L14. Arrays and Functions

1
L14. Arrays and Functions

• Functions with array
• parameters.
• Row and column vectors
• Built-Ins length, zeros, std
• Revisit rand, randn, max

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Row and Column Vectors

• gtgt v 1 2 3
• v
• 1 2 3
• gtgt v 1 2 3
• v
• 1
• 2
• 3

Observe semicolons
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zeros( , )

• gtgt x zeros(3,1)
• x
• 0
• 0
• 0
• gtgt x zeros(1,3)
• x
• 0 0 0

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rand( , )

• gtgt x rand(3,1)
• x
• 0.2618
• 0.7085
• 0.7839
• gtgt x rand(1,3)
• x
• 0.9862 0.4733 0.9028

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randn( , )

• gtgt x randn(1,3)
• x
• 0.2877 -1.1465 1.1909
• gtgt x randn(3,1)
• x
• 1.1892
• -0.0376
• 0.3273

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Normal Distribution withZero Mean and Unit STD
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Affirmations

• gtgt n 1000000
• gtgt x randn(n,1)
• gtgt ave sum(x)/n
• ave
• -0.0017
• gtgt standDev std(x)
• standDev
• 0.9989

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length

• gtgt v randn(1,5)
• gtgt n length(v)
• n
• 5
• gtgt u rand(5,1)
• gtgt n length(u)
• n
• 5

The length function doesnt care about row or
column orientation.
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Augmenting Row Vectors

• gtgt x 10 20
• x
• 10 20
• gtgt x x 30
• x
• 10 20 30
• gtgt

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Augmenting Column Vectors

• gtgt x 1020
• x
• 10
• 20
• gtgt x x 30
• x
• 10
• 20
• 30

Observe semicolons!
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Concatenating Row Vectors

• gtgt x 10 20
• x
• 10 20
• gtgt y 30 40 50
• y
• 30 40 50
• gtgt z x y
• z
• 10 20 30 40 50

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Concatenating Column Vectors

• gtgt x 10 20
• gtgt y 30 40 50
• gtgt z x y
• z
• 10
• 20
• 30
• 40
• 50

Observe semicolons!
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Application
Plot sine across 0,4pi and use the fact that
it has period 2pi.

• x linspace(0,2pi,100)
• y sin(x)
• x x x2pi
• y y y
• plot(x,y)

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x linspace(0,2pi,100) x x x2pi
linspace(2pi,4pi,100)
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The Empty Vector

• x
• for k150
• if floor(sqrt(k))sqrt(k)
• x x k
• end
• end
• x x

x 1 4 9 16 25 36 49
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Array Hints Errors
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Dimension Mismatch

• gtgt x 12
• x
• 1
• 2
• gtgt y 3 4
• y
• 3 4
• gtgt z xy
• ??? Error using gt plus
• Matrix dimensions must agree.

Cant add a row vector to a column vector
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Not a Syntax Error

• gtgt x rand(3)
• x
• 0.9501 0.4860 0.4565
• 0.2311 0.8913 0.0185
• 0.6068 0.7621 0.8214

You probably meant to say x rand(1,3) or x
rand(3,1).
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A Style Hint

• Assume n is initialized.

a zeros(1,n) for k1n a(k) sqrt(k) end
a for k1n a a sqrt(k) end
Better because it reminds you of the size and
shape of the array you set up.
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Error Out-ofRange Subscript

• gtgt x 10 20 30
• x
• 10 20 30
• gtgt c x(4)
• ??? Index exceeds matrix dimensions.

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This is OK

• gtgt x 10 20 30
• x
• 10 20 30
• gtgt x(4) 100
• x
• 10 20 30 100

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Forgot the Semicolon?

• gtgt x randn(1000000,1)

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Forgot the Semicolon?

• gtgt x randn(1000000,1)

Remember ctrl-C
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Question Time

• A 1
• while(length(A) lt 5)
• A length(A)1 A
• end
• A A
• Is this the same as
• A linspace(1,5,5) ?

A. Yes B. No
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No!

• Linspace
• 1 2 3 4 5
• Fragment
• 5
• 4
• 3
• 2
• 1

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Question Time

• x zeros(1,1)
• for k13
• x x x
• end
• y x(7)
• Will this cause a subscript out of
• bounds error?

A. Yes B. No
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No!

• How x changes
• After 1st pass 0 0
• After 2nd pass 0 0 0 0
• After 3rd pass 0 0 0 0 0 0 0 0
• So y x(7) makes sense.

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Polygon Transformations
Functions arrays
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A Polygon
(x1,y1)
(x2,y2)
Store xy-coordinates in vectors x and y.
(x5,y5)
(x3,y3)
(x4,y4)
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Operation 1 Centralize

• Move a polygon so that the centroid
• of its vertices is at the origin.

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Centralize
Before
After
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Centralize

• function xNew,yNew Centralize(x,y)
• n length(x)
• xBar sum(x)/n
• yBar sum(y)/n
• xNew x-xBar
• yNew y-yBar

Computes the vertices of the new polygon
Notice how length is used to figure out the
size of the incoming vectors.
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Operation 2 Normalize

• Shrink (enlarge) the polygon so that
• the vertex furthest from the
• (0,0) is on the unit circle.

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Normalize
Before
After
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Normalize

• function xNew,yNew Normalize(x,y)
• d max(sqrt(x.2 y.2))
• xNew x/d
• yNew y/d

Applied to a vector, max returns the largest
value in the vector.
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Operation 3 Smooth

• Obtain a new polygon by connecting
• the midpoints of the edges

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Smooth
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Midpoints
(c,d)
( (ac)/2 , (bd)/2 )
(a,b)
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Smooth

• function xNew,yNew Smooth(x,y)
• n length(x)
• xNew zeros(n,1)
• yNew zeros(n,1)
• for i1n
• Compute the mdpt of ith edge.
• Store in xNew(i) and yNew(i)
• end

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xNew(1) (x(1)x(2))/2yNew(1) (y(1)y(2))/2
(x2,y2)
(x1,y1)
(x5,y5)
(x3,y3)
(x4,y4)
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xNew(2) (x(2)x(3))/2yNew(2) (y(2)y(3))/2
(x2,y2)
(x1,y1)
(x5,y5)
(x3,y3)
(x4,y4)
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xNew(3) (x(3)x(4))/2yNew(3) (y(3)y(4))/2
(x2,y2)
(x1,y1)
(x5,y5)
(x3,y3)
(x4,y4)
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xNew(4) (x(4)x(5))/2yNew(4) (y(4)y(5))/2
(x2,y2)
(x1,y1)
(x5,y5)
(x3,y3)
(x4,y4)
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xNew(5) (x(5)x(1))/2yNew(5) (y(5)y(1))/2
(x2,y2)
(x1,y1)
(x5,y5)
(x3,y3)
(x4,y4)
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xNew(5) (x(5)x(1))/2yNew(5) (y(5)y(1))/2
(x2,y2)
(x1,y1)
(x5,y5)
(x3,y3)
(x4,y4)
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Smooth

• for i1n
• xNew(i) (x(i) x(i1))/2
• yNew(i) (y(i) y(i1))/2
• end
• Will result in a subscript
• out of bounds error when i is n.

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Smooth

• for i1n
• if iltn
• xNew(i) (x(i) x(i1))/2
• yNew(i) (y(i) y(i1))/2
• else
• xNew(n) (x(n) x(1))/2
• yNew(n) (y(n) y(1))/2
• end
• end

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Smooth

• for i1n-1
• xNew(i) (x(i) x(i1))/2
• yNew(i) (y(i) y(i1))/2
• end
• xNew(n) (x(n) x(1))/2
• yNew(n) (y(n) y(1))/2

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Proposed Simulation

• Create a polygon with randomly
• located vertices.
• Repeat
• Centralize
• Normalize
• Smooth

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(No Transcript)
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2D Random Walk
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Start in middle tile. Repeat until boundary
reached Pick a compass heading at
random. Move one tile in that direction.
1-by-1 tiles
North, East, South, West
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Function that Returns the Path
function x y RandomWalk2D(N) k 0 xc
0 yc 0 while abs(xc)ltN abs(yc)lt N
Take another hop. Update location
(xc,yc). k k 1 x(k) xc y(k) yc
end
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• k x(k) y(k)
• -------------
• 1 0 -1
• 2 0 0
• 3 -1 0
• 4 0 0
• 5 0 1
• 6 1 1
• 7 1 0
• 8 1 1
• 0 1

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• if rand lt .5
• if rand lt .5
• xc xc 1 East
• else
• xc xc - 1 West
• end
• else
• if rand lt .5
• yc yc 1 North
• else
• yc yc - 1 Sounth
• end
• end

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How Many Returns to (0,0)?

• x and y are n-vectors.
• How many times do we have
• (x(k),y(k)) (0,0)?
• m 0
• for k1length(x)
• if x(k)0 y(k)0
• m m 1
• end
• end