AMATH342 Study Guide - Midterm Guide: Orthogonal Polynomials, Lagrange Polynomial, Trapezoidal Rule

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

Document Summary

By h [(1 + h )n 1], n = 0, 1, 2, Lipschitz condition: kf (t, x) f (t, y)k kx yk x, y rd, t t0 where r+ is the lipschitz constant. If (0, ), f lipschitz continuous. 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,,, t /h kyn,h y(tn)k = 0. Theorem 1. 1: euler convergent. ken,hk h[exp(t ) 1] c. Peano kernel theorem, c = maxt r0,t0+t ky (t)k and the exact solution of y = ay is y(t) = eaty0. Euler"s method makes an error of o(h2) each step. Given a numerical method yn+1 = yn + hf (tn, yn), it is of order p if y(tn+1) yn+1 = o(hp+1). (euler = order 1) Trapezium rule: yn+1 = yn + 1 error o(h3), order 2.