matlab for the sciences - lagrange collegehome.lagrange.edu/.../matlab-week1-lecture5.pdf · matlab...

Two-dimensional PlottingThree-Dimensional Plotting

ExtrasExercises

MATLAB for the SciencesPlotting, Simple Arrays, and Special Functions

Jon M. Ernstberger

January 9, 2008

Jon M. Ernstberger MATLAB for the Sciences


ExtrasExercises

Plotting in MATLAB

Perhaps what makes MATLAB so wonderful is the ease ofgraphical output.

We all want to see pretty pictures!

In the past, for graphical output, you had two options:

Poor ASCII (DOS-type) graphics.Save program data and export it to another interpretingprogram.One is unprofessional and the other is time-consuming.

MATLAB’s Java-based implementation makes the plottingmuch simpler.

Good pictures can take mediocre results/research and impresspeople.



ExtrasExercises

Functions

Simple Plotting

You need two arrays of numbers to plot.

Example #1

x=[1 2 3]; % pick values for y=x;y=[1 2 3];plot(x,y); %plot y versus x

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 31

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3



ExtrasExercises

Functions

Defining Arrays....Differently

Until now we’ve explicitly defined an array element-wise. Now, wedefine the array using vector notation. Vector Notation

Example #2

x=0:2:100; %x=[0,2,4,...,98,100];y=100:-2:0; %y=[100,98,96,94,92,90,....6,4,2,0];

0 10 20 30 40 50 60 70 80 90 1000

10

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ExtrasExercises

Functions

...Array....Differently, cont.

Example #3

x=1:100; %x=[1,2,3,4,5,...,99,100];y=3*x; % y=[3,6,9,12,....297,300];

0 10 20 30 40 50 60 70 80 90 1000

50

100

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ExtrasExercises

Functions

...Array....Differently, cont.

Example #x4

x=1:100; %x=[1,2,3,4,5,...,99,100];y=x.^2; %y=[1,4,9,16,25,...9801,10000];

0 10 20 30 40 50 60 70 80 90 1000

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ExtrasExercises

Functions

MATLAB Functions

We just learned to do basic plotting. What about plottingwith more interesting data?

There are certain functions which take arrays (vectors) andoutput vectors.

Test the following:

x=-2*pi:pi/4:2*pi;y=sin(x);Figure(1) %trust me on this....plot(x,y);

How frequently is sin(x) being evaluated?



ExtrasExercises

Functions

MATLAB Functions, cont.

Now set the x-grid to π/2. Is this grid more or less coarse?

Now redo the whole thing:

x=-2*pi:pi/2:2*pi;y=sin(x);Figure(2) %what is this command doing?plot(x,y);



ExtrasExercises

Functions

MATLAB Functions, cont.

Now change the x-grid to π/16. Is this grid more or lesscoarse?

Try it again...

x=-2*pi:pi/16:2*pi;y=sin(x);Figure(3)plot(x,y);

Compare the three....

The less coarse (or more refined) grid yields the betterapproximate.

Remember that these are all approximates.



ExtrasExercises

Functions

Plotting

Until now, the plots are just pictures.

Graphics are only helpful if they add to a presentation.Otherwise, they detract.

The information in a picture must be clearly understood,otherwise readers gloss over the content.



ExtrasExercises

Functions

Text Labels

We use the xlabel command to generate labelscorresponding to what would be the ‘x-axis’.

xlabel(‘My text here’);

Similarly, the ylabel command generates labels thatcorrespond to the ‘y-axis’.

ylabel(‘The label for the y-axis goes here.’);

We use the title command in a similar fashion to put titlesat the top of the figure.

LATEXand MATLAB

You can use simple LATEX commands in MATLAB in yourlabels and title.

If you’re saving images for use in a LATEX document, don’tuse the title.



ExtrasExercises

Functions

Plotting Example, Part A

Try this:

x=-2*pi:pi/16:2*pi;plot(x,sin(x));xlabel(‘x’);ylabel(‘sin(x)’);

Still hard to see..

−8 −6 −4 −2 0 2 4 6 8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

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0.6

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1

x

sin

(x)



ExtrasExercises

Functions

Plotting Example, Part B

What if I want to increase the line thickness?plot(x,y,‘LineWidth’,3);

−8 −6 −4 −2 0 2 4 6 8−1

−0.8

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x

sin

(x)



ExtrasExercises

Functions

Plotting Example, Part C

Do a ‘help plot’.

What can I do if I want to change the color of the line to red?

−8 −6 −4 −2 0 2 4 6 8−1

−0.8

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x

sin

(x)



ExtrasExercises

Functions

Plotting Example, Font Sizes

Notice that font sizes are still way too small. I wouldn’t wantthose in a LATEX file.How do we fix this?

xlabel(‘x’,‘FontSize’,16);Same for the ylabel and title commands.

−8 −6 −4 −2 0 2 4 6 8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

sin

(x)

xJon M. Ernstberger MATLAB for the Sciences


ExtrasExercises

Functions

Plotting Example, Axis Sizes

To fix the font on the axes themselves, we issue the followingcommands

set(gca,‘FontSize’,16);

The gca stands for “Get Current Axes”.

Try it.



ExtrasExercises

Functions

Multiple Plots on the Same Figure

Can we plot multiple imageson the same figure? Youbet!

The plot command looks atvector combos.

x=-2*pi:pi/16:2*pi;plot(x,sin(x),‘r-x’,...x,cos(x),‘b-o’,...‘LineWidth’,3)

set(gca,‘FontSize’,16)legend(‘sin(x)’,‘cos(x)’);axis tight;

−6 −4 −2 0 2 4 6−1

−0.8

−0.6

−0.4

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0

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1

sin(x)

cos(x)



ExtrasExercises

Functions

Tips and Tricks

Use axis tight to zoom in the image as much as possible.

Plot black lines for axes (like the Cartesian Coordinate Plane).

Use a combination of plot techniques. Use ‘r-x’ and ‘b-o’ orsomething similar to compare actual data points.

You can use the built-in tools to change all of the properties.

You can save the image as a .eps file.



ExtrasExercises

3-D Plotting

MATLAB produces some superb three-dimensional graphics!x=-10:.1:10; and y=-10:.1:10;f (x , y) = x2y2



ExtrasExercises

Ways to Plot in Three Dimensions

Contour Plots

Surface Plots

Mesh Plots



ExtrasExercises

Contour Plots

Let’s set up ourfunctions

x=-10:.1:10;y=x;Z=(x.^2)’*(y.^2);contour(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Two-dimensional plot

The third dimensionis expressed viaconcentriccolor-coded curves.

x

y

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

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10

You can click on “Edit” and then “Figure Properties” to changethe contour color schemes.Jon M. Ernstberger MATLAB for the Sciences


ExtrasExercises

Contour Plot Variants

Run the program as contour3(x,y,Z) (shown left).

Run the program as contourf(x,y,Z) (shown right).

−10

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−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

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Bold colors are eye-catching and help readers to see informationmore clearly.



ExtrasExercises

Surface Plots

Same information

x=-10:10;y=x;Z=(x.^2)’*(y.^2);surf(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Creates a surface.

surf

surfc

surfl−10

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Figure: Three-dimensional surface plot.



ExtrasExercises

Surface Plots

Same information

x=-10:.1:10;y=x;Z=(x.^2)’*(y.^2);surfc(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Creates a surface.

surf

surfc

surflFigure: Surface plot with contours (surfc).



ExtrasExercises

Surface Plots

Same information

x=-10:10;y=x;Z=(x.^2)’*(y.^2);surfl(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Creates a surface.

surf

surfc

surfl−10

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Figure: Surface plot with lighting (surfl).



ExtrasExercises

Mesh Plots

Same information

x=-10:10;y=x;Z=(x.^2)’*(y.^2);mesh(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Creates a surface.

mesh

meshc

meshz

waterfall−10

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Figure: Three-dimensional mesh plot.



ExtrasExercises

Mesh Plots

Same information

x=-10:.1:10;y=x;Z=(x.^2)’*(y.^2);meshc(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Creates a surface.

mesh

meshc

meshz

waterfall

−10

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−10

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Figure: (meshc).



ExtrasExercises

Mesh Plots

Same information

x=-10:.1:10;y=x;Z=(x.^2)’*(y.^2);meshz(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Creates a surface.

mesh

meshc

meshz

waterfall

−10

−5

0

5

10

−10

−5

0

5

10

0

2000

4000

6000

8000

10000

Figure: (meshz).



ExtrasExercises

Mesh Plots

Same information

x=-10:10;y=x;Z=(x.^2)’*(y.^2);waterfall(x,y,Z);xlabel(‘x’);ylabel(‘y’);

Creates a meshgrid.

mesh

meshc

meshz

waterfall

−10

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Figure: Waterfall plot (waterfall).



ExtrasExercises

Ideas of Note

Don’t typically save as .fig image. Only MATLAB recognizesthis format.For 3-D plots, you can rotate the image to get just thepicture you want.Comment the plot command. What are you plotting? Whyare you plotting this?You can make your plots too accurate!



ExtrasExercises

Subplots

x=1:10;subplot(1,2,1)y=x;plot(x,y);subplot(1,2,2)y=x.^2;plot(x,y);

0 2 4 6 8 101

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This is great for presentations. However, in LATEX documentscouple a figure with an internal tabular environment where thetabular entries are graphics.



ExtrasExercises

Exercises

The results and discussion of the following should be included,presented, and discussed in a LATEXdocument.

1 For x=-10:.01:10;, plot the tangent of x versus x . Whatdoes the MATLAB image depict? What behaviors are youseeing? Clearly label all axes.

2 Plot the graph of

f (x , y) = xy2e−

„x2+y2

4

«.

Use a three-dimensional graph of appropriate type andcolor-scheme to relate the contours of the function. Giveattention to the grid refinement. Use any rotations or shifts togenerate an image which is most easily related.



ExtrasExercises

Exercises, cont.

3 Plot, in the same figure, the second, fourth, and sixth Taylorexpansion approximates to f (x) = sin(x). Use different linestyles (solid, dashed, .etc.). Clearly label axes and plottedfunctions. Which is the best approximate? Thoroughly discussthese results.

4 A simple harmonic oscillator has a closed form solution ofposition x(t) as a function of time t given by

x(t) = A cos (ωt + φ) .

Choose an amplitude A, frequency ω and phase φ and plotthe behavior of a simple (undamped) harmonic oscillator. Arethere examples of undamped harmonic oscillators that you candiscover? Does it make sense to do discuss negative t?


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