An Introduction to MatLab

Transcription

1 Introduction to MatLab 1 An Introduction to MatLab Contents 1. Starting MatLab Workspace and m-files Help Vectors and Matrices Objects Plots Statistics Import and Export of Data Appendix (Plots and Codes) Addition Physics: Solution of the time dependent Schrödinger-Equation... 27

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3 Introduction to MatLab 3 1. Starting MatLab Possibility 1 Possibility 2 Double-click the MatLab shortcut icon on your Windows desktop. >>cd 'work directory' >>edit explorer -> work directory create a new m-file click a 'm-file' with right mouse button open with, select program matlab.exe, activate 'open always with this program' double-click m-file (opens MatLab in the work directory) >>edit

4 4 2. Workspace and m-files ( a ) >> a = 5 stores value 5 in variable, prints result in workspace 5 ( a) ( prints value of variable a in workspace) >> a = 5; stores value 5 in variable >> a 5 >> >> t = ' hello world '; >> t ' hello world ' >> whos variable list ( opens editor with m-file ) ( calls function in m-file test) >> edit test test >> test example: our first MatLab program workspace editor >> edit test function test ' hello world ' F5 hello world Exercise Open MatLab Create a new m-file with name 'first' Type the program lines x = 1:20 (,sin( * /5)) plot x x pi

5 Introduction to MatLab 5 3. Help Examples for workspace help Examples for MatLab help menu help by >> help function >> help if >> help for >> help switch >> help whos Click with left mouse buttom: Menu Contents Using MatLab Mathematics - Contents Index Search 4. Vectors and Matrices Comand Output Comment ( 3) >> x = a 0 0 a abcd,,, are numbers (Integer, Real, π,e,i, >> x= a: d : b a a+ d a+ 2d b-d b >> x= a: b a a+ 1 a+ 2 >> x= [ a b c d] a b c d x 3 a+ 2d >> x = linspace( a, b, N) a a+ b b n elements >> x = linspace( a, b) b a b a N 1 N 1 ( ) b a b a a a elementes 99 b 99 b Comand Output Comment >> A= [1 2 3; 4 5 6] ; A( 2,1) x N 2 >> A= [ x ; y ] x, y R A R R Vektor y Vektor >> A= [ x y ] x y x R, y R A R ( ) Vektor Vektor x N N >> A= [ x' y' ] ( x' y' ) = x, y R A R R y ' M N M+ N N 2

6 6 Scalar produkt: Comand Comment >> x* y' N N = xnyn R, R R n >> A* x' M N N M = Amnxn R R, R R n >> A* B = AmnBnk n R R, R R R R ( n n) ( ) M N N K M K Point produkt: Comand Comment >> x.* y = x y N N R, R N R >> A.* B = A B R R, R R R R functions: Comand mn mn Comment ( 2 ( n )) ( ( sin mn )) N >> x.^2 = x R R ( ) ( ) M N M N M N >> sin A = A R R R R Exercise a) Which function is evaluated by N M N M N f = prod ( 1: N ) (>>'help prod' shows the definition and examples for 'prod') b) Study the results of the commands >> N = ones( 6) >> for n = 1:6, N ( n,1: n) = 1: n; end >> N >> f = prod( N,2) >> plot ( N ( 6,: ), f ',' b' ) >> xlabel (' N ') >> ylabel (' f ') >> title( ' Test 1' ) >> whos

7 Introduction to MatLab 7 c) Use A= and B= ci) Calculate the determinant of A (Menu help: MatLab - Using MatLab Matrices and linear algebra Inverses ) cii) Solve the linear system Ax = B (Menu help: MatLab - Using MatLab Matrices and linear algebra Solving linear )

8 8 5. Objects Example >> hnd = text 0,0,'hello world' x and y Position >> hnd=text(0,0,'hello world') hnd = >> get(hnd) Color = [0 0 0] EraseMode = normal Editing = off Extent = [ ] FontAngle = normal FontName = Helvetica FontSize = [10] FontUnits = points FontWeight = normal HorizontalAlignment = left Position = [0 0 0] Rotation = [0] String = hello world Units = data Interpreter = tex VerticalAlignment = middle BeingDeleted = off ButtonDownFcn = Children = [] Clipping = off CreateFcn = DeleteFcn = BusyAction = queue HandleVisibility = on HitTest = on Interruptible = on Parent = [ ] Selected = off SelectionHighlight = on Tag = Type = text UIContextMenu = [] UserData = [] Visible = on Get possibile properties: >> set ( hnd,'horizontalalignment' ) [ {left} center right ] Set possibile properties: >> set ( hnd, ' HorizontalAlignment ', ' center ', ' FontSize', 25) Exercise a) Rotate the string 'hello world' by 180 degrees (Rotation is a property of the object 'text'. b) Type in the command >> h= plot( [ ],[ 0 1 0] )

9 Introduction to MatLab 9 bi) What different line styles are possible bii) Change the line style and the line width biii) change the y-data to [ ]

10 10 6. Plots >>help plot >> x = linspace(0,2* pi); >> plot x,sin x ( ( )) (,sin( ),' :') [ ];.5*[ ]; (,,' y ') or (,,' ') ( Hnd is 'handle' for object) ( ) ( shows object properties) (,' ',' ',' ',10) ( changes object properties) >> plot x x b >> x = >> y = >> fill x y >> Hnd = fill x y y >> get Hnd >> set Hnd EdgeColor r LineWidth >> subplot n, m, k rows columns PlotNo >> hold >> hold on off ( overdraw) ( delete old objects when drawing a new one) >> try, delete() 1, end ( delete grafik window 1)

11 Introduction to MatLab 11 Curve fitting Data: x = ( x x x ) y = ( y y y ) fit: 1 2 N Y = a+ bx + cx. N get a, b and c with MatLab:\ a y' = ( 1 x' ( x.^2 )') b = M c A= M \ y' Y = M * A = A

12 12 Exercise Create a new m-file 'Plot1' Create the time series t=0..10,y=exp(at) with N elements; Reproduce the plots of Plot1 in the appendix If you don't know how to proceed, look into the program file (after the figure) Use the desktop help and the help menu to get an understanding of the statements

13 Introduction to MatLab Statistics Matlab help page: 'Using Matlab Mathematics Data analysis and statistics' Define an own statistic functions Example: Lineare correlation, r-value We consider two measured quantities The r-value is defind by ( x, y ) i { 1.. N } (7-1) i i Dat

14 14 i { 1.. N } ( ) ( x x)( y y) { } ( ) 2 2 i i { 1.. NDat} i 1.. NDat i [ ] Dat r = 1,1. x x y y It is a measure for the linear correlation between the two measured quantities. i i (7-2) Three examples for r-values. The closer the data pairs (x I,y i ) lie on a line the closer is the r-value to ±1. Fig. 7-1 MatLab formulation ( ) ( ) ( ) N = 10; Generte a data x= 1: N; y = rand(1, N); ( ) x1 = mean x ; Define r-value y1 = mean y; r = sum(( x- x1).*( y- y1)) /... ( sqrt( sum(( x - x1).^ 2))*... sqrt( sum(( y -y1).^2))); Exercise Create a new m-file 'Stat1' Reproduce the plots of Stat1 in the appendix If you don't know how to proceed, look into the program file (after the figure) Use the desktop help and the help menu to get an understanding of the statements

15 Introduction to MatLab Import and Export of Data Matlab help: 'Using matlab Development environment Importing and exporting -

16 16 Write time series in readable format to file 'TimeSeries.txt' ( ) N = 2^8; Generate time series x = 10* linspace(0,2* pi, N); c= (2*mod(2 : N + 1, 2) -1).*exp(-.5*(1: N)); y = c*sin((1: N)'* x); Y = fft( y) Fast Fourier transformation f _ hnd = fopen(' TimeSeries. txt ',' w'); for n = 1: length( x) fprintf ( f _ hnd,'%10.5 f ; %10.5 f ; %10.5 f \ n ',... xn ( ), yn ( ), Yn ( )) end fclose( f _ hnd) ( ) ( Out to file) Exercise Write the Data in m-file 'Stat1' into TimeSeries.Txt Create a new m-file 'ReadSeries.m' Write a code to read the data from TimeSeries.Txt and display them in a diagram Further exercises Create new m-files 'Plot2' and 'Plot3' Reproduce the plots in the appendix

17 Introduction to MatLab 17 Plot 1 9. Appendix (Plots and Codes)

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19 Introduction to MatLab 19 Statistics 1 Fortsetzung des Programms

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21 Introduction to MatLab 21 Plot 2

22 22 Plot 3

23 Introduction to MatLab 23

24 Addition Exercise (Fit of arbitrary functions) Create a 'random exponential function' by ( ( ( N) )) y = exp 3 sort ran 1: Approximate this function by z = αe β x Tipp: Approximate ( ) log ( ( )) by ( ) can display y x = y x z x = α + β x. Its an easy way to get α and β. The you z = αe β x.

25 Introduction to MatLab 25

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27 Introduction to MatLab Physics: Solution of the time dependent Schrödinger-Equation Equation and boundary conditions tψ = xψ, i 2 ψ H ( 0, t) = ψ ( π, t) = 0 ( Kasten) Solution 1 tψ = Hψ i ψ x t = e ψ x Basis (, ) iht (,0) 2 un x sin nx erfüllt Randbedingungen π ψ ( ) = ( ) ( ) N ( xt, ) = c ( tu ) ( x) n= 1 u = H u, mit H = u H u i cm t e c 0 oder c t e c ihmnt () = n( ) iht ˆ () = ( 0) Summary: n n t m mn m mn m n 2 un x sin nx π H ( ) = ( ) π u Hu dx mn m n 0 c t e c ψ = iht ˆ () = ( 0) N () t = c () t u ( x) n= 1 n Start function: 2 ( 0) ( ( π )) ( x) 2 ψ = e e c ( 0) π = ψ 0 ( 0) n ax ik x u dx n

28 28 Formulation with vectors and matrices ( M ) ( 0, π, ) ( 2) ( 1) x = linspace N dx = x x f = 1: M 2 2 ( M) ' ( ) π sin ( ) ( ) sin M mx ( ( )) n π m ( ) u = u = f x n = f x ( nm ) (,1) ( ) ( ) D = D = m = ones N f ( 2) ( ) ( ) ψ ( M ) (. ) π 2 ' mn = m n = m n 0 H u n u dx u D u dx ( ψ m ) 2 ( ( π 2 ) / 0 ) = = e. e ( 0) ( 0) x B ikx ( ) N 1 ( 1 ) ( ) ( 0' ) 0' ( 0 ( ) ( ) ) c 0 = uψ dx = unm ψm dx R R un x ψ ( x) dx

29 Introduction to MatLab 29 MatLab-Program