Here. (Disregard the points for the problems.)final_mock…
Archive  work
RSS feed for this sectionweek 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
\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)
Conjugate gradient method
Material:Guler, ch. 14.79
Conjugate directions. GramSchmidt algorithm. Conjugate directions algorithm. Performance estimates via spectral data.
Sample problems
 Consider the 4dimensional 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
\[
c_{ij}=a_{ij}+n_in_j.
\]
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

For which \(h\) the following system is feasible?
\[
\begin{array}{ccc}
&y&\leq 0\\
x&&\geq 0\\
x&y&\leq
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), …
Topology in Motion @ICERM Fall’16
If you are interested in Applied Topology, tend to plan far, far ahead and have some free time on your hands in Fall 2016, check this out, and let us know!
math285, week of october 27
Read the textbook, chapters 9.19.3.
Homework (due by Tuesday, 11.6):
 Find Fourier transform of the function on \((\pi,\pi)\) equal to \(0\) for \(\pi<x<0\) and to \(1\) for \(0\leq x<\pi\).
 Find cosine expansion of \(\sin(2x)\) on