Lishen He

CEE 470

Fall 2014

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What is Matlab? Comprehensive computer program to perform wide

range of mathematical operations

Easy to use programming language

No need to declare variables a = 5; %a is an int

A = [1 2 3 4 5]; %A is a vector

Capabilities

Graphics – 2D, 3D, color and lighting, animation

Computations – linear algebra, quadrature, ODES, etc.

External Interfacing – can interface with C and Fortran

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How Do I Get Matlab? Good News! Its

FREE for all U of I students.

Visit the CITES webstore: https://webstore.illinois.edu/Shop/product.aspx?zpid=1271

Must be logged into VPN

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References Handout

Introduction to MATLAB (CE-261, CE-361, CE-478)

Books

Getting started with MATLAB: A quick introduction for scientists and engineers, Rudra Pratap - Oxford University Press (Version 7 – older is fine too)

Online

Matlab Documentation

http://www.mathworks.com/help/techdoc/index.html

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Getting Started Windows

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Getting Started Windows

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Command Window:Good for small, “quick” computations

Getting Started Windows

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Workspace:Shows the active variables and their types, values and sizes

Getting Started Windows

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Command History:Saves all the commands typed on MATLAB’s command prompt(up arrow goes through this list)

Getting Started Windows

Change

directory

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Set working directory

Getting Started Windows

Type:

edit test1

in the

command

prompt

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Edit window – creates a .m file (test1.m). This is good for longer programs.

Getting Started Windows

Type:

figure

in the

command

prompt

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Figure window – Used to display graphics and plots

Basics of Matlab Basic data-type is an array

Can hold integers, doubles, matrices, strings, structures, cells

All automatic and internal

Defining an array: a = 5; %a is an int

A = [1,2,3,4,5]; %A is a vector

A = [1 2 sqrt(9); 4 5 6; 7 8 3*3]; %A is a matrix

Syntax note: semi-colon suppresses output, commas optional

Basics of Matlab Basic array operations:

Creating an array: A = [1:2:10]; %A is [1 3 5 7 9]

A = linspace(1,9,5) %A is [1 3 5 7 9]

A = ones(5,1); %A is a 5x1 vector of ones

A = zeros(2,3); %A is a 2x3 vector of zeros

A = eye(5); %A is the 5x5 identity matrix

Mathematical operations c = A*b; %matrix-vector multiplication

c = a*b; %vector-vector multiplication

c = a.*b; %vector component multiplication

c = A\b; %linear solve

Basics of Matlab Basic array operations (cont):

Matrix operations (HW 1): inv(A); %gives the inverse of A

det(A); %gives the determinant of A

norm(A,p); %gives the p-norm of A

chol(A); %gives the cholesky-factorization

lu(A); %Gauss-elimination factorization

tril(A); %lower triangular part of A

triu(A); %upper triangular part of A

eig(A); %eigenvalues of A

Note: for help type “help chol” in command prompt

Rows, Columns and Submatrices Accessing entries

A(3,4); %gives the 3rd row, 4th column of A

A(i,:); %gives the ith row of A

A(:,k); %gives the kth column of A

A([2:4],3) = ??

Resizing arrays

How to use the partition A for DOFs 1,2 and 4? A = A([1,2,4],[1,2,4]);

Same for vectors b = b([1,2,4]);

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Loops For

for i = 1:m

for j = 1:n

A(i,j) = 0;

end

A(i,i) = i;

end

While n = 0;

while n < 10

n = n + 1;

end

n

If-elseif-else if i>5

%do something

else if (i<=5) && (i > 2)

%do something else

else

%default

end

Break – can use, but bad coding practice (see: “help break”)

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Plotting x = linspace(0,5,20);

y = sin(x);

plot(x,y);

figure

%Creates a new figure

plot(x,y,’r’); %Colors

‘k’,’b’,’r’,’g’,’y’,’c’,’m’,’w’

plot(x,y,’--’);

%Linestyle – dashed line

plot(x,y,’b--’,’Linewidth’,3);

%Makes a thicker line

legend(‘sin x’)

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Creating a very simple .m File In the command prompt, edit test1.m

In the editor, function [x] = test1(A,b)

% Solve the system

x = A\b;

In the command prompt, C = [1 2 3; 4 5 6; 7 8 9];

d = [10; 11; 12];

[x] = test1(C,d)

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An Exercise… Write your own function to compute the cholesky

decomposition for a matrix using Matlab.

Why?

To solve systems of linear equations

Roughly twice as fast as LU decomposition!

In the command prompt, edit cholesky

In the editor, function [L] = cholesky(A)

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An Exercise… Cholesky decomposition

For the system, A x = b we want to factor A, to be of the form A = LLT , where L is lower triangular with positive diagonals.

Applicable when A is:

1. Square

2. Hermitian (symmetric and real)

3. Positive Definite

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An Exercise… In cholesky.m, check for square matrix

%check to make sure A is a square matrix by comparing

rows and columns of A

s = size(A);

if isequal (s(1,1),s(1,2)) == 0

disp('ERROR: A is not a square matrix. Please enter

a square matrix.')

return

end

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An Exercise… In cholesky.m, check for positive definite matrix

%need to check for positive definite matrix

E = eig(A);

if min(E) < 10^-6

disp('ERROR: A must be a positive definite matrix.')

return;

end

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An Exercise… In cholesky.m, check for symmetry

%check for symmetry

if isequal(A,A')== 0

disp('ERROR: A must be a symmetric matrix.')

return

end

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An Exercise… Now, starting the calculation of the terms, we need to

compute each coefficient using the following equations (see class notes):

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1

1

2j

r

jrjjii lal

1

1

1 j

r

jririj

jj

ij llal

l

An Exercise…

%Cholesky factorization loops

for i=1:s(1,2)

x = 0;

for r = 1:i-1

x = x + (L(i,r))^2;

end

L(i,i) = sqrt(abs(A(i,i)-x));

…..

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1

1

2j

r

jrjjii lal

L is not declared yet in first loop - why does this work?

An Exercise…

...

for j=i+1:s(1,2)

y = 0;

for r=1:i-1

y = y + L(i,r)*L(j,r);

end

L(j,i) = (1/L(i,i)*(A(i,j)- y));

end

end

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1

1

1 j

r

jririj

jj

ij llal

l

An Exercise… Let’s try! In the command prompt,

A = [2 0 1; 0 4 1; 1 1 3];

L = cholesky(A)

L = ??? (HW 1 Hint)

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Another Example: What about the transformation matrix? (HW 1 Hint)

% Build the transformation matrix Q as described

in class

% your code goes here...

%First row of Q

mag_a = sqrt((a(1,1))^2+(a(1,2))^2+(a(1,3))^2);

Q(1,1) = a(1,1)/mag_a;

Q(1,2) = ...

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Another Example: Building the transformation matrix

%First row of Q

mag_a = sqrt((a(1,1))^2+(a(1,2))^2+(a(1,3))^2);

Q(1,1) = a(1,1)/mag_a;

Q(1,2) = ...

%Last row of Q

c = cross(a,b);

mag_c = sqrt((c(1,1))^2+(c(1,2))^2+(c(1,3))^2);

Q(3,1) = c(1,1)/mag_c;

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Another Example: Building the transformation matrix

%Second row of Q

% your code here!

% Note: e3 is Q(3,:)

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'

1

'

3

'

2 eee

Printing results What is a “Diary file”?

% Create and output file of the command window

(before your work)

diary session1.out

% End saving output to the file

diary off

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Questions? Office Hours: Tuesdays 4:00-5:00 pm, Yeh 2312

Thursdays 5:00-6:00pm, Yeh 2312

Email: lhe10@illinois.edu

THANKS!

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