4.4 Composite Numerical Integration
Motivation: 1) on large interval, use Newton-Cotes formulas are not accurate.
2) on large interval, interpolation using high degree polynomial is unsuitable because of oscillatory nature of high degree
polynomials.
Main idea: divide integration interval into subintervals and use simple integration rule for each subinterval.
Example a) Use Simpson’s rule to approximate ∫
. b) Divide into . Use
Simpson’s rule to approximate ∫
, ∫
, ∫
and ∫
. Then approximate ∫
by adding approximations
for ∫
, ∫
, ∫
and ∫
. Compare with accurate value.
Solution:
a) ∫
( ) .
Error= | - | = 3.17143
b) ∫
∫
∫
∫
∫
( )
( )
(
)
( )
Error=| - | = 0.01807
b) is much more accurate than a).
