Download - American University of Beirut

A Class of Numerical Integration Rules With First Order
Derivatives
Mohamad Adnan AI-Alaoui"
Abstract
A novel approach to deriving a family of quadrature formulae is presented. The first member
of the new family is the corrected trapezoidal rule. The second member, a two-segment rule, is
obtained by interpolating the corrected trapezoidal rule and the Simpson one-third rule. The third
member, a three-segment rule, is obtained by interpolating the corrected trapezoidal rule and the
Simpson three-eights rule. The fourth member, a four-segment rule is obtained by interpolating
the two-segment rule with the Boole rule. The process can be carried on to generate a whole class
of integration rules by interpolating the proposed rules appropriately with the Newton-Cotes
rules to cancel Out an additional term in the Euler-MacLaurin error formula. The resulting rules
integrate correctly polynomials of degrees less or equal to n+3 if n is even and n+2 if n is odd,
where n is the number of segments of the single application rules. The proposed rules have
excellent round-off properties, close to those of the trapezoidal rule. Members of the new family
obtain with two additional fianctional evaluations the same order of errors as those obtained by
doubling the number of segments in applying the Romberg integration to Newton-Cotes rules.
Members of the proposed family are shown to be viable alternatives to Gaussian quadrature.
Key words: Numerical integration. Interpolation. Round-off error. Truncation error. Simpson's
rule. Trapezoidal rule. Boole's rule. Newton-Cotes rules. Gaussian quadrature. Romberg
integration.
* Tile author is with tile Department of Electrical and Computer Engineering, American University of Beirut,
Beirut, Lebanon. This work was supported in parl by The University Research Board of the American University
of Beirut.
25
I. Introduction
The problem of numerical integration, or quadrature, is that of estimating the number
b
I ( f ) = If(t)dt
(1)
O
with [a,b] finite, [5], [9-12], [14-16], [18-19], [22], [24], [26], [28-29].
The fundamental theorem of calculus proves that the definite integral of a function that has an
antiderivative exists and has a value equal to the difference of the values of the antiderivative
evaluated at the upper and lower limits of the integral. However, since most integrands do not
have antiderivatives expressible in terms of known functions, methods of approximating the
definite integrals are employed. There are also occasions for which the analytical form of the
integral is known but is too expensive to evaluate and it is cheaper to evaluate it using a
quadrature technique, polynomial approximation is often used, with f(t) replaced by an
approximating polynomial p(t). Among the most popular methods for approximating the
evaluation of the definite integrals are the trapezoidal rule and the Simpson rules.
To improve the approximation, the interval of integration is subdivided into smaller
subintervals, or segments, and multiple-application versions of the above rules, often called the
composite rules, are employed. Increasing the number of segments results in decreasing the error
until the round-off errors begin to dominate and the error begins to increase. In addition,
increasing the number of segments increases the computational effort. Hence, if high efficiency
and low errors are required, it is advisable to use the Romberg integration to obviate the
shortcomings of the traditional rules. The Romberg integration generalizes the Richardson's
extrapolation which consists of weighting the results obtained from using different numbers of
segments. This latter approach yields lower errors but does not necessarily achieve a higher
efficiency since the number of segments is not necessarily reduced drastically [5], [9-12], [14],
[16], [18-19], [22], [24], [26]. Members of the new family achieve both higher efficiency and
lower errors than those possible by using the multiple application Newton-Cotes rules. The roundoff properties of the proposed rules are close to those of the trapezoidal rule. In addition
polynomials of degrees less than or equal to five are integrated correctly by the two-segment and
three-segment rules while polynomials of degrees less than or equal to seven are integrated
correctly by the four-segment rule. Thus for four or less segments the members of the new class
yield error expressions that are better or equivalent to those obtained for Gaussian integration.
The new rules are competetive with the Romberg integration applied to the traditional NewtonCotes integration formulas, for with two additional functional evaluations they achieve what the
Romberg integration would achieve by doubling the number of segments. The examples show that
the new rules are competetive with Gaussian quadrature.
26
II. T h e B a s i c C o n c e p t
The author's interest in differentiators and integrators resulted in the design of analog and
digital differentiators and integrators that simulate numerical differentiation and integration [1-8].
The relationships between numerical and digital integrators was noted earlier by Hamming [1617]. In this paper, observations of the frequency responses of digital integrators are reflected in
the design of the proposed numerical integrators.
The basic concept for the development of the proposed numerical integration rules came
from observing that the ideal integrator absolute magnitude response versus frequency lies
between the responses of the trapezoidal rule and the Simpson rule [16], [6], [8]. The initial
research started with interpolating the trapezoidal and Simpson integration rules motivated by the
results in [8]. The final outcome of the research, however, is a class of integration rules that result
from the cancellation of a term in the Euler-Maclaurin error formula for each rule as compared to
the Newton-Cotes rules with the same number of segments. The first member of the class is the
corrected trapezoidal rule.
The first four members of the class are shown below for n = 1, 2, 3, 4. Where n
designates the number of segments and h = a____~).
(b Note that n is also used later to designate the
n
number of segments in the composite rules. All the following rules, including the derived rules,
have truncation errors of the form
E" = Chkf(k)(rl) + higher-order-wmls,
(2)
where C is a constant, k is an integer, and rl is in [a,h]. The rules assume that f ( t ) is
k continuously differentiable in [ a , b ] and thus are guaranteed to converge as n--+oo, where n
refers to the number of panels in a composite rule, provided that the norm of the derivatives
remain finite.
1) n = 1'
Ii((f) - (b-a------~[f(a)+ f (b)] +
2
4
h4(b-a)f(4)(q)
720
(b - a) 2
12
[ f ( l ) ( a ) - f(1)(b)]
(3)
+-higher - order" - terms.
2) n =2:
7h
16 a + b
I2 (f) = -i~[ f(a) + -~- f(--~--) + f(b)] + h2 "(') (a)- f(l) (b)]
l [t
(4)
h 6 (b - a ) f (6) ( r h )
9450
h 8 ( h --
a).f (8) (q2)
75600
+ higher- order - terms.
27
3) n=3"
3h 2
3h
13 ( f ) = ~-7[13f(a) + 2 7 f (a + h) + 2 7 f (b - h) + 13f (b)] + - ~ - [ f (1) (a) - f (1) (b)]
bU
3h 6 (b - a ) f (6) (rl)
11,200
~-higher - order - terms.
(5)
4) n = 4:
I4(f) =
2h 31
512
144 a+b
512
31
-~-[-~-f (a) +-~-i-f (a + h) +-~- f (2-~-) +-~-i-f (b- h) + :~ f (b)]
(6)
4h2 i f
+
63
(a) -
fO) (b)]
III. T h e T w o - S e g m e n t
+
12hS(b-a)f(8)(q) +higher-order-terms.
297,675
Intel~ration R u l e
The first error terms of the corrected trapezoidal integration rule and the Simpson
integration rules involve the fourth derivative and are of opposite signs. The proposed twosegment integration rule is obtained by combining the corrected trapezoidal rule with the Simpson
one-third rule. The error formulas for the resulting rule involve the sixth derivative while those
of the constituent rules involve the fourth derivative. In the following, the traditional rules and
their properties are presented in a) and b) while the proposed rule is developed in c).
a) The Corrected Trapezoidal Rule:
The simple trapezoidal rule is based on approximating f(t) of equation (1) by a straight line
(a,f(a)) and (b,f(b)). Adding the first error term to the trapezoidal rule results in the following
corrected trapezoidal rule [9], [11].
(b -- a) 2
(1
ii Or ) _ a(b )[ f ( a ) + f(b)] +
[f(l) (a) - f ) (b)],
(7)
2
12
where f(i)(13) denotes the ith derivative of f(t)evaluated at t = 13. The corrected composite
trapezoidal rule is obtained by breaking up the integral into a sum of integrals over small
subintervals and then applying the above rule to each of these smaller integrals. The resulting
corrected composite trapezoidal rule is
n-I
II Q]c) = h
Q]ci)--
(f0 + f . ) + - i - ~ - [ f ( l ) ( a ) - f ( 1 ) ( b ) ] ,
i=l
28
(8)
where n is an integer such that,
n>l,
h=
b-a
t .=a+/h
J
"
n
and
.//_. =f(tj__ )
The error of the composite corrected trapezoidal rule is [8]
E l" = IC/" ) - I[' ( f ) = f (4)(rl)h 4 (b - a) + higher - order - terms
720
(9)
for some 1"1in [a,b].
The error of the composite corrected trapezoidal rule could also be expressed by the following
asymptotic error formula [26]
n
n
h4
E l = l(:)- ]I (:)= 7~[.]~}3) ./o
f(3)] + higher- order- temps.
(10)
The above error formula may also be written as
El'
=
E: '4 + higher- order - terms.
(11)
b) T h e S i m p s o n Rule:
The Simpson one-third rule, which will be denoted simply as the Simpson rule, uses a quadratic
interpolating polynomial to approximate f(t) on [a,b] and results in
l.,.(f)=hlf(a).
+ b ' +f(h)1•
+4f~( a-~---J
(12)
where, h - (b - a)
2
h
/.'s.'(.f) = ~- [f0 + 4fl + 2../2 + 4f3 + 2 f4 +-.. +2.~,- 2 + 4.£,_ 1 + Z, 1"
The composite Simpson rule is given by
where n and k are integers such that,
b-a
n : 2 k , k_>l h , t i=a+jh
n
and
fj=f(tj).
The above equation can be written more compactly as
n/2
I; = ~ Z [f2./-2÷ 4.:2./-I+ .:2./]"
(I4)
./=1
29
(13)
The error of the composite Simpson rule is given as [6]
E~' = l ( f ) -
I~" ( / ` ) = -
h 4 (b - a ) f (4) (rl)
180
+ higher order terms,
(15)
for some ~ in [a, b].
The error could also be expressed by the following asyrnptotic error formula [9]
h4
E," = 180 [./,-(3)(b) -./-(3) (a)] + higher - order - terms.
(16)
The above equation could be expressed as
E~
=
~..h4 + higher order terms.
--s
(17)
c) The Two-Segment Rule, I~
The two-segment integration rule, I~ is obtained by combining the corrected composite
trapezoidal rule with the composite Simpson rule in such a fashion that the error contributions of
E~' and of E.~" cancel out. Note that n should be restricted to being even for meaningfial
interpolation, since n is always even for the Simpson rule.
The two-segment integration rule can be obtained as follows
n
(18)
n
I, =cd.,". + ( 1 - o 0 I , .
Solving for ot in the equation
~4
orE.. + ( I - o r ) E ,
h4
(19)
=0,
yields the value ot = 0.2, from the resulting solution.
The resulting composite two-segment rule is
ln=~k =02in=2k +0.8i,,=2k k > l .
2
•
s
I
'
(20)
--
The simple two-segment rule is obtained from the above composite rule with the value of n taken
as 2.
The error of the cornposite two-segment rule, E~ , may be expressed as
30
E 2n = 0 .2E,n + 0.8El.n
(21)
Thus for the composite two-segment rule the resulting error is obtained by adding the error
contributions of the higher order terms of the composite corrected trapezoidal rule ,[18,
p.302],[24, p. 117], to the error contributions of the higher order terms of the composite Simpson
rule [9]. The resulting asymptotic error formula is
0.2h6 [f(5)
_ f(5)
0.8h6 [f(5)
f(5)
E2- - (b)
(a)]
- (b) (a)].
(22)
1512
30,240
Simplifying the above equation and adding the contribution of the next higher term we obtain
_
En
6
h
(57
fo) (a)] 9 ~ 0 [f ( b ) -
h 8 [f(7) (b) - f(7~ (a)]
(23)
75600
This implies that the error should be reduced by a factor of 26=64, as h is halved.
An alternative form for the error is
h6 (b - a)f(6)('rll )
hS(b- a)f(8)(rh)
9450
75600
E~ =
for some rl~ 's in [a, b].
(24)
From this it is seen that E~ = 0 if fit) is a polynomial of degree _<5 . It should be noted that the
error term of equation (24) is the same as that of the Boole rule except for a constant multiplier.
The constant multiplier corresponding to the new rule is smaller than that of the Boole rule by a
factor of 20. The two-segment rule is clearly superior to its constituent rules as derived above and
as demonstrated by the following examples.
The simple form of the resulting two-segment rule is
I,(f)=7h[f(a)+16
15
a+b
h2[f(')(a)-~- f(--~--) + f(b)] +
f(')(b)],
(25)
where h = ( b - a)
2
The resulting composite rule is given by
,
n/2
7h Z
12 = 1~
[f2j-2
16
h2
+-~-.f2.i-I +f2jl+-~-[f(1)(a)-f(l)(b)]
,
j= 1
where n = 2k, k _>1,
h-
b-a
,
tj = a + . j #
and f j = f ( t j ) .
tl
Note that the simple rule is obtained from the composite new rule by using n = 2.
31
(26)
It is remarkable that this rule was derived by Cornelius Lanczos in 1956, [19] pp. 414-418, using
a different approach. The derivation presented in this paper is simpler and more direct than that of
Lanczos. Lanczos derivation is not widely known and most of the literature mention the Simpson
rule with end corrections using second order derivatives [5], [9-12], [14], [16], [18], [22], [24],
[26].
One factor that works against the use of high order Newton-Cotes formulas is that the higher
order formulas show greater fluctuation of the weights and larger round-off errors. It will be
shown that the round-off properties of the members of the proposed class are closer to the roundoff properties of the trapezoidal rule. The two-segment rule round-off properties are better than
those of the Simpson rule and close to those of the trapezoidal rule. This is to be expected since
the new rule is eighty percent trapezoidal. An estimate of the value of round-off error can be
measured by computing the sum of the square of the weights of f(t) in the integration formula
[12], [25]. The sum of the squares of the coefficients in the composite trapezoidal rule is
h2(n
; ) , while the sum of the squares of the coefficients in the composite Simpson rule is
h2(1~n
~-]=h2(1.1111n-
0.2222).Thesumofthesquaresofthecoefficientsinthe
composite new two-segment integation rule is h2(1.0044n
value corresponding to the composite trapezoidal rule.
- 0.4356), which is closer to the
IV. The Three-Segment Integration Rule
The proposed three-segment integration rule is obtained by combining the corrected trapezoidal
rule with the Simpson three-eights rule. The error formulas for the resulting new rule involve the
sixth derivative whereas those of the constituent rules involve the fourth derivative. In the
following the three-eights rule is presented in a) while the new rule is developed in b).
a) The Three-Eights Rule
The third of the Newton-Cotes rules is obtained by fitting a third order Lagrange polynomial to
four points. This rule is often called the Simpson three-eights rule and will be denoted simply as
the three-eights rule. The simple form of the rule is
13/8 (f)=3h8-~f (a)+3 f (a+h)+3 f (b-h)+ f (b)].
(27)
The composite three-eights rule is
n/3
I3/s
,, = Z
ij,; 3 + 3.~j_2
+ 3./;./_1 + ./;j],
j=l
where, n = 3k,
k _>1, that is n is restricted to be a multiple of 3,
h= b-a , t.=a+/h
n
.1
•
andf,=.l'(tj).
" ./
32
(28)
The error formula for the 3/8 rule is given by
,,
h 4 (b
E3/8 _
- a).1,.(4)(rl) + higher8O
order- terms,
(29)
for some ri in [ a, b ].
The error could also be expressed as
E~l/8 = ~3/8
L7"h4 +
higher- order- terms.
(30)
b) T h e T h r e e - S e g m e n t R u l e 13 :
The three-segment rule is obtained as follows
I~ = c d ~ / 8
(31)
+(1-a)I~.
Solving for oc from the equation
a - h 4 (b-a) f ( 4 ) (zT)+(l_ a )
h 4 (b-a)
80
720
yields the value ot = 0.1 from the solution.
The resulting asymptotic error formula
0.1h 6
E~ [f(~(b) - f(5~(a)]
3360
f ( 4 ) (V)=0 '
is
0.9h 6
_ _
[f(5)(b) - f(5)(a)]
30,240
(32)
(33)
Simplifying the above equation yields
E n _ 3h 6 [f(5)(b ) _ fcS)(a)] "
3 11,200"
- -
(34)
This implies that the error should be reduced by a factor of 36 if h is reduced to one third of its
value. An alternative form of the error is
3h 6(b - a)f (6) (rl)
E~ =
(35)
11,200
for s o m e r / i n [a,b]. From this it is seen that E~ = 0 if f(t) is a polynomial of degree less than or
equal to five.
It should be noted that the error term of equation (35) is the same as that o f the Boole rule except
for a constant multiplier. The new rule is smaller by a factor o f almost eight.
F o r n =3 the following simple form of the new rule is obtaind
qh
13 ( f ) = - " [ 1 3 f ( a )
80
"
+ 2 7 f ( a + h) + 2 7 f ( b - h) + 13 f ( b ) ] + ~'--Z--"[ f (1) (a) -.) 4"(1) (b)]. (36)
"
40 "
33
The composite form of the three-segment rule is
i3_.~j=,n
= S ''n/3 ~-~[13h3f3j_3 +27f3j_2 +27f3j_l +13f~j]+
where n is a multiple of three, h= b - a ,
n
t .=a+jh,
J
[f°)(a)-f(')(b)],
and
(37)
Jf~='f(tJ)"
Note that the simple form of the three-segment rule is obtained from the composite new rule by
letting n = 3.
The round-off properties of the three-segment rule are closer to those of the composite
trapezoidal rule, since the new rule is ninety percent trapezoidal. The sum of the squares of the
coefficients in the composite three-eights rule is h2(1.0313n
- 0.2813) while the sum of.the
squares of the coefficients of the composite three-segment rule is h 2(1.0003n - 0.4753). Thus
the three-segment rule has even better round-off properties than the two-segment rule.
V. T h e F o u r - S e g m e n t Integration Rule
The error terms of the two-segment rule and the Boole rule involve the sixth order derivative and
are of opposite signs. The four-segment integration rule is obtained by combining the twosegment rule and the Boole rule. The error formula for the resulting new rule involves the eighth
derivative. In the following the Boole rule properties and how it compares with the two-segment
and three-segment rules developed above is presented in a), while the four-segment rule is
developed in b).
a) The Boole Rule
The Boole rule is the fourth of the Newton-Cotes rules. It is obtained by fitting a fourth order
Lagrange polynomial to five points. The simple form of the rule is
2h
lB(.f) = _~517.f (a) + 32 .f (a +h) + l 2 "f (:-~--)
__a+b + 3 2 f ( b - h ) + 7 f ( b ) l .
(38)
The composite form of the Boole rule is
,,/4 2h
I~(f) = Z ~-[Tf4j_ 4 + 32f4j_ 3 + 12f4j_~ + 32f4j_ , + 7f4j],
j-|_
where n = 4k, k _> 1, that is n is restricted to be a multiple of 4,
b-a
h -- - - ,
11
tj = a +.jh and .fj = f ( t j ).
The error formula for the Boole rule, negecting higher order terms is given by
34
(39)
E~ -
2h6(b - a)
hS(b - a) f(8)
f(6) (1"1)a
(rl)
945
900
(40)
for some rl in [ a, b ].
The error could also be expressed by the following asymptotic error formula
EB-"
2h694--5
[ f ° ) ( b ) - f °) (a)] +
[f(7) ( b ) - f(7) (a)] '
(41)
The error terms of the two-segment rule and the Boole rule are of the same order. Comparing the
first error term of equation (24) with that of equation (40), it is found that the two error terms
differ only by a negative constant multiplier and that the error of the two-segment rule is smaller
in magnitude than that of the Boole rule by a factor of 20.
For the three-segment rule, comparing (35) with (40), it is found that the error of the new rule is
smaller in magnitude than that of the Boole rule by a factor of 7.9012.
b)The Four-Segment Rule, 14
The first error terms of the Boole rule, the two-segment and three-segment rules involve the sixth
derivative and are of opposite signs. A four-segment integration rule is obtained by combining the
Boole rule with the two-segment rule in such a fashion that the first error terms of equations (24)
and (40) cancel out. The derivation is similar to that carried out in equations (18)-(24). The
resulting new integration rule is given by
i~k = 20i.=4k
21 2
+
2~ I~=4k, k >
-- 1 "
(42)
The simple form of the rule is
2h [__~f
31 (a) + 5 1 2 f ( a + h ) + _ ~ _144
f ( T ) +a5+1h2 _ ~ _
i4 ( f ) = _4_~
31
•
(43)
4h2 - 0 )
+
[.] (a)-f(l)(b)],
63
where h - ( b - a)
4
35
The resulting composite rule is given by
n/4
2h 31
512 _
144 _
I~'((f)= 2 - z "~'[-~-f4j-4+ -~-.]4.i-3
+ -~-.]4j-2+
j=l
31
-2-i .t4 -1 + 5-A j]
(44)
4h 2
+ - 63
- [ f (1)(a)
wheren=4k,k>l,h---
512 _
(b)],
b-a
,
tj : a +.jh and f j = f ( t j ) .
tl
The resulting error formula is obtained from substituting equations (20) and (36) in (38), which
yields
n
g 4 =
12h8 (b -
a)f~(n)
(45)
297675
The error expression in (45) is of the same form as that of the six segment Newton-Cotes rule, but
smaller by a factor of ahnost 160.
The round-off properties of the four-segrnent rule are closer to the trapezoidal rule since it is
about ninety five percent a two-segment rule which in turn is ninety percent trapezoidal. The sum
of the squares of the coefficients of the Boole rule is h 2 ( 1 . 1 7 9 3 n
- 0."1936), while the sum of
the squares of the coefficients of the composite four-segment rule is h 2 ( 1 . 0 0 7 0 n
VI. Comparison
Quadrature
With
The
Romberg
Integration
and
- 0.47.18).
The
Gaussian
In comparison with the Romberg integration, it is found that the proposed rules, with two
additional functional evaluations, achieve error expressions of the same order as those achieved by
doubling the number of segments, ahnost doubling the number of functional evaluations, using the
Romberg integration.
The Newton-Cotes Integration rules are exact for polynomials of degree n + 1, if n is even and
for polynomials of degree 11 when n is odd, where n is the number of segments of the single
application rules. The rules of the new family are exact for polynomials of degree n +3 if n is
even and for polynomials of degree n +2 if n is odd. The Gaussian integration rules are exact for
polynomials of degrees < 2 n - 1, where n is the number of the nodes. The two-segment rule, i.e.
three nodes, is exact for polynomials of degrees < 5 which makes it equivalent to the Gaussian
36
rules with 3 nodes. The new four-segment rule, i.e. five nodes, is exact for polynomials of degree
_<7 which makes it equivalent to the Gaussian rules with four nodes.
Gaussian quadrature formulas of high order suffer even more than the Newton-Cotes formulas
from having high order derivatives in the error terms. Additionally, the sum of the squares of the
coefficients of the Gaussian quadrature formulas of high order is greater than that corresponding
to the composite new rules. This is because the higher formulas have the tendency to greater
fluctuation of the weights as the order increases [11],[25]. Thus, it is advisable to use composite
rules using Gaussian quadrature formulas of low order. That is, we can break up the interval into
subintervals and use a Gaussian formula in each subinterval. Note that we do not get the
advantage of having some of the abscissa common to two subintervals as in the case of the
Newton-Cotes formulas. In this case the new rules with their higher accuracy provide viable
alternatives to Gaussian quadrature. In the examples the proposed rules are indeed shown to be
competetive with the Gaussian quadrature formulas of high order.
VII. Examples
The computations were carried out using Mathcad 5.0 on an IBM-PC compatible 486-DX2
running at 66 Mhz. The machine epsilon is 2.77E-17. Mathcad has a maximum of 15 significant
digits, and all the computations were carried out using 15 significant digits.
Examples 1-3 verify the theoretical expectations of the proposed rules. Tables 1-3 summarize the
results. They show the results and the relative errors obtained from applying the Boole rule, the
Gauss-Legendre quadrature, the two-segment rule and the four-segment rule. It is to be noted
that n represents the number of nodes for the Gauss-Legendre quadrature and the number of
segments for the other rules. The results of the application of the three-segment rule is not shown
since they are similar to those of the two-segment rule. However, the three-segment rule should
be considered when low round-off error is desired, since its round-offerror properties are close to
those of the trapezoidal rule. The tables also display the Relative Error ,E n ,for each of the rules
where
Relative Error = True Value - Computed Value
True value
(46)
Example 1. This example is used by Smith [27]. In it the integrand is f(x)=cosx, and is integrated
over the interval [0, 1]. Thus the integral to be evaluated is
I = [lcos xdx
(47)
It is significant that with 64 segments the relative error is less than that obtained by using the
traditional trapezoidal and Simpson rules with 1000 segments or even 10,000 segments as
reported by Smith. For this example the proposed rules are clearly superior to the Gaussian
quadrature.
37
Example 2 . The integral to be evaluated is
1.4
I=• - -
1
a-4 l + x 2
dx
(48)
The value of the above integral is I = 2arctan(4) ~ 2.651635327336065.
Table 2 shows that the new rules converge in a manner similar to that of the Gaussian quadrature
with a slight edge for the proposed rules for small n.
Example 3. This example is used extensively by Atkinson [9]. In it the integrand is f(x)=eXcosx
and is integrated over the interval [0, 7t]. Thus the integral to be evaluated is
I =
e cos(x)dx.
(49)
(e" + 1)
The true value of I is I - - - - ~ - 1 2 . 0 7 0 3 4 6 3 1 6 3 8 9 6 3 .
Table 3 shows that although the
2
Gaussian integration is better for small n. The proposed rules give better results for large n.
VIII. Conclusion
This paper presents a novel class of numerical integration rules. The first member of the
class is the corrected trapezoidal rule. The second and third members of the class, a two-segment
and three-segment rules, are obtained by interpolating the corrected trapezoidal rule and the
Simpson rules so that their error O(h 4 ) terms cancel out. The resulting rules have an O(h 6) errors
and the errors are proportional to the sixth derivative. The fourth member of the new class is
obtained by interpolating the second member of the new class with the Boole rule so that their
O(h6)error terms cancel out and the new rule has an error O(h 8) and is proportional to the
eighth derivative. The process can be carried on to generate a whole class of new integration rules
by interpolating the new rules appropriately with the Newton-Cotes rules to cancel out an
additional term in the Euler-MacLaurin error formula.
The salient points of the proposed rules are:
1. The proposed rules, like the Newton-Cotes rules, are equal segment rules so they can be
applied where Gaussian rules would be inappropriate.
2. The proposed rules were shown to have excellent round-off error properties, close to those of
the trapezoidal rule. This make them viable alternatives to Gaussian quadrature, as demonstrated
by the examples. This is due in part to the fact that Gaussian quadrature formulas of high order
suffer from having high order derivatives in the error ten-ns even more than the Newton-Cotes
rules [9], [25].
38
3. The proposed rules obtain with two additional functional evaluations the same order of errors
as those obtained by doubling the number of segments in applying the Romberg integration to the
Newton-Cotes rules.
4. Applying the Romberg integration to the proposed rules could make them more competetive
with the Gaussian quadrature.
5. The proposed rules cannot be applied when the integrand first derivative has singularities at the
upper or lower limits. However, an approximation of the derivative may be used. Polya has
proved that for continuous functions with singularities in derivatives, the tapezoidal and Simpson
rules and others of similar types should converge to the correct integral [23]. Polya's remarks are
applicable to the new rules since they are obtained by interpolating the traditional rules. As an
alternative Davis shows, by a generalization of the Peano kernel formulation, that the traditional
integrals converge, and thus so do the new rules [12]. Lyness and Ninham show that for
integrands with algebraic and/or logarithmic singularities, it is often possible to obtain an
asymptotic error expansion [20].
6. The proposed rules exhibit rapid convergence for periodic integrands, this is due to the fact that
they are close to the trapeziodal rule in their properties. This was verified for the integral
I=
emS(X)dx and the results were omitted for brevity. Donaldson and Elliot had
~0
demonstrated the superiority of the trapezoidal rule for periodic integrands [ 13].
Acknowledgment
It is a pleasure to thank Professors Thomas Kailath and C. L. Nikias for providing the
atmosphere conducive to research by inviting me to spend the summers of 1991 and 1993 at
Stanford's Information Systems Lab. and the USC Signal and Image Processing Institute,
respectively. Thanks are also due to S. Maad, D. Matar and M. Shmaitelly for their help in
running the examples.
39
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40
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41
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