Mathematics of photonic crystals

Tutorial: An introdiuction to MATLAB

Dagmar Roth(Karlsruhe Institute of Technology)

Intensive Programme / Summer School Periodic Structures in Applied MathematicsGttingen, August 18 31, 2013

This project has been funded with support from the European Commission. This publication [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Grant Agreement Reference Number: D-2012-ERA/MOBIP-3-29749-1-6

Tutorial:

An introduction into

MATLAB

2MATLAB Overview

mathematical functions for linear algebra, statistics, Fourier analysis, numerical integration, differentiation, ordinary differential equations, and more fast execution of vector and matrix calculations no low-level administrative task such as declaring variables, specifying data types and allocating memory necessary (unlike in C(C++) or Fortran) built-in 2-D and 3-D plotting functions Many additional and useful toolboxes: symbolic math, parallel programming, partial differentail equations, optimization, interval analysis (INTLAB), ...

http://www.mathworks.com/products/matlab/

MATLAB is a high-level language and interactive environmentfor numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java.

3Example multiplication of a matrix A and a vector x

for(i=0; i

MATLAB's Main Window

Current Folder

Workspace: Shows assigned variables

Command Window: Execution of calculations, to call commands and functions

Command History:Lists all previously called commandsFile details

MATLAB Product Help

5>> a=1; b=2; c=a+bc = 3

Basics: In- and Output

Variables a and b are initialized (no prior declaration), sum of a and bis calculated and stored in variable c.

Semicolon after formula: Result is not displayedWithout semicolon the result is displayed

>> a=1, b=2.5, c=a+ba = 1b = 2.5000c = 3.5000

>> a=1; b=2; c=a+b;>>

In case that all formulas end withsemicolon, nothing is displayed.

Enter Key

6By adding three dots in the end of a line, input canbe continued to the next line.

>> a1=1; a2=2; a3=3; a4=4; ...a5=5; a6=6; a7=7; a8=8; ...a1+a2+a3+a4+a5+a6+a7+a8ans = 36

If no variable is initialized for a result, the expression ans = is displayed . You can use the command disp if you want to display a result without ans=.

>> a1=1; a2=2; a3=3; a4=4; ...a5=5; a6=6; a7=7; a8=8; ...disp(a1+a2+a3+a4+a5+a6+a7+a8) 36

Variables

The first character of a name of a variable should be an alphabetic character and the variable-name should be different from the name of MATLAB functions (or predefined variables) Names of variables are case sensitive, i.e. a capital letter is distinguished from a small letter Types of a variables do not need to be specified: In assignment, MATLAB does not distinguish between integers and real numbers with decimal point All initialized variables are listed in Workspace-window: Variables can easily be saved for later sessions or imported from saved data

8Special variables

In order to express some special numerical values orspecial matrices, some names are fixed as follows:

Name Meaning

pi Inf, inf infinity,i, j, 1i imaginary uniteps machine epsilon ( minimal satisfying 1+>1 ) eye(n) unit matrix of dimension nones(n,m) matrix (n rows, m columns) whose all elements are 1 zeros(n,m) matrix (n rows, m columns) whose all elements are 0NaN, nan Not a Numberrealmin the smallest positive floating point number which is expressible in a computerrealmax the greatest positive floating point number which is expressible in a computer

1

9If you substitute a different value for these special variables, then it is maintained until you use the clear command. Especially i, j are often used as variables in loops, so you must take care in case you use the imaginary unit.

>> pians = 3.141592653589793>> pi=3pi = 3>> 10*pians = 30>> clear pi>> 10*pians = 31.415926535897931

1 Kings 7,23

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

Perimeter=30Diameter=10 => pi=3

10

complex variables

>> z1=1+2*iz1 = 1 + 2i>> z2=3+4*jz2 = 3 + 4i>> z1+z2ans = 4 + 6i

Once a value is substituted for i , then you cannot use i for imaginary unituntil you use clear command in order to restore it.

>> i=2; z2=5+6*iz2 = 17>> clear i>> z2=5+6*iz2 = 5 + 6i

11

Change of displayed accuracy

>> x=1/3x = 0.3333>> format short e; xx = 3.3333e-001>> format long; xx = 0.33333333333333>> format long e; xx = 3.333333333333333e-001>> format hex; xx = 3fd5555555555555

short (default) fixed point format with 5 digitsshort e floating point number with 5 digitslong fixed point format with 15 digitslong e floating point number with 15 digitshex hexadecimal number short g, long g best of fixed or floating Point number with 5 or 15 digits, resp.

If you change the displayed accuracy using format, it is maintained unlessyou change it again.

Note that this does not affect computational accuracy

12

Useful commands and control key help command

You can check how to use some command as follows:>> help command

>> help format FORMAT Set output format. FORMAT with no inputs sets the output format to default. For float variables, the default is FORMAT SHORT. FORMAT does not affect how MATLAB computations are done. Computations on float variables are done in appropriate floating point precision, no matter how those variables are displayed. FORMAT SHORT and LONG do not affect the display of integer variables. FORMAT may be used to switch between different output display formats: FORMAT SHORT Scaled fixed point format with 5 digits. FORMAT LONG Scaled fixed point format with 15 digits for double FORMAT SHORT E Floating point format with 5 digits. FORMAT LONG E Floating point format with 15 digits for double FORMAT SHORT G Best of fixed or floating point format with 5 digits. FORMAT LONG G Best of fixed or fl. point format with 15 digits for double

how to useformat

13

The statement which starts with % in your original M-filecan be also displayed by the help command.For example, if the file flow.m has the following content,% Program to plot the solutionfunction [f] = flow(K);

then help command gives us the following output.

>> help flow Program to plot the solution

You had better add some appropriate comment in the beginningof your M-file in order to remember which kind of program it is.

14

lookfor commandYou can search for a keyword in predefined matlab functions and commands. For example, if you want to find functions and commands involving max then you may input lookfor max in the command line.

>> lookfor maxrandomc Complex random numbers in M+i*M with M=[min,max], default [-1,+1]random Random numbers in [min,max], default [-1,+1]Longprecision Sets/gets maximum precision for long number arithmeticmaxNumCompThreads controls the maximum number of computational threadsNamelengthmax Maximum length of MATLAB function or variable name.bitmax Maximum double precision floating point integer.intmax Largest positive integer value.realmax Largest positive floating point number.max Largest component.runtrackmm demonstrates using FMINIMAX with Simulink.dfildemo minimax Optimizationpdenanmx NaN protected maximum.hp2dmax [alpha,wm,theta,L,P,ejtheta] = hp2dmax(w1,w2,w)d2dmax [b,tm,psi,L,P,ejpsi] = d2dmax(t1,t2,theta)dhinfric [gopt,K,X1,X2,Y1,Y2] = dhinfric(P,r,gmin,gmax,tol,options)

Some functions or commands are displayed with simple explanations.

15

arrow key and control key

In MATLAB the command-history is preserved in the memory,you can use the arrow keys to call previous commands

, search in command historyx( , search in command history for commands starting with

x( only , Move right /left for one letter

Delete Delete the letter to the left of cursor Ctrl-D Delete the letter to the right of cursorCtrl-A Go to to beginning of the lineCtrl-E Go to to end of the lineCtrl-U Delete current lineCtrl-K Delete all letters from current position to the end of the line

16

Arrays

vector one dimensional arraymatrix two dimensional array

An index of an array must be a positive integer starting from 1.

>> x=[1 2 3]x = 1 2 3>> x'ans = 1 2 3>> y=[1;2;3]y = 1 2 3

row vector (also x=[1,2,3])

column vector is expressedadding .

Since semicolon expresses theend of a row, a column vectorcan also be generated like this.

17

There are other methods to generate vectors.

Specify the initial value, increment and last value with colon

>> x=[1:2:7]x = 1 3 5 7>> x=[1:4]x = 1 2 3 4

From 1 to 7 incrementing by 2

If the increment is 1 then itcan be omitted.

Specify the initial value, last value and the number of element using the function linspace

>> y=linspace(1,3,5)y = 1 1.5 2 2.5 3

Generate 5 elementsfrom 1 to 3 with equalspace.

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>> A=[1 2 3; 4 5 6]A = 1 2 3 4 5 6>> A'ans = 1 4 2 5 3 6>> a=A(:)a = 1 4 2 5 3 6

matrix with 2 rowsand 3 columns

A stands for the transposedmatrix of A (if A is real). (A complex: A' is the complexconjugate transpose of A)

transformation of two dimensional array to onedimensional array. Columns of the matrix arewritten into vector sequentially

19

Access to single elements of an array

x(k): k-th element of one dimensional array x.A(i,j): (i,j)-element of two dimensional array A.

>> x=[1 2 3]x = 1 2 3>> x(2)ans = 2>> A=[1 2 3;4 5 6]A = 1 2 3 4 5 6>> A(2,2)ans = 5>> A(5,5)??? Index exceeds matrix dimensions

An error code isdisplayed if youcalled anon-existing element.

Access to subarrays

x(k:l): subvector A(k:l,n:m): submatrix

A(:,m): gives the m-th column of A (column vector)A(n,:): gives the n-th row of A (row vector)

(Aij) kiln jm

( xi)kil

>>x=[1:7]x = 1 2 3 4 5 6 7>> x(3:5)ans = 3 4 5

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

21

Operations, Mathematical functions operations

formula expression in MATLAB

a b+a b-ab/a bba

a + b

a - b

a * b

a / b

a^b

22

comparison operation

formula expression in MATLAB

and

or

a < b

a b

a >= b

a == b

a ~= b

&

|

true: 1 false: 0

ab

ab

ab

ab

a=b

ab

23

Example:

>> a=1; b=[2;3]; A=[1 2;3 4]; B=[5 2; 3 5];>> a>0ans = 1>> b==2ans = 1 0>> A==Bans = 0 1 1 0>> A> A~=Bans = 1 0 0 1

24

element-wise operations

For example, for two matrices

>> A*B

gives

then replace * by .* as follows:

>> A .* Bans = 1 4 6 4

If you want to compute the matrix (element-wise operation)

A=1 23 4 B=1 22 1

AB=5 411 10aijbij

25

formula expression in MATLAB

a + b

a .* b

a ./ b

a .^ b

Example for vectors

a=a1 ,, an , b=b1 ,,bn

a1b1 ,, anbn a1b1 ,, anbn a1 /b1 , , an /bn a1b1 , ,anbn

26

Mathematical functions

Various mathematical functions are available in MATLAB.There are also many mathematical functions in each toolbox.Use the help command to understand their handling.

Basic mathematical functions

abs absolute valuesqrt square rootlog natural logarithmexp exponential functionreal real part of a complex variableimag imaginary part of a complex variableangle argument of a complex variableconj conjugate of a complex variable

27

Basic statistical functions

sum sum of elementsmean mean valuecov covariancecorrcoef correlation coefficientsort sort in ascending or descending orderprod product of elementscumsum cumulative sumcumprod cumulative product

trigonometric functions

sin sin(x)cos cos(x)tan tan(x)asinacosatansinh sinh(x)cosh cosh(x)

matrix functions

inv inverse matrixdet determinantlu LU decompositionchol Cholesky decompositionnorm (various) norms cond condition numbereig eigenvalues

etc

sin1 xcos1 xtan1 x

28

Branch, Loopif statement

if expression1 statement1elseif expression2 statement2else statement3end

or

if expression1, statement1elseif expression2, statement2else statement3end

elseif and else can be omitted.

29

Example

>> x=-1;>> if x>0, y=1; elseif x==0, y=0; else y=-1; end>> yy = -1

If x does not have any value then the following error messageappears:

??? Undefined function or variable 'x'.

30

Loop

Several program statements can be repeatedly executed in a loop. The variable controlling the repetition is called loop variable.

There are two types of loops.

To specify the number of iterations

To repeat a group of statements as long as a certain condition is satisfied

for

while

31

for

for loop variable = range of the loop variable statementend

>> s=0;>> for i=1:10 s=s+i; end;>> disp(s) 55

i: loop variable, iteration number is 10

>> s=0;>> for i=10:-1:1 s=s+i; end;>> disp(s) 55

If you want to execute the iterationconversely then these expressionsare available.

32

A loop variable can be incremented by a predefined value in each step:

>> s=0;>> for i=2:2:10 s=s+i; end;>> disp(s) 30

Loop variable i is set as even number.( i starts at 2, is incremented by 2 in each step and ends at 10 )

>> s=0;>> for i=10:-2:2 s=s+i; end;>> disp(s) 30

Converse version

>> v=[1 10 100];>> for k=v disp(k) end 1 10 100

vector can also serve as range for loop variable

33

The continue command is used when you want to go tothe next iteration of the loop, and the break command is used when you want to terminate the execution of the loop.

>> s=0;>> for k=[1 -1 2 -2 3 -3 4 -4 5 -5] if k5, break; end disp(s); end 1 3

If k is negative then go to next iteration.

If s is greater than 5,quit the loop.

continue and break

34

while

while condition statementend

>> a=10;>> while a>=1 a=a/2; disp(a) end 5 2.50000000000000 1.25000000000000 0.62500000000000

The commands within the loop are executed while a is greater than or equal to 1.

break and continue can be usedas in the for-loop.

35

Script and functionM-files can be scripts and functions , both are named ***.m.

script file: consisting of some commands

You can execute it by typing its file-name.

function file: consisting of definition for function

The name of the file is the name of the function.

You can use a function/script without any declaration if the folder containing the m-file is on your MATLAB-path. To add a folder:

File Set Path Add Folder

Exercise:

Write a script fibo.m, that computes all Fibonacci numberssmaller than 1000 and stores them in an array. Display the array.

Hint: If f=[1] is an array, the assignment f(2)=2 automatically increases its size

37

% An M-file to calculate Fibonacci numbersf=[1 1]; i=1;while f(i)+f(i+1)> fibo Columns 1 through 11 1 1 2 3 5 8 13 21 34 55 89 Columns 12 through 16 144 233 377 610 987

fibo.m

Size of array is increased automatically

38

Example of function

% Function to compute the sum and difference% of two real numbersfunction [wa,sa]=cal(a,b)wa=a+b; sa=a-b;

Definition of function

In the beginning of M-file, declare the function with in- and output parameters as follows:

function [list of output] = function-name(list of input)

cal.m

>> x=1; y=2;>> [c,d]=cal(x,y)c = 3d = -1

Exercise:

Modify fibo.m into a function, that computes all Fibonacci numberssmaller than some input value M, stores them in an array f andreturns f

40

% An M-file to calculate Fibonacci numbersfunction f=fibo(M)f=[1 1]; i=1;while f(i)+f(i+1)> f=fibo(1000)f = Columns 1 through 11 1 1 2 3 5 8 13 21 34 55 89 Columns 12 through 16 144 233 377 610 987

fibo.m

41

Global variables

function y=func(x)global a b cy=a*x^2 + b*x +c;

global a b ca=1; b=2; c=3;x=1;y=func(x);disp(y)

cal2.m func.m

The common variables between script and function can be declared as global variables.

>> cal2 6

Example

42

Measurement of elapsed timeThe elapsed time between tic and toc can be measured.

>> tic;A=rand(1000,1000);toc;elapsed_time = 0.078000

>> tic;A=rand(1000,1000);time=toc;disp(time); 0.07800000000000

elapsed time can be storedin a variable.

43

Vectorization, High-speed

Vectorization

Some tricks can speed up computations!

>> clear;tic;for j=1:1000 for k=1:1000 A(j,k) = sin(j)+cos(k); endendtoc;Elapsed time is 1.003845 seconds.

Try to optimize computation time.

Copy code into an m-file:MATLAB's code analyzer will give you hints!

Example:

44

>> clear;tic;A=zeros(1000,1000);for j=1:1000 for k=1:1000 A(j,k) = sin(j)+cos(k); endendtoc;Elapsed time is 0.112592 seconds.

Step 1

Add this line

In MATLAB arrays do not need to be declared in advance.However, resizing an array in a loop needs time, due to the allocation of memory in each iteration.In the above example, zeros(1000,1000) reserves all needed memoryat once (before the loop).

45

>> clear;tic;A=zeros(1000,1000);j=[1:1000];sinj = sin(j);for k=1:1000 A(j,k) = sinj+cos(k);endtoc;Elapsed time is 0.010881 seconds.

Step 2 (vectorization)

Treat row as vector

46

Graphics in 2 dimensions ezplot displays the graph of function

ezplot(f(x))

ezplot(f(x), [a,b])

ezplot(f(x,y))

ezplot(f(x,y), [a,b])

ezplot(f(x,y), [a,b,c,d])

ezplot(x(t)', 'y(t)) Plot x=x(t), y=y(t) in

ezplot(x(t)', 'y(t), [a,b])

Plot f(x) in

Plot f(x) in

Plot f(x,y)=0 in

Plot f(x,y)=0 in

Plot f(x,y)=0 in

Plot x=x(t), y=y(t) in

2x2

axb

2x , y22x , y2

ax , yb

axb,c yd

0t2

atb

47

plot 2D plot with vector data

>> x=linspace(0, 2*pi, 100);>> plot(x, sin(x));

Example

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

>> x=linspace(0, 2*pi, 100);>> plot(x, sin(x), x, cos(x));

plot(x,y) plots vector x versus vector y

48

When ezplot or plot is executed, the graph is displayed in a window named Figure 1.

Another call of ezplot/plot erases the graph.

Figure 1 Figure 2

>> x=linspace(0, 2*pi, 100);>> figure(1); plot(x, sin(x));>> figure(2); plot(x, cos(x));

In this way, two graphsare displayed in Figure 1and Figure 2 respectively.

49

The command hold on is useful when you want todisplay some graphs in the same Figure.(The command hold off is used to quit it.)

>> x=linspace(0, 2*pi, 100);>> plot(x,sin(x));>> hold on;>> plot(x,sin(2*x));>> plot(x,sin(3*x));

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sin( ), sin(2 ), sin(3 )x x xThe graphs of

are displayed in one Figure.

50

Some options in graphics

b blue . dot - solid lineg green o circle shape : dotted line r red x x shape -. chained line c cyan + plus shape -- broken line m magenta * star shape (none) no liney yellow s squarew white d diamond shapek black v triangle ^ inverted triangle < triangle (left direction) > triangle (right direction) p pentagon h hexagon

plot(x, y, s) 's' is specified as follows:

51

>> x=linspace(0, 2*pi, 20);>> plot(x, sin(x), 'r+:')

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

520 1 2 3 4 5 6 7

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

>> x=linspace(0, 2*pi, 20);>> plot(x, sin(x), 'r+:', x, cos(x), 'kd--')

53

>> x=linspace(0, 2*pi, 50);>> plot(x, sin(x), 'rv', 'MarkerSize', 5)

The size of marker can be specified by MarkerSize

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

>> x=linspace(0, 2*pi, 50);>> plot(x, sin(x), 'rv', 'MarkerSize', 10)

540 1 2 3 4 5 6 7

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Plot of Sin(x)

>> x=linspace(0, 2*pi, 100);>> plot(x, sin(x)); title('Plot of Sin(x)');

The title can be added by title(text) in upper part of the graph

Title

55

Graphics in 3 dimensions

02

46

810

0

2

4

6

8

10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

>> t=linspace(0,10,40);>> [x,y]=meshgrid(t,t);>> z=sin(x)+cos(y/2);>> mesh(x,y,z);

Additionally ezplot3 etc. Check via help ezplot3.

56

Variables: Save and Load

To save variables you can use

>> save filename

A file named filename.mat is created which saves all variables from workspace. Using

>> load filename

variables in filename.mat are imported into workspace

If you omit the file name after save command then all variables are saved in a file named matlab.mat (load restores data from matlab.mat)

Alternatively you can use the buttons Save and Import dataon top of workspace window

57

e.g. >> whos Name Size Bytes Class

A 100x100 80000 double array B 200x200 320000 double array fid 1x1 8 double array i 1x1 8 double array j 1x1 8 double array z 1x1 16 double array (complex)

Grand total is 50004 elements using 400040 bytes

>> save variable>> clear>> whos>> load variable>> whosName Size Bytes Class

A 100x100 80000 double array B 200x200 320000 double array fid 1x1 8 double array i 1x1 8 double array j 1x1 8 double array z 1x1 16 double array (complex)

Grand total is 50004 elements using 400040 bytes