Graphics Programs Reference
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so that Eq. (a) becomes
c
1
y
)
h
n
∂
F
∂
F
(
h
3
) d)
y
(
x
+
h
)
=
y
(
x
)
+
(
c
0
+
c
1
)
F
(
x
,
y
)
h
+
x
ph
+
qh
y
i
F
i
(
x
,
+
O
∂
∂
i
=
1
Comparing Eqs. (c) and (d), we find that theyare identical if
1
2
1
2
c
0
+
c
1
=
1
c
1
p
=
c
1
q
=
(e)
Because Eqs. (e) representthree equations in four unknownparameters, wecanassign
any valuetooneoftheparameters.Some of the popular choices and the names
associatedwith the resulting formulas are:
c
0
=
0
c
1
=
1
p
=
1
/
2
q
=
1
/
2
ModifiedEuler's method
c
0
=
1
/
2
c
1
=
1
/
2
p
=
1
q
=
1
Heun's method
c
0
=
1
/
3
c
1
=
2
/
3
p
=
3
/
4
q
=
3
/
4 Ralston's method
All these formulas areclassifiedas second-order Runge-Kuttamethods, with nofor-
mula having a numericalsuperiority over the others.Choosing the
modified Euler's
method
, wesubstitute the corresponding parameters into Eq. (a)toyield
F
x
y
)
h
h
2
,
h
2
F
(
x
y
(
x
+
h
)
=
y
(
x
)
+
+
y
+
,
(f )
This integration formula can beconveniently evaluatedbythe following sequence of
operations
K
1
=
h
F
(
x
,
y
)
h
F
x
2
K
1
h
2
,
1
K
2
=
+
y
+
(7.9)
y
(
x
+
h
)
=
y
(
x
)
+
K
2
Second-ordermethods are seldomusedincomputer application. Most program-
mers preferintegration formulasoforder four, which achieve a givenaccuracywith
less computational effort.
y'
(
x
)
Figure 7.2.
Graphical representation of modifiedEuler
formula.
f
(
x
+
h
/2,
y
+
K
1
/2)
h
/2
h
/2
f
(
x,y
)
x
x
x
+
h
Figure 7.2 displays the graphical interpretation of modifiedEuler'sformula fora
single differentialequation
y
=
,
y
). The first of Eqs. (7.9) yields an estimate of
y
at the midpoint of the panel by Euler'sformula:
y
(
x
f
(
x
+
h
/
2)
=
y
(
x
)
+
f
(
x
,
y
)
h
/
2
=
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