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Pawel Herman

Fö 1 Introduction, Matlab fundamentals

DN1212 Numeriska metoder och grundläggande programmering

Pawel HermanKTH/CB

15 januari 2013

Pawel Herman

• Matlab programming for engineering

• Matlab environment

• Matlab fundamentals – variables, arrays

• Matlab fundamentals – operators, basic commands and functions

Lecture outline

2Fö 1 Introduction, Matlab fundamentals

Pawel Herman

• Computational approach with the use of numerical methodsfacilitates• finding approximate solutions at low costs• rapid prototyping, modelling and verification• scope for automated analysis• approaching large and complex systems without the need for

excessive simplifications• effective communication of various aspects of data/solutions to

customers etc. (flexibility)

• Matlab accommodates for these requirements and has mostwidely distributed engineering and technical applications.

Motivation for programming in engineering

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• Matlab programming, introduction• Matlab environment• Fundamentals (variables, operators, basic functions)

Fö 1 Introduktion. Matlabrepetition

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• MATrix LABoratory – computer program to perform engineering and scientific computations

• It has its own programming/scripting language• interpreted high‐level language (like Java, Python, Perl etc.)• platform‐independence• relatively simple syntax with weak typing• IDE with debugger facilitates rapid development • extensive libraries of predefined functions in various domains• rich tools for device‐independent plotting

MATLAB can also be used in an interactive way, like a calculator.

Matlab... What is it?

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• MATrix LABoratory – computer program to perform engineering and scientific computations

• It has its own programming/scripting language• interpreted high‐level language (like Java, Python, Perl etc.)• platform‐independence• relatively simple syntax with weak typing• IDE with debugger facilitates rapid development • extensive libraries of predefined functions in various domains• rich tools for device‐independent plotting

However, execution in Matlab can be relatively SLOW.

Matlab... What is it?

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Desktop

Matlab environment, GUI



Start button

Current directory

Command history

Workspace

Command window

Current path (by default data is saved and loaded from the current directory)

Interactive work

Ctrl+C

UNIX‐type of navigation(TAB, ↑)

clc

clear

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Desktop

Matlab environment, GUI



Editor

Help window

Figure window

Algorithms, scripts, functions as .m‐files(see also diary)

closeclf

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Search path in Matlab, naming covention



via GUI Command window

addpath(path)

genpath(path/folder)

rmpath(path)

path2rc

savepath

editpath / pathtool

Priorities in search for variables and functionsIt is possible to check which version of a function is behind the function name.

which(’function_name’)

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GUI help Interactive help

Help and basic debugging



• interactive in the Command windowlookfor function_namehelp function_name

>> help logLOG Natural logarithm.

LOG(X) is the natural logarithm of the elements of X.Complex results are produced if X is not positive.

See also log1p, log2, log10, exp, logm, reallog.

Reference page in Help browserdoc log

Searchable database

Main information windowCorresponding entries

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Integrated editor‐debugger




breakpoint current status

run programplace/remove a breakpointremove all breakpointsdo not step into a functiondebug inside a functionleave the current functionstop execution / exit debug mode

comments following %

help info in first two lines

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PROBLEM: Inner matrix dimensions must agree!

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Additional features• evaluate selection• evaluate cell

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• Fundamental unit of data is array (scalar, vector or matrix)

• symbolic use as mathematical variables in expressionsavg_velocity = avg_distance / travel_time;

• they store intermediate values of calculations (ans)

• assign meaningful names to variables (Matlab is case‐sensitive)

Fundamental concepts – variables



[nrows,ncols] = size(array)

size(a) = [3, 2], numel(a)=6

size(b) = [1, 3], length(b)=3

size(c) = [3, 1], length(c)=3

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• Data types (weak typing convention)

Matlab variables



• No need to declare the type of variables since the data assigned decides about it– code is easily reusable but also more susceptible to bugs that are hard to track.

• Type casting allows for different interpretations of the same data.

• Memory vs precision trade-off.

int8 - 8 bit –128..127int16 -16 bit –32 768..32 767uint8 – unsigned int 8bit 0..255uint16 - unsigned16 bit 0..65 535

Type casting

d1 = 32; d2 = uint8(32); whos d1 d2

Name Size Bytes Classd1 1x1 8 double arrayd2 1x1 1 uint8 array

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• Predefined variables

• pi π (15 digits)

• i, j complex imarinary unit

• inf this represents machine infinity

• nan not‐a‐number as a result mathematically undefined operations, e.g. 0.0/0.0 or inf‐inf

• eps the smallest difference between two numbers that can be represented on the computer

• ans variable used to store the result of the most ecent expression

Matlab variables



1−

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• Notation, indexing

Array notation, access and initialisation



1 4 7 10 13 16

2 5 8 11 14 17

3 6 9 12 15 18

Matrix a

size(A)=[3, 6]3 rows

6 columns

a(2, 4) = 11; this is equivalent to a(11)=11;

row column 1D index2D index

One-based index in Matlab, i.e. the first row/column has index 1 (not 0 as in C)

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• Multidimensional matrix – a natural extension of a 2D case




1 22 0 13 9 8

2 5 8 11 14 17

8 6 9 12 15 18

a(: , : , 1) = [1 4 7 10 13 16; 2 5 8 11 14 17; 3 6 9 12 15 18]

3D matrix can be seen as an array of 2D matrices

6 4 7 1 21 6

2 5 8 11 14 17

3 6 9 12 15 18

9 6 17 31 5 3

0 5 8 11 14 17

7 6 9 12 15 18

1 4 7 10 13 16

2 5 8 11 14 17

3 6 9 12 15 18

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• Assigning values• [ ] is an empty array• If a = [ ] then matrix defined as a = [a, b] or a = [a; b] is the same as b this can be used to increment a vector or matrix with new values (see for‐loops)

• ; splits data into rows, e.g. a = [4, 9, 0, 1, 5; 0 1 2 6 7; 3 8 3 4 2];

• end function, a(2, 3:end) = [2 6 7]




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

243837621051094

a Here it corresponds tosize(a,2) = 5

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• Assigning values• [ ] is an empty array• If a = [ ] then matrix defined as a = [a, b] or a = [a; b] is the same as b this can be used to increment a vector or matrix with new values (see for‐loops)

• ; splits data into rows, e.g. a = [4, 9, 0, 1, 5; 0 1 2 6 7; 3 8 3 4 2];

• end function, a(2, 3:end) = [2 6 7]

• colon : for indexing and to refer to all elements,

a(1:2, :) = [4 9 0 1 5; 0 1 2 6 7]




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

243837621051094

a

Equivalent to 1 : size(a,2)

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• Initialisation• zeros(n), zeros(n, m) n x n (or n x m) matrix of zeros

• zeros(size(a)) matrix of zeros of the sime siz as matrix a

• ones(n), ones(n, m) n x n (or n x m) matrix of ones

• ones(size(a)) matrix of ones of the sime size as matrix a

• eye(n) identity matrix of size n x n• a : incr : b vector with the first value a and then a+incr,

a+2*incr until it is greater or equal to b, e.g.3 : 4 : 20 = [3 7 11 15 19]

• linspace(a, b, n) vector linearly spanning interval [a, b] with n elements, e.g.linspace(4, 16, 4) = [4 8 12 16]




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• Concatenation of vectors and matrices (dimension matters)

Matrix manipulations



These components eventually constitute subarrays.

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It is possible to access a set of array elements at the same time, either to select a subset of data for further processing

a = [1 2 7 4; 2 8 9 3; 1 6 2 -7] (3 x 4 array)a(1:2,1:3) = 1 2 7

2 8 9or .....

Subarrays and indexing



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a = [1 2 7 4; 2 8 9 3; 1 6 2 -7] (3 x 4 array)a(1:2,1:3) = 1 2 7

2 8 9or to set their values (either with an array of the same size as the target subarray:

If b = [3 2 1; 0 2 0], THEN a(1:2,1:3) = b;

alternatively, scalar value can be to used to set the entire subarray :

a(1:2,1:3) = 1, which imples a = [1 1 1 4; 1 1 1 3; 1 6 2 -7]




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a = [1 2 7 4; 2 8 9 3; 1 6 2 -7] (3 x 4 array)a(1:2,1:3) = 1 2 7

2 8 9

Do not forget that you can always use logic variables for indexing arrays, e.g.

a(a<0) = 0 (sets to 0 all elements of a that are negative in the beginning),

More explicitly, a( logical([1 1 1 0; 1 1 1 0; 0 0 0 0]))




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• Data input• loading data

– load(filename,’var1_name’,..,’varN_name’) or load filename to recover all variables– data can be read binary, ASCII or as Matlab format (it will be covered later on)

• input from the keyboard– val = input(’Please enter the value:’);– an additional parameter ’s’ converts the input value into string format

– user is prompted in the Command window

• import via ImportWizard or command line(mat‐type or ASCII data)

Matlab data



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• Data output• display (the role of ;)

– format short / long / short e / long e / compact / loose– disp (numerical and char strings; see int2str and num2str)– C‐style sprintf with a formatting string, sprintf(’Result = %f for %d’ trials, 2.3, 10)

• saving data– save(filename,’variable_name’) or save filename to save the entire workspace– data can be written binary in Matlab format or ASCII (it will be covered later on)

• graphical presentation (e.g. plot)x = 0 : 0.1 : 10; y = x.^2;plot(x,y);

Matlab data



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• plot function• figure and plot(x, y, ’k.-’)• multiple plots: plot(x1, y1, ’.-g’, x2, y2, ’--r’)• alternatively: plot(X, Y) plots columns in X vs. columns in Y• hold on, hold off can be used to control whether consecutive plots

should be made in the same figure

• fplot(’3*sin(2*x)+x^3‐2*x^2’, [‐0.5, 3.5]) for plotting functionalexpressions (compare with eval(’functional_expression’))

• figure and plot properties can be set in scripts (xlim, xlabel, title etc.)

• figures and their properties can be further modified in the Figurewindow (especially Insert and Tool menu)

Plotting Matlab data



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• Most common algebraic operators

element‐wise matrix operators

A + B addition (a+b)

A – B subtraction (a-b)

A .* B multiplication elem by elem (a*b) A * B matrix multiplication

A ./ B right division of elements (a/b) A / B right matrix division A * B-1

A .\ B left division of elements (b/a) A \ B left matrix division A-1 * B

A .^ B exponentiation

A .’ transpose A’ complex conjugate transpose

Fundamental concepts – operators



+ operations involving scalars

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• Most common logic operators (result in TRUE=1 or FALSE=0)

>, >= greater, greater or equal to

<, <= less, less or equal to

== equal to

~= different from

~ NOT

&, && AND

|, || OR

Fundamental concepts – operators



Operators are applied in order according to their priority (safe use of brackets)

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• [maxval, index]=max(A) [minval, index]=min(A)

• [index] = find(A) or [r, c] = find(A) all(A) any(A) sum(A)

• size(x) numel(x) length(x) reshape(x, r, c) isempty(x)

• round(x) ceil(x) floor(x) fix(x)

• char(x) double(x) int2str(x) num2str(x) str2num(x)

• abs(x) angle(x) conj(x) real(x) imag(x)

• setdiff(a,b) intersect(a,b) union(a,b) ismember(a,b)

• [y, xind] = sort(x) [y, xind, yind] = unique(x)

• Common mathematical functions: sin(x), cos(x), ...., exp(x), log(x), sqrt(x), mod(x,y), mean(x), std(x), var(x)

Common built‐in functions

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Fö 1 Introduction, Matlab fundamentals

Pawel Herman

• In summary• Matlab can be used both interactively and non‐interactively in the form

of scripted algorithms implemented in Matlab programming language• Array are a fundamental unit of stored information (variables)• Matlab allows for rapid prototyping and finding numerical, symbolic and

graphical solutions to engineering problems• It offers a plethora of built‐in ready‐to use functions

• What have we learnt today?• How to use the Matlab environment, GUI.• How to define variables and perform basic operations on arrays.• How to use interactive command line and simple M‐scripts.• How to graphically present data in Matlab (plotting basics).

Summary and recapitulation


• Matlab programming, introduction• Matlab environment• Fundamentals (variables, operators, basic functions)• Summary

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