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MATLAB - STUDENT EDITION!!!!
WHY USE MATLAB? 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
WHY IS MATLAB RELEVANT FOR YOU? 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
WHY IS LINEAR ALGEBRA INTERESTING? 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
WHAT IS A MATRIX?
A matrix is a set of elements, organized into rows and columns
rows
columns
mnm
n
aa
aaa
A
1
11111
MATRIX EXAMPLES
mnm
n
aa
aaa
A
1
11111
Patient
Gender,age,height, weight
VECTOR
nn
T Raaaa 121 ),...,,(
),( a 21 aa
Column vector (default):
12
1
n
n
R
a
a
a
a
Row vector:
VECTOR OPERATORS
Addition: Multiplication with scalar
Multiplication : Inner product (“row times column”)
Outer product (“column times row”)
)( 2211 nn
n
iii
T yxyxyxyxyx
),,( 2211 nn yxyxyxyx vw
V+w
Raaxaxxxaax ),,(),( 2121 v
av
€
yxT =
y1x1 y1x2 K y1xny2x1 y2x2 y2xn
ymx1 ymx2 ymxn
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
xyyx TT
TT xyyx
MATLAB: SCALARS, VECTORS
>> a = 5;>> b = [1 2 3 2 1];>> c = [1:0.25:2]’
c =1.001.251.501.752.00
>> c(5) = -10c = -10.00
1.251.501.752.00
>> a*b>> c = [5 10 15 10 5]
>> a-b
ans = [4 8 12 8 4]
NORM / LENGTH AND ORTHOGONALITY
222
21x n
T xxxxx >> x = [2 -5];>> norm(x)ans = 5.3852
)90cos(0 oT yx
2-Norm:
Orthogonal:
Orthonormal:
Orthogonal and ||x||=||y||=1
LOOK AT EXERCISE 1
MATRIX: BASIC OPERATIONS
Addition, Subtraction, Multiplication
hdgc
fbea
hg
fe
dc
ba
hdgc
fbea
hg
fe
dc
ba
dhcfdgce
bhafbgae
hg
fe
dc
ba
OPERATORS AND MATRIX OPERATIONS+ - 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.
ENTERING DATA - 2D MATRICES
>> c = [1 2 3; 4 5 6; 7 8 9]
c = 1 2 34 5 67 8 9
USEFUL MATRIX INITIALIZERS
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
SIMPLE OPERATORS IN MATLAB
>> 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;
OPERATORS AND MATRIX OPERATIONS
: “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)
INITIALIZING MATRICES
>> A = zeros (5,5)>> c = [5 10 15 10 5]
>> A[1,:] = c;
>> diag([1 2 3 4])1 0 0 00 2 0 00 0 3 00 0 0 4
>> eye(3)1 0 00 1 00 0 1
ACCESSING DATA - 2D ARRAYS
>> d = c(1:2,2:3)d =
4 75 8
>> e = [d d]
e = 4 7 4 75 8 5 8
>> f = [d; d]
f =4 75 84 75 8
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
ACCESSING DATA - 2D ARRAYS>> x = [3 2 0 1 5 4];>> minx = min(x)
minx = 0
>> y = sort(x)
y =0 1 2 3 4 5
>> find(x)
ans =1 2 4 5 6
>> i4 = find(x==4)
i4 =6
Functions for extracting data:
min, max, mean, median, sort
find
ARRAY VERSUS MATRIX OPERATIONS
>> a = [1 2 3 4 5]; >> b = a;>>>> c = a.*bc =
1 4 9 16 25
>> d = a*b’d =
55
Array operations include animplicit loop through theentire array.
The operation is repeated foreach element in the array.
The matrix operation does thelinear algebra operation forthe matrices.
SOME PREDEFINED VARIABLES
pi 3.14159265
i,j sqrt(-1) imaginary unit
eps 2.2204e-016 (small number, can also be set to a number)
inf infinity (from 1/0)
nan not-a-number (from 0/0)
THE DOT-M FILE
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);
THE SHAPE OF MATLAB PROGRAM>>[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 yx = [xmin:dx:xmax];y = [ymin:dy:ymax];
% Make a mesh out of x and y
Call to the example1 function
Comments and help file information
Definition of a function same name as the filenameNot always needed – scripts also possible
SOME STARTER TIPS 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
BASICS OF PROGRAMMING IN MATLAB
n = 5for 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 Loopdefault increments by 1can define other incrementfor i = 1:0.1:n
While Loopcontinues as long as boolean expression is true.
BASICS OF PROGRAMMING IN MATLAB
if a == 0b = a;
elseb = 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 - elsecan have several sequentialboolean tests
can use both else and elseifor just one.
SAVING DATA
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
INPUTTING DATA FROM FILE
load file of ascii data in columns>> load filenamedata loaded into variable filename>> A = load(‘filename’);data loaded into variable A
fscanf formatted ascii datafread binary data filefopen open files (for fscanf, fread)fclose closes files
PLOTTING
PLOTTING - GENERAL COMMANDS
figuresubplottitleXlabel,ylabelZlabeltextaxiscolorbarshadingsetgcagcf
creates a new figure windowdefines multiple plots on a single pagedefines plot title text X,y,x axes label text
puts text at specified coordinatesdefines the axis limitscreates a colorbardefines type of shading (faceted, flat, interp)use to define properties of the axis or figurereturns handle to current plot axisreturns handle to current figure
LOTS OF 2D GRAPHICS FUNCTIONS
Plotloglogsemilogxsemilogybarerrorbarfplothistpolarrosestairs
linear plot of lines, points or symbolslog-log scale plotlog (x-axis) - linear (y-axis) plotlog (y-axis) - linear (x-axis) plotLinear plot with error barsFunction plothistogrampolar coordinate plotangular histogram plotstairstep plot
3D GRAPHICS FUNCTIONS
plot3fill3contour,contour3pcolorquiver, quiver3stream, stream3mesh, meshc, meshz
surf, surfc, surfl
surfnormcylinder, sphere
Plot lines and points in 3D space Filled 3D polygons in 3D spacecontour data in 2D or 3D spacecheckerboard color plot in 2Dvector plot in 2D or 3DStream lines in 2D or 3DMesh surface in 3D, with contours
or zero plane.Shaded surface in 3DDisplay surface normal vectors.Generate a cylinder or sphere.
EXAMPLE: DOT-M FILES, FUNCTIONS, PLOTTING
Cosine Curve
EXAMPLE: DOT-M FILES, FUNCTIONS, PLOTTING
>> 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 yx = [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 pairZ = sin(X).*cos(Y);whos X Y Z
EXAMPLE: DOT-M FILES, FUNCTIONS, PLOTTINGminz = min(min(Z));maxz = max(max(Z));
save ex1dat X Y Z % saves variables in binary matlab format
% Plottingfigure; % 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 linesaxis([xmin-dx xmax+dx minz maxz]);
EXAMPLE: DOT-M FILES, FUNCTIONS, PLOTTING
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 labelsxlabel('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]);
EXAMPLE: DOT-M FILES, FUNCTIONS, PLOTTING
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')
EXAMPLE: DOT-M FILES, FUNCTIONS, PLOTTING
% 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
SIMPLE?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([-150 10])
SOME OTHER USEFUL COMMANDS
who lists all or specified variables currently defined
whos same as who, but also gives dimensions of arrays, size in bytes and type (e.g. double, complex)
clear clears all or specified variable from memory
keyboard within a dot-M file gives command control to the keyboard. Type return to continue.
ginput(N) Get N mouse clicks
IMAGES, PLOTS & MOUSE INPUTS
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
LEARNING MORE
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
YOUR FIRST ASSIGNMENT
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
MORE QUESTIONS?
Feel free to e-mail me at any time
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