Owner's Manual

Appendix B: Technical Reference 942
The Bogacki-Shampine 3(2) formula provides a result of 3rd-order accuracy
and an error estimate based on an embedded 2nd-order formula. For a
problem of the form:
y' = ƒ
(x, y)
and a given step size h, the Bogacki-Shampine formula can be written:
F
1
= ƒ(x
n
, y
n
)
F
2
= ƒ (x
n
+ h
1
2
, y
n
+ h
1
2
F
1
)
F
3
= ƒ (x
n
+ h
3
4
, y
n
+ h
3
4
F
2
)
y
n+1
= y
n
+ h (
2
9
F
1
+
1
3
F
2
+
4
9
F
3
)
x
n+1
= x
n
+ h
F
4
= ƒ (x
n+1
, y
n+1
)
errest = h (
5
72
F
1
ì
1
12
F
2
ì
1
9
F
3
+
1
8
F
4
)
The error estimate errest is used to control the step size automatically. For
a thorough discussion of how this can be done, refer to Numerical Solution
of Ordinary Differential Equations by L. F. Shampine (New York: Chapman
& Hall, 1994).
The TI-89 Titanium / Voyage™ 200 software does not adjust the step size to
land on particular output points. Rather, it takes the biggest steps that it can
(based on the error tolerance diftol) and obtains results for x
n
 x  x
n+1
using the cubic interpolating polynomial passing through the point
(x
n
, y
n
)
with slope F
1
and through (x
n+1
, y
n+1
) with slope F
4
. The interpolant is
efficient and provides results throughout the step that are just as accurate
as the results at the ends of the step.
Runge-Kutta Method
For Runge-Kutta integrations of ordinary differential
equations, the TI-89 Titanium / Voyage™ 200 uses the Bogacki-
Shampine 3(2) formula as found in the journal Applied Math
Letters, 2 (1989), pp. 1–9.
Bogacki-Shampine
3(2) Formula