Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2017 • Question 2

Question

Math 6420 • PDE Spring 2017 • Question 2

Let \(f\) be a real-valued and twice continuously differentiable on an interval \([a,b]\). Let \(\hat {x}\) be a simple zero of \(f\), i.e. \(f'(\hat {x})\ne 0\). Use the Banach Fixed Point Theorem to show that Newton’s method, defined by \[ x_{n+1}=x_n-\frac {f(x_n)}{f'(x_n)} \] converges to \(\hat {x}\) for any \(x_0\) sufficiently close to \(\hat {x}\).

Raw LaTeX

Let \(f\) be a real-valued and twice continuously differentiable on an interval \([a,b]\).
Let \(\hat{x}\) be a simple zero of \(f\), \textit{i.e.} \(f'(\hat{x})\ne0\). Use the Banach
Fixed Point Theorem to show that Newton's method, defined by
\[
  x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}
\]
converges to \(\hat{x}\) for any \(x_0\) sufficiently close to \(\hat{x}\).

Topic Tags

Banach and Normed Spaces Contraction Mapping and Nonlinear Equations Fixed-Point Iteration Linear Programming and Optimization