Lanza, Alessandro

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Almae Matris Studiorum Campus

UNIVERSITY OF BOLOGNA - FACULTY OF ENGINEERINGINTERNATIONAL MASTER COURSE IN CIVIL ENGINEERING - A.Y. 2012/2013INTRODUCTION TO NUMERICAL METHODSLAB 1:   MATLAB BASICSa)Working directly in the MATLAB Command Window, create the two matrices A =  −18661, B = 1 210 6−    , the column vector v = [1  2]T  and the scalar k = 2, check the existence (and the value, size, bytes and class) of the four created variables in the MATLAB Workspace, then compute:(ADDITION)                                         A + k    A + v    A + B     k + A    v + A    B + A(SUBTRACTION)                                A - k     A - v     A – B      k – A    v – A    B - A(MULTIPLICATION)                          A * k     A * v     A * B     k * A    v * A    B * A                                                              A .* k    A .* v    A .* B    k .* A   v .* A    B .* A(RIGHT and LEFT DIVISION)           A / k      A / v      A / B     A \ k     A \ v      A \ B                                                              A ./ k     A ./ v     A ./ B    A .\ k    A .\ v     A .\ B(EXPONENTIATION)                        v ^ k      A ^ k                                                             v .^ k     A .^ k                                                                                                             A        A          A(CONCATENATION)                        A || k     A || v       A || B       =         =          =                                                                                                          k         v          B(TRANSPOSITION)                           kT          vT          AT        PAY ATTENTION!!!  The operations are not all computable: why?b)Working directly in the MATLAB Command Window, create the 4x4 matrix (square matrix of order 4):A = −−−−−2123151681262109512462then:a) Create a vector  v  formed with the elements (entries) of the second row of  A;b) Compute and store in the variable  s  the sum of the elements of  v, after these have been divided (element by element) by the elements of the first column of A;c) Create a matrix  B  formed with all the elements between the second and the fourth column of A;d) Create a matrix  C  formed with all the elements of the first two rows and the last three columns of A;e) Create  At , the transpose of  A;Pag.  1 / 6UNIVERSITY OF BOLOGNA - FACULTY OF ENGINEERINGINTERNATIONAL MASTER COURSE IN CIVIL ENGINEERING - A.Y. 2012/2013INTRODUCTION TO NUMERICAL METHODSf) Create a vector  v  formed with the minimums of the elements in each column of  At  ;g) Create a vector v formed with the maximums of the elements in each row of  At  ;h) Create a vector v formed with the sums of the elements in each row of  At  ;i) Compute the sum of all the elements of A ;c)Create:      1)  a MATLAB script;          2)  a MATLAB function;       that, given a natural number  n , compute the sum  S  of the first  n  natural numbers by the formula   1( 1)2nkn nS k=+= =∑ .Use the script and the function to compute the sum  S  for different values of  n … Which is the difference between   script   and   function?d)An object is thrown vertically upwards with an initial velocity  v0 and reaches a height  h in a time  t, where 220gttvh −= . Write a MATLAB function that calculates (and provides as output) the time  t  required for reaching a given height  h  for a given value of  v0. The function takes as inputs  h, v0 and g.Test the function in the cases:    a)  h = 170 m, v0 = 60 m/sec e g = 9.81 m/s2                                          b)  h = 200 m, v0 = 60 m/sec e g = 9.81 m/s2 : What can you conclude?[case a) 4.4580 - 7.7744 ; case b) complex result ]e)After exploring (in the MATLAB Help) the following commands for the creation of special matrices: zeros zero matrix ones matrix of ones eye identity matrix diag diagonal matrix tril lower triangular matrix triu upper triangular matrix hilb Hilbert matrix vander Vandermonde matrix rand matrix of random numbers magic matrix with equal row, column and diagonal sumsthen write a MATLAB function that takes as input a natural number  n  and provides as output a square  n x n matrix  A = {ai,j}i,j = 1,…,n   defined as follows:,2 1 if 10 otherwisei j| i - j |  | i - j |a+ ≤= What can you say about the matrix?Pag.  2 / 6UNIVERSITY OF BOLOGNA - FACULTY OF ENGINEERINGINTERNATIONAL MASTER COURSE IN CIVIL ENGINEERING - A.Y. 2012/2013INTRODUCTION TO NUMERICAL METHODSf)f1)  After exploring (in the MATLAB Help) the potentialities of the command plot (in particular, how to plot with given color, style, marker), write a MATLAB script that plots in the same graph the four functions:310sin(2 )ln( )cos( ) log - 4x with [ 2,2]2x x xy x x y e y y x xx = = = = ∈ −  using different types of  line and a legend to distinguish the curves, axes labels, a title and the grid.f2)  Create a script that plots the functions in four different graphs in the same figure (command subplot).f3)  Create a script that plots the functions in four different figures.PAY ATTENTION TO THE DOMAIN for the third function!!!3D GRAPHICSMATLAB provides several functions that allow creating different types of three-dimensional plots: Line plots Surface plots Contour plotsLine plotsLines in three-dimensional space can be plotted by using the MATLAB function plot3; the syntax is:plot3(x,y,z)where  x, y, z  are vectors containing the  (x,y,z)  coordinates of a set of  3-D  points defining the line.Example: t = [0:0.1:10*pi] ;x = exp(-t/20) .* cos(t);   y = exp(-t/20) .* sin(t);   z = t;plot3(x,y,z);xlabel('x');  ylabel('y');  zlabel('z');Surface plotsA generic function of two variables  z = f(x,y)  represents a surface when plotted in three-dimensional space. The MATLAB functions mesh and surf allow to generate, respectively, a wireframe and a solid surface plot. Before being used, both functions require a grid of points in the xy plane (function f domain) to be created by means of the MATLAB function meshgrid, so that the z = f(x,y) function is then calculated in correspondence of the grid points. The syntax of the function meshgrid is:[X,Y] = meshgrid(x,y)After the grid has been generated, the surface plot can be created by the mesh (wireframe surface) or the surf (solid surface) function. The syntax is:mesh(X,Y,Z)                  surf(X,Y,Z)Examples:Pag.  3 / 6UNIVERSITY OF BOLOGNA - FACULTY OF ENGINEERINGINTERNATIONAL MASTER COURSE IN CIVIL ENGINEERING - A.Y. 2012/2013INTRODUCTION TO NUMERICAL METHODSa) using the MATLAB function mesh to visualize the two variables function z = f(x,y) = sin(x)cos(y):x = linspace(0,2*pi,50);y = linspace(0,pi,50);[X,Y] = meshgrid(x,y);Z = sin(X).*cos(Y);mesh(X,Y,Z);xlabel('x'); ylabel('y'); zlabel('z');grid on;b) using the Matlab function surf to visualize the two variables function z = f(x,y) = exp(-(x2+y2) / 3):[X,Y] = meshgrid(-3:.2:3 , -2:.2:2);Z = exp( -(X.^2+Y.^2)/3 );surf(X,Y,Z);xlabel('x'); ylabel('y'); zlabel('z');grid on;Contour plotsIn  topography,  three-dimensional  earth  surface  is  generally  represented  by  two-dimensional  maps,  called contour maps. These maps contain contour lines (or  isolines), that are lines joining points of equal elevation above a given level, such as the mean sea level. More generally, the shape of two variables functions can be  better understood by means of contour plots. In MATLAB, these plots can be created by using the function  contour. The syntax is:contour(X,Y,Z)The usage of this function is similar to that of mesh and surf, in the sense that a grid of points has to be preliminarily created by means of the MATLAB function meshgrid.Example: [X,Y] = meshgrid(-3:.2:3 , -2:.2:2);Z = exp( -(X.^2+Y.^2)/3 );contour(X,Y,Z);xlabel('x'); ylabel('y'); zlabel('z');grid on;g)After exploring (in the MATLAB Help) the potentialities of the command plot3, write a MATLAB script that plots the 3-dimensional curve  f(t) defined by the following parametric equations:cos( )( ) : ( )x a t tf t y a t sen tz b t= ⋅ ⋅= ⋅ ⋅= ⋅where a and b are real constants.Create a plot of the curve f(t) for each of the following four cases:a) b = 0.1Pag.  4 / 6set a = 1 and -10 π ≤ t ≤ 10 πRefer to the MATLAB Help to gather further information on functions that generate three-dimensional graphs ( doc graph3d )UNIVERSITY OF BOLOGNA - FACULTY OF ENGINEERINGINTERNATIONAL MASTER COURSE IN CIVIL ENGINEERING - A.Y. 2012/2013INTRODUCTION TO NUMERICAL METHODSb) b = 0.5c) b = - 0.1d) b = -5Compare the four obtained plots by means of the command subplot.h)  Consider the following function of two variables:221),(yxyxyxfz+++==Represent the function, that is the surface (hyperbolic paraboloid), in the domain –1 ≤ x ≤ 1 ,   -1 ≤ y ≤ 1, using the Matlab commands seen before (mesh, surf, contour, meshc, surfc …) and trying different graphics options (colormap, shading, axis…).i) Create the contour and surface plots for the following two variables function:z = f(x,y) = – x2+2xy+ 3y2This function has a critical point at (x=0 , y=0), but the point corresponds neither to a minimum nor to a maximum (inflectional point): which isolines correspond to this point?CREATION OF MOVIESMATLAB offers the possibility to store a series of plots and then to show them one after the other so as to obtain a real animation. The basic idea is to store tidily the plots within the columns of a proper matricial data structure. The MATLAB function movie, then, allows to show the animation.Firstly, the plots to be shown have to be generated (usually through a  for loop) and stored, by means of the MATLAB command getframe, in the columns of the matrix. It might be useful to set the axes limits through the command axis([xmin xmax ymin ymax]).Example:    x=linspace(0,pi);for k=1:5    plot(x,sin(k*x));    grid;    title('Sin(kx), k=1,2,3,4,5');    axis([0,pi,0,1]);    xlabel('x');    ylabel('y');    M(:,k)=getframe; % the current kth plot is stored in the kth column endThen, the movie is simply shown by the command:movie(M)It is worth noticing (in the Workspace panel) the large amount of memory required to store the movie (matrix M)!Pag.  5 / 6UNIVERSITY OF BOLOGNA - FACULTY OF ENGINEERINGINTERNATIONAL MASTER COURSE IN CIVIL ENGINEERING - A.Y. 2012/2013INTRODUCTION TO NUMERICAL METHODSj) Create a movie for the one variable function:2( ; )y f x a ax ax= = −with  a = 0,1,2,3,4,5,6,7,8,9  in the interval  ]5.1,5.0[−∈xk) Create a movie for the two variables function:)cos()sin();,( ykxkkyxfz pipi==with  k = 0,1,2,3,4,5  in the interval ]1,0[]1,0[),( ×∈yxPag.  6 / 6Refer to the Matlab Help to gather further information on functions that generate movies

INM12-13 LAB1 MATLAB BASICS

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INM12-13 LAB1 MATLAB BASICS

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Almae Matris Studiorum Campus

Almae Matris Studiorum Campus