University:
Boston University
Course:
CAS MA 411 | Advanced Calculus
Academic year:

2021
Views:

37

Pages:

18
Author:

Kenyon Arroyo

Introduction to MATLAB 17 If you have a group of commands that are expected to be executed repeatedly, it is convenient to save them in a file . . . >> edit mytrig.m % enter commands in editor window Select File/Save to save it as mytrig.m A script shares same memory space from which it was invoked. Define x, then use it in mytrig.m (mytrig can “see” x): >> >> a = 0.5000 b = 0.8660 Script works as if sequentially inserting the commands in mytrig.m at the >> Introduction to MATLAB • Declared with the key word 18 , with optional output parameters on the left and optional input on the right of =. All other parameters within function reside in function’s own workspace; deleted upon exiting the function. Use MATLAB editor to create file: >> edit average.m • Recommendation: saves file with name same as function name • It may be called from a script or another function • >> a= 2 >> Introduction to MATLAB 19 Scripts • Pros: - convenient; script’s variables are in same workspace as caller’s • Cons: - slow; script commands loaded and interpreted each time used - risks of variable name conflict inside & outside of script Functions • Pros: • Scope of function’s variables is confined to within function. No worry for name conflict with those outside of function. • What comes in and goes out are tightly controlled which helps when debugging becomes necessary. • Compiled the first time it is used; runs faster subsequent times. • Easily be deployed in another project. • Auto cleaning of temporary variables. • Cons: • I/O are highly regulated, if the function requires many pre-defined variables, it is cumbersome to pass in and out of the function – a script m-file is more convenient. Introduction to MATLAB >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> 20 % creates a special matrix; handy for testing matrix of (0) zeros(n,m) % creates ones(n,m) % creates matrix of (1) rand(n,m) % creates matrix of random numbers repmat(a,n,m replicates by rows and columns diag(M) % extracts the diagonals of a matrix M ) help elmat % list all elementary matrix operations ( or abs(x); % absolute value of exp(x); % to the x-th power fix(x); % rounds x to integer towards 0 log10(x); % common logarithm of x to the base 10 rem(x,y); % remainder of x/y mod(x, y); % modulus after division – unsigned rem sqrt(x); % square root of x sin(x); % sine of x; x in radians acoth(x) % inversion hyperbolic cotangent of x Introduction to MATLAB 21  Line plot  Bar graph  Surface plot  Contour plot  MATLAB tutorial on 2D, 3D visualization tools as well as other graphics packages available in our tutorial series. Introduction to MATLAB >> t = 0:pi/100:2*pi; >> y = sin(t); >> plot(t,y) 22 Introduction to MATLAB >> xlabel(‘t’); >> ylabel(‘sin(t)’); >> title(‘The plot of t vs sin(t)’); 23 Introduction to MATLAB >> y2 = sin(t-0.25); >> y3 = sin(t+0.25); >> plot(t,y,t,y2,t,y3) % make 2D line plot of 3 curves >> legend('sin(t)','sin(t-0.25)','sin(t+0.25',1) 24 Introduction to MATLAB 25 Generally, MATLAB’s default graphical settings are adequate which make plotting fairly effortless. For more customized effects, use the get and set commands to change the behavior of specific rendering properties. >> hp1 = plot(1:5) % returns the handle of this line plot >> get(hp1) % to view line plot’s properties and their values >> set(hp1, ‘lineWidth’) % show possible values for lineWidth >> set(hp1, ‘lineWidth’, 2) % change line width of plot to 2 >> gcf % returns current figure handle >> gca % returns current axes handle >> get(gcf) % gets current figure’s property settings >> set(gcf, ‘Name’, ‘My First Plot’) % Figure 1 => Figure 1: My First Plot >> get(gca) % gets the current axes’ property settings >> figure(1) % create/switch to Figure 1 or pop Figure 1 to the front >> clf % clears current figure >> close % close current figure; “close 3” closes Figure 3 >> close all % close all figures Introduction to MATLAB >> x = magic(3); % generate data for bar graph >> bar(x) % create bar chart >> grid % add grid for clarity 26 Introduction to MATLAB 27 • Many MATLAB utilities are available in both command and function forms. • For this example, both forms produce the same effect: • >> print –djpeg mybar % print as a command • >> print('-djpeg', 'mybar') % print as a function • For this example, the command form yields an unintentional outcome: • >> myfile = 'mybar'; • >> print –djpeg myfile • >> print('-djpeg', myfile) % myfile is defined as a string % as a command, myfile is treated as text % as a function, myfile is treated as a variable % i.e., ‘mybar’ is passed into print • Other frequently used utilities that are available in both forms are: • save, load Introduction to MATLAB >> Z = peaks; >> surf(Z) 28 % generate data for plot; peaks returns function values % surface plot of Z Try these commands also: >> shading flat >> shading interp >> shading faceted >> grid off >> axis off >> colorbar >> colormap(‘winter’) >> colormap(‘jet’) Introduction to MATLAB >> Z = peaks; >> contour(Z, 20) 29 % contour plot of Z with 20 contours >> contourf(Z, 20); % with color fill >> colormap('hot') % map option >> colorbar % make color bar 30 Introduction to MATLAB • Integration of cosine from 0 to π/2. • Use mid-point rule for simplicity. b m a i 1  cos( x)dx    a ih a ( i 1) h m cos( x )dx   cos(a  (i  12 )h)h i 1 mid-point of increment cos(x) h ; % range ; % # of increments ; % increment 31 Introduction to MATLAB % integration with for-loop tic m = 100; a = 0; % lower limit of integration b = pi/2; % upper limit of integration h = (b – a)/m; % increment length integral = 0; % initialize integral for i=1:m x = a+(i-0.5)*h; % mid-point of increment i integral = integral + cos(x)*h; end toc h a X(1) = a + h/2 b X(m) = b - h/2 32 Introduction to MATLAB % integration with vector form tic m = 100; a = 0; % lower limit of integration b = pi/2; % upper limit of integration h = (b – a)/m; % increment length x = a+h/2:h:b-h/2; % mid-point of m increments integral = sum(cos(x))*h; toc h a X(1) = a + h/2 b X(m) = b - h/2 Introduction to MATLAB 33 1. Write a program (with editor) to generate the figure that describe the integration scheme we discussed. (Hint: use plot to plot the cosine curve. Use bar to draw the rectangles that depict the integrated value for each interval. Save as plotIntegral.m 2. Compute the cosine integrals, from 0 to pi/2, using 10 different increment sizes (10, 20, 30, . . . , 100). Plot these 10 values to see how the solution converges to the analytical value of 1. Introduction to MATLAB a = 0; b=pi/2; % lower and upper limits of integration m = 8; % number of increments h = (b-a)/m; % increment size x= a+h/2:h:b-h/2; % m mid-points bh = bar(x,cos(x),1,'c'); % make bar chart with bars full width (1) and cyan (‘c’) hold % all plots will be superposed on same figure x = a:h/10:b; % use more points at which to evaluate cosine f = cos(x); % compute cosine at x ph = plot(x,f,'r'); % plots x vs f, in red % Compute integral with different values of m to study convergence for i=1:10 n(i) = 10+(i-1)*10; h = (b-a)/n(i); x = a+h/2:h:b-h/2; integral(i) = sum(cos(x)*h); end figure % create a new figure plot(n, integral) 34

Introduction To Matrix Laboratory Chapter 2 (Final)

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