Texas Instruments TI-89 User Manual - Page 946

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: F1 = ƒ(xn, yn) ( ) F2 = ƒ xn + h 12-- , yn + h 12-- F1 ( ) F3 = ƒ xn + h 34-- , yn + h 34-- F2 ( ) yn+1 = yn + h 29-- F1 + 13-- F2 + 49-- F3 xn+1 = xn + h F4 = ƒ (xn+1 , yn+1) ( ) errest = h 7--5-2-- F1 ì 1--1-2-- F2 ì 19-- F3 + 18-- 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 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 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. Appendix B: Technical Reference 947