dgesv reference implementation of LAPACK MATLAB (\) CPU time (s) Matrix dimension

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1 MATLAB Plan ffl Introduction to Matlab Matrices Mathematical expressions C vs. Matlab Working with Matlab ffl Graphics Functions of one variable Styles, labels & titles Functions of two variables ffl Matlab Scripts Script basics Selection Iteration & vectorization Functions Introductory Course on Scientific Programming Matlab Introductory Course on Scientific Programming Matlab Introduction Matlab is an integrated environment for computation, simulation and visualization. + High level programming language ( mathematical syntax") + Many application specific algorithms (optimisation, ODE's, PDE's, approximation, statistics,...) + Fast development of complex codes + Good support for visualization Slow (inefficient memory handling, limited compiler) Large memory requirements Not available on state of the art computers Working with matrices Matlab is designed for matrix computations and many functions accept matrices and/or vectors as arguments. m n-matrix with elements a ij is defined by A=[a,..., a n ;...; a m,..., a mn ] (The commas can be replaced by blanks.) Automatic type determination and storage allocation. The results of all computations are displayed by default. This output can be supressed by adding a trailing semicolon. >> A = [ 2 ; 3 4 ; 5,6 ] A = >> B = [ 5 ; 7 3]; >> size(a) % Return # of rows & cols 3 2

2 Working with matrices (cont.) The elements of a matrix can be accessed by A(i,j), where» i» m,» j» n. If i and/or j is a vector, a matrix is returned. Logical subscripting selects elements based on some logical and/or relational expression. zeros(m,n) and ones(m,n) functions returning matrices with all elements are 0 and respectively. Vectors can be created using the colon operator, min : step : max which produces an row vector. The complex conjugate transpose operator ' can be used to obtain a column vector. >> b = ( : 2 : 5)' b = 3 5 Matrix subscripting >> A = [ 2 ; 3 4] A = >> A(2,) % Element in row 2, column 3 >> A(:,2) % Second column 2 4 >> A(,:) % First row 2 >> B=[ 2 3 ; ; 7 8 9]; >> B(:2,2:3) % Extract submatrix Introductory Course on Scientific Programming Matlab Introductory Course on Scientific Programming Matlab Mathematical expressions Operator Effect +,-,*,/ as in math () specify evaluation order ^ n power left division ( linear solve") Operators work for matrices and scalars. Precedence as in mathematics. >> A = [0 5 5 ; ; 6 8 8]; >> x = ones(3,); >> y = 3*(A*x)' y = >> z = A y' z = Mathematical functions Complex arithmetics: >> x = +i; >> y = -2*i; >> z = x*y z = i (i = p unless redefined.) Matlab provides a large number of elementary math functions, e.g. cos, exp, sqrt, abs, log. Most elementary functions work with matrices, >> x = 0 : 2*pi/00 : 2*pi; >> y = sin(x); (pi = ß) For more information: >> help elfun (elementary math) >> help specfun (specialized math) >> help matfun (matrix)

3 Element-by-element operations An arithmetic operator preceded by a period (.) acts on the elements of an array and does not follow the rules of linear algebra. Polynomial interpolation Determine the polynomial p(x) passing through x 2 3 p(x) 3 Expression Z=X.*Y Z=X./Y Z=X.^p Example, Effect Z i;j ψ X i;j Y i;j Z i;j ψ X i;j =Y i;j Z i;j ψ X p i;j Mathematical solution Let p(x) = a + a 2 x + a 3 x 2, then C A a a 2 a 3 C A = 0 C A ) a = C A >> A=[ 2 ; 3 4]; >> A*A % Matrix multiplication >> A.*A % Element multiplication Matlab >> x = (:3)'; >> p = [;;3]; >> M = [ones(size(x)), x, x.^2]; >> a = M p a = 3-3 Introductory Course on Scientific Programming Matlab Introductory Course on Scientific Programming Matlab A least squares problem Determine a linear approximation l(x) to the polynomial p(x) passing through x 2 3 p(x) 3 Least Squares Solution Find coefficients in l(x) = b + b 2 x, such that the discrete norm kl(x) p(x)k 2 is minimized C vs. Matlab dgesv reference implementation of LAPACK MATLAB (\) Matlab >> x = (:3)'; >> p = [;;3]; >> M = [ones(size(x)), x]; >> b = M p b = >> norm(m*b-p,2) % Compute residual norm CPU time (s) Matrix dimension Full linear solve

4 CPU time (s) Introductory Course on Scientific Programming M Matlab Desktop Matlab a sample session Figur Scrip (emac A Christer christe@nad C vs. Matlab Optimized C code ( O2) MATLAB Number of unknowns Burgers' equation, upwind scheme (Computer exercise 5.3) Introductory Course on Scientific Programming Matlab Introductory Course on Scientific Programming Matlab Working with matlab Matlab desktop Typing matlab at the Unix prompt opens the Matlab desktop. Matlab is terminated by the command exit. Any command entered at the Matlab prompt is evaluated immediately. Working with matlab (cont.) Windows available on desktop (view menu): emacs For writing Matlab scripts (m-files) and functions (Lecture 3 ). Scripts are run from the Matlab prompt. Matlab 6 has its own combined editor/debugger. Figure window(s) Opened by Matlab for displaying graphics (Lecture 2 ). Command Window Command History Current Directory Workspace Launch Pad Help View Menu Matlab prompt Cut & paste to command window GUI to file system List current variables (with size) Help & tools sorted by toolbox Launch graphical help browser Windows can be resized and moved outside of desktop ( )

5 Some useful commands help doc clear save load clc format!cmd print Some useful commands plain text help graphical help browser Remove variables from workspace Save variables in workspace Load variables from file Clear screen Set output format Execute Unix command cmd Print figure window MATLAB 2 Some of these commands may need additional parameters. Matlab graphics Graphics in Matlab is displayed in the current figure window. Figure windows are opened by figure and closed by close. A figure is made active by figure(h), where H is the handle (integer returned when a window is opened). Matlab will open a figure window if one doesn't exist when a graphics function is called. 2D graphs Functions of one variable are typically displayed using plot(x, y,..., x n, y n); The elements of y i are plotted versus the elements of x i. The values are connected with straight lines (linear interpolation). Help on graphics: >> help graph2d (2D graphs) >> help graph3d (3D graphs) >> help specgraph (special, e.g. 4d") >> x = 0 : 2*pi/00 : 2*pi; >> y = exp(-abs(x-pi)); >> plot(x, y, x, sin(x))

6 2D graphs (cont.) The contents of the current figure are replaced when a new plot command is issued. The hold command can be used to add plots to an existing graph (c.f. previous slide), >> plot(x,y) >> hold on >> plot(x, sin(x)) >> hold off hold off returns to default mode. cla and clf can be used to clear the figure window. Curve styles plot(x, y, style,..., x n, y n, style n); Each curve style is a string of 3 characters: ffl Colour: e.g. 'b' (blue) or 'r' (red) ffl Linestyle: e.g. '-' (solid) or ':' (dotted) ffl Marker: e.g. 'o' (circles) or 's' (squares) See help plot for more styles. Zooming zoom on turns zoom on for the current figure. Left-clicking zooms in, right-clicking zooms out. For 2D graphs an area can be selected with the left mouse button. NB! cla, clf, hold works for 3D graphs as well. >> x = 0:2*pi/00:2*pi; y0=exp(-abs(x-pi)); >> y = sin(x); y2 = x/(2*pi); >> plot(x,y0,'bo-',x,y,'r',x,y2,'k:') Labels & Titles Command Effect xlabel('...') Adds text beside the x-axis title('...') Adds text at top of figure leg(...) Add leg to current graph >> x = 0 : 2*pi/00 : 2*pi; >> L = plot(x, cos(x), x, sin(x)); >> xlabel('x') >> ylabel('y(x)') >> title('trigonometric functions') >> leg(l, 'cos(x)', 'sin(x)') Special 2D graphs Logarithmic plots (base 0) For some plots logarithmic scale is more informative. semilogx(...) semilogy(...) loglog(...) Same interface as for plot. Polar plots polar(theta, r, style); Log. scale on x-axis Log. scale on y-axis Log. scale on x- and y-axes Polar coordinates with radius r and angle. The curve style is optional. Complex data By default only real part plotted. If a single complex array is passed to plot the imaginary part is plotted versus the real part.

7 Subplots Multiple plots can be displayed in the same figure by subplot. subplot(m, n, p) Divides the figure window into an m n-matrix of subplots", and makes subplot p active (numbered rowwise from ). Axis control The scaling and appearance of axes can be controlled, e.g. axis equal axis square grid axis(x 0 ;x ;y 0 ;y ) set aspect ratio equal make axis box square display grid lines set axis scaling >> x=0:2*pi/00:2*pi; >> subplot(2,2,); plot(x,sin(x),x,cos(x)) >> subplot(2,2,2); plot(cos(x),sin(x)) >> subplot(2,2,3); plot(exp(i*x)) >> subplot(2,2,4); polar(x,ones(size(x))) Titles & labels can be given for each subplot. >> x=0:2*pi/00:2*pi; >> subplot(2,,);plot(cos(x),sin(x)) >> grid >> subplot(2,,2);plot(cos(x),sin(x)) >> axis([ ]) >> axis square 3D Graphs: surface plots A function of two variables can be visualized by surf(x, Y, Z, C) where Z = Z(X; Y ) 2 R m n, X 2 R m n or X 2 R n (x-coords), Y 2 R m n or X 2 R m (y-coords) and C 2 R m n determines colour scaling Surface plots (cont.) C=Z is assumed if C not passed explicitly to surf. mesh has the same interface as surf, but plots a parametric mesh instead of a surface. surf and mesh require rectangular grids. The function [X,Y] = meshgrid(x,y) generates matrices X; Y from the vectors x; y. It is assumed that y = x if y is omitted. Use help graph3d for more information on 3D graphs. >> [X,Y]=meshgrid(-8:.5:8); >> R = sqrt(x.^2+y.^2) + e-6; >> subplot(2,,); mesh(x,y,sin(r)./r,r); >> title('mesh()') >> subplot(2,,2); surf(x,y,sin(r)./r); >> title('surf()')

8 Appearance of surface plots 3D Graphs: contour plots contour(x, Y, Z, V) plots the level sets of z(x; y), i.e. curves (μx; μy) : z(μx; μy) = constant >> surf(x,y,z) >> colormap(summer) >> shading interp >> colorbar('vert') shading: controls colour shading flat, interp or faceted (default) colorbar('...'): show colour scale 'vert' (vertical) or 'horiz' (horizontal) colormap: maps function value to colour colormap(a), A 2 R n 3 Rows interpreted as RGB-values (2 [0; ]) See help graph3d for predefined. X, Y and Z should be defined as for surf. V is either an integer specifying the number of curves or a vector specifying the constant values. It can be omitted. >> [X,Y] = meshgrid(0:.0:pi); >> H=contour(X,Y,cos(X.*Y)); >> clabel(h) clabel labels the plotted curves. Rotating & Printing Rotating rotate3d on turns on mouse-based 3D rotation. Pressing the mouse button allows the axis to be rotated. When the button is released the contents of the figure window are updated. view can be used to set the position from which a 3D graph is viewed, e.g. by view([x,y,z]). MATLAB 3 Printing graphics The current figure can be printed using print. The switch -Pprinter determines which printer to use (Unix & VMS only). print -ddevice figure to file. filename prints the current device specifies the format, e.g. psc (postscript colour, default) or jpeg.

9 Matlab scripts Matlab commands can be collected in scripts which are similar to C programs. Matlab scripts are written in a text editor, e.g. emacs or the Matlab editor/debugger. The file should have the suffix.m to be called from within Matlab. A script stored in file.m is executed by file in the Matlab prompt. Example, file mtx.m: A = [ ; ; ; ]; Script execution (Matlab): >> whos >> mtx >> whos Name Size Bytes Class A 4x4 28 double array Grand total is 6 elements using 28 bytes Matlab scripts (cont.) Each command in a script should be terminated by a new line, ; or,. A command can span several lines if a line is terminated by three periods (...). All text following a % is interpreted as a comment. Comments in the beginning of a file are interpreted as user help. Example, file mtx.m: % This script defines a matrix A = [ ; ;... Matlab: ; ]; >> help mtx >> This script defines a matrix. Selection : if if expression else The expression is true if the real part has all non-zero elements (i.e. complex matrix-valued). Selection : switch switch (switch-expression ) case case-expression case case-expression2 Some Operators <, <=, >, >= less than,... ==, ~= equal to, not equal to & Logical AND Logical OR ~ Logical NOT Example, [m,n] = size(a); if (m ~= n) disp('matrix A is not square')... otherwise Works as in C, but case-expression can be scalars and strings. There is no statement corresponding to break" in Matlab.

10 Iteration : while while expression Example, Newton-iteration on x cos x: x = 0; dx=; while (abs(dx) > e-0) f = x - cos(x); fprime = + sin(x); dx x = -f / fprime; = x + dx; fprintf('%20.2e %20.2e n', x, dx); Program execution: >> newton e e e e e e e e e e e e+00 Iteration : for for variable =matrix The loop body is executed once for each element in matrix. variable takes on the values of the matrix elements in column-wise order. >> A=[ 3.4 ; -5 0] A = >> for i=a, disp(i), Note: The colon operator should be used if possible to avoid memory allocation. for i=:0 better than I=:0; for i=i Vectorization Matlab is optimized for working with matrices. One should try to use matrices & vectors as much as possible. Example, outer product implementation of y = Ax (assuming A & x defined): [m,n] = size(a); y = zeros(m,); for j=:n for i=:m y(i) = y(i) + A(i,j)*x(j); 30.5 CPU-seconds for A 2 R y=a*x; requires 0.2 CPU-seconds. Simple loops can often be reformulated in terms of vectors. Matlab functions function [return-values ] = dummy (...) Functions in Matlab are m-files which can accept input arguments and return output values. More than one value can be returned. A function must be stored in a separate file, except local" functions which can be called from within another function only. A function is called by its filename. Example, file avg.m: function m = Donald_Duck(x) % Compute average of elements in x. m = sum(x) / length(x); Donald Duck is a dummy name which is never used. The function is called by avg.

11 Matlab functions (cont.) All arguments are passed by value in Matlab. Call by reference not possible. If a value is supposed to change it must be returned from the function. File increase.m: function i = increase(j) j = j+; i = j; Matlab: >> i=; >> increase(i); disp(i) Input & Output disp can be used for basic output in Matlab. It supports real and complex matrices, scalars,... More control can be obtained by using fprintf, fprintf(fid, format,...) The output (as specified by the format string) and variables are printed to file identifier fid, or the screen if fid is omitted. fprintf has limited support for matrices. It is possible to open files with fopen. >> i=increase(i); disp(i) 2 Example, returning several values: function [minval,maxval]=minmax(x) minval=min(x); maxval=max(x); Input from keyboard can be read using input(string ) The function input displays string, and returns the value entered. fscanf exists but can be used to read from files only. Input & Output (cont.) A number can be converted into a string by num2str. str2num performs the reverse conversion. Example (io.m): x = input('enter a vector: '); disp(['you entered a vector of length ',... num2str(length(x))]); fprintf('%e n', x); Matlab: >> io Enter a vector: (:5)' You entered a vector of length e e e e e+00 Executing functions by strings feval(str,...) evaluates the function whose name is given in the string str with arguments... sin(x) and feval('sin', x) equivalent. Can be used to pass functions to functions ( function pointers"). Example, solving y 0 (t) = y(t) 2, y(0) = ode45 solves initial value problems for ODE's. [T,Y] = ode45('odefile', [t0 t], y0). Y contains approximate solution at times in T 2 [t 0 ;t ]. Function yprime.m: function yp = yprime(t,y) yp = -y.^2; % (Vector notation to be on safe side)

12 Example: Solving an ODE Script solve.m: [T,Y] = ode45('yprime', [0,20], ); plot(t,y) xlabel('t'); ylabel('y(t)'); title('y''(t) = -y(t)^2, y(0)='); Storage class Variables in Matlab (and C) have local storage class by default. A variable in a function can be declared persistent to maintain its value between function calls (c.f. static in C). A persistent variable is initalized to the empty matrix []. Example, function test persistent i if (isempty(i)), i=; else i=i+; fprintf('this is function call %d n',i); Matlab: ode45 contains several function evaluations on the form feval(odefile, t, y). >> test This is function call >> test This is function call 2 >> test This is function call 3 Storage class It is sometimes convenient to have global variables. An identifier declared global prior to its initialization is added to the global symbol table. It can be used inside all functions declaring it. Example, Object orientation & GUI Matlab supports (emulated) object orientation. Messy to program since variables can not be passed by reference. Many routines for fast creation of GUI's. function dummy() global G disp(g) Matlab: >> global G >> dummy % Empty matrix, no output >> G=.0; >> dummy GUI for visualizing C computations (Colours screwed up when capturing image)

13 Introductory Course on Scientific Programming M home page Course home page for the introductory course is available at URL The html or you can enter and click your way there. A Christer christe@nad