AMATH342 Study Guide - Final Guide: Tridiagonal Matrix Algorithm, Rate Of Convergence, Gaussian Elimination

• School: University of Waterloo
• Department: Applied Mathematics
• Course: AMATH342
• Professor: Sander Rhebergen

Document Summary

Lipschitz condition: (cid:107)f (t, x) f (t, y)(cid:107) (cid:107)x y(cid:107) x, y rd, t t0 where r+ is the lipschitz constant. If (0, ), f lipschitz continuous. 1d lipschitz: |f (x) f (y)| |x y|, 1) if there exists (0, ), then. Lipschitz continuous, 2) for every pair of points (f (x), f (y)), the absolute value of the slope of the line connecting is not greater than . Euler"s method: yn+1 = yn + hf (tn, yn), y(t0) = y0, n = 1, 2, 3, . A numerical method is convergent if for every ode with f a lipschitz function, limh 0 maxn=0,1,,,(cid:98)t /h(cid:99) (cid:107)yn,h y(tn)(cid:107) = 0. Theorem 1. 1: euler convergent. (cid:107)en,h(cid:107) h[exp(t ) 1] c kernel theorem, c = maxt(cid:82) y(cid:48) = ay is y(t) = eaty0. Claim: (cid:107)en,h(cid:107) c h [(1 + h )n 1], n = 0, 1, 2, Euler"s method makes an error of o(h2) each step.