Apply Newton’s Method to find both roots of the function f(x) = 14xex−2 − 12ex−2 − 7×3 + 20×2 − 26x + 12 on the interval [0,3]. For each root, print out the sequence of iterates, the errors , and the relevant error ratio or that converges to a nonzero limit. Match the limit with the expected value M from Theorem 1.11 or S from Theorem 1.12.
Let f be twice continuously differentiable and f(r) = 0. If f’(r) ≠ 0, then Newton’s Method is locally and quadratic ally convergent to r. The error at step i satisfies
Assume that the (m + 1)-times continuously differentiable function f on [a,b] has a multiplicity m root at r. Then Newton’s Method is locally convergent to r, and the error at step i satisfies