NUMERICAL METHOD FOR
EVALUATING THE INTEGRABLE
FUNCTION ON A FINITE INTERVAL
RAJESH KUMAR SINHA
Department of Mathematics, National Institute of Technology, Patna-800005, Bihar, India
RAKESH KUMAR Dravidian University, Kuappm, Andhra Pradesh, India
Abstract:
The proposed method in this article is based on numerical method for evaluating the function on finite interval. Stress has been done on evaluation of Integrable function within a closed finite interval which provides the results based on Trapeziodal Rules, Simpson’s rule and also Newton-cote’s formula. In this paper, an alternative approach is attempted in order to discard the Taylor Polynomials as the approach follows the evaluation of derivative of function.
Keywords: Numerical method; Integrable function; Taylor series
1. Introduction
Numerical analysis is the study of algorithms that use numerical approximation for the problems of continuous mathematics. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation, modern numerical analysis does not seek exact answers, since the exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. It finds applications in all fields of engineering and the physical sciences. Ordinary differential equations appear in the movement of heavenly bodies (planets, stars and galaxies); optimization occurs in portfolio management; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In mathematics, an integrable function is a function whose integral exists.
Let f(x) be defined on finite interval (a, b), that is,
ba
dx
x
f
(
)
(1)
)
(
)
(
)
(
x
dx
F
b
F
a
f
b
a
Here an obvious approach is to replace the Integral
f
of equation (1) by approximating polynomial and integrating the Polynomial provided the Taylor polynomial is being discarded. The discard of the Taylor polynomial is being made since it requires the evaluation of derivatives of f. This means Taylor polynomial is not being considered when integration is of the function under finite interval (a, b)[1-3]. Thus this concept provides a study of inverse problem of differentiation. Using an Interpolations polynomial constructed at equally spaced pointsx
r
x
0
rh
, wheren
x
0
. Assuming, f(n+1) to be continuous on (x0, xn)From (1)
s nn n
f
n
s
h
f
n
s
f
s
f
sh
x
f
0 0 0 0 1 ( 1)
1
...
1
)
(
(2)
where
s
(
x
0,
x
n)
The paper is organized as follows section 2 covers the basic principle along with proposed method for evaluating integrable function. Section 3 covers the result and discussion.
2. Proposed Method
The general method for deriving Numerical differentiation formulae is to differentiate the Interpolating Polynomial. This formula provided a resulting corresponded to derivative forms. But the equation (2) follows with Newton’s forward difference formula; the method of derivation being the same with regard to some other formulae by replacing (x0, xn) by an appropriate larger interval when s lies outside the interval 0
s
1. Since under theassumption of given conceptions there can be constructed different Integration rule by choosing different values of n
in equation (2). With n = 0, the first derivative of f given by [4-6]
1
0
1
0 2
0
(
)
)
(
x
x
s
dx
f
s
h
hf
dx
x
f
(3)
Since there has been remark Newton’s forward difference for deriving function with difference operator
...
,
,
2
3
as first, second, third …… differences usually defined in the study of calculus of finite differences. There does not exists Newton forward differences rather other formulae such as Newton Backward differences and Sterling formula give the expressions corresponding to first, second …….. derivatives of the functions. Thus the required mathematical expressions corresponding to derivatives following with stated formulae is given by2 3
0 0 0 0
( 1) ( 1)( 2)
( ) ...
2 3
s s s s s
f x f s f f f
(4)
where x = x0 + sh
dx
ds
ds
x
df
dx
x
df
(
)
(
)
(5)
...
...
6
2
6
3
2
1
2
1
0 3 2
0 2
0
f
s
s
f
s
f
h
(6)
The above formula can be used for computing the value df x dx( ) for non- tabular values of x. For tabular values dx
...
4
1
3
1
2
1
1
)
(
0 4 0
3 0 2 0
0
f
f
f
f
h
dx
x
df
x x
Differentiating
...
24
22
36
12
6
6
6
1
)
(
0 4 2
0 3 0
2 2 2
2
f
s
s
f
s
f
h
dx
x
f
d
...
12
11
1
)
(
0 4 0
3 0 2 2 2
2
0
f
f
f
h
dx
x
f
d
x x
For computing higher derivatives may be obtained by successive differentiation. In a similar way, different formulae can be derived by starting with other interpolation formulae. From the above mathematical expression in stated formulae forward difference operator are being used in case of derivatives of different kinds under interval
x
x
0 andx
x
n. Sincex
x
0
sh
such thatdx
hds
; the required integrand on the right side of equation (3) iscontinuous function of s, where s does not change sign on (0,1) and
f
(
s)
. Using mean value theorem integral there is a numbers
s
with
s
(x
0, x
1)
such that
1
0
1
0 2
0
(
)
)
(
x
x
s
sds
f
h
hf
dx
x
f
replacing
by
s
1
0
)
(
2
1
)
(
0 2x
x
f
h
hf
dx
x
f
(7)
The first term on the right of equation (7) is the integration rule (also called a quadrature rule); the second term is the error term. Since the integral is approximated by the rectangle of width h= x1 – x0 , it is known as rectangular
quadrature rule. Putting n = 1 and integrating f(x) over (x0, x1),
1
0
1
0 3
0 0
2
)
(
2
1
)
(
x
x
ds
s
f
f
h
hf
dx
x
f
where
f
is assumed continuous on (x0, x1).Since the mean value theorem for Integrals sates that if f is continuous and
g
is Integrable on Interval (a, b) and g(x)
0 for a
x
b; there exist a number
(a, b) such that
ba b
a
dx
x
g
f
x
g
x
f
(
)
(
)
(
)
(
)
In this case
,
2
s
does not change sign on [0, 1],
)
(
12
)
(
2
)
(
3 1 0
1
0
f
h
f
f
h
dx
x
f
x
x
where
x x0, 1
and is usually distinct from the appearing in equation (7). This is thetrapezoidal rule
plus error term. The integral is approximated by 1 2h f
0 f1
.
The Trapezoidal rule is usually applied in a composite form. To estimate the integral of f over (a, b), we divide (a, b) into N sub-intervals of equal length h
ba
N. The end points of the sub-intervals are x1 a ih i, 0,1,...N, so that x0 a and xN b.Applying trapezoidal rule to each sub-intervals
[
s
i-1, x
i]
,
i
1
,
2
,...
.
N
. Distinguishing the numbers
occurring in the error terms, from (8))
(
12
)
(
2
..
...
)
(
12
)
(
2
)
(
3 1
1 3 1 0
0
N N
N x
x
f
h
f
f
h
f
h
f
f
h
dx
x
f
N
Maintaining the assumption that
f
is continuous on (a, b), combining the error terms),
(
12
)
2
...
2
2
(
2
)
(
3 1
2 1
0
f
Nh
f
f
f
f
f
h
dx
x
f
N Nb
a
for some
(a, b). This can be rewritten as).
(
12
)
(
)
(
2
)
....
(
)
(
2 0 0
1 1
0
f
h
x
x
f
f
h
f
f
f
h
dx
x
f
N N Nb
a
(9)
Since
x
N-x
0
Nh
, equation(9) reveals that the composite Trapezoidal rule consists of the composite rectangularrule plus the correction term
h f
N
f
0
2
. It can be observed that the rule is exact for every polynomial belonging to P1. Stressing on a generalization of the composite trapezoidal rule, that is, Gregory’s formula
mj
j j N
j j N
x
x
f
f
C
h
f
f
h
dx
x
f
N
0
0 1
0
...
)
(
(
1
)
),
(
)
(
0
(10)
where
0
1
)
1
(
ds
j
s
C
j j(11)
If
m
N
the rule uses values f(xi) with everyx
i
(x
0, x
N)
with m = 0, since C1=2
1
, equation(1)is simply the Trapezoidal rule.From (10), it can be stated that if m=2k or m=2k +1 then the rule equation (10) has error O(h2k+1) and integrates exactly every polynomial belonging to P2k+ 1. Thus it is desirable to use equation (10) with an even value of m. For
example, if m = 2 and N
2,
1
0
)
(
24
)
(
6
)
(
8
5
)
...
(
)
(
0 0 1 1 2 2x
x
N N
N
N
f
f
h
f
f
h
f
f
h
f
f
h
dx
x
f
Returning to equation (2), put n=3 and integrate f(x) over
(x
0, x
2)
. This time the error termcontains thebinomialcoefficient
4
s
which does change sign over
(x
0, x
2)
, However, it can be shown that, in this case the error termcan still be written in the form (using Mean Value Theorem)
2
0
) 4 ( 5 )
4 ( 5
90
4
s
ds
h
f
f
h
) 4 ( 5 2 1 0
90
)
4
(
3
)
(
2
0
f
h
f
f
f
h
dx
x
f
x
x
where
(x
0, x
2)
. This is called Simpson’s rule. Like the trapezoidal rule, Simpson’s rule is usually applied in composite form. Combining the error terms as we did for the trapezoidal rule, we obtain.)
4
2
...
4
2
4
(
3
)
(
0 1 2 3 2 2 2 1 22
0
N N
N x
x
f
f
f
f
f
f
f
h
dx
x
f
N
)
(
180
)
(
(4)4 0
2
f
h
x
x
N
where
(x
0, x
2N)
Furthermore, a remark may be given relating to study of composite form of Simpson’s rule that required even numbers of sub-intervals for which Chosen weight are applicable corresponding to values on n. by such stated conditions there is stressed in derivation of Simpson’s rule [7-9]. Since, the basis of Trapezoidal rule and Simpson’s rule in more generalized forms using weights wi, (i = 1, 2, 3, ….. n) the required integrand of the function f(x) over
the interval (x0, xn) is obtained as
n
i i x
x
ih
x
f
w
dx
x
f
n
0
0
)
(
)
(
0
(12)
where wi are chosen so that the rule is exact if f
Pn. These are called the closed Newton cotes formulae whichcontributes Simpson’s rule when different weights are evaluated in form of sub-intervals. In any Integration rule such as equation (12) the numbers wi are called the weight since the rule consist of a weighted sum of certain
function value. Now, to review on set formulae that is Newton Cotes, there arise the technical terms closed and open that refer to whether the end points x0 & xn are not included in the Simpson’s rule. Thus, the required form of Open
Newton Cotes may be obtain in the following expression is given by
n
i i x
x
ih
x
f
w
dx
x
f
n
0
0
)
(
~
)
(
0
To obtain the simplest Newton-Cotes rule we integrate equation (3) with n = 1 over the interval
(
x
0- h,x
0
h
)
such that the required expression for given integral is written as
1
1 3 0
2
2
)
(
0
ds
f
s
h
hf
dx
x
f
sh x
h x
As in the derivation of Simpson’s rule, the error term here is double some, since
2
s
change sign on [-1, 1], as the mean value theorem is not applicable here. However, again it is possible to write the error in the form
s
ds
h
f
f
h
3 1
1 3
3
1
2
Applying this rule to f on the interval [x0, x1],
f
h
h
x
hf
dx
x
f
x
x
(
)
1
2
24
3 0
1
0
(13)
where
(
x
0, x
1)
, this is known as mid-point rule [10-11]. Obviously the mid point rule is found more economical than the Trapezoidal rule on interval (x0, x1). Since both have accuracy of order h3 but equation (14)requires only one evaluation of r instead of two evaluation for the trapezoidal rule. This is misleading since the composite form (14) is given by
10
2 0
24
)
(
2
1
)
(
0
N
r
N r
x
x
f
h
x
x
h
x
f
h
dx
x
f
N
(14)
where
(x0, xN)Here N is required for evaluation of
f
that the function f(x) which only one fewer that the N + 1 evaluation required by the composite trapezoidal rule as stated above.3. Result and Discussion
Each Quadrature rule where the concept of weights exists, integration of the function over the interval (xn, xn+1) is
done where
x
n is defined asx
n
x
0
nh
. This relation holds for ordinary differential equation though it is conveniently use the backward difference form of interpolating polynomial constructed asx
x
n,
x
n-1,...,
x
n-m, where m
n. thus for s > 0, there is existing a mathematical relation used with backward difference operators
in general form given by
mj
s m m
m n
j j
n
f
m
s
h
f
j
s
sh
x
f
0
1 1
1
1
)
1
(
)
1
(
)
(
(15) Now, we are interested to integrate the equation (15) over the interval (
x
n,x
n1) such that the required integral value is obtained in terms of backward difference operator
following the condition of the change of the variablex
x
n
sh
, such thatdx
hds
then the required integral changes to
1
0
1
0
1 2
1
0
1
)
1
(
)
1
(
)
(
1
ds
f
m
s
h
ds
j
s
f
h
dx
x
f
m m m sm
j
n j j x
x n
n
ApplyingMean Value of theorem,
n mm m m
j
n j j x
x
f
b
h
f
b
h
dx
x
f
n
n
) 1 ( 1 2 0
1
)
(
Where
n
(
x
nm,
x
n1)
and
1
0
)
1
(
ds
j
s
b
j jIf it is sometimes appropriate to make use of f(xn+1) also and construct the interpolation polynomial at
m n n
n
x
x
x
x
1,
,...
...
-
10
2 2
2 1
1
2
)
1
(
)
1
(
)
(
m
j
s m m
m n
j j
n
f
m
s
h
f
j
s
sh
x
f
This represents equation (14) with n and m increased by 1. In this case,
x
n
x
x
n1 corresponds to-
1
s
0
. ApplyingMean Value of theorem,
1
0
) 2 ( 2 3 1
1
)
(
m
j
n m m m n
j j x
x
f
C
h
f
C
h
dx
x
f
n
n
where
n
(
x
nm,
x
n1)
and Cj as reported in [11] are the same numbers occuring in Gregory’s formula.4. Conclusion
The midpoint rule, Trapezoidal rule and Simpson’s rule is to introduce the technique and analysis of quadrature methods, Composite Simpson’s rule is easy to use and produces accurate approximation, unless the function oscillate is subinterval of finite. To minimize the number of nodes which maintain accuracy, study of Gaussian quadrature, Romberg integration was introduced to take advantage of the easily applied composite Trapezoidal rule and extrapolation
References
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[7] Bradi, Brain, (2009): A Friedly Introduction to Numerical Analysis, pp. 441-532, Pearson Education.
[8] Eliott, D.A. (1963): Chebyshev Series for the Numerical Solution of Fredholm integral equation, computer journal, Vol. 6, pp. 102. [9] Atkinson, K.E. (1978): An Introduction to Numerical Analysis, John Wiley & Sons.
[10] Beard, R.E. (1947). : Some notes on approximate product integration, J. Inst. Actur., 73, pp. 356-416.
[11] Baker, C.T.H. (1968): On the nature of certain quadrature formulas and their errors, SIAM. J. Numerical analysis, vol. 5, pp. 783-804.
Rajesh Kumar Sinha completed his Ph.D. in Application of Numerical Analysis and Postgraduate Programme in Mathematics & Computer Science. Currently, he is working as Assistant Professor in the Department of Mathematics, National Institute of Technology, Patna. In his total 17 years of teaching experience, he guided five M.Phil and two Ph.D students. His current research area includes Numerical Method, Fuzzy Mathematics and Graph Theory and Algorithms.
