Appendix B: Reference Information 573
8992APPBDOC TI
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89/TI
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92Plus:8992a
pp
bdoc (English) SusanGullord Revised:02/23/01 1:54 PM Printed:02/23/01 2:24 PM Page573 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:
F1 = ƒ(xn, yn)
F2 = ƒ (xn + h 1
2 , yn + h 1
2 F1)
F3 = ƒ (xn + h 3
4 , yn + h 3
4 F2)
yn+1 = yn + h ( 2
9 F1 + 1
3 F2 + 4
9 F3)
xn+1 = xn + h
F4 = ƒ (xn+1 , yn+1)
errest = h ( 5
72 F1 ì1
12 F2 ì1
9 F3 + 1
8 F4)
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
xn x xn+1 using the cubic interpolating polynomial passing through
the point (xn , yn) with slope F1 and through (xn+1 , yn+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.
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