Applied Numerical Methods Using MATLAB. Won Y. Yang
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Sometimes we need to see the inter‐relationship between two variables. Suppose we want to plot the lowest/highest temperature, respectively, along the horizontal/vertical axis in order to grasp the relationship between them. Let us try using the following command:
>plot(temp(:,1),temp(:,2),'kx') % temp(:,2) vs. temp(:,1) in black 'x'
This will produce a point‐wise graph like Figure 1.2a, which is fine. But if you replace the third input argument by 'b‐' or just omit it to draw a PWL graph connecting the data points like Figure 1.2b, the graphic result looks clumsy, because the data on the horizontal axis are not arranged in ascending or descending order. The graph will look better if you sort the data on the horizontal axis and also the data on the vertical axis accordingly and then plot the relationship in the PWL style by running the following MATLAB statements:
Figure 1.2 Examples of graphs obtained using the ‘ plot()’ command. (a) Data not arranged – pointwise, (b) data not arranged – interpolated linearly, and (c) data arranged along the horizontal axis.
>[temp1,I]=sort(temp(:,1)); temp2=temp(I,2); >plot(temp1,temp2)
This will yield the graph like Figure 1.2c, which looks more informative than Figure 1.2b.
We can also use the ‘ plot()’ command to draw a circle.
>r=1; th=[0:0.01:2]*pi; % [0:0.01:2] makes [0 0.01 0.02 .. 2] >plot(r*cos(th),r*sin(th)) >plot(r*exp(j*th)) % Alternatively,
Note that the ‘ plot()’ command with a sequence of complex numbers as its first input argument plots the real/imaginary parts along the horizontal/vertical axis.
The ‘ polar()’ command plots the phase (in radians) and magnitude given as its first and second input arguments, respectively (see Figure 1.3a).
Figure 1.3 Graphs drawn by using various graphic commands.
>polar(th,exp(-th)) % polar plot of a spiral
Several other plotting commands, such as ‘ semilogx()’ (with a base 10 logarithmic scale for the x‐axis), ‘ semilogy()’ (with a base 10 logarithmic scale for the y‐axis), ‘ loglog()’ (with a base 10 logarithmic scale for both the x‐axis and the y‐axis), ‘ stairs()’ (stairstep graph), ‘ stem()’ (discrete graph), ‘ bar()’ /‘ barh()’ (vertical/horizontal bar graph), and ‘ hist()’ (histogram), may be used to draw various graphs (shown in Figure 1.3). Readers may use the ‘ help’ command to get the detailed usage of each one and try running the following MATLAB script “nm01f03.m” to draw various types of graphs.
%nm01f03.m: plot several types of graph th=[0:0.02:1]*pi; subplot(521), polar(th,exp(-th)) % polar graph subplot(522), semilogx(exp(th)) % with a base 10 logarithmic scale for <i>x</i>-axis subplot(523), semilogy(exp(th)) % with a base 10 logarithmic scale for <i>y</i>-axis subplot(524), loglog(exp(th)) % with a logarithmic scale for <i>x</i>-/<i>y</i>-axis subplot(525), stairs([1 3 2 0]) % stairstep graph subplot(526), stem([1 3 2 0]) % discrete graph subplot(527), bar([2 3; 4 5]) % vertical bar graph subplot(528), barh([2 3; 4 5]) % horizontal bar graph y=[0.3 0.9 1.6 2.7 3 2.4]; subplot(529), hist(y,3) % histogram subplot(5,2,10), hist(y,0.5+[0 1 2])
Moreover, the commands ‘ sprintf()’, ‘ text()’, and ‘ gtext()’ are used for combining supplementary statements with the value(s) of one or more variables to construct a string and printing it at a certain location on the existing graph. For instance, let us run the following statements:
>f=1./[1:10]; plot(f) >n=3; [s,errmsg]=sprintf('f(%1d)=%5.2f',n,f(n)) >text(3,f(3),s) %writes the text string at the point (3,f(3)) >gtext('f(x)=1/x') %writes the input string at point clicked by mouse
The command ‘ ginput()’ allows you to obtain the coordinates of a point by clicking the mouse button on the existent graph. Let us try the following commands:
>[x,y,butkey]=ginput %get the x,y coordinates & # of the mouse button or ascii code of the key pressed till pressing the ENTER key >[x,y,butkey]=ginput(n) %repeat the same job for up to n points clicked
1.1.5 Three Dimensional (3D) Graphic Output
MATLAB has several three‐dimensional (3D) graphic plotting commands such as ‘ plot3()’, ‘ mesh()’, and ‘ contour()’. ‘ plot3()’ plots a two‐dimensional (2D) valued‐function of a scalar‐valued variable; ‘ mesh()’/‘ contour()’ plots a scalar valued‐function of a 2D variable in a mesh/contour‐like style, respectively.
Readers are recommended to use the ‘ help’ command for detailed usage of each command. Try running the above MATLAB script “nm01f04.m” to see what figures will appear (Figure 1.4).
%nm01f04.m: to plot 3D graphs t=0:pi/50:6*pi; expt= exp(-0.1*t); xt= expt.*cos(t); yt= expt.*sin(t); % dividing the screen into 2x2 sections clf subplot(521), plot3(xt,yt,t), grid on %helix subplot(522), plot3(xt,yt,t), grid on, view([0 0 1]) subplot(523), plot3(t,xt,yt), grid on, view([1 -3 1]) subplot(524), plot3(t,yt,xt), grid on, view([0 -3 0]) x=-2:.1:2; y=-2:.1:2; [X,Y] = meshgrid(x,y); Z =X.̂2 + Y.̂2; subplot(525), mesh(X,Y,Z), grid on %[azimuth,elevation]=[-37.5,30] subplot(526), mesh(X,Y,Z), view([0,20]), grid on pause, view([30,30]) subplot(527), contour(X,Y,Z) subplot(528), contour(X,Y,Z,[.5,2,4.5])
Figure 1.4 3D graphs drawn by using plot3(), mesh(), and contour().
1.1.6 Mathematical Functions
Mathematical functions and special reserved constants/variables defined in MATLAB are listed in Table 1.3.
Table 1.3 Functions and variables inside MATLAB.
| Function | Remark | Function | Remark |
cos(x)
|