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
Summary of Contents for Titanium TI-89
Page 9: ...Getting Started 6 TI 89 Titanium keys Ë Ì Í Ê ...
Page 34: ...Getting Started 31 2 or D 2 B u s i n e s s D B D B Press Result ...
Page 43: ...Getting Started 40 3 0 D B D D B D Press Result ...
Page 44: ...Getting Started 41 D 2 0 0 2 D B Scroll down to October and press Press Result ...
Page 58: ...Getting Started 55 Example Set split screen mode to TOP BOTTOM Press Result 3 B D ...
Page 70: ...Getting Started 67 ...
Page 175: ...Operating the Calculator 172 From the Keyboard ...
Page 456: ...Tables 453 ...
Page 527: ...Data Matrix Editor 524 ...
