CS370 Study Guide - Final Guide: Taylor Series, Lu Decomposition, Liberation Tigers Of Tamil Eelam

• School: University of Waterloo
• Department: Computer Science
• Course: CS370
• Professor: Yuying Li

Document Summary

Page 1: lte of 3rd order adams-bashforth. We want to determine the error in one step of this scheme. 12 yn+1 = yn + h f (tn, yn) 4. Replace rhs quantities with exact counterparts, and replace f (t, y(t)) with appropriate approxi- mate yn quantities: yn+1 = yn + Taylor expand rhs entries about time tn (just the ones not already at time tn). yn+1 = yn + 3 (cid:18) h2 y(cid:48)(cid:48)(cid:48)(tn) h3 (y(cid:48)(tn) hy(cid:48)(cid:48)(tn) + 12 y(cid:48)(cid:48)(cid:48)(cid:48)(tn) + o(h4) y(cid:48)(cid:48)(cid:48)(cid:48)(tn) + o(h4) (cid:18) (cid:19) (cid:19) Collect like terms: yn+1 = yn + hy(cid:48)(tn)(23/12 4/3 + 5/12) + h2y(cid:48)(cid:48)(tn)(4/3 10/12) + h3y(cid:48)(cid:48)(cid:48)(tn)( 4/6 + 20/24) +h4y(cid:48)(cid:48)(cid:48)(cid:48)(tn)(4/18 40/72) + o(h5) y(cid:48)(cid:48)(cid:48)(cid:48)(tn) + o(h5) yn + hy(cid:48)(tn) + y(cid:48)(cid:48)(tn) + h3. For our comparison, the taylor expansion of y(tn+1) is: y(tn+1) = yn + hy(cid:48)(tn) + y(cid:48)(cid:48)(tn) + h2. The local truncation error is the di erence between these two. Note that most of the terms cancel so we nd: