modifications of the formulas (5.1) and (5.2) is the same formula: xk+1 = xk −

4

uk f (xk ) f (xk ) , where uk = u(xk ) = 0 f (xk ) − f (xk − uk ) f (xk )

(5.1.1)

Analysis of convergence

Theorem 1 Let f be a real function. Assume that f (x) has first, second and third derivatives in the interval (a, b). If f (x) has a simple root an α ∈ (a, b) and x0 is sufficiently close to α, then the class of iterative methods (3) has second order of convergence and the classes (4), (5) of iterative methods has third order of convergence. Proof: a).First we shall discuss the convergence of the known Newton’s iterative formula. We denote: ϕ1 (x) = x − ff0(x) = x − u; δ1 = ϕ1 (x) − α and ε = x − α, then (x) 0

δ1 = ε −

0 0 f (x) εf (x) − f (x) A1 = = , where A1 = εf (x) − f (x), B1 = f (x) f 0 (x) f 0 (x) B1 0

Let α be a simple root of f (x), i.e. f (α) = 0, f (α) 6= 0. The Taylor expansions gives: 0

f (x) = f (α) + εf (α) + 0

Then A1 = εf (x) − f (x) =

ε2 (2) (α) 2 f

ε2 (2) (α) 2 f

+ O(ε3 )

and

0

0

f (x) = f (α) + εf (2) (α) + O(ε2 ). 0

+ O(ε3 ), and (B1 )−1 = (f (α))−1 + O(ε). Therefore

δ1 = C2 ε2 + O(ε3 ) where

C2 =

f (2) (α) . 2f 0 (α)

This equation establishes the second order of convergence. b). Order of convergence of the class of iterative formulas (3). Denote: A2 mf (x) εS − mf (x) ¡ ¢ and δ2 = ϕ2 (x) − α = 2 = , 0 i−1 S2 B2 x− m u i=1 f ¢ Pm 0 ¡ where S2 = i=1 f x − i−1 m u . Again using the Taylor expansion we get: ³¡ ¢ ¡ ¢ (2) ¢2 ´ 0 0 ¡ i−1 i−1 u = f (α) + ε − u f (α) + O ε − u f x − i−1 m m m ³¡ ¡ m+1−i ¢ (2) ¢2 ´ 0 2 , = f (α) + ε + O(ε ) f (α) + O ε − i−1 m m u ϕ2 (x) = x − Pm

3

(6)

from (6) we have u = ε − C2 ε2 , hence S2 =

m X

µ f

0

i=1

Therefore A2 =

f (2) (α) 2 ε 2

+ O(ε3 )

¶ 0 i−1 m + 1 (2) x− u = mf (α) + f (α) + O(ε2 ). m 2 0

for (B2 )−1 = (mf (α))−1 + O(ε), hence δ2 =

C2 2 ε + O(ε3 ). m

(7)

This equation establishes the second order of convergence. c). Observe the class of iterative formulas (4). Denote: mf (x) εS − mf (x) A3 ¢ , δ3 = ϕ3 (x) − α = 3 ¡ = , 0 2i−1 S B u f x − 3 3 i=1 2m

ϕ3 (x) = x − Pm where S3 = f

0

¡

Pm

x−

i=1

f

0

¢

2i−1 2m u

¡ x−

¢

2i−1 2m u

. We use Taylor expansion for:

¡ 0 = f (α) + ε − 0

= f (α) +

¢ (2) ¡ ¢2 f (3) (α) 2i−1 2i−1 (α) + + O(ε3 ) 2m u f 2 µ ε − 2m u ³ ´2 (3) ¶ 2(m−i)+1 (2) 2(m−1)+1 f (α) 2i−1 (2) f (α)ε + C f (α) + ε2 , 2 2m 2m 2m 2

from (6) we have u = ε − C2 ε2 , hence à ¶2 ! m µ 0 f (2) (α) f (2) (α) f (3) (α) X 2(m − i) + 1 ε2 + O(ε3 ). S3 = mf (α) + m ε+ m C2 + 2 2 2 2m i=1 0

We use (B3 )−1 = (mf (α))−1 + O(ε), hence by not difficult computation we obtain: Ã Ã !! m f (3) (α) 3 X 2 2 ε3 + O(ε4 ); C3 = δ3 = C2 + C3 (2(m − i) + 1) − 1 . 3 4m i=1 3!f 0 (α)

(8)

This equation establishes the third order of convergence. d). Observe the class of iterative formulas (5), denote: 2mf (x) εS4 − 2mf (x) A4 and δ4 = ϕ4 (x) − α = = , S4 S4 B4 ¢ Pm−1 0 ¡ 0 0 where S2 = f (α) + i=1 f x − 2i−1 2m u + f (x − u). We use Taylor expansion for: õ µ ¶ µ ¶ µ ¶2 (3) ¶3 ! 0 0 i i i f (α) i (2) Ti = f x − u = f (α) + ε − u f (α) + ε − u . +O ε− u m m m 2 m ϕ4 (x) = x −

Substitute in (3) Pm−1 0 0 S4 = f (α) + εf (2) (α) + ε2 f ³2(α) + 2 i=1 Ti + f (α) − u)f (2) (α) =´´ ... ³ + (εP (3) 0 m−1 ¡ m−i ¢2 f (α) (2) (2) 1 + 2 i=1 ε2 + O(ε3 ). =2mf (α) + mεf (α) + mC2 f (α) + 2 m 0

We use that (B4 )−1 = (2mf (α))−1 + O(ε), finally we obtain: Ã Ã ! ! m−1 3 X (2m − 3) 2 2 δ4 = C2 + C3 − 1 ε3 + O(ε4 ). (m − i) − m3 i=1 2m This equation establishes the third order of convergence. The theorem is proved.

4

(9)

Corollary 2 If the condition of the Theorem 1 is held then the iterative formulae (4.1) we have: δ=

¶ µ C3 ε3 + O(ε4 ), where C22 − 4

C3 =

f (3) (α) . 3!f 0 (α)

Proof:We use Theorem 1 in case m = 1 at equation (8). Corollary 3 If the condition of the Theorem 1 is held then the iterative formulae (5.1) we have: δ=

µ ¶ 1 C22 + C3 ε3 + O(ε4 ), where 2

C3 =

f (3) (α) . 3!f 0 (α)

Proof:We use Theorem 1 in case m = 1 at equation (9). Theorem 4 If the condition of the Theorem 1 is held then the iterative formulae (5.1.1) has third order of convergence. Proof: Denote ϕ5 (x) = x − u

µ ¶ f (x) f (x − u) =x−u 1+ , (f (x) − f (x − u)) (f (x) − f (x − u))

δ5 = ε − ϕ5 (x).

Using Taylor expansion we get: 0

0

f (x − u) = (x − u − α)f (α) + O((x − u − α)2 ) = C2 ε2 f (α) + O(ε4 ) = 0

and f (x) − f (x − u) = εf (α) + O(ε2 ). Therefore

f (x−u) f (x)−f (x−u)

ε2 (2) f (α) + O(ε4 ) 2

= C2 ε + O(ε3 ). From u = ε − C2 ε2 obtain

δ5 = ε − u(1 + C2 ε) = C2 ε2 − (ε − C2 ε2 )C2 ε + O(ε4 ) = C22 ε3 + O(ε4 ). This equation establishes the third order of convergence at iterative process (5.1.1). The theorem is proved.

5

Geometrical interpretation.

This paragraph gives geometrical interpretation of some of the above iterative processes. For the classes of iterative formulas (3),(4) and (5) can be said that each next approximation xk+1 of the root α is obtained as an intersection point of Ox ∩ l, where straight line l passes through the point (xk − uk , f (xk − uk )) and tan(l, Ox) is equals to corresponding sum of the value of the derivatives of the function(in denominator). We have denoted x∗k+1 an intersection point of Ox ∩ t where the straight line t is the tangent in point (xk , f (xk )) (in F ig.1 and F ig.2). 5.1.Iterative functions (4.2.1) At iterative process (4.2.1) each next approximation xk+1 is an intersection point of Ox ∩ l1 , where l1 is a straight line through the point (xk − uk , f (xk − uk )) and parallel to the straight line through the points (xk − 14 uk , f (xk − 14 uk )) and (xk − 43 uk , f (xk − 34 uk ))(1) the point xk+1 (in F ig.2). 5.2.Iterative functions (5.1) At iterative algorithm (5.1) each next approximation xk+1 is an intersection point of Ox ∩ l2 , where l2 is a straight line through the point (xk − uk , f (xk − uk )) and parallel to the tangent in point (xk − 12 uk , f (xk − 12 uk ))− the point xk+1 ( in F ig.1). 5.3.Iterative functions (5.1.1) The iterative process (5.1.1) known by the name Newton-Secant method is observed in [4] it determines each next approximation xk+1 as an intersection point of Ox ∩ l3 , where the line l3 is through the points (xk , f (xk )) and (xk − uk , f (xk − uk )) - the point xk+1 (in F ig.2).

5

6

Numerical experiments

We have done numerical experiments for different functions and initial points. All programs were written in Matlab. We compare five iterative procedures for computing the root of nonlinear equations. Table 1. Function f (x) 1) (x − 1)

x0

P20

i i=0 x

2) (x − 2)23 − 1

3) 4 +

P4 i=1

4) exp(x) − 1 +

5)

f 1)

2)

3)

4)

5)

p

x2i

P5 i=0

x2i+1

(x − 4)2 + 2 − x3 − 9

MN 1.8e-09 1.3e-12 4.6e-15 1.4e-09 5.2e-10 5.1e-14 3.9e-13 4.5e-10 7.8e-09 1.7e-09 -1.3e-07 8.6e-10 -2.6e-09 -1.6e-09 5.0e-09 1.4e-14 0 4.9e-10

2 2.5 3 10 3.5 4.5 5 -1 2 3 4 -6 2 3 6 -1 -3 -9

MN 18 23 27 51 13 25 29 99 10 14 15 19 13 17 24 5 6 8

V1 13 16 18 35 9 17 20 ND 10 12 12 14 11 13 18 3 4 6

Accuracy(f (xn )) V1 V2 V3 0 0 1.6e-09 0 0 5.4e-12 5.5e-12 5.1e-14 5.5e-09 9.3e-15 0 6.0e-09 1.6e-13 0 1.7e-10 0 0 1.2e-10 0 2.8e-09 0 ND 0 ND -8.4e-10 1.1e-08 7.3e-09 -7.9e-10 4.4e-09 -1.2e-08 -9.2e-09 -8.1e-10 7.1e-09 -1.2e-08 -1.1e-08 2.2e-09 8.8e-10 -6.5e-09 -1.1e-09 -1.1e-08 9.8e-10 8.4e-10 -6.0e-09 -1.0e-08 7.4e-10 -5.4e-11 -3.8e-12 -7.7e-13 0 -1.7e-15 0 -1.7e-15 1.8e-09 3.0e-10

iter V2 12 15 17 33 9 16 18 23 9 11 13 14 10 13 17 3 4 5

NS 0 0 4.6e-14 0 0 0 2.6e-09 1.4e-09 0 -3.3e-09 -6.3e-14 -1.7e-15 -8.9e-11 -4.2e-13 -5.2e-09 -3.8e-12 0 1.8e-09

V3 11 14 16 31 8 15 18 ND 9 10 12 14 10 13 17 3 4 5 MN 6.64 8.57 10.0 18.9 1.48 2.80 3.29 11.3 1.43 1.92 2.09 2.73 2.14 2.80 3.94 0.66 0.83 1.10

NS 12 15 17 33 9 16 18 33 6 7 9 11 7 10 14 3 4 5 V1 7.03 8.66 9.83 19.1 1.37 2.57 3.02 ND 1.86 2.31 2.25 2.75 2.48 2.85 4.01 0.54 0.70 1.09

Root α =1

=3

=1

≈ 0.90380796

≈ −1.49298702 time500 V2 8.57 10.7 12.1 23.6 1.64 2.96 3.29 4.39 2.09 2.52 3.01 3.35 2.73 3.62 4.66 0.64 0.88 1.09

V3 6.03 7.69 8.84 17.1 1.27 2.37 2.79 ND 1.70 1.93 2.31 2.80 2.25 2.91 3.84 0.60 0.76 0.94

NS 5.60 7.03 7.96 15.6 1.36 2.41 275 5.1 1.15 1.32 1.65 2.14 1.53 219 3.07 0.49 0.70 0.92

We introduce the notations: x0 - a initial point; iter - number of iterations; time500 - the execution time for 500 times execution; N D- Not defined; M N - Newton’s iterative formulae; V 1-Iterative formulae (5.1); V 2- Iterative formulae (5.2) for a = 4 and b = 4 ; V 3- Iterative formulae (4.1); N SIterative formulae (5.1.1) .

7

Conclusion

We have considered a few classes of iterative formulas and the discrete modifications for some of them. 6

In conclusion it should be said that from practical point of view the iterative algorithms (4.1) and (5.1.1) show the best results and that is shown in the numerical experiments. The reason for reduction of the execution time in these two algorithms is that the number of computation of the values of the derivative of the function is the smallest. The iterative process (5.1.1) known by the name of Newton-Secant method is a composition of Newton’s and Secant’s methods. Thus this method unites their positive qualities such as: at each step of the iteration the value of the derivative of the function is computed only once; in the beginning one initial point is chosen; the process is convergent (if the first and the second derivatives of the function do not change the signs); third order of convergence does not require computing the second or derivatives from higher order of the function. Have to be denote for the classes of iterative formulas (3),(4) and (5) that: with growing the value of m it grow the number of computation of the derivative of the function, too. Thus for higher values of m the index of efficiency is reduces sensitivity. For that reason the more interesting cases are when is small m(m < 3). In conclusion have to say that in practical view the best results shows the iterative algorithms (4.1) and (5.1.1)- that is shown in the numerical experiments. The reason for reduce the execution time by this two algorithms is the less number of computation the value of derivative of the function. The iterative process (5.1.1) known with name Newton-Secant method is composition of Newton’s and Secant methods. Thus this method unites the positive qualities of them, like: at each step of the iteration is compute only one time the value of derivatives of the function; in the beginning we chose one initial point; the process is converges(if the first and the second derivatives of the function are not change the signs); third order of convergence without require to compute the second or higher derivatives of the function.

Acknowledgment. This work was supported by the Shumen University under contract 24/9.05.2002. The author would like to thank prof. Milko Petkov for helpful suggestions.

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References [1] Weerakoon, S., Fernando, T.G.I. (2000) A Variant of Newton’s Method with Accelerated ThirdOrder Convergence, Applied Mathematics Letters, 13, 87–93. [2] Hasanov, V.I., Ivanov, I.G., Nedzhibov, G. A New Modification of Newton’s Method, In: Application of Mathematics in Engineering’27, Proc. of the XXVII Summer School Sozopol’01, pp. 278–286, Heron Press, Sofia 2002. [3] Sendov, B., Popov, V. (1976) Numerical Methods, Part I, Nauka i Izkustvo, Sofia, (in Bulgarian) [4] Traub, J.F. (1964) Iterative Methods for the Solution of Equations, Prentice Hall, Englewood Cliffs, New Jersey.

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