Texas Instruments TI-89 Voyage 200, User Manual

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Appendix B: Technical Reference 947
Bogacki-Shampine 3(2) Formula Bogacki-Shampine 3(2) Formula Bogacki-Shampine 3(2) Formula Bogacki-Shampine 3(2) Formula
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 , y
n
+ h F
1
)
F
3
=
ƒ
(
x
n
+ h , y
n
+ h F
2
)
y
n+1
= y
n
+ h
(
F
1
+ F
2
+ F
3
)
x
n+1
= x
n
+ h
F
4
=
ƒ
(
x
n+1
, y
n+1
)
errest = h
(
F
1
ì
F
2
ì
F
3
+ 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 (xn , yn) with slope F1 and through (x
n+1
, y
n+1
)
with slope F4. The interpolant is efficient and provides results throughout the step that
are just as accurate as the results at the ends of the step
.
1
2
---
1
2
---
3
4
---
3
4
---
2
9
---
1
3
---
4
9
---
5
72
------
1
12
------
1
9
---
1
8
---