This is a preprint of an article published in International Review of Automatic Control (I.RE.A.CO.), Vol.3, n. 4 July 2010
Generalizing Taylor’s theorem to approximate a function in powers of another function Hamed Shah-Hosseini Faculty of Electrical and Computer Engineering, Shahid Beheshti University,G.C., Tehran, Iran, emails:
[email protected],
[email protected]
Abstract – In this paper, Taylor’s theorem is generalized in such a way that a (real-valued) function is expressed in powers of another function. This proposed generalized theorem called “G-Taylor” includes several well-known theorems in Calculus as its special cases such as the Taylor’s formula, the Mean Value Theorem, Cauchy’s Mean Value Theorem, and NewtonRaphson’s method. Some applications of the G-Taylor’s formula are introduced through several examples. Moreover, the Newton-Raphson’s method is generalized in several ways such that not only the generalized versions converge faster but also they have larger radii of convergence. Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved.
Keywords: Numerical Analysis, Calculus, Taylor’s theorem, Root finding, Newton-Raphson’s method; Mean Value Theorem
Nomenclature ≈ : The symbol for almost equal f (.) : A function with one variable
(a, b ) : An open interval excluding a and b [a, b] : A closed interval including a and b
ε : A real number f ′(.) : The first derivative of f (.)
f ′′(.) : The second derivative of f (.)
f (n ) (.) : The nth derivative of f (.) n! : The factorial of n Rn (.) : The remainder term in the Taylor’s theorem D x : The derivative operator with respect to variable x d such that D x ≡ dx Tn (.) : The Chebyshev’s polynomial of degree n g 0 : The value of function g (.) at variable x0 Ln (.) : The natural logarithm function g ′′′(.) : The third derivative of g (.) x0 : A constant real number e : The Euler number e ≈ 2.71828
The error of the approximation is also expressed by the theorem. The theorem is named after the mathematician Brook Taylor, who stated it in 1712. The Taylor’s theorem is widely used in many scientific fields for different tasks such as Estimation, Optimization, and Approximation. In the Taylor’s formula, a function is approximated in a form of polynomial with an error. In other words, a function is expressed in powers of its argument. In this paper, a function is expressed in powers of another function, which extends the original Taylor’s formula to a new level. Throughout the paper, the proposed formula is called “G-Taylor” for generalized Taylor. It is reminded that only real-valued functions are considered in this paper. The paper is organized as follows. Next section mentions briefly some theorems that are used for proving the proposed G-Taylor’s formula. Section III introduces the proposed formula with its proof and includes some examples of how to use it. Section IV deals with finding the roots of a function using the G-Taylor’s formula. Some generalizations of the Newton-Raphson’s method are also proposed. Concluding remarks forms the final section of the paper.
II. I.
Introduction
In Calculus, the Taylor's theorem [1-3] is used to approximate any differentiable function near a given point by a polynomial whose coefficients are only dependent on the derivatives of the function at that point.
Copyright © 2010 Praise Worthy Prize S.r.l. - All rights reserved
Taylor’s Formula
Taylor’s formula may be used for approximating many functions. One simple example is to approximate function
e x near x = 0 with e x ≈ 1 + x +
x 2 x3 xn + + ⋅⋅⋅ + in 2! 3! n!
H. Shah-Hosseini
which only the first n+1 terms of the Taylor’s polynomial are used. In the following, Taylor’s theorem is reminded.
f(.) and g(.) are n+1 times differentiable in the open interval (a, b); then a point like ε exists in the open interval (a, b) such that
Taylor’s Theorem. If f(.) is a function which is n times continuously differentiable in the closed interval [a, b] and n + 1 times differentiable in the open interval (a, b), then it may be written
f (b) = f (a ) + (b − a) f ′(a) + ⋅⋅⋅ +
f (b) = f (a ) + (g (b) − g (a) ) ⋅
(b − a) 2 f ′′(a ) + 2!
1 f ′ ⋅ Dx + g ′(a) g ′ x = a
(g (b) − g (a) )3 ⋅
1 1 f ′ ⋅ D x ⋅ D x + g ′(a ) g ′ g ′ x =a
3!
(1) The term Rn(ε) is the remainder term and in Lagrange’s form it is :
(b − a ) n +1 ( n +1) (2) f (ε ) (n + 1)! Where ε is a number in the open interval (a, b). It is reminded that some other forms of remainder terms have also been suggested for Taylor’s theorem [1-3]. If the remainder term Rn(ε) has small value for an n > N , then the first N-terms of the polynomial obtained from Taylor’s theorem are a good approximation for the given function. In proving the proposed theorem, both the Rolle’s theorem and the Mean Value Theorem are needed. Therefore, both of them are mentioned here.
⋅⋅⋅ +
Rolle’s Theorem. If a function f is continuous in a closed interval [a, b], differentiable in the open interval (a, b), and f (a ) = f (b) , then there is a real number ε in the open interval (a, b) such that (3)
Mean Value Theorem. Let f be a continuous function in a closed interval [a, b], differentiable in the open interval (a, b), then there is a real number ε in the open interval (a, b) such that f (b) − f (a ) f ′(ε ) = (4) b−a It is seen that the Mean Value Theorem is a generalization of Rolle's theorem in which it is assumed that f(a) = f(b).
The following theorem is a generalization of Taylor’s formula or the proposed G-Taylor’s formula. In this formula, a function f(.) is approximated by powers of another function. G-Taylor’s theorem. If functions f and g are n-times continuously differentiable in the closed interval [a, b] and g′(x) nonzero for those x in [a, b], and also functions Copyright © 2010 Praise Worthy Prize S.r.l. - All rights reserved Automatic Control, Vol. 3, N. 2
n!
1 ⋅ g ′(a)
(5) where
Rn (ε ) =
(g (b) − g (a) )n+1 ⋅ 1 ⋅ (n + 1)! g ′(ε )
1 D x ⋅ D x 1 ⋅ ⋅ ⋅ D x f ′ ⋅ ⋅ ⋅ g ′ g′ g′ x =ε
(6)
in which n-terms of Dx are used in Rn(ε). Proof. According to the assumptions, g(b)≠g(a) because if it is supposed that g(b) = g(a), then the Mean Value Theorem states that if a function is continuous in [a, b] and differentiable in (a, b), there exists a point like ε in g (b) − g (a ) (a, b) such that g ′(ε ) = . But, g(b) = g(a). b−a Then, it is concluded that g ′(ε ) = 0 . This is a contradiction because in the assumptions g ′( x) ≠ 0 in [a, b], thus it demonstrates that g(b)≠g(a). Now, function B(x) is defined as follows:
B( x) = − f (b) + ( g (b) − g ( x) ) ⋅
(g (b) − g ( x) )n n!
III. The Proposed G-Taylor’s Formula
(g (b) − g (a))n ⋅
1 D x ⋅ D x 1 ⋅ ⋅ ⋅ D x f ′ ⋅ ⋅ ⋅ + Rn (ε ) g′ g′ g ′ x=a
Rn (ε ) =
f ′(ε ) = 0
(g (b) − g (a) )2 ⋅ 2!
(b − a) n ( n ) f (a ) + Rn (ε ) n!
f ′(a ) + g ′(a )
+
⋅
f ′( x) + ⋅⋅⋅ + g ′( x)
1 1 1 f ′ ⋅⋅⋅ ⋅ D x ⋅ D x ⋅ ⋅ ⋅ D x ′ ′ ′ g ( x) g g ′ g
(g (b) − g ( x) )n+1 ⋅ A n (n + 1)! (7)
The value of An should be determined to complete the conditions of the Rolle’s theorem. According to the assumptions of the proposed theorem, it is concluded that the function B(x) is continuous in the interval [a, b] and differentiable in the interval (a, b). It is
This is a preprint of an article published in International Review of
H. Shah-Hosseini
also easy to see that B(b)=0. By setting B(a)=0, we obtain : (n + 1)! ⋅ An = (g (b) − g (a))n+1
f ′(a ) − + − ⋅ + f ( b ) ( g ( b ) g ( a ) ) g ′(a ) ( g (b) − g (a) )n 1 ⋅ ⋅ ⋅ ⋅ ⋅ + n! g ′( a) D 1 ⋅ D 1 ⋅ ⋅ ⋅ D f ′ ⋅ ⋅ ⋅ x x g ′ x g ′ g ′ x =a
(8)
(g (b) − g (ε )) n!
f ′ + ⋅⋅⋅ + g ′ x =ε
⋅ f ( n ) (a ) +
(b − a )n+1 ⋅ f ( n+1) (a), (n + 1)!
a
