# Archive | work

## 1.19 Linear spaces

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}} \def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}} \def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)} \def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}} \def\ker{\mathbf{ker}}\def\im{\mathbf{im}}$$

1. Linear (or vector) spaces (over a field $$k$$ – which in our case will be almost always the complex numbers) are sets that satisfy the following standard list of properties:

• There exists

## solutions to exercises 1.17

Exercise 1: describe the curve $$1/z$$, where $$z=1+ti, ~~t\in (-\infty,\infty)$$.

Solution: apply $$\frac{1}{z} = \frac{\bar z}{|z|^2}$$ and get $$\frac{1}{1+ti} = \frac{1-ti}{1+t^2}$$, which is a parametric curve  of the form $$p(t) = x(t)+iy(t)$$ where $$x(t) = \frac{1}{1+t^2},~~~ y(t) = \frac{-t}{1+t^2}$$.…

## 1.17 Complex numbers

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}} \def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}$$

Complex numbers are expressions $$z=x+\i y$$, where $$\i$$ is the imaginary unit, defined by $$\i^2=-1$$. Note the ambiguity ($$-\i$$ would work just as well).

Complex numbers can be added subtracted, multiplied the usual way, with the …

## mock final

Here. (Disregard the points for the problems.)final_mock

## week 13

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

#### Interior point methods

Logarithmic barriers. Central path. Convergence and quality of solution estimates from duality. (Boyd, chapt.11).

#### Ellipsoid method

(Use Boyd’s notes).

#### Sample problems

• Find the central path for the

## week 12

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

Material:Guler, ch. 14.7-9

Conjugate directions. Gram-Schmidt algorithm. Conjugate directions algorithm. Performance estimates via spectral data.

#### Sample problems

• Consider the 4-dimensional space spanned by the polynomials
$p(x)=ax^3+bx^2+cx+d ## week 11 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ Gradient descent methods. Step size choices. Backtracking rule. Newton algorithm. Solving systems of nonlinear equations. Quadratic convergence, #### Sample problems • Compute first 5 iterations $$x_0=2,x_1,\ldots, x_5$$ of Newton method to solve ## programming exercise 2 $$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$ Set $$d=18$$. Consider the $$d\times d$$ matrix \[ C=\left(c_{ij}\right)_{i,j}$
where
$c_{ij}=a_{ij}+n_i-n_j.$
The base matrix of coefficients $$a_{ij}, 1\leq i,j\leq d$$ is given here; the modifier $$n_j$$ is …

## week 10

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

#### Simplex method

Tableau, pivot step, pivot rules. Exceptions. Phase one (finding a feasible point).

#### Sample problem

1. For which $$h$$ the following system is feasible?
\[
\begin{array}{ccc}
&y&\leq 0\\
x&&\geq 0\\
x&-y&\leq