Worksheet 5-1: Newton’s Method

Worksheet 5-1: Newton’s Method#

Download: CMSE382-WS5_1.pdf

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Worksheet 5-1: Q1#

Consider the function \(f(x,y) = 100x^4 + 0.01y^4\) and starting point \(\mathbf{x}_0 = (1, 1)^\top\).

  1. Compute the gradient \(\nabla f(\mathbf{x}_0)\).

  2. Compute the Hessian \(\nabla^2 f(\mathbf{x}_0)\).

  3. Find the Newton direction \(\mathbf{d}_0 = -[\nabla^2 f(\mathbf{x}_0)]^{-1} \nabla f(\mathbf{x}_0)\).

  4. Compute one step of Newton’s method: \(\mathbf{x}_1 = \mathbf{x}_0 + \mathbf{d}_0\).

Worksheet 5-1: Q2#

Consider the scalar function \(f(x) = \sqrt{1 + x^2}\).

Its derivatives are: $\(f'(x) = \frac{x}{\sqrt{1+x^2}}, \qquad f''(x) = \frac{1}{(1+x^2)^{3/2}}\)$

  1. Write the Newton update formula: \(x_{k+1} = x_k - \dfrac{f'(x_k)}{f''(x_k)}\).

  2. Simplify to show that \(x_{k+1} = -x_k^3\).

  3. For which starting values \(x_0\) does Newton’s method converge to \(x^* = 0\)?

  4. For which starting values does it diverge?

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