Scientific Computing with MATLAB2. Graphics for Research & Publication

Dra. K.-Y. Daisy FanDepartment of Computer ScienceCornell UniversityCornell UniversityIthaca, NY, USA

UNAM IIM 2012

Monte Carlo Approximation of

Sq area N L L

Throw N darts

LL/2

Sq. area = N = L L

Circle area = Ninin

= L2/4

4 N / N = 4 Nin / N

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Monte Carlo with N darts on L-by-L board

N= ___; L= ___; hits= 0;for k = 1:Nfor k = 1:N

% Throw kth dartx = rand(1)*L – L/2;x = rand(1) L L/2;y = rand(1)*L – L/2;% Count it if it is in the circle% Count it if it is in the circleif sqrt(x^2+y^2) <= L/2

hits = hits + 1;;end

end

4

myPi = 4*hits/N;

2-d array: matrix c

An array is a named collection of like data organized into

r

An array is a named collection of like data organized into rows and columns

A 2-d array is a table, called a matrixy , Two indices identify the position of a value in a matrix,

e.g.,gmat(r,c)

refers to component in row r, column c of matrix matp , Array index starts at 1 Rectangular: all rows have the same #of columns

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Rectangular: all rows have the same #of columns

Creating a matrix

Built-in functions: ones, zeros, rand E.g., zeros(2,3) gives a 2-by-3 matrix of 0sg , ( , ) g y

“Build” a matrix using square brackets, [ ], but the dimension must match up:p [x y] puts y to the right of x [x; y] puts y below x

4 0 3[ y] p y

[4 0 3; 5 1 9] creates the matrix [4 0 3; ones(1,3)] gives

4 0 3

5 1 9

4 0 3 [4 0 3; ones(3,1)] doesn’t work

4 0 3

1 1 1

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What will A be?

A= [0 0]A= [A’ ones(2,1)]A [0 0 0 0 A A]A= [0 0 0 0; A A]

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% Given an nr-by-nc matrix M.% What is A? for r= 1: nr

for c= 1: ncA(c,r)= M(r,c);

endend

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X-Y Plot

The basic command isplot(a b)plot(a,b)

which draws a line connecting the points ( (1) b(1)) ( (2) b(2)) ( ( ) b( )) (a(1),b(1)), (a(2),b(2)), … , (a(n),b(n)), where n is the length of vectors a and b.

Can add format specifications. E.g.,p gplot(a,b,’*r’)

draws red asterisks (*) at the data points instead of draws red asterisks (*) at the data points instead of using a line

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Color Options

White ‘w’Black ‘k’Black ‘k’Red ‘r’BlBlue ‘b’Green ‘g’Yellow ‘y’Magenta ‘m’gCyan ‘c’

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Making an x-y plot

a= [0 4 3 8]; % x-coordsb= [1 2 5 3]; % y-coordsb= [1 2 5 3]; % y-coordsplot(a, b, ‘-*’)

/ kx-values(a vector)

y-values(a vector)

Line/markerformat

6(a ecto )

4

5

2

3

0 2 4 6 8 100

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Making a figure with plot

figure % start a new graphics window

hold on % hold subsequent plot commands in current axeshold on % hold subsequent plot commands in current axes

plot(a,b)plot(c,d)

hold off % subsequent plot command refreshes axes--q p% erase previous items

close % close graphics windowclose % close graphics window

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Trajectory of a projectile – golf ball

Without air resistance (drag), the trajectory of a (drag), the trajectory of a golf ball in flight is a parabola. What is the parabola. What is the effect of drag?

Relevant equations for Relevant equations for simulating the golf ball in flight:flight:

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Trajectory of a projectile – golf ball

Without air resistance (drag), the trajectory of a (drag), the trajectory of a golf ball in flight is a parabola. What is the parabola. What is the effect of drag?

Relevant equations for Relevant equations for simulating the golf ball in flight:flight:

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Trajectory of a projectile – golf ball

Without air resistance (drag), the trajectory of a (drag), the trajectory of a golf ball in flight is a parabola. What is the parabola. What is the effect of drag?

Relevant equations for Relevant equations for simulating the golf ball in flight:flight:

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Simulate a golf ball in flight

Simulation begins at t = 0 and position (0,0) Simulation ends when the ball lands (y ≈ 0) or Simulation ends when the ball lands (y ≈ 0) or

when maximum simulation time is reachedPl h j Plot the trajectory

See the given file golfBall.m for the skeleton program, including given constant and parameters

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Simulate a golf ball in flight

Simulation begins at t = 0 and position (0,0) Simulation ends when the ball lands (y ≈ 0) or Simulation ends when the ball lands (y ≈ 0) or

when maximum simulation time is reachedPl h j Plot the trajectory

See the given file golfBall.m for the skeleton program, including given constant and parameters

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Simulate flight for different friction coefficient values

Modify golfBall to show the trajectories in one Modify golfBall to show the trajectories in one plot for different values of the friction coefficient:

k = 0, .02, … , .08

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Some format commands to use with plot

close all % close all graphics windowsaxis equal % same scaling for x, y axesaxis off % hide axesaxis off % hide axesaxis on % show axes

xlabel(‘text for labeling x-axis’)ylable(‘text for labeling y-axis’)title(‘text for plot title at top center’)( p p )

hold on % hold subsequent plot commands in current axeshold off % subsequent plot command refreshes axeshold off % subsequent plot command refreshes axes--

% erase previous items

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Draw a single line segment

a= 0; % x-coord of pt 1b= 1; % y-coord of pt 1b= 1; % y-coord of pt 1c= 5; % x-coord of pt 2d= 3; % y-coord of pt 2plot([a c], [b d], ‘-*’)

Line/markerformat

x-values(a vector)

y-values(a vector)

format

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( )

Making an x-y plot with multiple graphs (lines)My graphs

4 5

5

5.5

6My graphs

graph 1 namegraph 2 name

a= [0 4 5 8]; b= [1 2 5 3]; 3

3.5

4

4.5

y va

lues

[ ];f= [0 4 6 8 10];g= [2 2 6 4 3]; 1.5

2

2.5

plot(a,b,'-*',f,g,'c')legend('graph 1 name', 'graph 2 name')

0 1 2 3 4 5 6 7 8 9 101

x values

xlabel('x values')ylabel('y values')title('My graphs' 'Fontsize' 14)

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title('My graphs', 'Fontsize',14)

See also plotComparison.m

Simulate flight for different friction coefficient values

Modify golfBall to show the trajectories in one Modify golfBall to show the trajectories in one plot for different values of the friction coefficient:

k = 0, .02, … , .08

Note: Can create or lengthen a vector, e.g.,Note: Can create or lengthen a vector, e.g.,x = 3.1; % x is scalar 3.1x(4) = 2; % x is vector [3 1 0 0 2]x(4) = 2; % x is vector [3.1 0 0 2]

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Demonstrate commonly used2-d graphics

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Files for tomorrow:

ll d / df /www.cs.cornell.edu/~dfan/UNAM_IIM2012/popMexicoByStates.xlsx

A JPEG photo file, color or grayscale

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