MATLAB for Engineering Applications Ashok Krishnamurthy Siddharth Samsi Ohio Supercomputer Center...
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MATLAB for Engineering Applications Ashok Krishnamurthy Siddharth Samsi Ohio Supercomputer Center 1224 Kinnear Road Columbus, OH 43212 http://www.osc.edu/
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Transcript of MATLAB for Engineering Applications Ashok Krishnamurthy Siddharth Samsi Ohio Supercomputer Center...
MATLAB for Engineering ApplicationsAshok Krishnamurthy
Siddharth Samsi
Ohio Supercomputer Center
1224 Kinnear Road
Columbus, OH 43212
http://www.osc.edu/
2
Intro MATLAB
Table of Contents – Day One
• Overview
• Basic Interfaces
• Arrays, Matrices, Operators
• Programming
• Data I/O
3
Intro MATLAB
Table of Contents – Day Two
• Basic Data Analysis
• Numerical Analysis
• Graphics, Data Visualization, Movies
• Inter-language Programming
Overview
5
Intro MATLAB
MATLAB
• “MATrix LABoratory”
• Powerful, extensible, highly integrated computation, programming, visualization, and simulation package
• Widely used in engineering, mathematics, and science
• Why?
6
Intro MATLAB
MATLAB’s Appeal
• Interactive code development proceeds incrementally; excellent development and rapid prototyping environment
• Basic data element is the auto-indexed array• This allows quick solutions to problems that can
be formulated in vector or matrix form• Powerful GUI tools• Large collection of toolboxes: collections of topic-
related MATLAB functions that extend the core functionality significantly
7
Intro MATLAB
MATLAB Toolboxes
Signal & Image Processing Signal Processing Image Processing Communications Frequency Domain System Identification
Higher-Order Spectral Analysis System Identification Wavelet Filter Design
Control Design Control System Fuzzy Logic Robust Control μ-Analysis and Synthesis Model Predictive Control
Math and AnalysisOptimizationRequirements Management InterfaceStatisticsNeural NetworkSymbolic/Extended MathPartial Differential EquationsPLS ToolboxMappingSpline
Data Acquisition and ImportData AcquisitionInstrument ControlExcel LinkPortable Graph Object
8
Intro MATLAB
Toolboxes, Software, & Links
9
Intro MATLAB
MATLAB System
• Language: arrays and matrices, control flow, I/O, data structures, user-defined functions and scripts
• Working Environment: editing, variable management, importing and exporting data, debugging, profiling
• Graphics system: 2D and 3D data visualization, animation and custom GUI development
• Mathematical Functions: basic (sum, sin,…) to advanced (fft, inv, Bessel functions, …)
• API: can use MATLAB with C, Fortran, and Java, in either direction
10
Intro MATLAB
Online MATLAB Resources
• www.mathworks.com/
• www.mathtools.net/MATLAB
• www.math.utah.edu/lab/ms/matlab/matlab.html
• web.mit.edu/afs/athena.mit.edu/software/matlab/
www/home.html
• www.utexas.edu/its/rc/tutorials/matlab/
• www.math.ufl.edu/help/matlab-tutorial/
• www.indiana.edu/~statmath/math/matlab/links.html• www-h.eng.cam.ac.uk/help/tpl/programs/matlab.html
11
Intro MATLAB
References
Mastering MATLAB 7, D. Hanselman and B. Littlefield,Prentice Hall, 2004
Getting Started with MATLAB 7: A Quick Introductionfor Scientists and Engineers, R. Pratap, Oxford UniversityPress, 2005.
12
Intro MATLAB
Some More Resources
MATLAB Educational sites:
http://www.eece.maine.edu/mm/matweb.html
Yahoo! MATLAB Web site:
dir.yahoo.com/Science/mathematics/software/matlab/
Newsgroup: comp.soft-sys.matlab
Basic Interfaces
14
Intro MATLAB
Main MATLAB Interface
15
Intro MATLAB
Some MATLAB Development Windows
• Command Window: where you enter commands• Command History: running history of commands which is
preserved across MATLAB sessions• Current directory: Default is $matlabroot/work• Workspace: GUI for viewing, loading and saving MATLAB
variables• Array Editor: GUI for viewing and/or modifying contents of
MATLAB variables (openvar varname or double-click the array’s name in the Workspace)
• Editor/Debugger: text editor, debugger; editor works with file types in addition to .m (MATLAB “m-files”)
16
Intro MATLAB
MATLAB Editor Window
17
Intro MATLAB
MATLAB Help Window (Very Powerful)
18
Intro MATLAB
Command-Line Help : List of MATLAB Topics
>> helpHELP topics: matlab\general - General purpose commands.matlab\ops - Operators and special characters.matlab\lang - Programming language constructs.matlab\elmat - Elementary matrices and matrix manipulation.matlab\elfun - Elementary math functions.matlab\specfun - Specialized math functions.matlab\matfun - Matrix functions - numerical linear algebra.matlab\datafun - Data analysis and Fourier transforms.matlab\polyfun - Interpolation and polynomials.matlab\funfun - Function functions and ODE solvers.matlab\sparfun - Sparse matrices.matlab\scribe - Annotation and Plot Editing.matlab\graph2d - Two dimensional graphs.matlab\graph3d - Three dimensional graphs.matlab\specgraph - Specialized graphs.matlab\graphics - Handle Graphics.…etc...
19
Intro MATLAB
Command-Line Help : List of Topic Functions
>> help matfun Matrix functions - numerical linear algebra. Matrix analysis. norm - Matrix or vector norm. normest - Estimate the matrix 2-norm. rank - Matrix rank. det - Determinant. trace - Sum of diagonal elements. null - Null space. orth - Orthogonalization. rref - Reduced row echelon form. subspace - Angle between two subspaces.
…
20
Intro MATLAB
Command-Line Help : Function Help>> help det DET Determinant. DET(X) is the determinant of the square matrix X. Use COND instead of DET to test for matrix singularity. See also cond. Overloaded functions or methods (ones with the same
name in other directories) help laurmat/det.m Reference page in Help browser doc det
21
Intro MATLAB
Keyword Search of Help Entries
>> lookfor whonewton.m: % inputs: 'x' is the number whose
square root we seektestNewton.m: % inputs: 'x' is the number whose
square root we seekWHO List current variables.WHOS List current variables, long form. TIMESTWO S-function whose output is two times its
input.
>> whos Name Size Bytes Class Attributes ans 1x1 8 double fid 1x1 8 double i 1x1 8 double
22
Intro MATLAB
startup.m• Customize MATLAB’s start-up behavior• Create startup.m file and place in:
– Windows: $matlabroot\work– UNIX: directory where matlab command is issued
My startup.m file:
addpath e:\download\MatlabMPI\srcaddpath e:\download\MatlabMPI\examplesaddpath .\MatMPIformat short gformat compact eliminates extra blank lines in outputeliminates extra blank lines in output
Variables (Arrays) and Operators
24
Intro MATLAB
Variable Basics
no declarations neededno declarations needed
mixed data typesmixed data types
semi-colon suppresses output of the calculation’s result
semi-colon suppresses output of the calculation’s result
>> 16 + 24ans = 40
>> product = 16 * 23.24product = 371.84
>> product = 16 *555.24;>> productproduct = 8883.8
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Intro MATLAB
Variable Basics
complex numbers (i or j) require no special handling
complex numbers (i or j) require no special handling
clear removes all variables;
clear x y removes only x and y
clear removes all variables;
clear x y removes only x and y
save/load are used to
retain/restore workspace variables
save/load are used to
retain/restore workspace variables
>> clear>> product = 2 * 3^3;>> comp_sum = (2 + 3i) + (2 - 3i);>> show_i = i^2;>> save three_things>> clear>> load three_things>> whoYour variables are:comp_sum product show_i >> productproduct = 54>> show_ishow_i = -1
use home to clear screen and put cursor at the top of the screen
use home to clear screen and put cursor at the top of the screen
26
Intro MATLAB
MATLAB Data•
The basic data type used in MATLAB is the double precision array
• No declarations needed: MATLAB automatically allocates required memory
• Resize arrays dynamically
• To reuse a variable name, simply use it in the left hand side of an assignment statement
• MATLAB displays results in scientific notationo Use File/Preferences and/or format function to change default
short (5 digits), long (16 digits) format short g; format compact (my preference)
27
Intro MATLAB
Variables Revisited
• Variable names are case sensitive and over-written when re-used
• Basic variable class: Auto-Indexed Array– Allows use of entire arrays (scalar, 1-D, 2-D, etc…) as
operands– Vectorization: Always use array operands to get best
performance (see next slide)
• Terminology: “scalar” (1 x 1 array), “vector” (1 x N array), “matrix” (M x N array)
• Special variables/functions: ans, pi, eps, inf, NaN, i, nargin, nargout, varargin, varargout, ...
• Commands who (terse output) and whos (verbose output) show variables in Workspace
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Intro MATLAB
Vectorization Example*
>> type slow.mtic;x=0.1;for k=1:199901 y(k)=besselj(3,x) +
log(x); x=x+0.001;endtoc;>> slowElapsed time is 17.092999 seconds.
*times measured on this laptop
>> type fast.mtic;x=0.1:0.001:200;y=besselj(3,x) + log(x);toc;>> fastElapsed time is 0.551970 seconds.
Roughly 31 times faster without use of for loop
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Intro MATLAB
Matrices: Magic Squares
This matrix is called a “magic square”
This matrix is called a “magic square”
Interestingly, Durer also dated this engraving by placing 15 and 14 side-by-side in the magic square.
Interestingly, Durer also dated this engraving by placing 15 and 14 side-by-side in the magic square.
30
Intro MATLAB
Durer’s Matrix: Creation
» durer1N2row = [16 3 2 13; 5 10 11 8];» durer3row = [9 6 7 12];» durer4row = [4 15 14 1];» durerBy4 = [durer1N2row;durer3row;durer4row];» durerBy4
durerBy4 =
16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
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Intro MATLAB
Easier Way...
durerBy4 =
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
» durerBy4r2 = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]
durerBy4r2 =
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
32
Intro MATLAB
Multidimensional Arrays>> r = randn(2,3,4) % create a 3 dimensional array filled with
normally distributed random numbers
r(:,:,1) =
-0.6918 1.2540 -1.4410
0.8580 -1.5937 0.5711
r(:,:,2) =
-0.3999 0.8156 1.2902
0.6900 0.7119 0.6686
r(:,:,3) =
1.1908 -0.0198 -1.6041
-1.2025 -0.1567 0.2573
r(:,:,4) =
-1.0565 -0.8051 0.2193
1.4151 0.5287 -0.9219
randn(2,3,4): 3 dimensions, filled with normally distributed random numbers
randn(2,3,4): 3 dimensions, filled with normally distributed random numbers
“%” sign precedes comments, MATLAB ignores the rest of the line
“%” sign precedes comments, MATLAB ignores the rest of the line
33
Intro MATLAB
Character Strings
>> hi = ' hello';>> class = 'MATLAB';>> hihi = hello>> classclass =MATLAB>> greetings = [hi class]greetings = helloMATLAB>> vgreetings = [hi;class]vgreetings = helloMATLAB
semi-colon: join verticallysemi-colon: join vertically
concatenation with blank or with “,”concatenation with blank or with “,”
34
Intro MATLAB
Character Strings as Arrays>> greetings
greetings =
helloMATLAB
>> vgreetings = [hi;class]
vgreetings =
hello
MATLAB
>> hi = 'hello'
hi =
hello
>> vgreetings = [hi;class]
??? Error using ==> vertcat
CAT arguments dimensions are not consistent.
note deleted space atbeginning of word;results in error
note deleted space atbeginning of word;results in error
35
Intro MATLAB
yo =
Hello
Class
>> ischar(yo)
ans =
1
>> strcmp(yo,yo)
ans =
1
String Functions
returns 1 if argument is a characterarray and 0 otherwise
returns 1 if argument is a characterarray and 0 otherwise
returns 1 if string arguments are thesame and 0 otherwise; strcmpi ignores case
returns 1 if string arguments are thesame and 0 otherwise; strcmpi ignores case
36
Intro MATLAB
Set FunctionsArrays are ordered sets:
>> a = [1 2 3 4 5]a = 1 2 3 4 5>> b = [3 4 5 6 7]b = 3 4 5 6 7
>> isequal(a,b)ans = 0>> ismember(a,b)ans = 0 0 1 1 1
returns true (1) if arrays are the same size and have the same values
returns true (1) if arrays are the same size and have the same values
returns 1 where a is in b and 0 otherwise
returns 1 where a is in b and 0 otherwise
37
Intro MATLAB
>> durer = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]
durer = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
>> % durer's matrix is "magic" in that all rows, columns,>> % and main diagonals sum to the same number>> column_sum = sum(durer) % MATLAB operates column-wise
column_sum = 34 34 34 34
Matrix Operations
MATLAB also hasmagic(N) (N > 2)function
MATLAB also hasmagic(N) (N > 2)function
38
Intro MATLAB
Transpose Operator>> % to get the row sums, we'll use the transpose operator>> % (an apostrophe)
>> durer'ans = 16 5 9 4 3 10 6 15 2 11 7 14 13 8 12 1
>> row_sums = sum(durer')'row_sums = 34 34 34 34
39
Intro MATLAB
Diagonal Elements
>> durerdurer = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
>> diag(durer) % diag plucks out the diagonal elementsans = 16 10 7 1
>> sum(diag(durer))ans = 34
40
Intro MATLAB
The Other Diagonal…>> durerdurer = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
>> fliplr(durer) % “flip left-right”ans = 13 2 3 16 8 11 10 5 12 7 6 9 1 14 15 4>> sum(diag(fliplr(durer)))ans = 34
41
Intro MATLAB
Matrix Subscripting>> durerdurer = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
>> diag_sum = durer(1,1) + durer(2,2) + durer(3,3)diag_sum = 33>> durer(4,4) = pidurer = 16.0000 3.0000 2.0000 13.0000 5.0000 10.0000 11.0000 8.0000 9.0000 6.0000 7.0000 12.0000 4.0000 15.0000 14.0000 3.1416
>> durer(4,4) = 1
42
Intro MATLAB
Colon Operator (Vector Creation)
>> 1:5 % use the colon operator to create row vectorsans = 1 2 3 4 5
>> 1:0.9:6 % you can vary the increment (0.9 in this case)
ans = 1.0000 1.9000 2.8000 3.7000 4.6000
5.5000
The last element is always less than or equal to the upper limit
43
Intro MATLAB
Colon Operator (Indexing)>> sum(durer(1:3,4)) % sums first three
% elements of column 4
ans =
33
>> sum(durer(:,end)) % a lone colon is ALL
% elements, end is % the last element
ans =
34
44
Intro MATLAB
The “Dot Operator”• By default and whenever possible MATLAB will
perform true matrix operations (+ - *). The operands in every arithmetic expression are considered to be matrices.
• If, on the other hand, the user wants the scalar version of an operation a “dot” must be put in front of the operator, e.g., .*. Matrices can still be the operands but the mathematical calculations will be performed element-by-element.
• A comparison of matrix multiplication and scalar multiplication is shown on the next slide.
45
Intro MATLAB
Dot Operator Example
>> A = [1 5 6; 11 9 8; 2 34 78]
A =
1 5 6
11 9 8
2 34 78
>> B = [16 4 23; 8 123 86; 67 259 5]
B =
16 4 23
8 123 86
67 259 5
46
Intro MATLAB
Dot Operator Example (cont.)>> C = A * B % “normal” matrix multiply
C =
458 2173 483
784 3223 1067
5530 24392 3360
>> CDOT = A .* B % element-by-element
CDOT =
16 20 138
88 1107 688
134 8806 390
47
Intro MATLAB
Two Division Operators
• Right divide (familiar version) a/b– What happens: a is divided by b– Right operand “goes into” left operand
• Left divide a\b– What happens: b is divided by a– Left operand “goes into” right operand– Behavior depends on operands (scalar vs. matrix)
• Both operators work with matrices (of course). More later on what is actually calculated …
• Comparison of the use of / and \ on next slide
48
Intro MATLAB
Using the Division Operators
>> x = 53.0;>> y = 22.5;
>> x/y
ans = 2.3556
>> x\y
ans = 0.4245
>> (x/y)^(-1)
ans = 0.4245
For matrix operands, A\B is the solution to
Ax = B obtained by Gaussian elimination.
Read “Arithmetic Operators + - * / \ ^ ’ ”
in “MATLAB Function Reference”:
Help Search for: division
49
Intro MATLAB
Easy 2-D Graphics>> x = [0: pi/100: pi]; % [start: increment: end]
>> y = sin(x);
>> plot(x,y), title('Simple Plot')
50
Intro MATLAB
Adding Another Curve
Line color, style, marker type, all within single quotes; type
>> doc LineSpec
for all available line properties
Line color, style, marker type, all within single quotes; type
>> doc LineSpec
for all available line properties
>> z = cos(x);
>> plot(x,y,'g.',x,z,'b-.'),title('More complicated')
51
Intro MATLAB
Lab 1
• Create a row vector called X whose elements are the integers 1 through 9.• Create another row vector called Temp whose elements are:
15.6 17.5 36.6 43.8 58.2 61.6 64.2 70.4 98.8• These data are the result of an experiment on heat conduction through an iron bar.
The array X contains positions on the bar where temperature measurements were made. The array Temp contains the corresponding temperatures.
• Make a 2-D plot with temperature on the y-axis and position on the x-axis.• The data shown in your plot should lie along a straight line (according to physics)
but don’t because of measurement errors. Use the MATLAB polyfit function to fit the best line to the data (use >> hold on; for multiple plots in same figure). In other words use polyfit to determine the coefficients a and b of the equation
T = ax + b• Lastly, we can calculate a parameter called chi-square (χ2) that is a measure of
how well the data fits the line. Calculate chi-square by running the MATLAB command that does the following matrix multiplication:
>> (Temp-b-a*X)*(Temp-b-a*X)'
52
Intro MATLAB
Lab 2• Write a MATLAB command that will generate a column vector called
theta. theta should have values from –2π to 2π in steps of π/100.• Generate a matrix F that contains values of the following functions in
the columns indicated: Column 1: cos(θ) Column 2: cos(2θ)(1 + sin(θ2) Column 3: e -0.1|θ|
• Evaluate each of the above functions for the θ values in the theta vector from above.
• Plot each of the columns of F against theta. Overlay the three plots, using a different color for each.
• Create a new column vector called maxVect that contains the largest of the three functions above for each theta. Plot maxVect against theta.
• Create a column vector called maxIndex that has the column number of the maximum value in that row.
Programming
54
Intro MATLAB
• MATLAB m-file Editor– To start: click icon or enter edit command in
Command Window, e.g., >> edit test.m• Scripts and Functions• Decision Making/Looping
– if/else– switch – for and while
• Running Operating System Commands
Outline
55
Intro MATLAB
You can save and run the file/function/script in one step by clicking here
You can save and run the file/function/script in one step by clicking here
Tip: semi-colons suppress printing, commas (and semi-colons) allow multiple commands on one line, and 3 dots (…) allow continuation of lines without execution
Tip: semi-colons suppress printing, commas (and semi-colons) allow multiple commands on one line, and 3 dots (…) allow continuation of lines without execution
m-file Editor Window
56
Intro MATLAB
Scripts and Functions
• Scripts do not accept input arguments, nor do they produce output arguments. Scripts are simply MATLAB commands written into a file. They operate on the existing workspace.
• Functions accept input arguments and produce output variables. All internal variables are local to the function and commands operate on the function workspace.
• A file containing a script or function is called an m-file• If duplicate functions (names) exist, the first in the
search path (from path command) is executed.
57
Intro MATLAB
function [a b c] = myfun(x, y)b = x * y; a = 100; c = x.^2;
>> myfun(2,3) % called with zero outputsans = 100>> u = myfun(2,3) % called with one outputu = 100>> [u v w] = myfun(2,3) % called with all outputsu = 100v = 6w = 4
Functions – First Example
Write these two lines to a file myfun.m and save it on MATLAB’s path
Write these two lines to a file myfun.m and save it on MATLAB’s path
Any return value which is not stored in an output variable is simply discarded
Any return value which is not stored in an output variable is simply discarded
58
Intro MATLAB
Example: deLaunay Triangulation
• Have a set of random (x,y) points and want to connect them together to make a triangular grid
• The deLaunay algorithm creates a set of triangles such that no (other) data points are contained within the area or perimeter of any given triangle.– Creates an “orthogonal” set of triangles– The resulting grid is useful as a coordinate system
• Used in scatter pattern analysis:– Position of debris resulting from an explosion– Establish properties of the explosion: its original location,
strength, parts specifications, …
59
Intro MATLAB
Interactive Session>> x = randn(1,12); % generates 12 normally distributed numbers
>> y = randn(1,12);
>> z = zeros(1,12); % trimesh (used below) needs three arguments
>> plot(x,y,'o');
>> tri = delaunay(x,y);
>> hold on, trimesh(tri,x,y,z), hold off; % plot triangles
>> hidden off %reveal all hidden points
>> title(‘deLaunay Triangulation')
60
Intro MATLAB
MATLAB script: mydelaunay.m
% deLaunay triangulation% ----------------------% You must have variables x, y, and z instanced% in the workspaceplot(x,y,'o');tri = delaunay(x,y);hold on, trimesh(tri,x,y,z), hold off; % plot triangleshidden off % reveal all hidden points!title(‘deLaunay triangulation')
61
Intro MATLAB
Using the mydelaunay script>> x = randn(1,12); % generates 12 normally distributed numbers
>> y = randn(1,12);
>> z = zeros(1,12); % trimesh (used below) needs three arguments
>> mydelaunay
62
Intro MATLAB
Function: Header/Help Comments
function angles = ortho(a,b,c)%ortho function input: Three vectors each with 3 elements% The output is a 3-element array containing the% angles between each pair of input vectors. The% output elements are respectively:% angle between a and b% angle between b and c% angle between a and c% Typical use or ortho is to determine if a,b,c form an% orthogonal basis set that spans 3-D space.
All initial comment lines are displayed when help is used on a function
NOTE: This function should be saved in a file named ortho.m
H1 (help 1) line displayed
when using lookfor
63
Intro MATLAB
Function: Body
anorm = norm(a); % Local Variables
bnorm = norm(b); % Calculate vector lengths
cnorm = norm(c);
ab = dot(a,b); % Calculate Dot Products
bc = dot(b,c);
ac = dot(a,c);
cosy_ab = ab/(anorm*bnorm); % Calculate cosine of
cosy_bc = bc/(bnorm*cnorm); % included angles
cosy_ac = ac/(anorm*cnorm);
angles(1) = convert2deg(acos(cosy_ab)); % Create output
angles(2) = convert2deg(acos(cosy_bc));
angles(3) = convert2deg(acos(cosy_ac));
return
64
Intro MATLAB
Using Your ortho Function
>> a = [1 2 3];>> b = [4 5 6];>> c = [7 8 9];>> ortho(a,b,c)ans = 12.9332 3.4470 16.3801>> a = [22 0 0];>> b = [0 5 0];>> c = [0 0 13];>> ortho(a,b,c)ans = 90 90 90
65
Intro MATLAB
Getting ortho Function Help>> help ortho ortho function input: Three vectors each with 3 elements The output is a 3-element array containing the angles between each pair of input vectors. The output elements are respectively: angle between a and b angle between b and c angle between a and c Typical use or ortho is to determine if a,b,c form an orthogonal basis set that spans 3-D space.
>> help sin SIN Sine. SIN(X) is the sine of the elements of X. See also asin, sind.
66
Intro MATLAB
Function Syntax Summary
• If the m-file name and function name differ, the file name takes precedence
• Function names must begin with a letter
• First line must contain function followed by the most general calling syntax
• Statements after initial contiguous comments (help lines) are the body of the function
• Terminates on the last line or a return statement
67
Intro MATLAB
Function Syntax Summary (cont.)
• error and warning can be used to test and continue execution (error-handling)
• Scripts called in m-file functions are evaluated in the function workspace
• Additional functions (subfunctions) can be included in an m-file
• Use which command to determine precedence, e.g.,
>> which titleC:\MATLAB71\toolbox\matlab\graph2d\title
68
Intro MATLAB
Variable Argument Lists
• varargin / varargout allow variable numbers of input / output function arguments
• Used only inside function m-files
• Must be declared as the last input / output argument
• Declarations must be typed in lowercase
69
Intro MATLAB
Variable Argument Lists (cont.)• Consider the following function m-file:
function myplot(x, varargin)plot(x, varargin{:})
• All input arguments beginning with the second one are collected into the variable varargin so the function call:
myplot(x.^2,'color',[.5 .7 .3],'linestyle',‘o’)
results in varargin being a 1-by-4 cell array with the values ‘color’, [.5 .7 .3], ‘linestyle’ and ‘o’
70
Intro MATLAB
Variable Argument Lists (cont.)• Consider the m-file:
function [s, varargout] = mysize(x)
nout = max(nargout,1) - 1;
s = size(x);
for k = 1:nout, varargout(k) = {s(k)}; end
• The following
>> [s,rows,cols] = mysize(rand(4,5))
returns
s = [4 5], rows = 4, cols = 5
pack all output valuesinto varargout cell array
pack all output valuesinto varargout cell array
nargout: number of outputarguments in function call
nargout: number of outputarguments in function call
71
Intro MATLAB
if/elseif/else Statement>> A = 2; B = 3;>> if A > B 'A is bigger' elseif A < B 'B is bigger' elseif A == B 'A equals B' else error('Something odd is happening') endans =B is bigger
72
Intro MATLAB
switch Statement
>> n = 8n = 8>> switch(rem(n,3)) case 0 m = 'no remainder' case 1 m = ‘the remainder is one' case 2 m = ‘the remainder is two' otherwise error('not possible') endm =the remainder is two
73
Intro MATLAB
for Loop>> for i = 2:5
for j = 3:6
a(i,j) = (i + j)^2
end
end
>> a
a =
0 0 0 0 0 0
0 0 25 36 49 64
0 0 36 49 64 81
0 0 49 64 81 100
0 0 64 81 100 121
74
Intro MATLAB
while Loop
>> b = 4; a = 2.1; count = 0;>> while b - a > 0.01 a = a + 0.001; count = count + 1; end>> countcount = 1891
75
Intro MATLAB
A Performance Tip
Input variables are not copied into the function
workspace, unless
– If any input variables are changed, the variable will be copied
– Avoid performance penalty when using large arrays by extracting only those elements that will need modification
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Intro MATLAB
MATLAB’s Search Path
• Is name a variable?
• Is name a built-in function?
• Does name exist in the current directory?
• Does name exist anywhere in the search path?
• “Discovery functions”: who, whos, what, which, exist, help, doc, lookfor, dir, ls, ...
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Intro MATLAB
Changing the Search Path• The addpath command adds directories to the MATLAB search
path. The specified directories are added to the beginning of the search path.
• rmpath is used to remove paths from the search path
>> path
MATLABPATH
E:\MATLAB\R2006b\workE:\MATLAB\R2006b\work\f_funcsE:\MATLAB\R2006b\work\na_funcsE:\MATLAB\R2006b\work\na_scriptsE:\MATLAB\R2006b\toolbox\matlab\generalE:\MATLAB\R2006b\toolbox\matlab\ops
>> addpath('c:\');>> matlabpath
MATLABPATH
c:\E:\MATLAB\R2006b\workE:\MATLAB\R2006b\work\f_funcsE:\MATLAB\R2006b\work\na_funcsE:\MATLAB\R2006b\work\na_scriptsE:\MATLAB\R2006b\toolbox\matlab\generalE:\MATLAB\R2006b\toolbox\matlab\ops
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Intro MATLAB
Common OS Commands
• ls / dir provide a directory listing of the current directory
• pwd shows the current directory
>> ls
. .. sample.m
>>
>> pwdans =e:\Program Files\MATLAB\R2006b\work>>
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Intro MATLAB
Running OS Commands
• The system command can be used to run OS commands• On Unix systems, the unix command can be used as well• On DOS systems, the corresponding command is dos
>> dos('date')The current date is: Thu 01/04/2007 Enter the new date: (mm-dd-yy) ans = 0
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Intro MATLAB
Lab 1
• Create, perhaps using for-loops, a synthetic “image” that has a 1 in the (1,1) location, and a 255 in the (128,128) location, and i + j - 1 in the i, j location. This we'll refer to as the ”diagonal gray'' image. Can you manage to do this without using for-loops?
• Display the image using (we’ll assume you placed your image in a matrix named a) image(a); colormap(gray). (Don’t worry if this doesn’t work exactly the way you expect. Colormaps can be tricky!)
• Now convert your code to a MATLAB script• Test your script to insure that it produces the same results as
the ones obtained interactively.
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Lab 2
• Write a MATLAB function that implements Newton’s iterative algorithm for approximating the square root of a number.
• The core of Newton’s algorithm is that if last is the last approximation calculated, the next (improved) approximation is given by
next = 0.5(last +(x/last)) where x is the number whose square root you seek.• Two other pieces of information are needed to implement the algorithm.
The first is an initial guess at the square root. (A typical starting value might be 1.0, say). The second is the accuracy required for the approximation. You might specify you want to keep iterating until you get an approximation that is good to 5 decimal places for example.
• Your MATLAB function should have three input arguments: x, the initial guess, and the accuracy desired. It should have one output, the approximate square root of x to the desired accuracy.
Data I/O
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Intro MATLAB
Loading and Saving Workspace Variables
• MATLAB can load and save data in .MAT format
• .MAT files are binary files that can be transferred across platforms; as much accuracy as possible is preserved.
• Load: load filename OR A = load(‘filename’)
loads all the variables in the specified file (the default name is MATLAB.MAT)
• Save: save filename variables
saves the specified variables (all variables by default) in the specified file (the default name is MATLAB.MAT)
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Intro MATLAB
ASCII File Read/Write
load and save can also read and write ASCII files with
rows of space separated values:
• load test.dat –ascii
• save filename variables
(options are ascii, double, tabs, append)
• save example.dat myvar1 myvar2 -ascii -double
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Intro MATLAB
ASCII File Read/Write (cont.)
• dlmread
M = dlmread(filename,delimiter,range);
reads ASCII values in file filename that are separated by delimiter into variable M; most useful for numerical values. The last value in a line need not have the delimiter following it.
range = [R1 C1 R2 C2] (upper-left to lower-right corner)
• dlmwrite
dlmwrite(filename,A,delimiter);
writes ASCII values in array A to file filename with values separated by delimiter
• Useful with spreadsheet data
range of data to be readrange of data to be read
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Intro MATLAB
More ASCII File Read• textread
[A, B, C, ...] = textread[‘filename’, ‘format’];
[A, B, C, ...] = textread[‘filename’, ‘format’, N];
[...] = textread[..., ‘param’, ‘value’, ...];
• The type of each return argument is given by format (C-style conversion specifiers: %d, %f, %c, %s, etc…)
• Number of return arguments must match number of conversion specifiers in format
• format string is reused N times or entire file is read if N not given• Using textread you can
– specify values for whitespace, delimiters and exponent characters– specify a format string to skip over literals or ignore fields
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Intro MATLAB
textread Example
• Data file, tab delimited:
• MATLAB m-file:
• Results:
‘param’,’value’ pairs
use doc textread for available param options
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Intro MATLAB
Import Wizard
Import ASCII and binary files using the Import Wizard. Type uiimport at the Command line or choose Import Data from the File menu.
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Intro MATLAB
Low-Level File I/O Functions
• File Opening and Closing– fclose: Close one or more open files – fopen: Open a file or obtain information about open files
• Unformatted I/O– fread: Read binary data from file – fwrite: Write binary data to a file
• Formatted I/O– fgetl: Return the next line of a file as a string
without line terminator(s) – fgets: Return the next line of a file as a string with line terminator(s) – fprintf: Write formatted data to file
– fscanf: Read formatted data from file
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Intro MATLAB
Low-Level File I/O (cont.)
• File Positioning– feof: Test for end-of-file – ferror: Query MATLAB about errors in file input
or output – frewind: Rewind an open file – fseek: Set file position indicator – ftell: Get file position indicator
• String Conversion– sprintf: Write formatted data to a string – sscanf: Read string under format control
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Intro MATLAB
File Open (fopen)/Close (fclose)• fid = fopen(‘filename’, ‘permission’);
• status = fclose(fid);
File identifiernumber
Name offile
Permissionrequested:‘r’, ’r+’‘w’, ’w+’‘a’, ’a+’
0, if successful-1, otherwise
File identifier numberor ‘all’ for all files
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Intro MATLAB
Formatted I/O• fscanf: [A, count] = fscanf(fid,format,size);
• fprintf: count = fprintf(fid, format, A,...);
Dataarray
Numbersuccessfully
read
Fileidentifiernumber
Amount of data to read:
n, [n, m], Inf
Formatspecifier
Numbersuccessfully
read
Fileidentifiernumber
Formatspecifier
Dataarray(s) to
write
fscanf and fprintf are similar to C version but vectorized
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Intro MATLAB
Format String Specification
%-12.5einitial %character width and
precision
conversionspecifier
alignment flag
Specifier Description %c Single character %d Decimal notation (signed) %e Exponential notation %f Fixed-point notation %g The more compact of %e or %f %o Octal notation (unsigned) %s String of characters %u Decimal notation (unsigned) %x Hexadecimal notation ...others...
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Intro MATLAB
Other Formatted I/O Commands• fgetl: line = fgetl(fid);
reads next line from file without line terminator
• fgets: line = fgets(fid);
reads next line from file with line terminator
• textread: [A,B,C,...] = textread('filename','format',N)
reads N lines of formatted text from file filename
• sscanf: A = sscanf(s, format, size);
reads string under format control
• sprintf: s = sprintf(format, A);
writes formatted data to a string
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Intro MATLAB
Binary File I/O• [data, count] = fread(fid, num, precision);
• count = fwrite(fid, data, precision);
Dataarray
Numbersuccessfully
read
Fileidentifiernumber
Amount to read n, [n, m],...
‘int’, ‘double’, …
array towrite
Numbersuccessfully
written
Fileidentifiernumber
‘int’, ‘double’, …
fread and fwrite are vectorized
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Intro MATLAB
File Position Commands
• feof: tf = feof(fid); tests for end of file
• fseek: status = fseek(fid, offset, origin); sets the file position
• ftell: position = ftell(fid); gets the file position
• frewind: frewind(fid);rewinds the file
• ferror: message = ferror(fid); inquire about file I/O status
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Intro MATLAB
File I/O Example
fid = fopen('asciiData.txt','r');i = 1;while ~feof(fid) name(i,:) = fscanf(fid,'%5c',1); year(i) = fscanf(fid,'%d',1); no1(i) = fscanf(fid,'%d',1); no2(i)=fscanf(fid,'%d',1); no3(i)=fscanf(fid,'%g',1); no4(i)=fscanf(fid,'%g\n'); i=i+1;endfclose(fid);
Data file MATLAB m-file to read it
MATLAB outputSince a tab counts as one character in MATLAB,
you must use spaces after the name field in the
data file (else get “Tom 1” in name output, etc...)
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Intro MATLAB
File I/O Example (Alternative)• Cell arrays (storage mechanism for dissimilar kinds of
data) offer a very flexible alternative• Avoid the nuances and pitfalls of counting spaces and
tabs• Create a cell array to store the field name by using curly
braces after variable name:
name{i} = fscanf(fid, ‘%s’, 1);
%s (string) format specifier can be used here with cell array
%s (string) format specifier can be used here with cell array
curly braces are cell array constructors
curly braces are cell array constructors
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Intro MATLAB
Specialized File I/O Commands
• hdf: HDF interface • imfinfo: Return information about a graphics file • imread/imwrite: Read/Write image from graphics file • wk1read/wk1write: Read/Write a Lotus123 WK1
spreadsheet file into a matrix • xlsread/xlswrite: Read/Write a matrix to a Excel
spreadsheet file • urlread: read data from a URL
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Intro MATLAB
uigetfile: Interactively Get a Filename
• [filename, pathname, filterindex] = uigetfile(‘Filterspec’, ‘DialogTitle’);
Example:>> f = uigetfile('*.jpg;*.bmp;*.gif;*.tif','Specify Graphics File:')
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• Multidimensional MATLAB arrays• Access elements using textual field designators• Create structures by using periods (.):
>> class.name = ‘MATLAB’;
>> class.day1 = ‘2/27/07’;
>> class.day2 = ‘2/28/07’;
>> class
class =
name: ‘MATLAB’
day1: ‘2/27/07’
day2: ‘2/28/07’
Structures
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Intro MATLAB
• Structures are arrays (no surprise)• Fields can be added one at a time:
>> class(2).name = ‘MPI’;
>> class(2).day1 = ‘TBA’;
>> class(2).day2 = ‘TBA’;
• Can also use a single statement:
>> class(2) = struct(‘name’,‘MPI’,...
‘day1’,‘TBA’,‘day2’,‘TBA’)
Manipulating Structures
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• Consider the simple structure
>> exam.name = ‘Jim Kirk’;
>> exam.score = 79;
>> exam(2).name = ‘Janice Lester’;
>> exam(2).score = 89;
>> [exam.score]
ans =
79 89
Manipulating Structures (cont.)
square brackets produce a numeric row vector
square brackets produce a numeric row vector
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Manipulating Structures (cont.)• Can also create a cell array using curly braces:
>> {exam.name}
ans =
'Jim Kirk' 'Janice Lester'
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Lab 1
• The data file DataIO_lab1.dat is a binary data file containing a 256 X 256 image. The data is stored in row order with each pixel value being a double value.
• Read the data in DataIO_lab1.dat into a 256 X 256 real array, and display it as a gray scale image. Here are some suggestions to help you:– Preallocate the array that will hold the image data.
– Use the function fopen to open the file and get a file handle.
– Since the data is binary, use the function fread to read the data; if you want you can read it 256 values at a time.
– To display the data array as an image, use the MATLAB command image– If the resulting image is rotated, you can use the transpose operator to take care of
that.
– Close the file using fclose.– Scale the array so that all pixel values are between 1 and 64
• What are you looking at? (FUN: experiment with non-gray colormaps …)
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Lab 2
• The data file DataIO_lab2.csv is an ASCII data file that consists of comma separated real values. There are 3000 rows and 16 columns of data. The first column corresponds to sampling instants and the next 15 columns correspond to vibration data collected from a shaker table.
• Write a short m-file to read the data in the DataIO_lab2.csv file and assign it to two variables: t which is a 3000 X 1 array containing the sampling instants and x which is a 3000 X 15 array containing the data on all the channels. You can use fscanf (in conjunction with fopen) or dlmread to read the data.
• Plot x(:,1) through x(:,15) against t.
• MATLAB provides another way of reading ASCII data files: textread. Use textread to read the data in DataIO_lab2.csv and assign it to the variables t and x.
Basic Data Analysis
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Basic Data Analysis
• Basic, and more advanced, statistical analysis is easily accomplished in MATLAB.
• Remember that the MATLAB default is to assume vectors are columnar.
• Each column is a variable, and each row is an observation.
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Intro MATLAB
Vibration Sensors Data
Each column is the raw rpm sensor data from a different sensor used in an instrumented engine test. The rows represent the times readings were made.
Each column is the raw rpm sensor data from a different sensor used in an instrumented engine test. The rows represent the times readings were made.
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Intro MATLAB
Plotting the Data
Note that in this case the plot command generates one time-series for each column of the data matrix
Note that in this case the plot command generates one time-series for each column of the data matrix
>> plot(rpm_raw)>> xlabel('sample number - during time slice');>> ylabel('Unfiltered RPM Data');>> title(‘3 sequences of samples from RPM sensor’)
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Average of the Data:
Applying the mean function to the data matrix yields the mean of each column
Applying the mean function to the data matrix yields the mean of each column
But you can easily compute the mean of the entire matrix (applying a function to either a single row or a single column results in the function applied to the column, or the row, i.e., in both cases, the application is to the vector).
But you can easily compute the mean of the entire matrix (applying a function to either a single row or a single column results in the function applied to the column, or the row, i.e., in both cases, the application is to the vector).
1
2
>> mean(rpm_raw)
ans = 1081.4 1082.8 1002.7
>> mean(mean(rpm_raw))
ans = 1055.6
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The mean Function
But we can apply the mean function along any dimension
But we can apply the mean function along any dimension
So we can easily obtain the row means
So we can easily obtain the row means
3
>> help mean 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]>> mean(rpm_raw, 2)
ans = 1045.7 1064.7 1060.7 1055 1045
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max and its Index
We can compute the max of the entire matrix, or of any dimension
We can compute the max of the entire matrix, or of any dimension
1 2 MAX Largest component. For vectors, MAX(X) is the largest
element in X. For matrices, MAX(X) is a row vector containing the
maximum element from each column. For N-D arrays, MAX(X) operates along
the first non-singleton dimension. [Y,I] = MAX(X) returns the indices of
the maximum values in vector I. If the values along the first non-
singleton dimension contain more than one maximal element, the index
of the first one is returned.
>> max(rpm_raw)ans = 1115 1120 1043
>> max(max(rpm_raw))ans = 1120
>> [y,i] = max(rpm_raw)y = 1115 1120 1043i = 8 2 17
max along the columnsmax along the columns
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min
>> min(rpm_raw)ans = 1053 1053 961
>> min(min(rpm_raw))ans = 961
>> [y,i] = min(rpm_raw)y = 1053 1053 961i = 22 1 22
min along each columnmin along each column
min of entire matrixmin of entire matrix
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Standard Deviation, Median, Covariance>> median(rpm_raw) % median along each columnans = 1080 1083.5 1004>> cov(rpm_raw) % covariance of the dataans = 306.4 -34.76 32.192 -34.76 244.9 -165.21 32.192 -165.21 356.25>> std(rpm_raw) % standard deviation along each columnans = 17.504 15.649 18.875>> var(rpm_raw) % variance is the square of stdans = 306.4 244.9 356.25
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Data Analysis: Histogram
HIST Histogram. N = HIST(Y) bins the elements of Y into 10 equally spaced containers and returns the number of elements in each container. If Y is a matrix, HIST works down the columns. N = HIST(Y,M), where M is a scalar, uses M bins. N = HIST(Y,X), where X is a vector, returns the distribution of Y among bins with centers specified by X. The first bin includes data between -inf and the first center and the last bin includes data between the last bin and inf. Note: Use HISTC if it is more natural to specify bin edges instead.
. . .
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Histogram (cont.)>> hist(rpm_raw) %histogram of the data
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Intro MATLAB
Histogram (cont.) >> hist(rpm_raw, 20) %histogram of the data
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Intro MATLAB
Histogram (cont.)>> hist(rpm_raw, 100) %histogram of the data
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Data Analysis: Sorting
1
2
3
>> help sort SORT Sort in ascending or descending order. For vectors, SORT(X) sorts the elements of X in ascending order. For matrices, SORT(X) sorts each column of X in ascending order. For N-D arrays, SORT(X) sorts the along the first non-singleton dimension of X. When X is a cell array of strings, SORT(X) sorts the strings in ASCII dictionary order. Y = SORT(X,DIM,MODE) has two optional parameters. DIM selects a dimension along which to sort. MODE selects the direction of the sort 'ascend' results in ascending order 'descend' results in descending order The result is in Y which has the same shape and type as X.
[Y,I] = SORT(X,DIM,MODE) also returns an index matrix I. If X is a vector, then Y = X(I). If X is an m-by-n matrix and DIM=1, then for j = 1:n, Y(:,j) = X(I(:,j),j); end
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Sorting Data (cont.)
1
>> magic(4)ans = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1
>> sort(magic(4))ans = 4 2 3 1 5 7 6 8 9 11 10 12 16 14 15 13
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Sorting Data (cont.)
2
>> magic(4) >> sort(magic(4),2)ans = ans = 16 2 3 13 2 3 13 16
5 11 10 8 5 8 10 11 9 7 6 12 6 7 9 12 4 14 15 1 1 4 14 15
>> sort(magic(4),1)ans = 4 2 3 1 5 7 6 8 9 11 10 12 16 14 15 13
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Sorting Data (cont.)
3
>> magic(4)ans = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1
>> [y i] = sort(magic(4))y = i = 4 2 3 1 4 1 1 4 5 7 6 8 2 3 3 2 9 11 10 12 3 2 2 3 16 14 15 13 1 4 4 1
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Intro MATLAB
Bin Average Filtering FILTER One-dimensional digital filter. Y = FILTER(B,A,X) filters the data in vector X with the filter described by vectors A and B to create the filtered data Y. The filter is a "Direct Form II Transposed" implementation of the standard difference equation: a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(n-nb) - a(2)*y(n-1) - ... - a(na+1)*y(n-na)
>> filter(ones(1,3), 3, rpm_raw) ans = 359 351 335.67 719 724.33 667 1088.3 1081.7 1001 1084 1091.7 1004.7 1081 1073 1006.7
This example uses an FIR filter to compute a moving average using a window size of 3
This example uses an FIR filter to compute a moving average using a window size of 3
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Filtered Data Plot
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Intro MATLAB
Fast Fourier Transform (FFT)
• fft is one of the built-in functions in MATLAB
• The fft function can compute the discrete Fourier transform of any arbitrary length sequence. fft incorporates most known fast algorithms for various lengths (e.g. power of 2)
• Not all lengths are equally fast
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Intro MATLAB
Discrete Fourier Transform Definition
•
21
0
21
0
[ ] [ ]
1[ ] [ ]
j knNN
n
j knNN
k
X k x n e
x n X k eN
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Intro MATLAB
fft and fftshift
0 2
N=11
-
1 11
0
N=11
After fftshift
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Example: FFT of sine Wave in Noise
>> fs = 1000;>> t = [0:999]*(1/fs);>> x = sin(2*pi*250*t);>> X = fft(x(1:512));>> noise = 0.8*randn(size(x));>> xn = x + noise;>> XnMag = fftshift(20*log10(abs(fft(xn(1:512)))));>> XnMagPf = XnMag(256:512);>> frq = [0:length(XnMagPf) - 1]'*(fs/length(XnMag));>> plot(frq, XnMagPf)>> xlabel('freq. (Hz)');>> ylabel('Mag. (db)');
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Frequency Spectrum
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Intro MATLAB
Lab 1
• Load the data_analysis_lab1.mat file into the MATLAB workspace. This will produce an array variable called grades containing grades on an exam between 0 and 100.
• Calculate the average and standard deviation of the grades.• Plot a histogram of the grades using 100 bins.• We want to compare the histogram with a Gaussian distribution.• Write you own MATLAB Gaussian function M-file which returns a value y
using the following formula
y=exp(-[x-m]2/2σ2)
where m is the average and σ is the standard deviation of the distribution. Your function should have input arguments x,m, and σ.
• On the histogram plot also plot a Gaussian distribution of the grades using the calculated average and standard deviation.
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Lab 2
• Load the file data_analysis_lab2.mat. Since this is a .mat file, you should be able to load it easily using the load command.
• Your workspace should now contain a single variable x. x is 3000 points long and consists of the sum of 3 sine waves. The sampling frequency is 1000 Hz.
• Plot the first 0.33 seconds of x. You may find it convenient to create a second array (say called time) that has the time values corresponding to the samples in x.
>> Fs = 1000; %Sampling frequency >> time = [0:length(x)-1]’*(1/Fs); % time index• fft is a built-in function in MATLAB. We can compute and plot the
magnitude of the FFT of x to identify the frequencies of the sine waves. >> X = fft(x);• X is a complex valued array that is the FFT of x. We can compute the
magnitude of the FFT by taking the absolute value of X. >> Xmag = abs(X); >> plot(Xmag);
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Lab 2 (cont.)
• The plot of Xmag shows 6 components, and also we have only index values not real frequency values along the abscissa. Six components show up because the FFT is evaluated over positive and negative frequencies. Also, the frequencies are “wrapped around”. We can take care of the wrap around using the fftshift function.
>> Xmag = fftshift(Xmag);• Next, we can generate a suitable frequency axis for plotting Xmag.
>> frq = [0:length(Xmag)-1]’*(Fs/length(Xmag)) – (Fs/2);
>> plot(frq, Xmag);• Can you see the 3 frequency components (in the positive freq. part
of the axis)? Zoom into the plot either using the axis command or the interactive zoom button on the figure’s toolbar and determine the frequencies of the 3 components.
Numerical Analysis
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Intro MATLAB
Overview
• IEEE double precision numbers• Numerical Linear Algebra
– Solving linear equations (Ax b)– Condition number– Matrix factorizations– Eigenvalues and eigenvectors– Singular value decomposition
• Solving ODE’s• Numerical integration• Root finding• Nonlinear optimization
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Intro MATLAB
• Fundamental data type in MATLAB is a double precision value in ANSI/IEEE Standard 754 format:
• Roundoff: eps = 2-52 ≈ 10-16
• Underflow: realmin = 2-1022 ≈ 10-308
• Overflow: realmax = (2 - eps) * 21023 ≈ 10308
• eps, realmin and realmax are built in variables in MATLAB.
IEEE Double Precision Numbers
s
s E (11 bits) f (52 bits)
A numeric value is represented as: (-1)s (1.f) 2 (E-1023)
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Intro MATLAB
Solving Linear Equations
Consider the set of equations Ax = b•A is an n x m matrix, x is an m x 1 vector and b is
an n x 1 vector
• The rank of a matrix is the number of independent rows (or columns). Rank can be checked using the MATLAB command rank
• Equations are• consistent if rank(A) = rank([A b])• independent if rank(A) = n
Existence of solution
Existence of solution
Uniqueness of solution
Uniqueness of solution
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Intro MATLAB
Linear Equations, n = m
• When A is square (i.e.,
n = m) and the equations are independent and consistent, the unique solution can be found using the \ operator.
• MATLAB finds the solution using a LU decomposition of A.
>> A = [1 2 3; 4 5 6; 7 8 0]A = 1 2 3 4 5 6 7 8 0>> b = [366; 804; 351]b = 366 804 351>> [rank(A) rank([A b])]ans = 3 3>> x = A\bx = 25 22 99
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Linear Equations, n < m
• When the number of equations is less than the number of unknowns (i.e., n < m), usually an infinite number of solutions exist.
• \ finds the solution with no more than rank(A) non-zero elements.
• pinv can be used to find the solution with min ||x||.
>> A = [2 3 4; 1 1 1]; b = [4;5];>> x = A\bx = 8 0 -3>> x1 = pinv(A)*bx1 = 7.1667 1.6667 -3.8333>> sqrt([sum(x.^2) sum(x1.^2)])ans = 8.544 8.2966
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Intro MATLAB
Example: Force Required to Move Object
Force, F
Unit mass, velocity at time t = 0 is 0. Force on object is F(t).
• F(t) = xj, j - 1 < t < j, j = 1, …, 10
• Want total distance moved in 10 s to be 1.
• Want velocity at t =10 s to be 0.
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Intro MATLAB
Example (cont.)
• This leads to the underdetermined set of equations
1 10
,
19 / 2 17 / 2 15/ 2 1/ 2,
1 1 1 1
1,
0T
Ax b
A
x x x b
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Intro MATLAB
Example (cont.)
>> x1 = A\bx1 = 0.11111 0 0 0 0 0 0 0 0 -0.11111
>> x2 = pinv(A) * b
x2 = 0.054545 0.042424 0.030303 0.018182 0.0060606 -0.0060606 -0.018182 -0.030303 -0.042424 -0.054545
>> [norm(x1) norm(x2)]ans = 0.15713 0.1101
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Linear Equations, n > m
• When there are more equations than unknowns (i.e., n > m), usually no solution exists.
• \ can be used to find the least squares solution, i.e., the x that minimizes ||Ax-b||2
>> A = [2 -1; 1 1; 6 -1];>> b = [2; 5; -5];>> [rank(A) rank([A b])]ans = 2 3>> x = A\bx = -0.094595 2.4459
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Example: Fit Polynomial to Data
Assume e is measurement noise, and that we have nmeasurements of x and y. This leads to an overdeterminedset of equations:
Assume we can model data as
0 1p
py a a x a x e
1 1 0 11
1
p
pn n p n
x x a y
x x a y
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Example (cont.)
3 2ˆ 1.64 5.3 7.69 3y x x x
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Condition of a Matrix
• Consider Ax = b. If A changes by a small amount A, how large is the change in the solution x?
• ||x||/||x|| < k(A) ||A||/||A||
(A) is the condition number of A
• (A) is calculated using the MATLAB command cond(A)
• Consider A essentially singular if (A) > 1/eps
>> A = [1 1; 1 1.01]>> b = [2; 2.01];>> x = A\bx = 1 1>> A1 = [1 1.005; 1 1.01];>> x1 = A1\bx1 = -0.01 2>> cond(A)ans = 402.01
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Matrix Factorizations: lu
lu: factors a square matrix A into the product of a permuted lower triangular matrix L and an upper triangular matrix U such that A = LU.
Useful in computing inverses, Gaussian elimination.
>> A = [1 2 -1; 1 0 1; -1 2 1];
>> [L, U] = lu(A)L = 1 0 0 1 -0.5 1 -1 1 0U = 1 2 -1 0 4 0 0 0 2>> L * Uans = 1 2 -1 1 0 1 -1 2 1
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Matrix Factorizations: chol
chol: factors a symmetric, positive definite matrix A as RTR, where R is upper triangular.
Useful in solving least
squares problems.
>> A = [2 -1; 1 1; 6 -1];>> B = A'*AB = 41 -7 -7 3>> R = chol(B)R = 6.4031 -1.0932 0 1.3435>> R'*Rans = 41 -7 -7 3
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Eigenvalues and Eigenvectors: eig
eig: computes the eigenvalues, i and eigenvectors, xi of a square matrix A..
• i and xi satisfy
Axi = i xi
• [V,D] = eig(A) returns the eigenvectors of A in the columns of V, and the eigenvalues in the diagonal elements of D.
>> A = [1 -1 0; 0 1 1; 0 0 -2];>> [V, D] = eig(A)V = 1 1 -0.10483 0 0 -0.31449 0 0 0.94346D = 1 0 0 0 1 0 0 0 -2>> [A*V(:,3) D(3,3)*V(:,3)]ans = 0.20966 0.20966 0.62897 0.62897 -1.8869 -1.8869
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Singular Value Decomposition: svdsvd: factors an n x m
matrix A as A = USVT, where U and V are orthogonal matrices, and S is a diagonal matrix with singular values of A.
• Useful in solving least squares problems.
>> A = [2 -1; 1 1; 6 -1];>> [U,S,V] = svd(A)U = -0.32993 0.47852 -0.81373 -0.12445 -0.87653 -0.46499 -0.93577 -0.052149 0.34874S = 6.4999 0 0 1.3235 0 0V = -0.98447 -0.17558 0.17558 -0.98447
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Pseudoinverse: pinv
pinv: The pseudoinverse of an n x m matrix A is a matrix B such that
BAB = B and
ABA = A
• MATLAB uses the SVD of A to compute pinv.
• Useful in solving least squares problems.
>> A = [2 -1; 1 1; 6 -1];>> B = pinv(A)B = -0.013514 0.13514 0.14865 -0.36486 0.64865 0.013514>> A*B*Aans = 2 -1 1 1 6 -1>> B*A*Bans = -0.013514 0.13514 0.14865 -0.36486 0.64865 0.013514
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More Matrix Math in MATLAB
• det(A): computes determinant
• inv(A): computes inverse
• expm(A),logm(A), sqrtm(A): computes exponential, logarithm and square root of A
• polyvalm(p,A): evaluate matrix polynomial, p(A).
• lscov(A, b, V): computes least square solution with known covariance
• lsqnonneg(A,b): non-negative least squares
• norm(A): computes matrix norm
• orth(A), null(A): finds a basis for the range and null space of A
• qr(A): orthogonal-triangular decomposition of A
• subspace(A,B): computes angle between subspaces defined by A and B
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Ordinary Differential Equations
• MATLAB has a collection of m-files, called the ODE suite to solve initial value problems of the form
M(t,y)dy/dt = f(t, y)
y(t0) = y0
where y is a vector.
• The ODE suite contains several procedures to solve such coupled first order differential equations.
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Steps in ODE Solution Using MATLAB• Express the differential equation as a set of first-order ODEs
M(t,y)dy/dt = f(t,y)
• Write an m-file to compute the state derivative
function dydt = myprob(t, y)
• Use one of the ODE solvers to solve the equations
[t, y] = ode_solver(‘myprob’, tspan, y0);
Timeindex
Solutionmatrix
ODEsolver
ODE filefor
derivatives
Solutiontime span
[t0 tf]
Initialconditions
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ODE Suite Solvers
• ode23: explicit, one-step Runge-Kutta low-order solver
• ode45: explicit, one-step Runge-Kutta medium order solver. First solver to try on a new problem
• ode113: multi-step Adams-Bashforth-Moulton solver of varying order
• ode23s: implicit, one-step modified Rosenbrock solver of order 2
• ode15s: implicit, multi-step numerical differentiation solver of varying order. Solver to try if ode45 fails or is too inefficient
Non-stiff equations Stiff equations
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Example : van der Pol Equation
• Equation is
d2x/dt2 - (1-x2)dx/dt + x = 0
• Convert to first order ODEs using
y1 = x, y2 = dx/dt
dy1/dt = y2
dy2/dt=(1-y12)y2-y1
function dydt = vdpol(t,y)
%
% van der Pol equation
mu = 2;
dydt = [y(2);mu*(1- … y(1)^2)*y(2)-y(1)];
ODE File vdpol.m
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van der Pol Equation Solution>> tspan = [0 20];>> y0 = [2; 0];>> [t, y] = ode45('vdpol', tspan, y0);>> plot(t, y(:,1), t, y(:,2), '--');
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More on ODE Solvers
• There are a number of different options in specifying the ODE file. Check HELP on odefile for details.
• odeset and odeget can be used to set and examine the ODE solver options.
• Can find events (such as max/min/zero, crossings etc.) in the solution.
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Numerical Integration
• trapz: Trapezoidal integration
• quad: Adaptive, recursive Simpson’s Rule for quadrature
• quadl: Adaptive, recursive Newton-Coates
8-panel rule
• dblquad: Double integration using quad or quadl
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Integration Example: humps Function
>> x = linspace(-1,2,150);>> y = humps(x);>> plot(x,y)>> format long>> trapz(x,y) % 5-digit accuracyans = 26.344859225225534
>> quad('humps', -1, 2) % 6-digit accuracyans = 26.344960501201232
>> quadl('humps', -1, 2) % 8-digit accuracyans = 26.344960471378968
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Root Finding and Minimization
• roots: finds roots of polynomials
• fzero: finds roots of a nonlinear function of one
variable
• fminbnd, fminsearch: finds maxima and
minima of functions of one and several variables
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Example of Polynomial Roots
p(x)=x3+4x2-7x-10
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Example of Roots for Nonlinear Functions
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Example of Function Minimization
>> p = [1 0 -2 -5];>> x = linspace(0,2,100);>> y = polyval(p,x);>> plot(x,y)>> fminbnd('x.^3-2*x-5',0,2)ans = 0.8165
>> polyval(p,ans)ans = -6.0887
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Lab 1
• Consider the set of equations Ax=b where A is an 8x8 matrix given by
A(i,j)=1.0+(|i-j|/8.0)½
• and b is a 8x1 array given by
b(i)=i• Solve for x using:
– The \ operator
– The MATLAB pinv function
– The MATLAB inv function
– LU Decomposition
• How do your answers compare?
• For best performance, evaluate the matrix A without using any for loops
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Lab 2
• Use numerical integration to integrate 1/(1+x2) from 0 to 1. The result is analytically calculated to be /4.
• Use the following three MATLAB functions:– trap()– quad()– quadl()
and compare the accuracy of your numerical result with the exact value.
• Use quad or quadl to get the most accurate result possible with MATLAB. How accurate is it?
Graphics, Data Visualization & Movies
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Overview• Plots
– Simple plots– Subplots (Multiple Axis Regions)– Mesh plots (Colored wire-frame view of surface)– Surface Plots– Patches– Contour Plots– Visualization
• Images– Indexed images– Intensity images– Truecolor images– Reading and writing images
• Movies
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Basic XY Plot
>> x = [0:pi/100:pi];
>> y = sin(x);
>> plot(x,y), title('Simple Plot')
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Multiple Curve Plots
Line color, style, marker type, all within single quotes
Line color, style, marker type, all within single quotes
>> z = cos(x);
>> plot(x,y,'g.',x,z,'b-.'), title('More Complicated')
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Plot Power: Contour & 3-D Mesh
To save:
print -djpeg myfigure.jpg
use help print for options
To save:
print -djpeg myfigure.jpg
use help print for options
>> t = 0:pi/25:pi;
>> [x,y,z] = cylinder(4*cos(t));
>> subplot(2,1,1)
>> contour(y)
>> subplot(2,1,2)
>> mesh(x,y,z)
>> xlabel('x')
>> ylabel('this is the y axis')
>> text(1,-2,0.5,...
'\it{Note the gap!}')
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Subplots
Used to display multiple plots in the same figure window, subplot(m,n,i) subdivides the window into m-by-n subregions
(subplots) and makes the ith subplot active for the current plot
>> subplot(2,3,1)>> plot(t, sin(t), 'r:square')>> axis([-Inf,Inf,-Inf,Inf])
>> subplot(2,3,3)>> plot(t, cos(t), 'g')>> axis([-Inf,Inf,-1,1])
>> subplot(2,3,5)>> plot(t, sin(t).*cos(t), 'b-.')>> axis([-Inf,Inf,-Inf,Inf])
4 5 6
2
31
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Mesh Plots• MATLAB defines a surface by the z-coordinates of points above a rectangular
grid in the x-y plane • Plot is formed by joining adjacent defining points with straight lines• Surface plots are used when matrices are too large to visualize numerically,
and also to graph functions of two variables• Use to generate a colored wire-frame view of a surface displayed in a 3-D
view• Only the lines connecting the defining points are colored
>> figure(2);>> [X,Y] = meshgrid(-16:1.0:16);>> Z = sqrt(X.^2 + Y.^2 + 5000);>> mesh(Z)
•mesh(Z) generates a wireframe view of matrix Z, where Z(i,j) define the height of a surface over the rectangular x-y grid:
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Surface Plots
•surf(Z) generates a colored faceted 3-D view of the surface.– By default, the faces are quadrilaterals, each of constant color,
with black mesh lines– The shading command allows you to control the view
>> figure(2);>> [X,Y] = meshgrid(-16:1.0:16);>> Z = sqrt(X.^2 + Y.^2 + 5000);>> surf(Z)
>> shading flat
>> shading interp
Default: shading faceted
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Surface Plots: Colormaps
>> colormap hot
>> colormap gray
>> colormap cool
>> colormap pink
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More Surface Plots
>> meshc(Z)
>> meshz(Z)
>> surfl(Z)
>> pcolor(Z)
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Patches• A patch is a graphics object which contains one or more polygons.
• The polygons don’t have to be connected• Useful for modeling real-world objects such as missiles and tanks• Use the patch function to display a patch
• One way to define a patch is to specify Faces and Vertices
Vertex 1 0 0 0Vertex 2 1 0 0Vertex 3 1 1 0Vertex 4 0 1 0Vertex 5 0.25 0.25 1Vertex 6 0.75 0.25 1Vertex 7 0.75 0.75 1Vertex 8 0.25 0.75 1
Face 1 1 2 3 4Face 2 5 6 7 8Face 3 1 2 6 5Face 4 2 3 7 6Face 5 3 4 8 7Face 6 4 1 5 8
Vertices, v Faces, f
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Patches: MATLAB code>> v = [0 0 0; 1 0 0 ; 1 1 0; 0 1 0; 0.25 0.25 1; 0.75 0.25 1;
0.75 0.75 1; 0.25 0.75 1];>> f = [1 2 3 4; 5 6 7 8; 1 2 6 5; 2 3 7 6; 3 4 8 7; 4 1 5 8];>> % Code to make top figure on previous slide>> patch('Vertices', v, 'Faces', f, 'FaceVertexCData', hsv(6),
'FaceColor', 'flat')>> view(3)>> axis square>> grid on>> clf>> % Code to make bottom figure on previous slide>> patch('Vertices', v, 'Faces', f, 'FaceVertexCData', hsv(8),
'FaceColor', ‘interp’)>> view(3)>> axis square>> grid on
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Contour Plots• Use to create, display, and label isolines determined by one or more
matrices•contour(Z) generates isolines from values given by a matrix Z and
displays it in 2-D•contour3(Z) generates isolines from values given by a matrix Z and
displays it in 3-D
>> Z = peaks;>> contour(Z,40)>>
>> Z = peaks;>> contour3(Z,40)>>
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More Contour Plots
>> Z = peaks;>> [C, h] = contour(Z, 10);>> clabel(C, h);>> title('Labeled Contour')
>> Z = peaks;>> [C, h] = contourf(Z, 10);>> title('Filled Contour')>>
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Visualization: Light• Technique for adding photo-realistic appearance to a graphical scene• Use light to create lighting effects in MATLAB in conjunction with
the following three important properties– Color– Style– Position
>> set(L1, 'Color', 'g')
>> set(L1, 'Position', [-1, -1, 1]) >> set(L1, 'Style', 'local')
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MATLAB Lighting Code
>> % This code creates the upper left figure on previous slide
>> [X, Y, Z] = sphere(64);>> h = surf(X, Y, Z);>> axis square>> reds = zeros(256, 3);>> for i=1:256 reds(i, 1) = (i-1)/255; end>> colormap(reds)>> shading interp>> L1 = light('Position', [-1, -1, -1]);>> lighting phong>> set(h, 'AmbientStrength', 0.75);>> set(h, 'DiffuseStrength', 0.5);
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Visualization: Viewpoint• Use view to specify the viewpoint by defining azimuth and elevation with
respect to the origin
x
-y
z
Azimuth
Elevation
y
• MATLAB defaults– For 2-D plots, azimuth = 0o elevation = 90o
– For 3-D plots, azimuth = -37.5o elevation = 30o
Default View >> view(-37.5, 60);
>> view(0, 90); >> view(-37.5, 90);
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Visualization: Camera Properties
>> set(gca,'CameraPosition',[-800,-800,13])Default View
>> set(gca,'CameraTarget',[0,0,2])
>> set(gca,'CameraUpVector',[0,1,0])
>> set(gca,'CameraViewAngle',30) >> set(gca,'Projection','perspective')
• Use the set command to modify parameters associated with a graphics object. In this case, the Camera Properties
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Camera Default Properties• MATLAB defaults:
– CameraPosition: Position adjusted such that the orientation of the scene is the standard MATLAB 2-D or 3-D view
– CameraTarget: Center of plot box
– CameraUpVector: y-direction for 2-D views and z-direction for 3-D views
– CameraViewAngle: Angle adjusted such that scene fills the position rectangle
– Projection: orthographic
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Indexed Images• Consists of a data matrix, I and a colormap matrix, C– C is an m-by-3 matrix, with each row specifying the R, G, and B
components of a single color– Values in C are floating point numbers in the range [0, 1]– Color of each pixel is determined by using the corresponding value
of I as an index into the colormapR G B
0
1 0
C
12
m
10
I
0.5 0.5 0.5
1 0.35 0.25
10
1
1 0
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Intensity Images• Consists of a data matrix, I, whose values represent intensities within some
range.– For double-precision data, the intensity values are in the range [0, 1], where 0
represents black, and 1 represents white. Values in between 0 and 1 represent shades of gray
• Use the following to display intensity images. >> imagesc(I, [0, 1]); colormap(gray);
– The second input argument [0, 1] to imagesc specifies the desired intensity range. I is displayed by first mapping the first value in the range to the first colormap entry, and second value in the range to the last colormap entry. Values in between are mapped linearly.
• To automatically map the minimum value in I to the first colormap entry, and the maximum value in I to the last colormap entry, do the following.
>> imagesc(I); colormap(gray);
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Truecolor Images (RGB Images)• Consist of a m-by-n-by-3 data array, I, containing the R, G, and B components for each
individual pixel– I(:, :, 1) is the red component of the image– I(:, :, 2) is the green component of the image– I(:, :, 3) is the blue component of the image
• To display a truecolor image, do the following >> image(I)
– Truecolor images do not use colormaps
HumVee(:, :, 1)
HumVee(:, :, 2)
HumVee(:, :, 3)
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Summary: Commands to Display Images
Use the following to display an Indexed image.
>> image(I); colormap(map)
Use the following to display an Intensity image>> imagesc(I); colormap(map);
Use the following to display a Truecolor image
>> image(I);
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Reading Images
• MATLAB can read images of various formats including
– BMP, HDF, JPEG, PCX, TIFF, XWD• Use function imread to read image files
–imread reads indexed, intensity, and truecolor images
– Images are read into a uint8 matrix of appropriate size
•imread automatically determines the format of the image based on information in the header
– You can specify a format as an optional second argument
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MATLAB Code for Reading Images
>> Crusader = imread(’Crusader.jpg');
>> image(Crusader)
>> whos Crusader
Name Size Bytes Class
Crusader 186x250x3 139500 uint8 array
Grand total is 139500 elements using 139500 bytes
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Writing Images
• MATLAB can write images of various formats including the following– BMP, HDF, JPEG, PCX, TIFF, XWD
• Use function imwrite to write image files– imwrite writes indexed, intensity, and truecolor images– Images are written as a uint8 matrix (converted if necessary) of appropriate size
along with colormaps (if necessary) and headers•imwrite determines the format from extension of filename. You can specify an
optional format if extension is absent or to force a particular format
Use imfinfo(filename) to get information on an image file
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Writing Images: MATLAB code>> Abrams = imread(‘Abrams.jpg');
>> image(Abrams)
>> whos Abrams
Name Size Bytes Class
Abrams 511x640x3 981120 uint8 array
Grand total is 981120 elements using 981120 bytes
>> % Write out tank as gray image
>> AbramsGray = rgb2gray(Abrams);
>> colormap gray;
>> image(AbramsGray)
>> imwrite(AbramsGray, gray, 'Abrams.bmp');
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Creating Movies in MATLAB• MATLAB movies are stored in an array of movie frames. For example, in a
movie array M, the ith frame is M(i).
• A movie frame is a structure having the fields "cdata" and "colormap" which contain the image data in a uint8 matrix and the colormap in a double matrix. Movie frames can be created by following commands– getframe returns a movie frame by taking a snapshot of the current
axis. For example, F=getframe;– im2frame converts an indexed image into movie format. For example, F=im2frame(A,MAP) returns the frame as an indexed image matrix A and a colormap MAP.
A MATLAB movie array can be played back by the movie command. movie(M,N,FPS) plays the movie M for N times at FPS frames per second. The default if FPS is omitted is 12 fps.
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Movie Preparation & Play
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MATLAB movie ↔ AVI format
• movie2avi(M,FILENAME,PARAM,VALUE,PARAM,VALUE...) creates an AVI file from the MATLAB movie M using the specified parameter settings. Available parameters are– FPS - The frames per second for the AVI movie. The default is 15 fps. – COMPRESSION - A string indicating the compressor to use. For example, ‘Indeo3’,
‘Indeo5’, ‘Cinepak’, ‘MSVC’, or ‘None’. – QUALITY - A number between 0 and 100. Higher quality numbers result in higher
video quality and larger file sizes. The default is 75. – KEYFRAME - For compressors that support temporal compression, this is the number
of key frames per second. The default is 2 key frames per second.– COLORMAP - An M-by-3 matrix defining the colormap to be used for indexed AVI
movies. – VIDEONAME - A descriptive name for the video stream. This parameter must be no
greater than 64 characters long. The default name is the filename.
• M = aviread(FILENAME) reads the AVI movie FILENAME into a movie array M.
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figure(1)numframes=16;
% gca: “get current axis”; returns handle to the% current axes for the current figureset(gca, ‘NextPlot', 'replacechildren')axis equal % fix the axes for k=1:numframes plot(fft(eye(k+16))); % eye: Identity matrix A(k)=getframe;endmovie(A)
Example that Illustrates the Use of Movies to Visualize the Various Powers of the N-th Root of
Unity, exp(2pi / n)
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aviread
• Refer to Example 1 in movie documentation (doc
movie) to create frame array F using getframe then:
>> movie2avi(F, ‘wave.avi’)
>> M = aviread(‘wave.avi’)
>> movie(M)
•This can create some very large files!
•aviread can only read Type-2 Digital Video AVI files
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Lab 1
Show views from various angles of the surface defined by the following function: z = |x| * exp(-x2-y2) * y in MATLAB, use: Z = abs(X) .* exp(-X .^ 2 – Y .^ 2) .* Y; Define an x-y grid with x and y in [-2, 2] with increments of 0.2. Show a total of 6 views in the same figure. Camera parameters for each view should appear in the title for the sub-image. Detailed Instructions Use meshgrid to define the grid ([X, Y] = meshgrid(-2:0.2:2, -2:0.2:2);) Use “subplot” to get multiple plots in the same figure Use surfc for the first three plots, and surf for the remaining three plots Use the “shading” and “view” functions to achieve desired appearance and view Set the title for each view using the “title” function Use “axis tight” to make the surface fill the extent of the axes for each view Use “axis vis3d” to preserve aspect ratio for different views Set the size and position of the figure window using the “set” function.
Inter-Language Programming
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MEX Basics
• MEX stands for MATLAB EXecutable
• MEX files are C and FORTRAN programs that are callable from MATLAB after compiling
• Why?
– Pre-existing C/FORTRAN programs can be called from MATLAB without rewriting codes in MATLAB
– Computations that do not run fast enough in MATLAB, such as for loops, can be coded in C or FORTRAN for efficient implementation.
– Access to hardware such as A/D, D/A converters, GPIB hardware, serial/parallel port, etc.
– Protect intellectual property
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MEX ProcedureProcedures for working with MATLAB’s MEX mechanism
1. Prepare the C or Fortran MEX program according to MATLAB external interfacing rules
2. Compile the C or FORTRAN MEX program using MATLAB command “mex”
3. mex in turn makes use of external C or FORTRAN compilers
4. Call the compiled MEX function in the same way as calling any MATLAB function
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MEX Include File : mex.h
• Must be included in all MEX Files source code
• Defines the prototypes of all mex* API functions (e.g. mexErrMsgTxt() in our example )
mex* functions used to set up program tasks
• Includes matrix.h, which in turn defines the prototypes of all mx* API functions (e.g. mxGetN() in our example)
mx* functions used for data variables
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Gateway Function : mexFunction()
void mexFunction (int nlhs,
mxArray *plhs[],
int nrhs,
const mxArray *prhs[])
• Equivalent to main() in C programs
• Has 4 arguments
Number of LHS arguments
Number of RHS arguments
Pointer to array of of LHS argument
pointers
Pointer to array of of RHS argument
pointers
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Sample Problem: Scaling• Simple example: A C function that takes its input,
multiplies each element by 2, and then returns the “scaled-up” values
• Straightforward C task chosen so that MEX requirements can be emphasized
• Resulting C MEX file is general purpose in nature
• How the function might be used in MATLAB:
>> [a b] = timestwo(x,y);
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mexFunction in Scaling Example
>> a = timestwo(x)
nrhs = 1
prhs x (matrix)
nlhs = 1
plhs a
void mexFunction( int nlhs, mxArray *plhs[],int nrhs, const mxArray *prhs[])
In MATLAB:
In timestwo.c:
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MEX Procedure1. Compile timestwo.c using “mex” command in MATLAB
2. Use timestwo.mexw32 like any MATLAB command
>> mex timestwo.c>> dir *.mexw32
timestwo.mexw32
Compiling mex program, timestwo.c
A platform-specific binary is generated after compiling
>> [a b] = timestwo(5,6)a = 10b = 12
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Intro MATLAB
timestwo.cinclude “mex.h”
Gateway function and
its 4 arguments
Processing the input
parameters
Dynamically allocate
memory for the output
array
#include “mex.h” // do not forget this
void mexFunction(int nlhs, mxArray *plhs[],
int nrhs, const mxArray *prhs[]) {
double *p,*n; // pointers to output and input arrays
int i,j,m,n; // indices and dimension variables
if (nrhs != nlhs) // Error Check
mexErrMsgTxt(“Number of input and output args differ");
for(i = 0; i < nrhs; i++) {
m = mxGetM(prhs[i]); n = mxGetM(prhs[i]); // get dims
plhs[i] = mxCreateDoubleMatrix(m,n,mxREAL);
data1 = mxGetPr(prhs[i]); // retrieve input
data2 = mxGetPr(plhs[i]); // create pointer to output
for(j = 0; j < m*n; j++ {
data2[j] = 2 * data1[j];
}
}
}
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