Texas Instruments TI-89 Titanium User Manual

Appendix B: Technical Reference 920
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
c x c 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

Texas Instruments TI-89 Titanium Specifications

General

Brand	Texas Instruments
Model	TI-89 Titanium
Category	Calculator
Language	English