Talk by Daniel Zwillinger
Raytheon
13 June 2001

This file contains: (1) a list of slide titles (hyperlinked), and (2) the contents of each slide.
A pdf version of the talk is available here

Matlab Overview

Daniel Zwillinger, PhD

13 June 2001

Sudbury 1-1-623 telephone 978-440-1660
Daniel_I_Zwillinger@raytheon.com

http://www.mathtable.com/zwillinger/talks/20010613/

Abstract

This course will expose users to the Matlab software language. The language will be described at a high level. Matlab's capabilities (data types, programming constructs, functions, toolboxes, and graphics) and how users tend to use them will be discussed. A discussion of when to use Matlab will be given. A demonstration will be given.

Outline

Classes of computer languages
Matlab: A linear algebra language
Programming
Functions
Graphics
Local facilities
Conclusion

Classes of computer languages

Database: mySQL, Oracle, ...
Numerical: Basic, FORTRAN, C, $\fbox {Matlab}$ , ...
Symbolic: Lisp, Maple, Mathematica, ...
Web: HTML, PHP, XML, ...
Low level: assembler, machine language, ...
Text processing: sed, awk, perl, python, ...
Typesetting: Troff, TEX, L^ATEX, ...

Matlab: MATrix LABoratory

First introduced at Stanford University in 1979
Initially an interactive shell to FORTRAN routines
MathWorks was formed to market Matlab
Webb & Wilson, Dr. Dobb's Journal, Jan 1999
``Like every other scripting language, Matlab began as a simple way to do powerful things, and it has become a not-so-simple way to do very powerful things.''

Matlab: MATrix LABoratory

Powerful engineering environment and language, useful for problem solving, data analysis, modeling and visualization
Runs on nearly every operating system
More than 400 books in 17 languages
Diverse and powerful built-in functions
- linear algebra
- polynomials
- Fourier analysis
- differential equations
- GUI builder
- Movies & sound

Matlab: A linear algebra language

Underlying data structure is a multi-dimensional array (e.g., scalar, vector, or matrix)

2 $\begin{bmatrix} 4 & 5 & 6\end{bmatrix}$ $\begin{bmatrix} 8\\ 9\\ \end{bmatrix}$ $\begin{bmatrix} 11 & 12 & 13\\ 21 & 22 & 23\\ \end{bmatrix}$

Programming I

1.: Use interactively or as programming language
(interpreted or compiled)
2.: Can link to other languages (e.g., compiled C code)
3.: Large number of included examples
4.: Conditionals, looping, functions, globals, etc
5.: Sophisticated debugger, profiler
6.: GUI development tools

Programming II

1.

Object oriented capabilities

2.

Variable number of input and output arguments

3.

Case sensitive variables

4.

Operator precedence

(a): arithmetic (+, -, *, /, etc.)
(b): relational (==, <, >, etc)
(c): logical (AND, OR, NOT, etc)

5.

Memory partitioned into ``workspaces''

6.

``Toolboxes'' contain collections of functions

7.

(Optional) Space allocation for data structures

Many tools for matrix manipulation I

>> A=zeros(2,2)
A =
     0     0
     0     0
>> B=ones(3,2)
B =
     1     1
     1     1
     1     1
>> C=eye(3)
C =
     1     0     0
     0     1     0
     0     0     1
>> D=[1 2; 3 4]
D =
     1     2
     3     4

Many tools for matrix manipulation II

>> s= 1:4
s =
    1    2    3    4
>> s2= 1:2:7
s2 =
    1    3    5    7
>> s3= 1:.5:2
s3 =
  1.0000  1.5000  2.0000

>> r=rand(2,3)
r =
   0.47   0.85   0.20
   0.42   0.53   0.67
>> r(:,1)
ans =
   0.47
   0.42
>> r(:,1)'
ans =
   0.47   0.42

Many tools for matrix manipulation III

>> A=zeros(2,2);
>> B=ones(3,2);
>> C=[ [A;B], [B+5;A-7] ]
C =
     0     0     6     6
     0     0     6     6
     1     1     6     6
     1     1    -7    -7
     1     1    -7    -7

>> C(:,[1 4])
ans =
     0     6
     0     6
     1     6
     1    -7
     1    -7

Functions extend naturally to higher dimensional objects

>> log( 1 )
ans =
     0
>> log( [1 2] )
ans =
         0    0.6931
>> log( [1 2; 0 NaN] )
Warning: Log of zero.
ans =
         0    0.6931
      -Inf       NaN

Function examples

Say u=[1 2 3], then

Input		Output
`u<3`		`[1 1 0]`
`all(u<3)`		`0`
`any(u<3)`		`1`
`find(u<3)`		`[1 2]`

Easy high level manipulations

Example: find change in eigenvalues when the identity matrix is slightly perturbed

>> a = eye(4) + 0.01*rand(4,4) 
a =
    1.0095    0.0089    0.0082    0.0092
    0.0023    1.0076    0.0044    0.0074
    0.0061    0.0046    1.0062    0.0018
    0.0049    0.0002    0.0079    1.0041
>> eig(a)
ans =
   1.0232          
   1.0009 + 0.0046i
   1.0009 - 0.0046i
   1.0023

Linear algebra operations

To solve Ax=b (when A is invertible), write
     x=inv(A) * b
or
     [L,U,P]=lu(A)
     x=inv(U) * inv(L) * P * b
or
     x=Ab
or
     ...

Solving systems of equations (Ax=b)

Write x=Ab (even if A is not invertible!)

>> A=rand(3,2)
A =
    0.7095    0.1897
    0.4289    0.1934
    0.3046    0.6822

>> b=rand(3,1)
b =
    0.3028
    0.5417
    0.1509
>> soln=A\b
soln =
    0.6387
   -0.0121

Various matrix extensions

$\bullet$ Sparse matrices

>> A=speye(100000,100000);
>> A2=2*A;
>> A2(4,5)=5;
>> nnz(A2)
ans =
       100001

$\bullet$ Multidimensional arrays

>> r=rand(2,2,3)
r(:,:,1) =
    0.1389    0.1987
    0.2028    0.6038
r(:,:,2) =
    0.2722    0.0153
    0.1988    0.7468
r(:,:,3) =
    0.4451    0.4660
    0.9318    0.4186

Vectorized operations are fast

Need a vector of $\sin(t)$ for $0\le t \le 10$

>> i=0;
>> for t=0:0.001:10
      i=i+1;
      y(i)=sin(t);
   end
>> time1=toc
time1 =
    0.1936

>> t=0:0.001:10;
>> y=sin(t);
>> time2=toc
time2 =
    0.0111
>> time1/time2
ans =
   17.4677

Special variables

`ans`	most recent result
`eps`	machine epsilon
`flops`	total floating point ops during session
`i,j`	$\sqrt{-1}$
`inf`	$\infty$
`NaN`	not-a-number
`pi`	$\pi$
`realmax`	largest positive floating point number
`realmin`	smallest positive floating point number

Graphics I (two-dimensional graphics)

>> x=0:.1:10;
>> y=sin(x);
>> plot(x,y)

$\resizebox {1in}{!}{\includegraphics{fig.matlab.ex1.eps}}$

>> data=[2 6 4];
>> text={'a','b','c'};
>> pie(data,text)

$\resizebox {1in}{!}{\includegraphics{fig.matlab.ex2.eps}}$

Graphics II (three-dimensional graphics)

>> [X,Y]=... 
    meshgrid(-2:.2:2,-2:.2:3);

>> Z = X.*exp(-X.^2-Y.^2);
>> [C,h] = contour(X,Y,Z);
>> clabel(C,h)

$\resizebox {1in}{!}{\includegraphics{fig.matlab.ex4.eps}}$

>> t=0:.1:10;
>> x=sin(t);
>> y=cos(t);
>> z=x'*y;
>> meshc(x,y,z);

$\resizebox {1in}{!}{\includegraphics{fig.matlab.ex3.eps}}$

Graphics III (matrix visualization)

>> format +
>> A=random(5,15)-1/2
A =
++--+++-+++++-+
+-+---+-++-+++-
-++--+-+--+-+--
+-+--+---+++---
-+-++-----+--+-

>> B=zeros(10,10)
>> for i=1:9
>>    B(i,i+1)=2;
>> end
>> spy(B)

$\resizebox {1in}{!}{\includegraphics{fig.matlab.ex5.eps}}$

Graphics IV

Dozens of graphic styles
Lighting schemes
- ambient light
- diffuse reflection
- specular reflection
- specular exponent
- specular color reflectance
Movies: store frames as columns of a matrix
Can create fly-bys and other animation features
Easy GUI creation

Functions

$\bullet$ Usually, each function in its own file:

file run.m

$\fbox {\begin{minipage} {0.88in} \texttt{function a=run(b)}\\ $\vdots$\\ \texttt{d=2*b}\\ \texttt{c=foo(d)}\\ $\vdots$\\ \texttt{a=log(c)} \end{minipage}}$

file foo.m

$\fbox {\begin{minipage} {0.98in} \texttt{function e=foo(f,g)}\\ $\vdots$\\ \texttt{[h,i]=bar(f)} \\ $\vdots$\\ \texttt{e=sqrt(h)} \end{minipage}}$

file bar.m

$\fbox {\begin{minipage} {1.19in} \texttt{function [j,k,l]=bar(m)}\\ $\vdots$\space \\ \texttt{j=m} \\ \texttt{k=2*m}\\ \texttt{l=m*m} \end{minipage}}$

$\bullet$ Have variables nargin and nargout

$\bullet$ Frequently have many short files

Simulink

Simulink is a companion to MATLAB
Useful for modeling dynamic systems
Provides GUI for building/using block diagrams
Models are hierarchical
Similar to National Instrument's LabView, except building blocks are Matlab functions

Toolboxes

Available from the MathWorks and other sources
http://www.mathtools.net/MATLAB/toolboxes.html lists more than 200

1.
Communications Toolbox
2.
Control System Toolbox
3.
Data Acquisition Toolbox
4.
Filter Design Toolbox
5.
Financial Derivatives Toolbox
6.
Financial Toolbox
7.
Fuzzy Logic Toolbox
8.
Image Processing Toolbox
9.
Instrument Control Toolbox
10.
Instrument Control Toolbox
11.
LMI Control Toolbox
12.
Mapping Toolbox
13.
Neural Network Toolbox
14.
Optimization Toolbox
15.
Robust Control Toolbox
16.
Signal Processing Toolbox
17.
Spline Toolbox
18.
Stateflow Coder Toolbox
19.
Statistics Toolbox
20.
Symbolic Math Toolbox
21.
System Identification Toolbox
22.
Wavelet Toolbox

Local facilities

Matlab available for UNIX and Microsoft platforms
http://nesystemsengineering.rsc.ray.com/Tools/Matlab/sysmatlabtools.htm contains
- Binary conversion tools
- Calculations-data summary tools
- Clustering algorithm tools
- Clutter tools
- Coordinate transformations
- Data filtering tools
- Dave Shnidman detection models
- Detection models tools
- File & matrix processing
- Filter design tools
- General signal processing
- General tools
- Label plots
- Missile & radar tools
- Other general plotting tools
- Probability routines
- Scale factors rise time
- Smith chart tools
- Swerling detection models
- Target jammer noise samples
- Thresholds statistics image processing
- Transfer function conversions

Octave (free Matlab look-alike)

http://www.octave.org (UNIX and Microsoft)

GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with Matlab. It may also be used as a batch-oriented language.

Octave has extensive tools for solving common numerical linear algebra problems ...It is easily extensible and customizable via user-defined functions written in Octave's own language, or using dynamically loaded modules written in C++, C, Fortran, or other languages.

GNU Octave is also freely redistributable software. You may redistribute it and/or modify it under the terms of the GNU General Public License (GPL) as published by the Free Software Foundation.

Conclusion

Use Matlab for problems involving linear algebra

1.
Algorithmic design
2.
Data analysis & visualisation
3.
Detailed design
4.
End-to-end performance
5.
Fast prototyping
6.
Modeling & simulation
7.
Sensitivity analysis
8.
Trade studies
9.
Web/GUI interaction
10.
Typically not for real-time operation

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