SCIENTIFIC COMPUTING 7P-100-SCI

Transcription

1 Engineering cycle, 2nd year 1 SCIENTIFIC COMPUTING 7P-100-SCI Lesson 1: General introduction and reminder on Matlab Hervé Sauer, Charles Bourassin-Bouchet, Mondher Besbes Ludivine Emeric, Léo Wojszvzyk

2 Scientific Computing 2 Some practical points 8x4h tutorials (1 per week) in the teaching unit «Systems S1» Final exam: 1h1/2, December 11 th 2018 Course 0235 on Libres Savoirs: A complete Matlab manual + PPT of the lessons The tutorials and their corrections + The SupOptique toolbox Campus license : Install the latest version of Matlab for free on your personal PC/Mac (details on Libres Savoirs) The computer sciences rooms are open 24/24h, 6/7 days (closed on sundays)

3 Scientific Computing 3 What is Matlab MATLAB: MAtrix LABoratory Programming language + integrated development environment (IDE) for scientific computing Can be used for: data processing, numerical simulations, computer code prototyping Created in the late 70 s by Clive Moler, Prof. of numerical analysis and linear algebra in the University of New-Mexico ~1 million users worldwide Used by 60% of IO former students 2000 per license + ~600 per extra toolbox

4 Scientific Computing 4 Why scientific computing? Learn how to solve complex scientific and engineering problems with a computer Scientific computing (M1 course) Applied mathematics with Matlab and Comsol Initiation to Matlab (L3 course) Programming with Matlab Outline: How to use vectors, matrices, 2D graphics (Reminder) and 3D graphics (New) Advanced programming in Matlab Linear least-squares and singular-value decomposition (SVD) Nonlinear least-squares and optimisation 2D Fast Fourier Transform and applications Ordinary differential equations & Partial derivative differential equations Introduction to Comsol Multiphysics + some surprises

5 Scientific Computing 5 Why scientific computing? Learn how to solve complex scientific and engineering problems with a computer Scientific computing (M1 course) Applied mathematics with Matlab and Comsol Initiation to Matlab (L3 course) Programming with Matlab Outline: How to use vectors, matrices, 2D graphics (Reminder) and 3D graphics (New) Advanced programming in Matlab Linear least-squares and singular-value decomposition (SVD) Nonlinear least-squares and optimisation 2D Fast Fourier Transform and applications Ordinary differential equations & Partial derivative differential equations Introduction to Comsol Multiphysics + some surprises

6 Scientific Computing 6 Graphical interface 1. The Command window: Use it as a calculator 2. The script editor: Write Script/Livescripts or functions 3. The workspace: List of the currently used variables

7 Scientific Computing 7 Use the help When using Matlab in your future life, the help will always be available, the teacher won t! Get familiar with the help!

8 Scientific Computing 8 Introduction to Livescript Type in the command «CopieTravail LiveScript_Exemple1_ENG.mlx» (This command copies the file in your current directory and then opens it in the editor). 1. Run successively sections 1 to 4 in the Live Script, and pay attention to the code and the obtained results 2. Run section 5, modify the code in a suitable way then run it again. 3. Answer to the questions in sections 6 and 7 It s your turn!

9 Scientific Computing 9 Organizing a project in Matlab The commands in a Script (or in the Command Window) call functions that perform a specific task. C main.c int main() { Arg = Myfunction(arg1,arg2); } source.c (+ source.h) int Myfunction(int arg1, int arg2) { }

10 Scientific Computing 10 Organizing a project in Matlab The commands in a Script (or in the Command Window) call functions that perform a specific task. C main.c int main() { Arg = Myfunction(arg1,arg2); } Matlab My_script.m %Calls the function Myfunction() Arg = Myfunction(arg1,arg2); source.c (+ source.h) int Myfunction(int arg1, int arg2) { } Myfunction.m function Arg = Myfunction(arg1,arg2)

11 Scientific Computing 11 Functions File Poynting.m function P = Poynting( E, B) % Poynting: Evaluate the Poynting vector of an electromagnetic field % USE: P=Poynting(E,B) % E: Complex 3-component vector: the complex amplitude of the electrical field vector (Ex,Ey,Ez) in V/m. % B: Complex 3-component vector: the complex amplitude of the magnetic field vector (Bx,By,Bz) in tesla. % P: Real 3-component vector, giving the componant (rpx,rpy,rpz) % of the real part of the complex Poynting vector in W/m². % Cf. [Jackson, Classical Electrodynamics, 3rd Ed, Wiley 1998] mu0 = 4*pi*1E-7; % Vacuum permeability (H/m) P = real(cross(e,conj(b)))/(2*mu0); % cross computes the cross product end

12 Scientific Computing 12 Functions File Poynting.m Variable which contains the output of the function Name of the function function P = Poynting( E, B) % Poynting: Evaluate the Poynting vector of an electromagnetic field % USE: P=Poynting(E,B) % E: Complex 3-component vector: the complex amplitude of the electrical field vector (Ex,Ey,Ez) in V/m. % B: Complex 3-component vector: the complex amplitude of the magnetic field vector (Bx,By,Bz) in tesla. % P: Real 3-component vector, giving the componant (rpx,rpy,rpz) % of the real part of the complex Poynting vector in W/m². % Cf. [Jackson, Classical Electrodynamics, 3rd Ed, Wiley 1998] mu0 = 4*pi*1E-7; % Vacuum permeability (H/m) P = real(cross(e,conj(b)))/(2*mu0); % cross computes the cross product end Variables which contains the inputs of the function

13 Header comments Heading Scientific Computing 13 Functions File Poynting.m Variable which contains the output of the function Name of the function function P = Poynting( E, B) % Poynting: Evaluate the Poynting vector of an electromagnetic field % USE: P=Poynting(E,B) % E: Complex 3-component vector: the complex amplitude of the electrical field vector (Ex,Ey,Ez) in V/m. % B: Complex 3-component vector: the complex amplitude of the magnetic field vector (Bx,By,Bz) in tesla. % P: Real 3-component vector, giving the componant (rpx,rpy,rpz) % of the real part of the complex Poynting vector in W/m². % Cf. [Jackson, Classical Electrodynamics, 3rd Ed, Wiley 1998] mu0 = 4*pi*1E-7; % Vacuum permeability (H/m) Variables which contains the inputs of the function Local variable P = real(cross(e,conj(b)))/(2*mu0); % cross computes the cross product end

14 Header comments Heading Scientific Computing 14 Functions File Poynting.m Variable which contains the output of the function Name of the function Variables which contains the inputs of the function function P = Poynting( E, B) % Poynting: Evaluate the Poynting vector of an electromagnetic field % USE: P=Poynting(E,B) % E: Complex 3-component vector: the complex amplitude of the electrical field vector (Ex,Ey,Ez) in V/m. % B: Complex 3-component vector: the complex amplitude of the magnetic field vector (Bx,By,Bz) in tesla. % P: Real 3-component vector, giving the componant (rpx,rpy,rpz) % of the real part of the complex Poynting vector in W/m². % Cf. [Jackson, Classical Electrodynamics, 3rd Ed, Wiley 1998] Local variable mu0 = 4*pi*1E-7; % Vacuum permeability (H/m) Definition of the output argument of the function P = real(cross(e,conj(b)))/(2*mu0); % cross computes the cross product end optional

15 Scientific Computing 15 Call to a function Number of output arguments Number of input arguments Syntax 0 0 Myfunction() OR Myfunction 0 1 Myfunction(x) 1 0 y = Myfunction several 1 [y1,y2,,yn] = Myfunction(x) several several [y1,y2,,yn] = Myfunction(x1,x2,x3, xn) It s your turn! Download the tutorial available on : -Libres Savoirs => cours 0235 Calcul Scientifique (sous-rubrique «Énoncés des TD en PDF») -on the local drive S:\Matlab\ÉnoncésTD_CalcSci

16 Scientific Computing 16 Conditional expressions if condition end if condition else end if condition1 elseif condition2 else end What is a condition? A statement which can be true of false

17 Scientific Computing 17 Conditional expressions if condition end if condition else end if condition1 elseif condition2 else end What is a condition? A statement which can be true of false Comparison operators Some testing functions Logical operators <, <=, >, >=, ==, ~= isnan(),isinf(),isempty(),isreal() &&,, ~ Example: x = 4; y = 3; x>2 && y==1 is equal to is different from isinf(1/0) v = []; isemtpy(v) AND OR NOT

18 Scientific Computing 18 The for and while loops k : loop variable for k = 1:N end Row vector if N<1, the loop is not executed while condition end while condition if condition2 break end end Stops the while loop while condition if condition2 return end end Stops the function

19 Scientific Computing 19 3D graphics In this 2 nd part, we ll learn how to: plot 3D curves change a colormap display images (grayscale, RGB) in different file format (.png,.tif ) How to plot a function z = f(x,y) 1st case: the values of z are known z = [ ; ; ]; mesh(z); OR surf(z); xlabel( x ) ylabel( y ) zlabel( z )

20 Scientific Computing 20 How to plot a function z = f(x,y) 2nd case: f is known and you need to calculate z for specific intervals of x and y e.g. z = sinc(x) sinc(2y) with x [ 4,4] and y [ 3,3] evaluate f for each couple (x,y) of the grid y 3 (3;2) -4 (-3;-1) (1;-2) 4 x -3

21 Scientific Computing 21 How to plot a function z = f(x,y) 2nd case: f is known and you need to calculate z for specific intervals of x and y e.g. z = sinc(x) sinc(2y) with x [ 4,4] and y [ 3,3] evaluate f for each couple (x,y) of the grid y 3 (3;2) z = sinc(x).*sinc(2*y) -4 (-3;-1) (1;-2) 4 x -3 How to generate such matrices? Take 2min. and propose a solution

22 Scientific Computing 22 How to plot a function z = f(x,y) 2nd case: f is known and you need to calculate z for specific intervals of x and y e.g. z = sinc(x) sinc(2y) with x [ 4,4] and y [ 3,3] evaluate f for each couple (x,y) of the grid y 3 (3;2) z = sinc(x).*sinc(2*y) -4 (-3;-1) (1;-2) 4 x -3 X = ones(ny,1)*x; OR [X,Y] = meshgrid(x,y); Y = y. *ones(1,nx); transpose

23 Scientific Computing 23 x = linspace(-4,4,nx); y = linspace(-3,3,ny); Transposition A = M. transposed [X,Y] = meshgrid(x,y); OR X = ones(ny,1)*x; Y = y. *ones(1,nx); A = M Conjugate/Hermitian transposed z = sinc(x).*sinc(2*y) mesh(x,y,z) It s your turn! Plot the function z = sinc( x 2 + 2y 2 ) with x [ 3,3] and y [ 2,2] 1. Use the functions mesh, surf, surfl, surfc, especially try: surfl(x,y,z) shading interp colormap( copper ) axis equal 2. Try different colormaps (find the suitable command or use the GUI)

24 Grayscale images Scientific Computing 24 To load an image: IM = imread( LeProf.tif ); OR imread( moond.tif ); colormap( bone ) image(im) Any problem?

25 Grayscale images Scientific Computing 25 8 bit image (255 levels) To load an image: IM = imread( LeProf.tif ); OR imread( moond.tif ); 63 levels in this colormap colormap( bone ) image(im) Any problem?

26 Grayscale images Scientific Computing 26 8 bit image (255 levels) To load an image: IM = imread( LeProf.tif ); OR imread( moond.tif ); 63 levels in this colormap colormap( bone ) image(im) Any problem? It s your turn! 1. Find a matlab function that solves this problem 2. Find two matlab functions that display the image histogram

27 Grayscale images Scientific Computing 27 8 bit image (255 levels) To load an image: IM = imread( LeProf.tif ); OR imread( moond.tif ); 63 levels in this colormap colormap( bone ) image(im) Any problem? It s your turn! 1. Find a matlab function that solves this problem imagesc 2. Find two matlab functions that display the image histogram imhist, imtool Use imagesc instead of image

28 RGB and indexed images Scientific Computing 28 RGB image = combination of 3 grayscale images (nxmx3 table) IM = imread( flowers.jpg ); imagesc(im); imshow(im); imshow(im(:,:,1)); It s your turn! Find an RGB image on the internet and display it with the function affrvb(im) (nxmx3 table)

29 RGB and indexed images Scientific Computing 29 RGB image = combination of 3 grayscale images (nxmx3 table) IM = imread( flowers.jpg ); imagesc(im); imshow(im); imshow(im(:,:,1)); It s your turn! Find an RGB image on the internet and display it with the function affrvb(im) (nxmx3 table) Indexed images contain a matrix and a colormap [IM,LUT] = imread( m83.tif ); OR imread( canoe.tif ); OR imread( LogoIOGS.png ); colormap(lut) imagesc(im)