Information Technology Reference
In-Depth Information
A typical choice of
n
i
is
.b a/=2
i
.If
is too small, the adaptive algorithm can
run for an unacceptable long time (or forever if
is so small that round-off errors
destroy the convergence of the integral as
n
grows). We should therefore return when
n
exceeds a prescribed large value. Algorithm
6.6
lists the pseudo code.
Algorithm 6.6
Adaptive Integration: Repeated Evaluations.
adaptive integration
(
a
,
b
,
f
,
I
,
,
n
max
)
r D I.a; b; f; n
0
/
for
n D n
1
;n
2
;n
3
;:::
s D I.a; b; f; n/
if
js rj
then
return
s
else
r D s
end if
if
n>n
max
then
print error message, return
s
end if
end for
Another strategy for developing adaptive rules is to base the integration on an
unequal spacing of the evaluation points
x
i
in the interval
Œa; b
. In the case of
the trapezoidal rule, we introduce evaluation points
x
i
,where
x
0
D a
,
x
n
D b
,
and
x
i
<x
iC1
,
i D 0; n 1
. The trapezoidal rule for unequal spacing can be
expressed as
Z
b
f.x/dx
n
X
iD1
1
2
.x
i
x
i1
/.f .x
i1
/C f.x
i
// :
(6.5)
a
This can be directly expressed in pseudo code. In case
f
is expensive to evaluate,
we could develop an optimized version where
f
is evaluated only once at every
point:
Z
b
f.x/dx
n
X
iD1
1
2
.x
i
x
i1
/.f .x
i1
/ Cf.x
i
//
(6.6)
a
DC
1
2
.x
i
x
i1
/.f .x
i1
/ C f.x
i
//C
(6.7)
1
2
.x
iC1
x
i
/.f .x
i
/C f.x
iC1
// C
(6.8)
