The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 397

6.6.1 Newton-Cotes Quadrature

Consider the one-dimensional case. With the Newton-Cotes 5 method, the n integration

points is to be evaluated are established at the onset. Usually,

n equally spaced integration points are chosen over the integration interval [ a, b ]. The
weighting factors W ( n )
i

ξ

i at which the function F

(ξ )

i

are determined by replacing F (ξ ) by a polynomial p (ξ ) obtained

from the Lagrangian interpolation, i.e.,

n

p (ξ ) = N i (ξ ) F (ξ i ) (6.112)

i = 1

with N i (ξ ) defined in Eq. (6.64) and note that p (ξ i ) = F (ξ i ) . The intention is to replace F (ξ )

with an approximating function that is relatively simple to integrate. Then use a

p (ξ ) d ξ

as an approximation to b
a

(ξ ) ξ

F d . Integration of Eq. (6.112) over the interval [ a, b ] gives

b b

n n

W ( n )
i

p (ξ ) d ξ = N i (ξ ) d ξ F (ξ ) = F (ξ ) (6.113)

i i

a a

i = 1 i = 1

Thus,

b

W ( n )
i

C ( n )
i

= N i (ξ ) d ξ = ( b − a ) (6.114)

a

where C ( n )
i

are the “weights” for the integration. Equation (6.113) is called Newton-Cotes

quadrature .
This method permits the integral of a polynomial of order n

−

1 to be evaluated exactly.

(ξ ) −

Thus, if F 1, Eq. (6.113) gives the exact result and the error

is zero. Furthermore, it can be shown that when n is odd, Eq. (6.113) permits a polynomial
of order n to be integrated exactly. If F

is a polynomial of order n

(ξ ) is not a polynomial, there will be an error R n in

using Eq. (6.113) to evaluate b
a

F (ξ ) d ξ .If n is even,

b
a n (ξ )

F ( n ) (

r )

R n = d ξ (6.115a)

n !

where (ξ ) = (ξ − ξ )(ξ − ξ ) ··· (ξ − ξ ) ,r is a point in [ a, b ], and F ( n ) ( r ) is the n th derivative

n 1 2 n

of F (ξ ) .If n is odd,

b
a ξ

F ( n ) ( r )

R n = (ξ ) d ξ (6.115b)

n

( n + 1 ) !

, the smaller the error. Table 6.6 lists the weights C ( n )
i

Usually, the higher the order of N i (ξ )

and errors for n = 2 to 6 over the integration interval [ a, b ]. Note that formulas for n = 3

and n = 5 have the same order of accuracy as the formulas for n = 4 and n = 6, respectively.

For this reason, the odd formulas with n = 3 and n = 5 are used in practice.

5 Roger Cotes (1682-1716), the son of an English minister, received his BA from Cambridge in 1702 and MA in 1706,
when he was given a professorship in astronomy and natural philosophy. For almost 4 years he assisted Newton
in the preparation of the second edition of Newton's Principia . His death of fever at 33 caused Newton to comment
“Had Cotes lived we might have known something.” Cotes wrote a paper on Newton's differential method in
which he describes how to compute the area under a curve. The modern version of this is called the Newton-Cotes
method . He proposed a technique similar to least squares for representing observed data. This preceded the efforts
of Gauss (1795) and Legendre (1806).