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Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 4372 – 4376 www.elsevier.com/locate/procedia
Advanced in Control Engineering and Information Science
The covered operation of grey logarithmic function Qiao-Xing Li a, Jun-Fang Wub a* b
a School of Management, Lanzhou University, No.222 Tianshui South Road, Lanzhou, 730000, China Department of Physical Education, Lanzhou University, No.222 Tianshui South Road, Lanzhou, 730000, China
Abstract Grey function is an important part of grey mathematics and the core is the covered operation. On the basis of operational rules of grey number, we proposed the computable formula of the function-covered set of grey logarithmic function. An example is given to illustrate the operational effectiveness.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011] Open access under CC BY-NC-ND license.
Keywords: grey mathematics; grey logarithmic function; covered operation; function-covered set; grey system theory; system engineering
1. Introduction Grey system theory has been an effective tool to control the uncertain system with poor information (see [1,2] etc.). Because the complex system often induce poor information, we can not get the true value of data. However, we can obtain its boundary by using the correct investigation methods. In grey system theory, the true value of data is unknown, then the data is called grey number and its boundary is the number-covered set (see [2,3] etc.). Since the theory was proposed, it has been applied in many fields such as control engineering and computer science (see [4,5,6], etc.). Grey mathematics is the mathematical foundation of grey system theory. The papers [2,3] have discussed the covered operation of grey number. Paper [7] gave the general definition of grey function and proposed grey basic elementary function. By using the operational rules of grey number, papers [2,8] proposed grey matrix and gave the
* Corresponding author. Tel.: +86-731-8912466; fax: +86-731-8910402. E-mail address:
[email protected];
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.821
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computable formulas of the function-covered set of inverse of grey matrix. However, the computable formula of grey logarithmic function has not been obtained yet. In this paper, we give the computable formula by using the operational rules of grey number, and the result shows that the function-covered set of grey logarithmic function can be obtained by finite computational steps. Then we can forecast or control the development of objective things which approximately obey the rule of logarithmic function even if the information is poor. The rest of the paper is organized as follows. We introduce some mathematical results in section 2. The computable formula of grey logarithmic function is proposed and an example shows the feasibility of our method in sections 3. At last, we have a conclusion. 2. Preliminaries In this section, we introduce some mathematical results which can be seen in many textbooks. Lemma 2.1 Supposing that
∞
∑u n =0
lim (un +1 / un ) = ω < 1 holds, then
n → +∞
n
is a positive series, i.e., un ≥ 0 ∞
∑u n =0
n
(n = 1,2,...) , and the equation
is convergent and we get the equation lim un = 0 .
Lemma 2.2 Supposing that the power series
n → +∞
∞
∑ a (x − x ) n=0
n
0
n
satisfies lim an +1 / an = n → +∞
ρ,
where
0 < ρ < ∞ , then the series is convergent within the interval x − x0 < 1 / ρ . Lemma 2.3 Supposing that the function f (x) exists continuous derivative of higher order from 1 to n+1 within the neighborhood δ ( x0 ) of the point x0 , then ∀x0 ∈ δ ( x0 ) , there at least exists one point ξ = x0 + θ ( x − x0 ) , where θ ∈ (0,1) , and we get Taylor Formula below: f ( n ) ( x0 ) f ( n +1) (ξ ) ( x − x0 ) n + ( x − x0 ) n +1 . n! (n + 1)! Lemma 2.4 Supposing that the function f (x) exists continuous derivative with any order within the interval ( x0 − r , x0 + r ) , where r is a real number, then for all x ∈ ( x0 − r , x0 + r ) , we have f ( x) = f ( x0 ) + f ' ( x0 )( x − x0 ) + ... +
f ( n ) ( x0 ) f ( x) = f ( x0 ) + f ' ( x0 )( x − x0 ) + ... + ( x − x0 ) n + ... n! ( n +1) f (ξ ) iff lim Rn ( x ) = lim ( x − x0 ) n +1 = 0 , where ξ = x0 + θ ( x − x0 ) and θ ∈ (0,1) . n→∞ n → ∞ ( n + 1)! Lemma 2.5 Supposing that c is a positive real number, then the equation lim (cn / n!) = 0 holds. n →∞
y = log a x is the set of real number (−∞,+∞) , where a > 0 and a ≠ 1 . The function is monotonously decreasing when 0 < a < 1 and monotonously increasing when a > 1 . The logarithmic function can be denoted by the series and we state them below. Apparently, the domain of logarithmic function
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Lemma 2.6 Supposing that that
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y = log a x is the logarithmic function, where a > 0 and a ≠ 1 , and
x0 > 0 is a real number, then for all real number x which satisfies x − x0 < x0 , y = log a x can
be denoted by the following Taylor series:
log a x =
ln x0 1 (−1) n −1 + ( x − x0 ) + ... + ( x − x0 ) n + ... . n ln a x0 ln a n( x0 ) ln a
3. The covered operation of grey logarithmic function In reality, some objective things can be modelled by logarithmic function. However, we only utilized the determinative form before. Because of the cognitive limitation of human being and the complexity of objective things, we can not get the true value of parameter, then the model is uncertain. By using the correct investigation methods, we can obtain the range of parameter when the information correctly reflects the objective thing. In this situation, the parameter is a grey number and the range is the numbercovered set (see [2,3], etc.). Paper [7] only gave the definition of grey elementary function but the form of function-covered set is not computable. On the basis of the covered operation of grey number, we will propose the computable formula of the function-covered set of grey logarithmic function in this section. Theorem Supposing that a (⊗) and y (⊗) = log a ( ⊗) x are respectively grey number and grey
a(⊗) , where [a] ∈ (0,1) or [a] ∈ (1,+∞) , and that x0 > 0 is a real number, then for all x ∈ (0,2 x0 ) and any number ε > 0 , there must exist an integer N and the function-covered set [ y] = log[ a] x of y (⊗) is as follows:
logarithmic function, and that [a ] = [ a1 , a2 ] is the number-covered set of
ln x0 1 (−1) N −1 [ y] = + ( x − x0 ) + ... + ( x − x0 ) N + [−ε ,+ε ] . N ln[a] x0 ln[a] N ( x0 ) ln[a]
Proof: Supposing that
n
n
n
n
n y01 = min{[(−1) n−1 (n − 1)!] /[ x0 ln a1 ], [(−1) n−1 (n − 1)!] /[ x0 ln a2 ]}
and
n y02 = max{[(−1) n−1 (n − 1)!] /[ x0 ln a1 ], [(−1) n−1 (n − 1)!] /[ x0 ln a2 ]}
then from [7], we get grey derivative y ( n)
y (⊗) |x = x0 = [(−1)
and
n −1
( n)
(n = 1,2,...) , (n)
(⊗) | x = x0 and its number-covered set [ y ] |x = x0 below:
−n
(n − 1)! x0 ] / ln a(⊗) −n
n n [ y ( n ) ] |x = x0 = [(−1) n −1 (n − 1)! x0 ] /[ln a] = [ y01 , y02 ].
From Lemma 2.6, we get grey Taylor series of y (⊗) = log a ( ⊗ ) x below:
ln x0 (−1) n −1 ( x − x0 ) n x − x0 + + ... + + ... , y (⊗) = n ln a(⊗) x0 ln a(⊗) nx0 ln a(⊗) and the function-covered set [ y ] is as follows: ln x0 (−1) n −1 ( x − x0 ) n x − x0 [ y] = + + ... + + ... n ln[a] x0 ln[a ] nx0 ln[a]
(1).
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From Lemma 2.4, we get
x − x0 n +1 (−1) n ( ) =0, n →∞ n → ∞ ( n + 1) ln a (⊗) ξ where ξ = x0 + θ ( x − x0 ) and θ ∈ (0,1) . lim Rn ( x)(⊗) = lim
Supposing that Rn ' ( x)(⊗) = [(−1)
n x −x R'n +1 ( x)(⊗) = × 0 R'n ( x)(⊗) n +1 x0
n −1
n
( x − x0 ) n ] /[nx0 ln a (⊗)] , then we get
(n = 1,2,...) .
Supposing that a ∈ [ a1 , a2 ] is the only potential true number of a (⊗) , then we have o
R'on ( x) =
(−1) n −1 ( x − x0 ) n R'on +1 ( x) x − x0 and < < 1, n R'on ( x) x0 nx0 ln a o
where R 'n ( x ) is the only potential true number of R'n ( x )(⊗) ( n = 1,2,...) . o
On the other hand, for all
ε > 0 , there exists an integer N below:
N = int( x0 /[ε ( x0 − x − x0 ) ln a ]) ,
where ln a = min{ ln a1 , ln a2 } and int( x ) is an integer number that is not larger than x, then
R'on ( x) − 0 = ( x − x0 / x0 ) n /[n ln a o ] < 1 /(n ln a) < [ε ( x0 − x − x0 )] / x0 holds true when n>N, and we have
R'on ( x) + R'on +1 ( x) + ... ≤ R'on ( x) × [1 + ( x − x0 / x0 ) + ...) = x0 R'on ( x) /( x0 − x − x0 ) ≤ ε , so the theorem holds from Theorem 2.6 of paper [6] and Eqn.(1). By using the theorem above, we can get the function-covered set
[ y ] of y (⊗) = log a ( ⊗ ) x when N
is obtained. However, the natural number N is dependent on the variable x, and we can not get N when x is close to 0 or 2x0. In real-world application, we also need to get the range of the variable x, for example, x ∈ [ x1 , x2 ] ⊂ (0,2 x0 ) . The following algorithm is to calculate the function-covered set of grey logarithmic function. Step 1: Get the number-covered set [a]=[a1,a2] which satisfies [ a ] ⊂ (0,1) or [a ] ⊂ (0,+∞) . Choose one point x0 and estimate the range of the variable x ∈ [ x1 , x2 ] ⊂ (0,2 x0 ) . Determine a level of accuracy
ε > 0;
Step 2: Calculate ln a = min{ ln a1 , ln a2 } and xc = min{x0 − x1 − x0 , x0 − x2 − x0 } , and let
N = int[ x0 /(εxc ln a )] ; Step 3: Get [ y ] =
x − x0 (−1) N ( x − x0 ) N +1 ln x0 + + ... + + [−ε ,+ε ] and Stop. ln[a] x0 ln[a] ( N + 1)( x0 ) N +1 ln[a]
In order to illustrate the proposed method, we give an example below. Example: Supposing that a (⊗) is a grey number and [a]=[20,23] is its number-covered set, then the steps to calculate the function-covered set of y (⊗) = log a ( ⊗ ) x are below: Step 1: Get x0=10 and
x ∈ [6,13] ⊂ (0,20) , and let ε = 0.1 ;
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Step 2: Get n=5; Step 3: The function-covered set [ y ] = log[ a ] x of log a ( ⊗ ) x is below:
ln10 ( x − 10) 2 ( x − 10)3 ( x − 10) 4 x − 10 [ y] = + − + − ln[20,23] 10 ln[20,23] 200 ln[20,23] 3000 ln[20,23] 40000 ln[20,23] +
( x − 10)5 ( x − 10) 6 − + [−0.1,0.1] 500000 ln[20,23] 60000000 ln[20,23]
= [0.7344,0.7686] + [0.0319,0.0334]( x − 10) − [0.0016,0.0017]( x − 10) 2 + [1.0631 × 10−4 ,1.1127 × 10−4 ]( x − 10)3 − [7.9732 × 10−6 ,8.3452 × 10−6 ]( x − 10) 4 + [6.3786 × 10−7 ,6.6762 × 10−7 ]( x − 10)5 − [5.3155 × 10−8 ,5.5635 × 10−8 ]( x − 10)6 + [−0.1,0.1] . 4. Conclusion We get the computable formula of the function-covered set of grey logarithmic function. The example illustrates the effectiveness of the method. The result means that the development of objective things which approximately obey the rule of logarithmic function can be forecasted or controlled even if the information is poor. Acknowledgements This work is supported by the Fundamental Research Funds for the Central Universities (No. 11LZUJBWZY056). References [1] Deng J. L.. Elements on grey theory. Wuhan: Huazhong University of Science and Technology Press; 2000, p. 61–86. [2] Li Q. X.. The foundation of the grey matrix and the grey input-output analysis. Applied Mathematical Modelling 2008;32:267–291. [3] Li Q. X. and Liu S. F.. Some results about grey mathmatics. Kybernetes 2009;38:297–305. [4] Li Q. X.. The grey departmental input-output analysis. The Journal of Grey System 2011;23:101–112. [5] Zhu J. R., Wang J. and Lei J. T.. Grey predictive control of stress on trauma section during union of fracture. Journal of Grey System 2011;23:15–24. [6] Chou J. R. and Tsai H. C.. On-line learning performance and computer anxiety measure for unemployed adult novices using a grey relation entropy method. Information Processing and Management 2009;45:200–215. [7] Li Q. X.. The grey elementary functions and their grey derived functions. The Journal of Grey System 2008;20:245–254. [8] Li Q. X.. Grey dynamic input-output analysis. Journal of Mathematical Analysis and Application 2009;359:514–526.
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