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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 …

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JACT

coming soon.…

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mock final

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

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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
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week 12

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

Conjugate gradient method

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
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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
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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 …

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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
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Topologically constrained models of statistical physics.

\(\def\Real{\mathbb{R}}
\def\Int{\mathbb{Z}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\def\blob{\mathcal{B}}
\)

The Blob

Consider the following planar “spin model”: the state of the system is a function from \(\Int^2\) into \(\{0,1\}\) (on and off states). We interpret the site \((i,j), …

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CDC’15

Slides are here.…

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