# Worksheet 5-1: Newton's Method

Download: [CMSE382-WS5_1.pdf](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: $\mathbf{x}_1 = \mathbf{x}_0 + \mathbf{d}_0$.

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## 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?