Graphics Programs Reference
In-Depth Information
7.6
Bulirsch-Stoer Method
Midpoint Method
The midpointformula of numerical integration of
y
=
F
(
x
,
y
) is
2
h
F
x
y
(
x
)
y
(
x
+
h
)
=
y
(
x
−
h
)
+
,
(7.25)
It is a second-order formula,like the modifiedEuler'sformula. We discuss ithere
because it is the basisofthepowerful
Bulirsch-Stoer method
, which is the technique
of choice in problems wherehigh accuracyis required.
y'
(
x
)
Figure
7.3.
Graphical repesentation of
the midpoint
f
(
x,y
)
formula.
h
h
x
x
x
-
h
x
+
h
Figure 7.3 illustrates the midpointformula forasingle differentialequation
y
=
f
(
x
,
y
). The change in
y
over the two panels shown is
x
+
h
y
(
x
)
dx
y
(
x
+
h
)
−
y
(
x
−
h
)
=
−
x
h
which equals the area under the
y
(
x
)curve. The midpoint method approximates this
areabythe area2
hf
(
x
,
y
) of the cross-hatchedrectangle.
H
h
Figure 7.4.
Mesh usedinthe midpoint method.
x
x
0
x
1
x
2
x
3
x
n
- 1
x
n
Considernowadvancing the solution of
y
(
x
)
=
,
=
x
0
to
x
0
+
H
with
the midpointformula. We divide the intervalofintegrationinto
n
stepsoflength
h
F
(
x
y
)from
x
=
H
/
n
each, as shown in Fig. 7.4, and carry out the computations
y
1
=
y
0
+
h
F
0
y
2
=
y
0
+
2
h
F
1
y
3
=
y
1
+
2
h
F
2
(7.26)
.
y
n
=
y
n
−
2
+
2
h
F
n
−
1
Here we used the notation
y
i
=
y
(
x
i
) and
F
i
=
F
(
x
i
,
y
i
). The first of Eqs. (7.26) uses
the Euler formula to “seed” the midpoint method; the other equations are midpoint
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