Dan Witzner Hansen

 Interactive System  Data processing  Programming  Visualization  Basic element is an array  Easy vector and matrix manipulation  Lots of built in functions and code libraries  Implicit loops for arrays  Many preexisting functions  Filtering, converting, sorting, finding, means, minimum.  Lots of online material

 Avoid using too many libraries (and convert between formats)  Easy to see the connection between code and mathematics (Matrix of doubles is the primitive type)  Good plotting facilities (also for other courses)  Programming is a little different from what you have tried before – in some sense a little more primitive :D

 It has many applications in many diverse fields. (computer graphics, chemistry, biology, differential equations, economics, business,...)  It strikes a nice balance between computation and theory.  Essential the course

A matrix is a set of elements, organized into rows and columns rows columns

Patient Gender,age,height, weight

Column vector (default): Row vector:

 Addition:  Multiplication with scalar  Multiplication :  Inner product (“row times column”)  Outer product (“column times row”) v w V+w vav

>> a = 5; >> b = [ ]; >> c = [1:0.25:2]’ c = >> c(5) = -10 c = >> a*b >> c = [ ] >> a-b ans = [ ]

>> x = [2 -5]; >> norm(x) ans = Norm: Orthogonal: Orthonormal: Orthogonal and ||x||=||y||=1

 Addition, Subtraction, Multiplication

+ - addition, subtraction / ^ matrix multiplication, division, power * [ 2 x 3] * [3 x 2] = [2 x 2] [ ] define matrix.*./.^ array multiplication, division, power (arrays must be the same size) n’ transpose, complex conjugate ; separate rows, non-echo end of line.

>> c = [1 2 3; 4 5 6; 7 8 9] c =

zeros(n,m) an n x m matrix of zeros ones(n,m) an n x m matrix of ones rand(n,m) an n x m matrix of uniformly distributed numbers randn(n,m) an n x m matrix of normally distributed numbers

>> c = [1 2 3; 4 5 6; 7 8 9]; >> d = [1 1 1; 2 2 2; 3 3 3]; Res = c*d’; M = c.*d; Add = c+d;

: “all elements” of row, column or between 2 numbers. ( ) specify elements of a defined vector/matrix / A/B = (B'\A')'B'\A ^z = x^y, x or y must be a scalar. X = A\B is the solution to the equation A*X = B (more about this next week)

>> A = zeros (5,5) >> c = [ ] >> A[1,:] = c; >> diag([ ]) >> eye(3)

>> d = c(1:2,2:3) d = >> e = [d d] e = >> f = [d; d] f = Extracting part of a matrix ( ) specify elements of an array, separates dimension : entire row or column Combining Arrays Horizontal concatenation no punctuation vertical concatenation ; separates arrays Vectors of indices

>> x = [ ]; >> minx = min(x) minx = 0 >> y = sort(x) y = >> find(x) ans = >> i4 = find(x==4) i4 = 6 Functions for extracting data: min, max, mean, median, sort find

>> c = [1 2 3; 4 5 6; 7 8 9; ; 3 3 3; 4 4 4]’ c = >> size (c) 3 6 >> d = reshape(c, 3,3,2) d(:,:,1) = d(:,:,2) = >> size(d) ans = 3 3 2

>> a = [ ]; >> b = a; >> >> c = a.*b c = >> d = a*b’ d = 55 Array operations include an implicit loop through the entire array. The operation is repeated for each element in the array. The matrix operation does the linear algebra operation for the matrices.

pi i,j sqrt(-1) imaginary unit eps e-016 (small number, can also be set to a number) inf infinity (from 1/0) nan not-a-number (from 0/0)

 Commands can be entered one at a time on the command line within MATLAB  A series of commands (functions) can be entered into a dot-M file (name.m).  To run the set of command, just type the name of the dot-M file:  dot-M file namet.m  execute commands >> name [a,b,c,d] = someFunction(x,y,z);

>>[A,B]= example1('Example 1',0.1,0,5,0.2,0,5); >> help example1 % example1.m % Comment lines start with the percent sign % This is an example function which takes 3 parameters modelname is a string with the % name of the model to be used in the title line. dx and dy are the intervals for the % x and y variables, which range in value from from xmin to xmax or ymin to ymax. function [Value1,Value2]=example1(modelname,dx,xmin,xmax,dy,ymin,ymax); % Define x and y x = [xmin:dx:xmax]; y = [ymin:dy:ymax]; % Make a mesh out of x and y

 MATLAB is case sensitive:  a and A are different.  Use a ; at the end of the line to stop commands from echoing to the screen  Use   keys to scroll through previously entered commands and rerun.  Help / helpwin will tell you how to use a function and what it does. (e.g. >> help plot or >>help ops ). Help on toolbar

n = 5 for i = 1:n a(i) = i + 2; end test = 1; cnt = 1; while (test ~= 0) b(cnt) = 1/test; test = test - 0.1; cnt = cnt + 1; end For Loop default increments by 1 can define other increment for i = 1:0.1:n While Loop continues as long as boolean expression is true.

if a == 0 b = a; else b = 1/a; end if and(x > xmin,x < xmax) disp(‘x is within limits’); elseif (x > xmax) disp(‘x is too large’); elseif (x < xmin) disp(‘x is too small’); end If - elseif - else can have several sequential boolean tests can use both else and elseif or just one.

 save  creates matlab.mat with all variables in workspace  save fname  creates fname.mat with all variables  save fname var1 var2  saves only variables var1 & var2  save fname.txt var1 -ascii -tabs -double  saves var1 as a table of ASCII numbers called fname.txt

load file of ascii data in columns >> load filename data loaded into variable filename >> A = load(‘filename’); data loaded into variable A fscanf formatted ascii data fread binary data file fopen open files (for fscanf, fread ) fclose closes files

figure subplot title Xlabel,ylabel Zlabel text axis colorbar shading set gca gcf creates a new figure window defines multiple plots on a single page defines plot title text X,y,x axes label text puts text at specified coordinates defines the axis limits creates a colorbar defines type of shading (faceted, flat, interp) use to define properties of the axis or figure returns handle to current plot axis returns handle to current figure

Plot loglog semilogx semilogy bar errorbar fplot hist polar rose stairs linear plot of lines, points or symbols log-log scale plot log (x-axis) - linear (y-axis) plot log (y-axis) - linear (x-axis) plot Linear plot with error bars Function plot histogram polar coordinate plot angular histogram plot stairstep plot

plot3 fill3 contour,contour3 pcolor quiver, quiver3 stream, stream3 mesh, meshc, meshz surf, surfc, surfl surfnorm cylinder, sphere Plot lines and points in 3D space Filled 3D polygons in 3D space contour data in 2D or 3D space checkerboard color plot in 2D vector plot in 2D or 3D Stream lines in 2D or 3D Mesh surface in 3D, with contours or zero plane. Shaded surface in 3D Display surface normal vectors. Generate a cylinder or sphere.

Cosine Curve

>> example1('Example 1',0.1,0,5,0.2,0,5); >> help example1 % example1.m % Comment lines start with the percent sign % This is an example function which takes 3 parameters modelname is a string with the % name of the model to be used in the title line. dx and dy are the intervals for the % x and y variables, which range in value from from xmin to xmax or ymin to ymax. function example1(modelname,dx,xmin,xmax,dy,ymin,ymax); % Define x and y x = [xmin:dx:xmax]; y = [ymin:dy:ymax]; % Make a mesh out of x and y [X,Y] = meshgrid(x,y); % takes vectors and repeats x for every y and y for every x. % Define a function for every x, y pair Z = sin(X).*cos(Y); whos X Y Z

minz = min(min(Z)); maxz = max(max(Z)); save ex1dat X Y Z % saves variables in binary matlab format % Plotting figure; % opens a new figure window. subplot(2,3,1) % 6 plots in figure split into 2 rows and columns % 1- first plot is in top-left plot(X(1,:),Z(1,:),'r') % plot all the points in the % first column as a red line % (also b, g, y, c, m, k, w). title(['Plot 1: ' modelname]); xlabel('X'); ylabel('Z'); grid;% Plots grid lines axis([xmin-dx xmax+dx minz maxz]);

subplot(2,3,2) plot(Y(:,2),Z(:,2),'gx'); % plots points as green x's % (type 'help plot' for full list). vlab = input(‘Enter label: ‘,’s’); % get data from command line hold on; % allows you to overlay more % symbols, lines etc... plot(Y(:,4),Z(:,4),'ro'); plot(Y(:,6),Z(:,6),'m*'); title('Plot 2: plot symbols'); text(2,0.5,vlab);% add text labels xlabel('Y'); ylabel('Z'); grid; axis([ymin-dy ymax+dy minz maxz]); subplot(2,3,3) plot(Y,Z,'.-'); % plot all rows of X versus all rows of Z % -- dashed line, also -solid, : dotted, -.dash-dot % can also combine symbols and line 'o-; title('Plot 3: whole array'); xlabel('Y'); ylabel('Z'); grid; axis([ymin-2*dy ymax+2*dy 2*minz 2*maxz]);

subplot(2,3,4) pcolor(X,Y,Z); shading flat; colorbar('horiz'); title('Plot 4: pcolor'); xlabel('X'); ylabel('Y'); subplot(2,3,5) meshc(X,Y,Z); shading flat; colorbar('horiz'); title('Plot 5: meshc'); xlabel('X'); ylabel('Y'); zlabel('Z') subplot(2,3,6) surfl(X,Y,Z); shading interp; colorbar; set(gca,'Zlim',[-2 2]); view(15,20); title('Plot 6: surfl'); xlabel('X'); ylabel('Y'); zlabel('Z')

% Save data to ascii file as column data [nx,ny] = size(X); X = reshape(X,nx*ny,1); Y = reshape(Y,nx*ny,1); Z = reshape(Z,nx*ny,1); dat = [X Y Z]; keyboard; save ex1.asc dat -ascii Cosine Curve

figure(1) ; clf t = linspace(0, 2*pi, 512) ; [u,v] = meshgrid(t) ; a = -0.2 ; b =.5 ; c =.1 ; n = 2 ; x = (a*(1-v/(2*pi)).* (1+cos(u)) + c).* cos(n*v) ; y = (a*(1-v/(2*pi)).* (1+cos(u)) + c).* sin(n*v) ; z = b*v/(2*pi) + a*(1-v/(2*pi)).* sin(u) ; surf(x,y,z,y) shading interp axis off axis equal colormap(hsv(1024)) material shiny lighting gouraud lightangle(80, -40) lightangle(-90, 60) view([ ])

wholists all or specified variables currently defined whossame as who, but also gives dimensions of arrays, size in bytes and type (e.g. double, complex) clearclears all or specified variable from memory keyboardwithin a dot-M file gives command control to the keyboard. Type return to continue. ginput(N)Get N mouse clicks

 Reading and saving images easy (imread, imshow)  Often want to draw on top of an image  Hold (on/off) >> I = imread(‘GroundFloor.bmp’) >> figure(1) >> imshow(I); >> [X,Y,Z] = ginput(2) >> hold on >> plot(X,Y,’rx’,’MarkerSize’,10,’LineWidth’,3) >> hold off

 Remember the help / helpwin commands  MATLAB help pages:  describe how to use functions and gives examples  provides theory behind more complex functions  Getting Started  MATLAB Functions Listed by Category  MATLAB Functions Listed Alphabetically  MATLAB Reference Guide, User’s Guide  same as help pages, but in print -- sometimes nice to flip through when you’re not sure what you’re looking for

 Try to get Matlab running  Take a look at a Matlab primer  Watch some movies – if you have not done so already  Next week:  More linear algebra  DON’T LET THAT DISCURAGE YOU

Feel free to me at any time