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\title{HW1 }
\author{Fall 2016. Optimization methods\\
Due September 6, 2016}
\date{}
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\maketitle
Approximate the minimum of unimodal function $f(x)$ where $x$ is a real number $x \in [a,b]$,
approximate or locate
$x_* = \mbox{argmin}f(x)$, 
using
\newline
(a) Fibnacci search method (5 steps),
\newline
(b) Golden ratio search method (5 steps),
\newline
(c) Newton method with modifications (compute 3 steps). If Newton method
fails, explain why and suggest an improvement.

Use the following unimodal test functions $f(x)$
\begin{eqnarray}
f_1(x) = -.2\, x ^4 + x^2 -x, \quad  x \in [-1; 1], \quad x_
0 = -1. \\
f_2(x) = 1 -\exp(-x^2),\quad  x\in [-3; 3],\quad  x_0 = -2.5.  \\
f_3(x) =
\sqrt{ x^2 + .3}, \quad  x\in [-3; 3], \quad x_0 = -2. 
%% f_4(x) = \max \left\{
%% \sqrt{2 - x}, 
%% \sqrt{2 + x} \right\}, \quad  x\in [-2; 2] , \quad x_0 = -1.5. 
\end{eqnarray}
Plot the functions, list the points of evaluation.

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