% a matlab demonstaration by Carl de Boor to Ron's CS412, spring 99. $ highly recommended! % this is a cleaned-up and heavily edited diary of the 04feb99 session of % CS412. It is worthwhile reading through, particularly in order to catch % the changes, particularly the Vandermonde example ... >> % Matlab's characteristics: interactive (hence interpretative), matrix and vector arithmetic, handy graphics, extensible + adaptible, relying strongly on default values for easy work. >> % online information comes in many forms. Try them all! >> tour % nontechnical overviews >> demos % try the first few matlab demos, about basic stuff >> helpdesk % even gets you the official (typeset) manuals >> helpwin % this is the quick way; it gets you a second window showing % groups of commands, lets you click on groups and on specific % commands >> help mean % gives information about a specific command >> MEAN Average or mean value. For vectors, MEAN(X) is the mean value of the elements in X. For matrices, MEAN(X) is a row vector containing the mean value of each column. For N-D arrays, MEAN(X) is the mean value of the elements along the first non-singleton dimension of X. MEAN(X,DIM) takes the mean along the dimension DIM of X. Example: If X = [0 1 2 3 4 5] then mean(X,1) is [1.5 2.5 3.5] and mean(X,2) is [1 4] See also MEDIAN, STD, MIN, MAX, COV. % Basic objects: m-by-n matrices % vectors: m-by-1 or 1-by-n (ambiguity can be troublesome at times) % scalars: 1-by-1 % entries can be scalars (numbers) or characters (giving rise to strings). % INTERACTIVE: gives back results unless semicolon is used: >> a = [1 2 3] a = 1 2 3 >> a = [1 2 3]; >> % many ready-made matrices: >> zeros(3) ans = 0 0 0 0 0 0 0 0 0 >> ones(2,3) ans = 1 1 1 1 1 1 >> eye(2) ans = 1 0 0 1 >> rand(2,3) ans = 0.9501 0.6068 0.8913 0.2311 0.4860 0.7621 >> 0:4 ans = 0 1 2 3 4 >> 4:-1:0 ans = 4 3 2 1 0 >> x = linspace(0,2*pi,21) x = Columns 1 through 7 0 0.3142 0.6283 0.9425 1.2566 1.5708 1.8850 Columns 8 through 14 2.1991 2.5133 2.8274 3.1416 3.4558 3.7699 4.0841 Columns 15 through 21 4.3982 4.7124 5.0265 5.3407 5.6549 5.9690 6.2832 >> x(11) ans = 3.1416 >> % is that really pi??? Use more explicit format: >> format long >> x(11) ans = 3.14159265358979 % If you really want to know what's in the machine, use hexadecimals: >> format hex >> x(11) ans = 400921fb54442d18 >> % not so easy to read :-) even when you use a familiar number: >> 1 ans = 3ff0000000000000 % fortunately, you don't usually have to worry about how numbers are % actually represented. >> format short % many other ready-made matrices, e.g.: >> magic(4) ans = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 >> % MAKE YOUR OWN matrices, using brackets: >> A = [1 2 3; 2 4 6] A = 1 2 3 2 4 6 >> % Note use of semicolon to separate rows. Could also do >> A = [1 2 3 2 4 6] A = 1 2 3 2 4 6 >> b = [4;2] b = 4 2 % can even put together matrices to make bigger matrices: >> C = [b A] C = 4 1 2 3 2 2 4 6 >> D = [ A b;b' 1 -1] D = 1 2 3 4 2 4 6 2 4 2 1 -1 >> % the empty matrix: >> [] ans = [] >> % If you want to know more about a variable, use whos >> whos ans Name Size Bytes Class ans 0x0 0 double array Grand total is 0 elements using 0 bytes >> % Note that ans is the default variable % you can ask other questions: >> why Damian wanted it. >> why A system manager wanted it that way. >> why The tall and bald bald mathematician told me to. >> % ARITHMETIC OPERATIONS come in two flavors: >> % MATRIX, e.g., >> b'*A ans = 8 16 24 >> % POINTWISE, e.g., >> A.*A ans = 1 4 9 4 16 36 >> % basic ops: +, -, *, /, ^, \ % note the backslash: A\B tries to give you X such that A*X = B % very handy for solving a system of linear equations, data fitting, etc. % PLOTTING >> plot(A) % matlab tries to make sense of this input; given a matrix, it % plots all its columns, each as a sequence of values at integer % points, using straight line connects as the default, and % different default colors for different columns >> % here is A again, to compare against the picture: >> A A = 1 2 3 2 4 6 % Here is a more standard plot, using the point sequence x we generated % earlier: >> plot(x,sin(x)) % We can become ever more specific about the plot details: >> plot(x,sin(x),'-or') >> plot(x,sin(x),'-or','linewidth',2) >> title('plot of sine on [0,2pi]') >> xlabel('note the use of line-fill AND a marker') >> % many other plotting commands: mesh, surf, contour, etc.. >> % question from class: how about plotting just some data points? >> % For fun, here is a way of getting some points interactively: >> [x,y] = ginput >> % (I had trouble with this, eventually realizing that, when finished, % I must hit while the cursor is in the figure window.) x = 0.7097 0.6106 0.4286 0.2304 0.2166 y = 0.8655 0.3070 0.2602 0.3947 0.7281 >> plot(x,y,'kx') % question from class: % can I label those points, e.g., with their coordinates? >> help text TEXT Text annotation. TEXT(X,Y,'string') adds the text in the quotes to location (X,Y) on the current axes, where (X,Y) is in units from the current plot. If X and Y are vectors, TEXT writes the text at all locations given. If 'string' is an array the same number of rows as the length of X and Y, TEXT marks each point with the corresponding row of the 'string' array. etc >> text(x(1),y(1),sprintf('(%g, %g)',x(1),y(1))) % question from class: can I put a curve through those data? >> s = spline(x,y); >> hold on >> xx = linspace(min(x),max(x),51); >> plot(xx, ppval(s,xx)) % matlab commands come in many different ways. One way to classify them is: % % SCALAR commands, like abs or sin that expect a scalar as input and return % a scalar as output. When applied to a vector or matrix, they return % the vector or matrix obtained by applying the command to each entry. >> sin(x) ans = 0.6516 0.5734 0.4156 0.2284 0.2149 % VECTOR commands, like min, max, sum, prod, cumsum, cumprod, sort, diff % that act on vectors and return a scalar or a vector. When applied to % a matrix, they work on each of that matrix' columns (unless told % otherwise by an additional input argument). >> min(A) ans = 1 2 3 >> min(A,[],2) ans = 1 2 % (the empty matrix here is a kludge; the first version of the min command % allowed the version min(A,B) which applies min pointwise, i.e., returns % the matrix of the same size as A and B whose i,j entry is the smaller of % A(i,j) and B(i,j). In particular, the command >> min(A,2) ans = 1 2 2 2 2 2 >> % since, by another nice default: If matlab expects a matrix of a certain size and is given a scalar instead, it simply makes up the matrix of that expected size all of whose entries equal that scalar. ) >> cumprod(1:4) ans = 1 2 6 24 >> % MATRIX commands, like eig or norm , operate on matrices . % matlab commands are built-in or M-files. Tell the difference by trying to % print them on the screen using type: >> type min min is a built-in function. >> type mean function y = mean(x,dim) %MEAN Average or mean value. % For vectors, MEAN(X) is the mean value of the elements in X. For % matrices, MEAN(X) is a row vector containing the mean value of % each column. For N-D arrays, MEAN(X) is the mean value of the % elements along the first non-singleton dimension of X. % % MEAN(X,DIM) takes the mean along the dimension DIM of X. % % Example: If X = [0 1 2 % 3 4 5] % % then mean(X,1) is [1.5 2.5 3.5] and mean(X,2) is [1 % 4] % % See also MEDIAN, STD, MIN, MAX, COV. % Copyright (c) 1984-98 by The MathWorks, Inc. % $Revision: 5.13 $ $Date: 1997/11/21 23:23:55 $ if nargin==1, % Determine which dimension SUM will use dim = min(find(size(x)~=1)); if isempty(dim), dim = 1; end y = sum(x)/size(x,dim); else y = sum(x,dim)/size(x,dim); end % M-files come in two flavors: scripts and functions. % SCRIPTS are just abbreviations; calling on a script (by typing its name % on a single line) has exactly the same effect as typing in all the lines % of the script one by one. % FUNCTIONS are essentially different in that they start their own work % space. This is very convenient since the author doesn't have to worry about % what variables are already in the workspace. It does mean that the function % needs to be given input and (usually) the only thing one knows about its % activities is via the output it provides. % The function mean printed above is a typical example; it has two input % arguments, and (only) one output argument. Note that all its assignment % statements end in a semicolon since there's no use having those intermediate % results printed out. % Note the initial block of comments (identified by the percent sign). This % block is printed on the screen in response to the help command: >> help min MIN Smallest component. For vectors, MIN(X) is the smallest element in X. For matrices, MIN(X) is a row vector containing the minimum element from each column. For N-D arrays, MIN(X) operates along the first non-singleton dimension. [Y,I] = MIN(X) returns the indices of the minimum values in vector I. If the values along the first non-singleton dimension contain more than one minimal element, the index of the first one is returned. MIN(X,Y) returns an array the same size as X and Y with the smallest elements taken from X or Y. Either one can be a scalar. [Y,I] = MIN(X,[],DIM) operates along the dimension DIM. When complex, the magnitude MIN(ABS(X)) is used. NaN's are ignored when computing the minimum. Example: If X = [2 8 4 then min(X,[],1) is [2 3 4], 7 3 9] min(X,[],2) is [2 and min(X,5) is [2 5 4 3], 5 3 5]. See also MAX, MEDIAN, MEAN, SORT. >> the functions you write all should start with such a help block. % WORKSPACE: To find out the content of the current workspace, use who: >> who Your variables are: A D ans s y C a b x >> % ... i.e., all the variables we've introduced earlier. For more detail >> % use whos: >> whos x Name Size Bytes Class x 5x1 40 double array Grand total is 5 elements using 40 bytes >> % INLINE functions: there is an additional way to construct a function % helpful for quick experimentations: >> f = inline('3*x - 4') f = Inline function: f(x) = 3*x - 4 >> % You use it just like the built-in functions: >> f(4/3) ans = 0 >> % If you want them to work entrywise on matrices, you have to vectorize them: >> f = vectorize(f) f = Inline function: f(x) = 3.*x-4 % You can make up your own commands, by putting appropriate function M-files % into the current directory or into a directory in the matlab search path. % When matlab comes across a name, it checks for it: % first in the workspace, % then among the built-in functions, % then in the current directory, % then in the directories in the matlab search path. >> % language constructs: for, while, if-else, switch-case, etc. >> % for a=b, ..., end >> for j = eye(3) j' >> end ans = 1 0 0 ans = 0 1 0 ans = 0 0 1 >> while 1 disp('I must go on'), pause(1) >> end I must go on I must go on I must go on etc % to TERMINATE a run-away calculation, use CONTROL-c . >> % if condition, ...., else, ..., end >> % condition needs relational operators: ==, ~=, <, etc, &, |, ~, all, any >> % MAKING NEW MATRICES FROM OLD >> % repmat >> x = 1:3; >> repmat(x,2,3) ans = 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 >> % A(b,c) gives the matrix whose (i.j) entry is A(b(i),c(j)): >> A A = 1 2 3 2 4 6 >> A(:,2:3) >> ans = 2 3 4 6 >> A([1 1],[3 1 1]) ans = 3 1 1 3 1 1 >> % Note the use of : to indicate that ALL rows are to be taken. >> >> % reshape(A,m,n) >> reshape(A,3,2) ans = 1 4 2 3 2 6 % question from class: how can I print out the diary, right from matlab? % Close the diary, by saying >> diary off % then, assuming you are on a unix machine, you could use the standard system % command lpr (that prints to the default printer), as long as you remember % the name of the file (feb04 in our case) >> !lpr feb04 >> % NOTE the use of the exclamation point to tell matlab that this is a % system command. % If you want to edit it first, you could, in this way, evoke your favorite % editor to work on the diary file. % If you'd rather do it later, be sure to remember the directory in which the % file resides. Use to pwd command to help you: >> pwd ans = \matlab\work >> ls feb04 % I ran out of time before I could do the following Vandermonde example: use vector and matrix operations, for efficiency (avoiding for-loops since matlab is interpretive). Example: generate the Vandermonde matrix V(i,j) = t(i)^(j-1), i=1:n; j=1:k: for i=1:n for j=1:k V(i,j) = t(i)^(j-1); end end Several complaints: (i) many copies of increasingly larger matrices V are generated; response: start with a V of the right size: V = ones(n,k); % first column is already correct! (ii) if I know t(i)^(j-1), then get t(i)^j from it just by multiplying by t(i) for i=1:n for j=2:k V(i,j) = V(i,j-1)*t(i); end end (iii) since Matlab is interpretive, avoid loops by using vector and matrix operations: inner loop can be replaced by for i=1:n V(i,:) = t(i).^(0:k-1); end which is quite expensive, or by for i=1:n V(i,:) = cumprod([1,repmat(t(i),1,k-1)]); end or can replace outer loop by for j=2:k V(:,j) = V(:,j-1).*t(:); end or can replace both loops by V = repmat(t(:),1,k).^repmat(0:k-1,n,1); (which is quite expensive), or by V = cumprod( [ones(n,1) repmat(t(:),1,k-1)], 2); which is much faster, but may be hard to understand..