eecs 110 recitation #3 ning xia northwestern university
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EECS 110Recitation #3
Ning XiaNorthwestern University
1 102 3 4 5 6 7 8 9
1
0.5
0
y = 1 x area
steps(low,hi,N) fsteps(recip,low,hi,N)
def recip(x): return 1.0/x
finteg(f,low,hi,N)
Numerical IntegrationLab 2: Tuesday
1 102 3 4 5 6 7 8 9
1
0.5
0
1 102 3 4 5 6 7 8 9
1
0.5
0
low = 1.0 hi = 10.0
y = 1 x area
steps(low,hi,N)
N == 9 == total number of steps (rectangles)
fsteps(recip,low,hi,N)
def recip(x): return 1.0/x
finteg(f,low,hi,N)
Numerical Integration
def fracSteps(N):
return [ x/N for x in range(N) ] 0 1 41
424242
def steps(low,hi,N):
return [ low + (hi-low)*x for x in fracSteps(N) ]0 1 6
77710 10 15
def fsteps(f,low,hi,N):
return [ f(x) for x in steps(low,hi,N) ]
def finteg(f,low,hi,N): return sum(fsteps(f,low,hi,N))*(hi-low)/float(N)
x values
fractional steps
y values
integral: heights
10 16
1610width
An example closer to home
......25 26 27 28 502423220
An overworked NU student (S) leaves a pub after a “late-night” breakfast and, each moment, randomly
stumbles toward Dorms (W) or toward Michigan Lake (E)
Dorms
Michigan Lake
(E)
(W)
Platt
Write a program to model and analyze! this scenario...
Hw2 Pr2
S
Once the student arrives at the dorms or the Michigan Lake, the trip is complete.
The program should then print the total number of steps taken.
An example closer to home
......25 26 27 28 502423220
Dorm Michigan Lake
(E)
(W)
Platt
Write a program to model and analyze! this scenario...
Hw2 Pr2
S
rs() rwPos(s, nsteps) rwSteps(s, low, hi)
take a random step of +1 or -1
take nsteps random steps starting at s
take random steps starting at s until you reach either low or hi
An overworked NU student (S) leaves a pub after a “late-night” breakfast and, each moment, randomly
stumbles toward Dorms (W) or toward Michigan Lake (E)
Once the student arrives at the dorm or the Michigan Lake, the trip is complete.
The program should then print the total number of steps taken.
rwPos(start,nsteps)
• start = position of sleepwalker• nsteps = number of random steps taken by
walker• Returns the position of the walker after nsteps
random steps
• Call to rs() + recursion
rwSteps(start, low, hi)
• start = position of sleepwalker• low = lowest position he can be at• hi = highest position he can be at• Returns the number of steps until walker
reaches low or hi
• Call to rs() + recursion.
Random walk analysis
• Average final signed-displacement- Statistics on the output of rwPos()
• Average final squared signed-displacement
• Both take as input the number of steps of the random walker and the number of statistical runs
Python's Etch-a-Sketch
A new human-computer interface?
from turtle import *
reset()
left(90)
forward(50)
right(90)
backward(50)
down() or up()
color('green')
width(5)
done()
and lots more!
for turtle help
degrees!
states if the pen draws or not
http://networks.cs.northwestern.edu/EECS110-s14/misc/TurtleDirections.htm
hw2pr3 spiral (I)
100
90
81
72.9
spiral( initLength, angle, multiplier )
close-up of innermost part of the spiral…
spiral( 100, 90, 0.9 )
hw2pr3 spiral (II)• You can draw it inside out (stop at 1000 pixels) or
outside in (stop at 1 pixel)• Function takes as input length of segment to
draw, angle and a scaling factor for the length of the segment
• Uses recursion• Each call to spiral:– Draw line– Change orientation by angle– Update length of segment by scaling– Check base case– Recall spiral
hw2pr3 svTree (I)
svTree( trunkLength, levels )
svTree( 100, 4 )
and more! (if you want)
hw2pr3 svTree (II)
• Function takes as input a segment length and the number of tree levels
• Uses recursion• Each call to svTree:• Decrease number of levels• Draw part of tree at this level• Check base case• Call recursively svTree twice to draw the left
and right subtrees• Don’t forget to change the angles before
The Koch curve (I)
snowflake( 100, 0 )snowflake( 100, 1 ) snowflake( 100, 2 )
snowflake( 100, 3 ) snowflake( 100, 4 ) snowflake( 100, 5 )
The Koch curve (II)
• snowFlake gets as input the segment size and the number of levels to draw
• Helpful to have a function which draws the sides of the triangle.
• The side drawing function:• If reached final level, draw full side• Otherwise just call the side drawing function
on the 4 sides that we just broke into
Good Luck !
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