Texas Instruments PLUS TI-89 - Manual (Page 590)

Appendix B: Reference Information 573
8992APPB DOC TI-89/TI-92 Plus:8992appb doc (English) Susan Gullord Revised: 02/23/01 1:54 PM Printed: 02/23/01 2:24 PM Page 573 of 34
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 / TI-92 Plus 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 / TI - 92 Plus 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