BSM510 Numerical Analysis

Transcription

1 BSM510 Numerical Analysis Introduction and Matlab Fundamentals Manar Mohaisen Department of EEC Engineering

2 Lecture Content Introduction to MATLAB 2

3 Introduction to MATLAB MATLAB 3

4 Scalars >> x = 5; x 5 >> y = sin(pi/4); x >> z = power(2, 5); % 2 to power 5 z 32 >> data = 5/2 + 5; data 7.5 >> d = 5 + j3; d 5 + sqrt(-1)*3 d = d / 2; d j1.5 >> scale = sqrt(sin(0.5*pi*real(d))); scale sqrt(sin(3/4*pi)) = i

5 Vectors >> x = [1 3 5]; x is a row vector of size 13 >> y = x.^2; % y[i] = x[i]*x[i] y [1 9 25] >> p = sum(y); p is the total power of x >> vec = x. ; vec is a column vector = [1 3 5] T % transpose operator >> q = [5, 3, -1.1]; % equivalent to [ ] q is a row vector of size 13 >> v = [-9; 4; 16]; v is a column vector of size 31 >> sr = sqrt(v); sr [3i; 2; 4], with i = sqrt(-1) 5

6 Vectors contd. >> re = [ ]. ; >> im = [ ]. ; >> co = re + sqrt(-1)*im; co [7+i 3+i3 2+i -1] T ; coct = co. ; cot = [7+i 3+i3 2+i -1] cort = co ; cot = [7-i 3-i3 2-i -1] >> mul = re.*im; mul = [7; 9; 2; 0] >> div = 7*im./re; % transpose % conjugate transpose % element-wise multiplication div = [1; 7; 3.5; 0] >> une = ones(1, 5); zero = zeros(1, 4); une [ ]; zero = [ ]; 6

7 Vectors contd. >> index = 1:2:9; index [ ] >> range = 5:-0.5:3; range [ ] >> a = [ ]; b = [ ]; >> y = a*b. ; y 28.5 >> z = a*b; % vector multiplication Wrong: inner matrix dimensions must agree >> x = a(1, 1:2); w = [a(1) a(3) a(2)]; z=a(4:-1:1); x [1-1]; w [1 5-1]; z = [ ]; >> v1 = max(a); v2 = min(a); v3 = var(a); >> v4 = mod(b, 2); v4 [ ] 7

8 Vectors contd. >> x = [ ]; >> Y = diag(x); >> q = prod(x); q = -15 % product of the elements of x >> y = find(x >= 2); % indices of elements >= 2 y [2 3] Y >> num = prod(size(y)); % the number of elements >= 2 num >> y = [ ]; >> m = [x y]; z = [x; y]; m [ ] z

9 Vectors example Plot the following function using commands in the command window f() x 1 e 2 x2/2 >> x = [-5:0.1:5]; % range of ind. variable >> f = 1/sqrt(2*pi) * exp(-x.^2./ 2); % dependent variable >> plot(x, f, linewidth, 1.5) % 2d plot >> xlabel( x ) >> ylabel( f(x) ) >> grid on % add the grid to the figure f(x) x 9

10 Vectors example Plot the following function in the range [-5, 5] with increment 0.05 fx () sin(2 x)cos( x) x2 1 >> x = [-5:0.05:5]; % range of ind. variable >> f = sin(2*pi*x).*cos(pi*x); % divide expression into 1 >> f = f./(x.^2 + 1); % and 2 >> plot(x, f, linewidth, 1.1) % 2d plot >> axis([ ]; %axis([xmin xmax ymin ymax]) >> xlabel( x ) >> ylabel( f(x) ) >> grid on % add the grid to the figure 0.5 f(x) x 10

11 Vectors creating functions (m files) The function should have the following form Programmed in the edit window, then it is called in the command window or in other function or m file function [list of outputs] = functionname(list of inputs) % function s body % Example 1 function [val] = NormalF(x, a, b) % returns the value of the pdf of gaussian RV at x % a is the variance and b is the mean % Val = 1/sqrt(2*pi*a) * exp(-(x-b)^2 / (2*a) ) Example 2 function [vecp] = vecpower(v) % returns the total power of the vector v v = v(:); % v becomes column vector vecp = v *v; % works for complex vectors as well 11

12 Vectors example function [va vn] = acceleration(m, cd) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % inputs: m (kg) % cd (kg/m) % outputs: va analytical results % vn numerical results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g = 9.81; % acceleration due to gravity t = 0:0.5:12; % time range va = sqrt(g*m/cd).*tanh(sqrt(g*cd/m).*t); % analytical velocity vn = zeros(1, length(t)); % initiate numerical v to zero for ii = 2 : 1 : length(t) vn(ii) = vn(ii-1) + (g - cd/m*vn(ii-1)^2) * (t(ii)-t(ii-1)); end plot(t, va, '-o') % plot the analytical results hold % hold the plot to add other curve plot(t, vn, 'r') % plot the numerical results xlabel('time (s)'); % set the x label ylabel('v (m/s)'); % set the y label grid on % show the grid v (m/s) analytical numerical time (s) 12

13 Vectors random values % Gaussian numbers >> x = randn(1, 4); % Gaussian random values w. mean=0, var=1 >> y = randn(1, 4)+2; % Gaussian random values w. mean=2, var=1 >> z = 2*randn(1, 4)-2; % Gaussian random values w. mean=-2, var=4 >> w = (randn(1, 4) >= 0); % generates binary numbers with equal prob % equal probability integers >> x = randint(1, 4); % binary values >> y = randint(1, 4, [-2 2]); % random int in [-2,2] ( ) % any real value with any probability >> a = randsrc(1, 4, [1 3]); % random numbers 1 or 3 (same prob) >> b = randsrc(1, 4, [1 3; ]);% prob(1) = 0.4, prob(3) = 0.6 % real random numbers (uniform distribution) >> g = rand(1, 4); % real numbers in [0, 1] with equal prob >> h = rand(1, 4) 0.5; % real numbers in [-0.5, 0.5] with equal prob % try to check the help on those functions using the help command >> help rand 13

14 Matrices A vector is a special case of matrices that has either dimension = 1 All what we introduces about vectors still applied to matrices >> x = [2 4; 5 1] % 2x2 matrix, the ; separates rows >> y = [5 1 7; 3 1 0]; % 2x3 matrix >> z = randn(5); % 5x5 matrix with N(0,1) entries >> w = diag(z); % 5x1 vector (diagonal elements of z) >> a = transpose(x); >> u = inv(x); >> detx = det(x); >> tracex = trace(x); >> vec = x(:, 1); >> vec1 = x(2, :); % same as x or x. for real % inverse of x % determinant % trace of x % vec is the first column of x % vec1 is the second row of x >> pary = y(:, 1:2); % pary = [5 1; 3 1] >> prodm = x*y; % matrix multiplication 14

15 Matrices contd. >> x = [2 4; 5 1] % 2x2 matrix >> y = x.^2; % y = [4 16; 25 1] >> z = x^2; >> U = trace(x *x); >> logv = log(x); >> w = sqrtm(x); % z = x*x; % what is this? % element-wise log (natural log) % w*w = x >> m = sqrt(x); % m(i,j) = sqrt(x(i, j) >> [a b] = size(x); % a=# of rows, b=#of columns >> n = prod(size(x)); % # of elements in x >> T = reshape(x, 1, 4); % T = [ ] >> U = x(:, [2 1]); % exchange columns 1 and 2 >> M = x*y; >> V = x.*y; % #of col. in x = #of rows in y % x and y are with same dimensions >> P = sum(sum(x.*x)); % total power of x 15

16 Input-output Input >> x = input( display message ) Example: n = input( enter your weight (kg): ) Output >> disp( message to display ) Example: disp( error, please try again! ) >> fprintf( format, values or variables); Example: fprintf( your weight is %f kg, 125.3); Notes» This functionality is same as printf in c» fprintf can be used for both file/screen display in MATLAB 16

17 If statement Loops if (condition 1) statements elseif (condition 2) statements else statements end for index = start:step:end statements end Example if (x > 0) disp( x is positive) elseif (x < 0) disp( x is negative ) else disp( x = 0 ) end Example y(1) = -5; for ii = 2 : 1 : 100 y(ii) = y(ii-1) + ii^2 end 17

18 Relational operators == equal ~= not equal < less than > larger than <=,>= less/larger than Logical operators ~ not & and or 18

19 Anonymous functions Allows you to create a function without creating an M-file. Format: fhandle of args) expression >> a = 5; >> b = 3; >> f1 a*x^2 + b*y^2 >> f1(2, 3) ans = 47 >> a = 10; % does not affect the result >> f1(2, 3) ans = 47 19

20 Function functions A function that takes a function as an argument Example: M-file function favg = funcafvg(f, a, b, n) % inputs: f is a function to be evaluated % [a,b,n] interval limits and # of points x = linspace(a, b, n); Y = f(x); favg = mean(y); Command window >> f1 5*x^2 + 3*x 2; >> funcavf(f1, 0, 12, 60); 20

21 Lecture Summary We introduced an overview of MATLAB 21