An Introduction to Scientific Computing using MatLab

Dr. Xianghua Xie, x.xie@swansea.ac.uk, http://csvision.swan.ac.uk

Introduction

}  Matlab is a program for doing numerical computation. It was originally designed for solving linear algebra type problems using matrices.

}  It’s name is derived from MATrix LABoratory.

}  Matlab is is widely used as a platform for developing tools for Machine Learning, Computer Vision, Computational Analysis, Artificial Intelligence …

Many of the following slides are adapted from Ian Brooks (Leeds)

MATLAB User Environment

Workspace/Variable Inspector

Command History

Command Window

Getting help

There are several ways of getting help: Basic help on named commands/functions is echoed to the command window by: >> help command-name A complete help system containing full text of manuals is started by: >> doc

Getting help

System Environment

}  Windows }  MATLAB installed in e.g. c:\matlab6.5 }  Your code…anywhere convenient (e.g. h:\matlab)

}  Linux (Environment network) }  MATLAB installed in e.g. /apps/matlab }  Your code in e.g. /home/username/matlab }  Your environment configuration in e.g. ~/.matlab

startup.m

}  The script $matlab_root\toolbox\local\matlabrc.m is always run at startup – it reads system environment variables etc, and initialises platform dependent settings. If present it will run a user defined initialisation script: startup.m

}  Linux: /home/user/matlab/startup.m }  Windows: $matlab_root\toolbox\local\startup.m

}  Use startup.m for setting paths, forcing preferred settings, etc.

%------------------------------------- % Matlab startup file for IMB's laptop %------------------------------------- %-- add paths for my m-files -- addpath d:/matlab addpath d:/matlab/bulk2.5 addpath d:/matlab/bulk2.6 addpath d:/matlab/coastal addpath d:/matlab/lidar addpath d:/matlab/ndbc addpath d:/matlab/page addpath d:/matlab/sections addpath d:/matlab/sharem addpath d:/matlab/wavelet addpath d:/matlab/LEM addpath d:/matlab/GPSbook addpath d:/matlab/FAAM addpath d:/matlab/FAAM/winds addpath d:/matlab/faam/bae %-- add netCDF toolbox -- addpath c:/matlab701/bin addpath c:/matlab701/bin/win32 addpath d:/matlab/netcdf addpath d:/matlab/netcdf/ncfiles addpath d:/matlab/netcdf/nctype addpath d:/matlab/netcdf/ncutility

%-- add path for generic data -- addpath d:/matlab/coastlines % coastline data addpath d:/cw96/flight_data/jun02 % raw cw96 addpath d:/cw96/flight_data/jun07 % aircraft data addpath d:/cw96/flight_data/jun11 addpath d:/cw96/flight_data/jun12 addpath d:/cw96/flight_data/jun17 addpath d:/cw96/flight_data/jun19 addpath d:/cw96/flight_data/jun21 addpath d:/cw96/flight_data/jun23 addpath d:/cw96/flight_data/jun26 addpath d:/cw96/flight_data/jun29 addpath d:/cw96/flight_data/jul01 addpath d:/cw96/runs % run definitions for cw96 flights %---------------------------------------------------------------------- %-- set default figure options -- set(0,'DefaultFigurePaperType','a4') % this should be the default in EU anyway set(0,'DefaultFigurePaperUnits','inches') % v6 defaults to cm for EU countries set(0,'DefaultFigureRenderer','painters') % v7 default OpenGL causes problems

Example startup.m file

The WORKSPACE

}  MATLAB maintains an active workspace, any variables (data) loaded or defined here are always available.

}  Some commands to examine workspace, move around, etc:

>> who Your variables are: x y

who : lists variables in workspace

>> whos Name Size Bytes Class x 3x1 24 double array y 3x2 48 double array Grand total is 9 elements using 72 bytes

whos : lists names and basic properties of variables in the workspace

>> pwd ans = D:\ >> cd cw96\jun02 >> dir . 30m_wtv.mat edson2km.mat jun02_30m_runs.mat .. 960602_sst.mat edson_2km_bulk.mat

pwd, cd, dir, ls : similar to operating system

>> x=[1 2 3] x = 1 2 3 >> x=[1,2,3] x = 1 2 3 >> x=[1 2 3 4]; >> x=[1;2;3;4] x = 1 2 3 4

When defining variables, a space or comma separates elements on a row.

A newline or semicolon forces a new row; these 2 statements are equivalent. NB. you can break definitions across multiple lines.

}  1 & 2D arrays are treated as formal matrices }  Matrix algebra works by default:

>> a=[1 2]; >> b=[3 4]; >> a*b ans =

11 >> b*a ans =

3 6 4 8

1x2 row oriented array (vector) (Trailing semicolon suppresses display of output)

2x1 column oriented array

Result of matrix multiplication depends on order of terms (non-cummutative)

>> a=[1 2] A =

1 2 >> b=[3 4]; >> a.*b ans =

3 8 >> c=a+b c =

4 6

Matrix addition & subtraction operate element-by-element anyway. Dimensions of matrix must still match!

No trailing semicolon, immediate display of result

Element-by-element multiplication

>> A = [1:3;4:6;7:9] A = 1 2 3 4 5 6 7 8 9 >> mean(A) ans = 4 5 6 >> sum(A) ans = 12 15 18

Most common functions operate on columns by default

INDEXING ARRAYS

}  MATLAB indexes arrays: }  1 to N }  [row,column]

[1,1 1,2 . 1,n

2,1 2,2 . 2,n

3,1 3,2 . 3,n . . .

m,1 m,2 . m,n]

n

m

>> A = [1:3;4:6;7:9] A = 1 2 3 4 5 6 7 8 9 >> A(2,3) ans =

6 >> A(1:3,2) ans =

2 5 8

>> A(2,:) ans =

4 5 6

The colon indicates a range, a:b (a to b)

A colon on its own indicates ALL values

THE COLON OPERATOR

}  Colon operator occurs in several forms }  To indicate a range (as above) }  To indicate a range with non-unit increment

>> N = 5:10:35

N =

5 15 25 35

>> P = [1:3; 30:-10:10]

P =

1 2 3

30 20 10

Basic Operators

+, -, *, / : basic numeric operators \ : left division (matrix division) ^ : raise to power ’ : transpose (of matrix) – flip along diagonal Example: Least square optimisation

>> X=[1,1;3,1;5,1;7,1]; >> y=[2,2,4,5]'; >> a=inv(X'*X)*X'*y;

SAVING DATA

}  MATLAB uses its own platform independent file format for saving data – files have a .mat extension }  The ‘save’ command saves variables from the workspace to a

named file (or matlab.mat if no filename given) }  save filename – saves entire workspace to filename.mat }  save var1 var2 … filename – saves named variables to

filename.mat

}  By default save overwrites an existing file of the same name, the –append switch forces appending data to an existing file (but variables of same name will be overwritten!) }  save var1 var2 filename -append

}  Data is recovered with the ‘load’ command }  load filename – loads entire .mat file }  load filename var1 var2 …– loads named variables

}  load filename –ascii – loads contents of an ascii flatfile in a variable ‘filename’. The ascii file must contain a rectangular array of numbers so that it loads into a single matrix.

}  X=load(‘filename’,’-ascii’) – loads the ascii file into a variable ‘X’

}  save var1 filename –ascii – saves a single variable to an ascii flat file (rectangular array of numbers)

}  Script & function

Scripts

}  It is possible to achieve a lot simply by executing one command at a time on the command line

}  It is easier & more efficient to save extended sets of commands as a script – a simple ascii file with a ‘.m’ file extension, containing all the commands you wish to run.

}  The script is run by entering its filename (without .m extension)

}  In order for MATLAB to see a script it must be: }  In the current working directory, or }  In a directory on the current path

Functions

}  A MATLAB function is very similar to a script, but: }  Starts with a function declaration line }  May have defined input arguments }  May have defined output arguments

function [out1,out2]=function_name(in1)

}  Executes in its own workspace: it CANNOT see, or modify variables in the calling workspace – values must be passed as input & output variables

function [x1,x2,x3]=powers1to3(n)

% calculate first three powers of [1:n]

% (a trivial example function)

x1=[1:n];

x2=x1.^2;

x3=x1.^3;

>> [x1,x2,x3]=powers1to3(4) x1 = 1 2 3 4 x2 = 1 4 9 16 x3 = 1 8 27 64

>> [x1,x2]=powers1to3(4) x1 = 1 2 3 4 x2 = 1 4 9 16

If fewer output parameters used than are declared, only those used are returned.

}  There are many functions built-in or supplied with MATLAB. }  A few very basic functions are built in to the matlab executable }  Very many more are supplied as m-files; the code can be

viewed by entering: >> type function_name

}  Supplied m-files can be found, grouped by category, in $matlab_root/toolbox/catagory

Function Help

}  MATLAB uses the ‘%’ character to start a comment – everything on line after ‘%’ is ignored

}  A contiguous block of comment lines following the first comment, is treated as the ‘help’ text for the function. This block is echoed to screen when you enter >> help function_name This is very useful…use it when writing code!

>> help epot MATLAB FUNCTION EPOT Calculates theta_e by Bolton's formula (Monthly Weather Review 1980 vol.108, 1046-1053) usage: epot=epot(ta,dp,pr) outputs epot : equivalent potential temperature (K) inputs ta : air temp (K) dp : dew point (K) pr : static pressure (mb) IM Brooks : july22 1994

e.g. help text from a function to calculate equivalent potential temperature

function epot=epot(ta,dp,pr) % MATLAB FUNCTION EPOT % Calculates theta_e by Bolton's formula (Monthly Weather % Review 1980 vol.108, 1046-1053) % % usage: epot=epot(ta,dp,pr) % % outputs % epot : equivalent potential temperature (K) % inputs ta : air temp (K) % dp : dew point (K) % pr : static pressure (mb) % % IM Brooks : july22 1994 % ensure temperatures are in kelvin ta(ta<100)=ta(ta<100)+273.15; dp(dp<100)=dp(dp<100)+273.15; % where dewpoint>temp, set dewpoint equal to temp dp(dp>ta)=ta(dp>ta); % calculate water vapour pressure and mass mixing ratio mr=mmr(dp,pr); vap=vp(dp); % calculate temperature at the lifing condensation level TL=(2840./(3.5*log(ta) - log(vap) - 4.805))+55; % calculate theta_e epot=ta.*((1000./pr).^(0.2854*(1-0.28E-3*mr))).* ... exp((3.376./TL - 0.00254).*mr.*(1+0.81E-3*mr));

What it does

How it’s called

What input and outputs are (and units!)

Good coding practice, note what and why…you will forget

If, else, elseif

if condition statements; elseif condition statements; else statements; end

Generic form:

if A<0 disp(‘A is negative’) elseif A>0 disp(‘A is positive’) else disp(‘A is neither’) end

example

switch, case

switch statement case value1 statements; case value2 statements; case value2 statements; otherwise statements; end

Generic form:

switch A*B case 0 disp(‘A or B is zero’) case 1 disp(‘A*B equals 1’) case C*D disp(‘A*B equals C*D’) otherwise disp(‘no cases match’) end

example

for & while loops

for n=firstn:dn:lastn statements; end

FOR: Generic form:

for n = 1:10 x(n)=n^2; end

example

N.B. loops are rather inefficient in MATLAB (and IDL), the example above would execute much faster if vectorized as

>> x=[1:10].^2;

If you can vectorize code instead of using a loop, do so.

while condition statements; end

WHILE: Generic form:

n = 1; while n <= 10 x(n)=n^2; n=n+1; end

example

strings

}  MATLAB treats strings as arrays of characters }  A 2D ‘string’ matrix must have the same number of characters

on each row!

>> name = [‘Ian’,’Brooks’] name =

IanBrooks >> name=[‘Ian';‘Brooks'] ??? Error using ==> vertcat All rows in the bracketed expression must have the same number of columns >> name=[‘Ian ';‘Brooks'] Name =

Ian Brooks

Cell arrays

}  Cell arrays are arrays of arbitrary data objects, as a whole they are dimensioned similar to regular arrays/matrices, but each element can hold any valid matlab data object: a single number, a matrix or array, a string, another cell array…

}  They are indexed in the same manner as ordinary arrays, but with curly brackets

}  Also support Structure Array

>> X{1}=[1 2 3 4]; >> X{2}=‘some random text’ X = [1x4 double] 'some random text'

}  Graphical output

Basic Plotting Commands

}  figure : creates a new figure window }  plot(x) : plots line graph of x vs index

number of array }  plot(x,y) : plots line graph of x vs y }  plot(x,y,'r--')

: plots x vs y with linetype specified in string : 'r' = red, 'g'=green, etc '-' solid line, '- -' dashed, 'o'

>> plot(glon,glat) >> xlabel('Longitude') >> ylabel('Latitude') >> title('Flight Track : CW96 960607')

Subplots

}  subplot(m,n,p) : create a subplot in an array of axes

>> subplot(2,3,1);

>> subplot(2,3,4);

m

n

P=1 P=2 P=3

P=4

>> peaks;

Peaks is an example function, useful for demonstrating 3D data, contouring, etc. Figure above is its default output. P=peaks; - return data matrix for replotting…

>> P = peaks; >> contour(P)

>> contourf(P,[-9:0.5:9]); >> colorbar

>> colormap cool

Object select Add/edit text

Add arrow & line zoom

3D rotate