CompSciWeek13

Numerical Integration + Binary Output Format

• Van der Pol Oscillator: <math>\ddot u - \mu (1-u^2) \dot u + u = 0</math>
  • Implementations in OpenOffice Spreadsheet, GNU ODE, and Python
  • Perturbed initial conditions, Lyapunov exponents, and large deviation principles
• Working well with others: text
• Working with machines: binary
  • This is preferable when you have lots and lots of data
  • The relevant numpy methods are x.tofile("file") and x = fromfile("file") -- but save("file", x) and x = load("file") are preferred
  • inspecting binary formats with od
• Lecture Slides

GNU ODE Integrator

vanderpol.ode

<source lang="haskell"> mu = %MU

u = 1.0 y = 0.0

u' = y y' = mu*(1-u*u)*y - u

print t, u, y, y! every 10 from 50 step 0, 100 </source>

You can do the substitution with sed and run all dynamics with:

 for((i=0;i<10;i++)); do
   sed -e "s|%MU|$i|" vanderpol.ode >vanderpol-$i.ode
   ode -f vanderpol-$i.ode >out-$i.dat </dev/null
 done

Plotting with Gnuplot

<source lang="gnuplot"> set term postscript enh color lw 3 "Times-Roman" 28 set out 'van.eps'

set xlabel "u" set ylabel "y" plot 'out-0.dat' u 2:3 w l title "0", \

    'out-1.dat' u 2:3 w l title "1", \
    'out-4.dat' u 2:3 w l title "{/Symbol m} = 4"

</source>

Run with:

gnuplot plot_van.gplot

Working with binary

Example: ELF header

od -t x1 -t c -N 8 /bin/bash

Binary comes in lots of units

8 bits	= 1 byte
4 bits	= 1 nibble
1 byte	= 1 ascii character
2 bytes	= 1 short = 1 word
4 bytes	= 1 32-bit int = 1 float
8 bytes	= 1 64-bit int = 1 double = 1 float complex
16 bytes	= 1 long double = 1 double complex

Byte ordering on most modern 64-bit processors is little-endian (Intel, AMD). For more info, see [1].

In an imaginary system where base 10 numbers are encoded by 4-bit nibbles, a 4-digit representation of the number 123 is

3 = 0011
2 = 0010
1 = 0001
0 = 0000

The binary together would read:

0011 0010 0001 0000

Since we have to use 4 digits, the most significant digit is set to all zeros, and goes last. The little end (least significant byte) goes first. Notice that within the 4-bit nibbles, the least-significant bit is still on the right. So the number is encoded using this left-to right (within a digit) and bottom to top (digits within a word) order.

Real binary encoding is just like the above, except that each 'digit' is a byte, so every 8 bits are in order, but the bytes are reversed.

Basis Functions

• Constructing B-splines - tensor method
• Representing the differentiation operator
• Solving PDEs using the implicit method
• Lecture Slides