1
COMPUTER SCIENCE 349A
Handout Number 33
RICHARDSON'S EXTRAPOLATION
(Section 22.2.1)
--
the technique of combining two different numerical approximations that
depend on a parameter (usually a stepsize
h
) in order to obtain a new approximation
having a smaller truncation error.
Let
M
denote some value to be computed, for example,
′
f
(
x
0
)
or
′′
f
(
x
0
)
or
f
(
x
)
dx
.
a
b
∫
Let
N
1
(
h
)
denote a formula (that depends on a parameter
h
that can take on different
values) for computing an approximation to
M
, and suppose that the form of the truncation
error of this formula is a known infinite series in powers of
h
.
For example, the most
common case is that the truncation error is
O
(
h
2
) and is an infinite series with only even
powers of
h
, that is,
(1)
{
4
4
4
4
3
4
4
4
4
2
1
L
3
2
1
)
(
is
error
truncation
6
3
4
2
2
1
stepsize
using
ion
approximat
computed
1
value
exact
2
)
(
h
O
h
h
K
h
K
h
K
h
N
M
+
+
+
+
=
where the values
K
i
are some (possibly unknown) constants.
The parameter
h
can be
any positive value, but as
h
→
0, the truncation error
→
0 ;
that is,
N
1
(
h
)
→
M
.
If (1) holds, then using a stepsize of
h
/2
,
(2)
L
+
+
+
+
⎟
⎠
⎞
⎜
⎝
⎛
=
64
16
4
2
6
3
4
2
2
1
1
h
K
h
K
h
K
h
N
M
.
In order to obtain an
O
(
h
4
)
approximation to
M
, we need to determine a linear
combination of equations (1) and (2) in which the
O
(
h
2
) terms cancel out; this will occur
if we compute
)
1
(
)
2
(
4
−
×
,
which gives
L
−
−
−
−
⎟
⎠
⎞
⎜
⎝
⎛
=
−
16
15
4
3
)
(
2
4
4
6
3
4
2
1
1
h
K
h
K
h
N
h
N
M
M
,

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