MATH119 Study Guide - Polynomial, Interpolation, Ibm System P

• School: University of Waterloo
• Department: Mathematics
• Course: MATH119
• Professor: Michael Dunphy

Document Summary

L(x) = f (a) + f(cid:48)(a)(x a) Find a, b where f (a) < 0 && f (b) > 0. a + b: if f ( ) < 0 then a = a + b. Iterative formula: xn+1 = xn f (xn) f(cid:48)(xn) Fixed-point iteration: isolate for x, solve xn+1 = g(xn), repeat until answer is same twice in a row (to required decimal places), if divergence occurs, try isolating x di erently and repeat steps 2-3. Newton forward di erence forumla y = y0 + (x x0) h. 2y0 + + (x x0)(x x1)(x xn+1) n!hn. Lagrange linear interpolation formula f (x) (x x1) (x0 x1) f1 + (x x0) (x1 x0) f2. Pn,x0(x) = f (k)(x0) k! (x x0)k + rn(x) n(cid:88) k=0. Pn,0(x) = n(cid:88) k=0 f (k)(0) k! xk + rn(x) f (x) = f (0) + f(cid:48)(0) x + f(cid:48)(cid:48)(0) x2. |rn(x)| = | x0 (x t)n (cid:90) x.