matplotlib Code

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\begin_body

\begin_layout Title
Python Workshop Problems
\end_layout

\begin_layout Standard
\begin_inset LatexCommand \tableofcontents{}

\end_inset


\end_layout

\begin_layout Section
Quicksort
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: lists, recursion.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
: implement the quicksort algorithm.
\end_layout

\begin_layout Subsubsection*
Code skeleton
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
codelist{skel/qsort_skel.py}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Wallis' 
\begin_inset Formula $\pi$
\end_inset


\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: arbitrary size integers, simple function definitions.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
: define a simple function that implements the (slowly converging) Wallis
 infinite product approximation to 
\begin_inset Formula $\pi$
\end_inset

:
\begin_inset Formula \[
\pi(n)=2\prod_{i=1}^{n}\frac{4i^{2}}{4i^{2}-1}\]

\end_inset


\end_layout

\begin_layout Subsubsection*
Code skeleton
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
codelist{skel/wallis_pi_skel.py}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Trapezoid rule
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: basic array slicing, functions as first class objects.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
: define a trapezoid rule integrator:
\begin_inset Formula \[
I=\int_{a}^{b}f(x)dx\approx\frac{1}{2}\sum_{i=1}^{N}(x_{i}-x_{i-1})(y_{i}+y_{i-1})\]

\end_inset

where 
\begin_inset Formula $y_{i}=f(x_{i})$
\end_inset

, 
\begin_inset Formula $x_{o}=a$
\end_inset

, 
\begin_inset Formula $x_{N}=b$
\end_inset

.
 
\end_layout

\begin_layout Standard
Consider both the case where the input is given as a pair of arrays 
\begin_inset Formula $(x,y)$
\end_inset

 and where the input is 
\begin_inset Formula $(f,a,b,N)$
\end_inset

.
 Write two separate functions.
\end_layout

\begin_layout Subsubsection*
Code skeleton
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
codelist{skel/trapezoid_skel.py}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Numerical integration and root finding
\end_layout

\begin_layout Standard

\series bold
Illustrates:
\series default
 functions as first class objects, use of the scipy libraries.
\end_layout

\begin_layout Standard

\series bold
Task:
\series default
 Find 
\begin_inset Formula $t$
\end_inset

 such that
\begin_inset Formula \[
\int_{0}^{t}f(s)\, ds=u\]

\end_inset

for a known, monotonically increasing 
\begin_inset Formula $f(s)$
\end_inset

 and a fixed 
\begin_inset Formula $u$
\end_inset

.
 The example below uses 
\begin_inset Formula $f(s)=s\cdot\sin^{2}(s)$
\end_inset

 and 
\begin_inset Formula $u=1/4.$
\end_inset


\end_layout

\begin_layout Subsubsection*
Code skeleton
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
codelist{skel/quad_newton_skel.py}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Wrapping Fortran codes
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: wrapping existing Fortran libraries, improving the call interface of Fortran
 codes.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
: write a Fortran subroutine 
\family typewriter
cumsum(x,y,n)
\family default
 that computes the cumulative sums of entries of 
\begin_inset Formula $x$
\end_inset

 and stores them in 
\begin_inset Formula $y$
\end_inset

:
\begin_inset Formula \[
y_{j}=\sum_{i=1}^{j}x_{i},\; j=1,\ldots,n.\]

\end_inset


\end_layout

\begin_layout Subsubsection*
Code skeleton
\end_layout

\begin_layout Standard
The following skeleton contains a wrapper of another Fortran routine, a
 simple Fibonacci number calculation, to illustrate the f2py syntax.
 A Makefile and a test file are included as well.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
codelist{skel/fortran_wrap/fib3.f}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Univariate polynomials, root finding
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: poly1d objects, convolutions, plotting.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
:
\end_layout

\begin_layout Subsubsection*
Code skeletonq
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
codelist{skel/polyroots1d_skel.py}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Image processing and the FFT
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: 2-d image denoising, use of the scipy FFT library, array manipulations,
 image plotting.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
:
\end_layout

\begin_layout Subsubsection*
Code skeleton
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
codelist{skel/fft_imdenoise_skel.py}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Bessel functions
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: Special functions library, array manipulations to check recursion relation.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
:
\end_layout

\begin_layout Subsection*
Code skeleton
\end_layout

\begin_layout Standard
MISSING
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\begin_layout Section
Monte Carlo integration
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: random number generation.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
: compute an approximation to 
\begin_inset Formula $\pi$
\end_inset

 via Monte Carlo integration.
\end_layout

\begin_layout Subsection*
Code skeleton
\end_layout

\begin_layout Standard
MISSING
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\begin_layout Section
Speed optimizations using 
\family typewriter
weave
\end_layout

\begin_layout Standard

\series bold
Illustrates
\series default
: weave.blitz, weave.inline.
\end_layout

\begin_layout Standard

\series bold
Task
\series default
: ...
\end_layout

\begin_layout Subsection*
Code skeleton
\end_layout

\begin_layout Standard
MISSING
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\end_body
\end_document