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CS205 – Class 14
Covered in class: 1, 2
Readings: 8.1 to 8.3
1.
Numerical quadrature
approximate
()
b
a
I
fxd
x
=
∫
for a given
f
a.
These
f
’s might be arbitrarily difficult to compute and only available by running a
program.
b.
General approach : Subdivide
[,]
ab
into n intervals
1
[, ]
ii
x
x
+
with
0
x
a
=
and
n
x
b
=
and
consider each subinterval separately
2.
NewtonCotes quadrature
for each subinterval
1
x
x
+
, choose
n
equally spaced points and
use k1 degree polynomial interpolation to approximate the integral
a.
Exact on polynomials of degree q=k1 when k is even, as expected
b.
Exact on polynomials of degree q=k when k is odd, from symmetric cancellation
i.
c.
local accuracy
an exact method on q degree polynomials has a local error that scales
like
2
k
Oh
+
in each subinterval where
h
is the length of the subinterval
d.
global accuracy
since there are
( )
1
subintervals, the total error scales like
1
k
+
i.
doubling the number of subintervals, sends
2
hh
→
, and reduces the error by
(½)
q+1
ii.
order of accuracy is q+1
e.
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 Fall '07
 Fedkiw

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