Graphics Programs Reference
In-Depth Information
The coefficients appearing in these formulas are not unique. The tables belowgive the
coefficients proposedbyCash and Karp
18
which areclaimed to be an improvement
overFehlberg'soriginal values.
i
A
i
B
i j
C
i
D
i
37
378
2825
27 648
1
−
−−− − −
1
5
1
5
2
−− − −
0
0
3
10
3
40
9
40
250
621
18 575
48384
3
−
−
−
3
5
3
10
9
10
6
5
125
594
13 525
55 296
4
−
−
−
11
54
5
2
70
27
35
27
277
14 336
5
1
−
−
−
0
7
8
1631
55296
175
512
575
13824
44275
110592
253
4096
512
1771
1
4
6
Table 7.1.
Cash-Karp coefficients for Runge-Kutta-Fehlberg formulas
The solutionis advancedwith the fifth-order formulainEq. (7.19a). The fourth-
order formulais used only implicitlyinestimating the truncation error
6
E
(
h
)
=
y
5
(
x
+
h
)
−
y
4
(
x
+
h
)
=
(
C
i
−
D
i
)
K
i
(7.20)
i
=
1
Since Eq. (7.20) actually applies to the fourth-order formula, ittendstooverestimate
the errorinthe fifth-order formula.
Note that
E
(
h
) is avector, its components
E
i
(
h
) representing the errors in the
dependent variables
y
i
. This brings up the question: what is the errormeasure
e
(
h
)
that we wish to control? There is nosingle choice that works well in all problems. If
we wanttocontrol the largest componentof
E
(
h
), the errormeasure wouldbe
e
(
h
)
=
max
i
|
E
i
(
h
)
|
(7.21)
18
J.R. Cash and A.H. Carp,
ACM Transactions on Mathematical Software
16
, 201-222 (1990).
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