BSM510 Numerical Analysis - mohaisen.net · Introduction and Matlab Fundamentals Korea University...

Introduction and Matlab Fundamentals

Korea University of Technology and Education (KUT)

BSM510

Numerical Analysis

Manar Mohaisen

Department of EEC Engineering

Lecture Content

Introduction to MATLAB

2Korea University of Technology and Education (KUT)

Introduction to MATLAB

MATLAB


Introduction to MATLAB - contd.

Scalars♦ >> x = 5;

x 5

♦ >> y = sin(pi/4);

x 0.7071

♦ >> 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 2.5 + j1.5

♦ >> scale = sqrt(sin(0.5*pi*real(d)));

scale sqrt(sin(3/4*pi)) = i0.8409



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.’; % transpose operator

vec is a column vector = [1 3 5]T

♦ >> q = [5, 3, -1.1]; % equivalent to [5 3 -1.1]

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)



Vectors – contd.♦ >> re = [7 3 2 -1].’;

♦ >> im = [1 3 1 0].’;

♦ >> co = re + sqrt(-1)*im;

co [7+i 3+i3 2+i -1]T;

♦ coCT = co.’; % transpose

coT = [7+i 3+i3 2+i -1]

♦ coRT = co’; % conjugate transpose

coT = [7-i 3-i3 2-i -1]

♦ >> mul = re.*im; % element-wise multiplication

mul = [7; 9; 2; 0]

♦ >> div = 7*im./re;

div = [1; 7; 3.5; 0]

♦ >> une = ones(1, 5); zero = zeros(1, 4);

une [1 1 1 1 1]; zero = [0 0 0 0];



Vectors – contd.♦ >> index = 1:2:9;

index [1 3 5 7 9]

♦ >> range = 5:-0.5:3;

range [5 4.5 4 3.5 3]

♦ >> a = [1 -1 5 3]; b = [0.5 3 5 2];

♦ >> y = a*b.’; % vector multiplication

y 28.5

♦ >> z = a*b;

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 = [3 5 -1 1];

♦ >> v1 = max(a); v2 = min(a); v3 = var(a);

♦ >> v4 = mod(b, 2);

v4 [0.5 1 1 0]



Vectors – contd.♦ >> x = [1 5 3 -1];

♦ >> Y = diag(x);

♦ >> q = prod(x); % product of the elements of x

q = -15

♦ >> y = find(x >= 2); % indices of elements >= 2

y [2 3]

>> num = prod(size(y)); % the number of elements >= 2

■ num 2

♦ >> y = [7 8 4 1];

>> m = [x y]; z = [x; y];

m [1 5 3 -1 7 8 4 1]


1 0 0 00 5 0 00 0 3 00 0 0 1

Y

1 5 3 17 8 4 1

z


Vectors – example

♦ Plot the following function using commands in the command window


>> 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

2/21( )2

xf x e

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

x

f(x)


Vectors – example

♦ Plot the following function in the range [-5, 5] with increment 0.05


2sin(2 )cos( )

1( ) x x

xf x

>> 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([-5 5 -0.8 0.8]; %axis([xmin xmax ymin ymax])

>> xlabel(‘x’)

>> ylabel(‘f(x)’)

>> grid on % add the grid to the figure

-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.5

0

0.5

x

f(x)


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

Example 1

Example 2


function [list of outputs] = functionName(list of inputs)

% function’s body

%

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) )

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


Vectors – example 3


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

0 2 4 6 8 10 120

10

20

30

40

50

60

time (s)

v (

m/s

)

analytical

numerical


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] (-2 -1 0 1 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; 0.4 0.6]);% 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


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); % same as x’ or x.’ for real

♦ >> u = inv(x); % inverse of x

♦ >> detX = det(x); % determinant

♦ >> traceX = trace(x); % trace of x

♦ >> vec = x(:, 1); % vec is the first column of x

♦ >> vec1 = x(2, :); % vec1 is the second row of x

♦ >> parY = y(:, 1:2); % parY = [5 1; 3 1]

♦ >> prodM = x*y; % matrix multiplication



Matrices – contd.♦ >> x = [2 4; 5 1] % 2x2 matrix

♦ >> y = x.^2; % y = [4 16; 25 1]

♦ >> z = x^2; % z = x*x;

♦ >> U = trace(x’*x); % what is this?

♦ >> logV = log(x); % element-wise log (natural log)

♦ >> w = sqrtm(x); % 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 = [2 5 4 1]

♦ >> U = x(:, [2 1]); % exchange columns 1 and 2

♦ >> M = x*y; % #of col. in x = #of rows in y

♦ >> V = x.*y; % x and y are with same dimensions

♦ >> P = sum(sum(x.*x)); % total power of x



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



If statement

Loops


if (condition 1)

statements

elseif (condition 2)

statements

else

statements

end

if (x > 0)

disp(‘x is positive)

elseif (x < 0)

disp(‘x is negative’)

else

disp(‘x = 0’)

end

for index = start:step:end

statements

end

y(1) = -5;

for ii = 2 : 1 : 100

y(ii) = y(ii-1) + ii^2

end

Example

Example


Relational operators

Logical operators


== equal

~= not equal

< less than

> larger than

<=,>= less/larger than

~ not

& and

| or


Anonymous functions

♦ Allows you to create a function without creating an M-file.

♦ Format: fhandle = @(list of args) expression


>> a = 5;

>> b = 3;

>> f1 = @(x,y) a*x^2 + b*y^2

>> f1(2, 3)

ans =

47

>> a = 10; % does not affect the result

>> f1(2, 3)

ans =

47


Function functions

♦ A function that takes a function as an argument

♦ Example:


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);

>> f1 = @(x) 5*x^2 + 3*x – 2;

>> funcavf(f1, 0, 12, 60);

M-file

Command window

Lecture Summary

We introduced an overview of MATLAB


BSM510 Numerical Analysis - mohaisen.net · Introduction and Matlab Fundamentals Korea University...

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