We can continue to generate higher-order Newton-Cotes methods, but the cost is needing to use more and more points, and increasingly complex coefficients schemes. An alternative strategy is to break the interval [a,b] into the elemental patches that we used to define the lower-order methods that we studied in section 4.3, and integrate over the patches and add up the results.
-
Composite scheme: divide the interval [a,b] into n patches
appropriate to a particular method, apply that method on the patches, then add
the results to approximate the integral
.
Composite Simpson's Rule: dividing the interval [a,b] into 2n+1
points 
, with stepsize 
we obtain n
panels: on each we use elemental Simpson's, so that we get the following 
matrix that multiplies the vector of points 
:
The estimate is then given by
The product 
is constant, so we can do that product once and
for all:
I call this the ``1-4-2-4-2-step'': just one of the classic dances that arise in numerical analysis.
Composite Trapezoidal Rule: is derived similarly, only using the weights
The estimate is then given by
(not nearly such an interesting dance...).
Theorem 4.5: Let 
, 
, and
. Then 
for which
Composite Midpoint rule:
Theorem 4.6: Let , n be even, 
, and
for 
. Then for which
Example: #12, p. 204
The derivation of the error terms is not too very complicated: for example, for Simpson's rule we make an ``elemental error'' of the form
on each panel. Hence the total error for the interval is
which can be written as
by taking 
such that 
is the mean value of the error terms:
Now, to look at this error term in a slightly different way,
is a Riemann sum for the integral
so that
so that
(Compare to Beth's clever approach on #21 of the section 3.4 homework.) Similar tricks works for the other elemental rules.
Example: #16, p. 205
Example: #20, p. 205