introduction to matlab programming
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TRANSCRIPT
Zen and the Art of MatLab
Damian Gordon
Hard work done by :
Daphne Gilbert &
Susan Lazarus
Introduction to MatLab
• MatLab is an interactive, matrix-based system for numeric computation and visualisation
• MATrix LABoratory
• Used in image processing, image synthesis, engineering simulation, etc.
References
• “Mastering MatLab” Duane Hanselman, Bruce Littlefield
• “The MatLab Primer” http://www.fi.uib.no/Fysisk/Teori/KURS/WRK/mat/mat.html
• “The MatLab FAQ” http://www.isr.umd.edu/~austin/ence202.d/matlab-faq.html
Printed Circuit Board
Specific Bond Selected
Bond Shape Estimated
MATLAB Command Window
To get started, type one of these: helpwin, helpdesk, or demo. For product information, type tour or visit www.mathworks.com. »
» help
HELP topics:
Creating Variables
>> varname = 12
varname =
12
>> SS = 56; N = 4; Tot_Num = SS + N
Tot_Num =
60
Operations
+ Addition
- Subtraction
* Multiplication
^ Power
\ Division
/ Division
• Add vars• Subtract vars
Multiplication • Raise to the power• Divide vars (A div B)• Divide vars (B div A)
Creating Complex Numbers
>> 3 + 2i
>> sqrt(9) + sin(0.5)*j
ans =
3.0000 + 0.4794i
Num = sqrt(9) + sin(0.5)*j
real(Num)
imag(Num)
Entering Matrices (1)
>> A = [1 2 3; 4 5 6; 7 8 9]
OR
>> A = [
1 2 3
4 5 6
7 8 9 ]
Entering Matrices (2)
• To create an NxM zero-filled matrix
>> zeros(N,M)• To create a NxN zero-filled matrix
>> zeros(N)• To create an NxM one-filled matrix
>> ones(N,M)• To create a NxN one-filled matrix
>> ones(N)
Entering Matrices (3)
• To create an NxM randomly-filled matrix (which is uniformly distributed)
>> rand(N,M)
• To create an NxM randomly-filled matrix (which is normally distributed)
>> randn(N,M)
Complex Matrices
• To enter a complex matrix, you may do it in one of two ways :
>> A = [1 2; 3 4] + i*[5 6;7 8]
OR
>> A = [1+5i 2+6i; 3+7i 4+8i]
MATLAB Command Window
» who
Your variables are:
a b c
» whos Name Size Bytes Class
a 8x8 512 double array b 9x9 648 double array c 9x9 648 double array
Grand total is 226 elements using 1808 bytes
Matrix Addition» A = [ 1 1 1 ; 2 2 2 ; 3 3 3]
» B = [3 3 3 ; 4 4 4 ; 5 5 5 ]
» A + B
ans =
4 4 4
6 6 6
8 8 8
Matrix Subtraction» A = [ 1 1 1 ; 2 2 2 ; 3 3 3]
» B = [3 3 3 ; 4 4 4 ; 5 5 5 ]
» B - A
ans =
2 2 2
2 2 2
2 2 2
Matrix Multiplication» A = [ 1 1 1 ; 2 2 2 ; 3 3 3]
» B = [3 3 3 ; 4 4 4 ; 5 5 5 ]
» A * B
ans =
12 12 12
24 24 24
36 36 36
Matrix - Power» A ^ 2
ans =
6 6 6
12 12 12
18 18 18
» A ^ 3
ans =
36 36 36
72 72 72
108 108 108
Matrix Transpose
A =
1 1 1
2 2 2
3 3 3
» A'
ans =
1 2 3
1 2 3
1 2 3
Matrix Division
Left Division \
x = A\B (is A*x=B)
>> A = rand(4)
>> B = rand(4)
>> C = A \ B
=> A * C = B
Right Division /
x=A/B (is x*A=B)
>> A = rand(4)
>> B = rand(4)
>> C = A / B
=> C * A = B
Matrix Operations
+ Addition
- Subtraction
* Multiplication
^ Power
‘ Conjugate Transpose
\ Left Division
/ Right Division
• Add matrices• Subtract matrices
Matrix Multiplication • Raise to the power• Get transpose• x = A\B (is A*x=B)• x=A/B (is x*A=B)
Philosophiae Naturalis Principia Matlabematica
Damian Gordon
Getting to Matlab
Magic Matrix
MAGIC Magic square.
MAGIC(N) is an N-by-N matrix constructed from the integers
1 through N^2 with equal row, column, and diagonal sums.
Produces valid magic squares for N = 1,3,4,5,...
Identity Function
>> eye (4)
ans =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Upper Triangle Matrix
» a = ones(5)
a =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
» triu(a)
ans =
1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1
Lower Triangle Matrix
» a = ones(5)
a =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
» tril(a)
ans =
1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1
Hilbert Matrix» hilb(4)
ans =
1.0000 0.5000 0.3333 0.2500
0.5000 0.3333 0.2500 0.2000
0.3333 0.2500 0.2000 0.1667
0.2500 0.2000 0.1667 0.1429
Inverse Hilbert Matrix» invhilb(4)
ans =
16 -120 240 -140
-120 1200 -2700 1680
240 -2700 6480 -4200
-140 1680 -4200 2800
Toeplitz matrix.
TOEPLITZ
TOEPLITZ(C,R) is a non-symmetric Toeplitz matrix having C as its
first column and R as its first row.
TOEPLITZ(R) is a symmetric (or Hermitian) Toeplitz matrix.
-> See also HANKEL
Summary of Functions
• magic• eye(4)• triu(4)• tril(4)• hilb(4)• invhilb(4) • toeplitz(4)
- magic matrix
- identity matrix
- upper triangle
- lower triangle
- hilbert matrix
- Inverse Hilbert matrix
- non-symmetric Toeplitz matrix
Dot Operator
• A = magic(4); b=ones(4);
• A * B
• A.*B
• the dot operator performs element-by-element operations, for “*”, “\” and “/”
Concatenation
• To create a large matrix from a group of smaller ones
• try– A = magic(3)– B = [ A, zeros(3,2) ; zeros(2,3), eye(2)]– C = [A A+32 ; A+48 A+16]– Try some of your own !!
Subscripts
• Row i and Column j of matrix A is denoted by A(i,j)
• A = Magic(4)
• try– A(1,4) + A(2,4) + A(3,4) + A(4,4)
• try– A(4,5)
The Colon Operator (1)
• This is one MatLab’s most important operators
• 1:10 means the vector– 1 2 3 4 5 6 7 8 9 10
• 100:-7:50– 100 93 86 79 72 65 58 51
• 0:pi/4:pi– 0 0.7854 1.5708 2.3562 3.1416
The Colon Operator (2)
• The first K elements in the jth column is– A(1:K, j)
• Sum(A(1:4, 4)) is the sum of the 4th column
or
• Sum(A(:, 4)) means the same
matlABBA
The Colon Operator (1)
• This is one MatLab’s most important operators
• 1:10 means the vector– 1 2 3 4 5 6 7 8 9 10
• 100:-7:50– 100 93 86 79 72 65 58 51
• 0:pi/4:pi– 0 0.7854 1.5708 2.3562 3.1416
The Colon Operator (2)
• The first K elements in the jth column is– A(1:K, j)
• Sum(A(1:4, 4)) is the sum of the 4th column
or
• Sum(A(:, 4)) means the same
Deleting Rows and Columns (1)
• Create a temporary matrix X
• X=A;
• X(:, 2) = []
• Deleting a single element won’t result in a matrix, so the following will return an error
• X(1,2) = []
Deleting Rows and Columns (2)
• However, using a single subscript, you can delete – a single element– sequence of elements
So X(2:2:10) = []
gives
• x = 16 9 2 7 13 12 1
The ‘FIND’ Command (1)
>> x = -3:3
x =
-3 -2 -1 0 1 2 3
K = find(abs(x) > 1)
K =
1 2 6 7
The ‘FIND’ Command (2)
A = [ 1 2 3 ; 4 5 6 ; 7 8 9]
[i, j] = find (A > 5)
i =
3 3 2 3
j =
1 2 3 3
Special Variables
• ans• pi• eps• flops• inf• NaN• i,j• why
- default name for results
- pi
- “help eps”
- count floating point ops
- Infinity, e.g. 1/0
- Not a number, e.g. 0/0
- root minus one
- why not ?
LOGO Command
The ‘PLOT’ Command (1)
>> X = linspace(0, 2*pi, 30);
>> Y = sin(X);
>> plot(X,Y)
>> Z = cos(X);
>> plot(X,Y,X,Z);
The ‘PLOT’ Command (2)
>> W = [Y ; Z]
>> plot (X,W)
Rotate by 90 degrees
>> plot(W,X)
The Meditations ofMatlab Aurelius
The ‘PLOT’ Command (1)
>> X = linspace(0, 2*pi, 30);
>> Y = sin(X);
>> plot(X,Y)
>> Z = cos(X);
>> plot(X,Y,X,Z);
The ‘PLOT’ Command (2)
>> W = [Y ; Z]
>> plot (X,W)
Rotate by 90 degrees
>> plot(W,X)
PLOT Options
>> plot(X,Y,’g:’)
>> plot(X,Y,’r-’)
>> plot(X,Y,’ko’)
>> plot(X,Y,’g:’,X,Z,’r-’,X,Y,’wo’,X,Z,’c+’);
PLOT Options
>> grid on
>> grid off
>> xlabel(‘this is the x axis’);
>> ylabel(‘this is the y axis’);
>> title(‘Title of Graph’);
>> text(2.5, 0.7, ’sin(x)’);
>> legend(‘sin(x)’, ‘cos(x)’)
SUBPLOT Command• Subplot(m,n,p)
– creates a m-by-n matrix in and plots in the pth plane.
subplot(2,2,1)
plot(X,Y)
subplot(2,2,2)
plot(X,Z)
subplot(2,2,3)
plot( X,Y,X,Z)
subplot(2,2,4)
plot(W,X)
Specialised matrices
• compan • gallery• hadamard• hankel • pascal • rosser • vander • wilkinson
• Companion matrix• Higham test matrices• Hadamard matrix• Hankel matrix• Pascal matrix.• Classic symmetric
eigenvalue test problem
• Vandermonde matrix• Wilkinson's eigenvalue
test matrix
Polynomials
Polynomials are represented as
row vectors with its coefficients in
descending order, e.g.
X4 - 12X3 + 0X2 +25X + 116
p = [1 -12 0 25 116]
Polynomials
The roots of a polynomial are found as follows
r = roots(p)
roots are represented as a column vector
Polynomials
Generating a polynomial from its roots
polyans = poly(r)
includes imaginary bits due to rounding
mypolyans = real(polyans)
Polynomial Addition/Sub
• a = [1 2 3 4]
• b = [1 4 9 16]
• c = a + b
• d = b - a
Polynomial Addition/Sub
• What if two polynomials of different order ?
X3 + 2X2 +3X + 4
X6 + 6X5 + 20X4 - 52X3 + 81X2 +96X + 84
• a = [1 2 3 4]
• e = [1 6 20 52 81 96 84]
• f = e + [0 0 0 a] or f = e + [zeros(1,3) a]
Polynomial Multiplication
• a = [1 2 3 4]
• b = [1 4 9 16]
Perform the convolution of two arrays !
• g = conv(a,b)
g =
1 6 20 50 75 84 64
Polynomial Division• a = [1 2 3 4]• g = [1 6 20 50 75 84 64]
Perform the deconvolution of two arrays !
[q,r] = deconv(g, a)
q = {quotient}
1 4 9 16
r = {remainder}
0 0 0 0 0 0 0 0
Polynomial Differentiation
• f = [1 6 20 48 69 72 44]
• h = polyder(f)
h =
6 30 80 144 138 72
Polynomial Evaluation
x = linspace(-1,3)
p = [1 4 -7 -10]
v = polyval(p,x)
plot (x,v), title(‘Graph of P’)
Rational Polynomials
• Rational polynomials, seen in Fourier, Laplace and Z transforms
• We represent them by their numerator and denominator polynomials
• we can use residue to perform a partial fraction expansion
• We can use polyder with two inputs to differentiate rational polynomials
An Anthropologist on Matlab
Data Analysis Functions (1)
• corrcoef(x)
• cov(x)
• cplxpair(x)
• cross(x,y)
• cumprod(x)
• cumsum(x)
• del2(A)
• diff(x)
• dot(x,y)
• gradient(Z, dx, dy)
• Correlation coefficients
• Covariance matrix
• complex conjugate pairs
• vector cross product
• cumulative prod of cols
• cumulative sum of cols• five-point discrete Laplacian
• diff between elements
• vector dot product
• approximate gradient
Data Analysis Functions (2)
• histogram(x)
• max(x), max(x,y)
• mean(x)
• median(x)
• min(x), min(x,y)
• prod(x)
• sort(x)
• std(x)
• subspace(A,B)
• sum(x)
• Histogram or bar chart
• max component
• mean of cols
• median of cols
• minimum component
• product of elems in col
• sort cols (ascending)
• standard dev of cols
• angle between subspaces
• sum of elems per col
Symbolic Math Toolbox
Symbolic Expressions
• ‘1/(2*x^n)’
• cos(x^2) - sin(x^2)
• M = sym(‘[a , b ; c , d]’)
• f = int(‘x^3 / sqrt(1 - x)’, ‘a’, ‘b’)
Symbolic Expressions
diff(‘cos(x)’)
ans =
-sin(x)
det(M)
ans =
a*d - b * c
Symbolic Functions
• numden(m) - num & denom of polynomial
• symadd(f,g) - add symbolic polynomials
• symsub(f,g) - sub symbolic polynomials
• symmul(f,g) - mult symbolic polynomials
• symdiv(f,g) - div symbolic polynomials
• sympow(f,’3*x’) - raise f^3
Advanced Operations
• f = ‘1/(1+x^2)’
• g = ‘sin(x)’
compose(f,g) % f(g(x))
ans=
1/(1+sin(x)^2)
Advanced Operations
• finverse(x^2)
ans =
x^(1/2)
• symsum(‘(2*n - 1) ^ 2’, 1, ‘n’)
ans =
11/3*n + 8/3-4*(n+1)^2 + 4/3*(n+1)^3
Symbolic Differentiationf = ‘a*x^3 + x^2 + b*x - c’
diff(f) % by default wrt x
ans =
3*a^2 + 2*x + b
diff(f, ‘a’) % wrt a
ans =
x^3
diff(f,’a’,2) % double diff wrt a
ans =
0
Symbolic Integrationf = sin(s+2*x)
int(f)
ans =
-1/2*cos(s+2*x)
int(f,’s’)
ans =
-cos(s+2*x)
int(f, ‘s’, pi/2,pi)
ans=
-cos(s)
Comments &Punctuation (1)
• All text after a percentage sign (%) is ignored
>> % this is a comment
• Multiple commands can be placed on one line separated by commas (,)
>> A = magic(4), B = ones(4), C = eye(4)
Comments &Punctuation (2)
• A semicolon may be also used, either after a single command or multiple commands
>> A = magic(4); B = ones(4); C = eye(4);
• Ellipses (…) indicate a statement is continued on the next line
A = B/…
C
SAVE Command (1)
>> save• Store all the variables in binary format in a file
called matlab.mat
>> save fred• Store all the variables in binary format in a file
called fred.mat• >> save a b d fred• Store the variables a, b and d in fred.mat
SAVE Command (2)
>> save a b d fred -ascii• Stores the variables a, b and d in a file called
fred.mat in 8-bit ascii format
>> save a b d fred -ascii -double• Stores the variables a, b and d in a file called
fred.mat in 16-bit ascii format
Load Command
• Create a text file called mymatrix.dat with– 16.0 3.0 2.0 13.0– 5.0 10.0 11.0 8.0– 9.0 6.0 7.0 12.0– 4.0 15.0 14.0 1.0
• “load mymatrix.dat”, create variable mymatrix
M-Files
• To store your own MatLab commands in a file, create it as a text file and save it with a name that ends with “.m”
• So mymatrix.mA = […
16.0 3.0 2.0 13.0
5.0 10.0 11.0 8.0
9.0 6.0 7.0 12.0
4.0 15.0 14.0 1.0];
• type mymatrix
IF Condition
if condition {commands}end
If x = 2output = ‘x is even’end
WHILE Loop
while condition {commands}end
X = 10;count = 0;while x > 2x = x / 2;count = count + 1;end
FOR Loop
for x=array {commands}end
for n = 1:10x(n) = sin(n);end
A = zeros(5,5); % preallocfor n = 1:5for m = 5:-1:1A(n,m) = n^2 + m^2;enddisp(n)end
Creating a function
function a = gcd(a,b)
% GCD Greatest common divisor
% gcd(a,b) is the greatest common divisor of
% the integers a and b, not both zero.
a = round(abs(a)); b = round(abs(b));
if a == 0 & b == 0
error('The gcd is not defined when both numbers are zero')
else
while b ~= 0
r = rem(a,b);
a = b; b = r;
end
end
Quick Exercise (!)
• Consider Polynomial Addition again :
how would you write a program that takes in two polynomials and irrespective of their sizes it adds the polynomials together ? Given that the function length(A) returns the length of a vector.
Answers on a postcard to : [email protected]
oh, and while you’re here anyhow, if you have a browser open, please go to the following sites :
http://www.the hungersite.com
http://www.hitsagainsthunger.com
Flash Gordon and the Mud Men of Matlab
Quick Exercise (!)
• Consider Polynomial Addition again :
how would you write a program that takes in two polynomials and irrespective of their sizes it adds the polynomials together ? Given that the function length(A) returns the length of a vector.
Answers on a postcard to : [email protected]
oh, and while you’re here anyhow, if you have a browser open, please go to the following sites :
http://www.the hungersite.com
http://www.hitsagainsthunger.com
Creating Programs
Title : Program.m
function out = program(inputs)
% PROGRAM
<code>
Know Thyself
• Where am I ?– pwd
• Get me onto the hard disk– cd C:
• Where am I now ?– pwd
• Get me to where I know– cd ..
Quick Answer (!)
function c = mypoly(a,b)
% MYPOLY Add two polynomials of variable lengths
% mypoly(a,b) add the polynomial A to the polynomial
% B, even if they are of different length
%
% Author: Damian Gordon
% Date : 3/5/2001
% Mod'd : x/x/2001
%
c = [zeros(1,length(b) - length(a)) a] + [zeros(1, length(a) -
length(b)) b];
Recursion
function b = bart(a)
%BART The Bart Simpson program writes on the blackboard
% program, Bart writes the message a few times
% and then goes home to see the Simpsons
if a == 1
disp('I will not....');
else
disp('I will not skateboard in the halls');
bart(a - 1);
end
Curve Fitting
• What is the best fit ?– In this case least squares curve fit
• What curve should be used ?– It depends...
POLYFIT : Curve Fitting
• polyfit(x,y,n) - fit a polynomial
• x,y - data points describing the curve
• n - polynomial order
• n = 1 -- linear regression
• n = 2 -- quadratic regression
Curve Fitting Example
x = [0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1];
y = [-.447 1.978 3.28 6.16 7.08 7.34 7.66 9.56 9.48 9.30 11.2];
polyfit(x,y,n)• n = 1
p = 10.3185 1.4400• n = 2
p = -9.8108 20.1293 -0.0317
y = -9.8108x2 + 20.1293x - 0.0317
Curve Fitting Example
xi = linspace(0,1,100);
z = polyval(p,xi)
plot(x,y,'o',x,y,xi,z,':');
Interpolation - 1D
• t = interp1(x,y,.75)t =
9.5200
also• interp1(x,y,.75,’spline’)• interp1(x,y,.75,’cubic’)
Interpolation - 2D
• interp2(x,y,Z,xi,yi,TYPE)
TYPE =
'nearest' - nearest neighbor interpolation 'linear' - bilinear interpolation 'cubic' - bicubic interpolation 'spline' - spline interpolation
Fourier Functions
• fft• fft2• ifft• ifft2• filter• filter2• fftshift
• Fast fourier transform• 2-D fft• Inverse fft• 2-D Inverse fft• Discrete time filter• 2-D discrete tf• shift FFT results so -ve
freqs appear first
Tensors
• See ‘Programs’
• ‘Tensors’
• ‘Tensors.html’
3D Graphics
T = 0:pi/50:10*pi;
plot3(sin(t), cos(t), t);
3D Graphics
title('Helix'), xlabel('sin(t)'),
ylabel('cos(t)'), zlabel('t')
grid
3D Graphics
• Rotate view by elevation and azimuth
view(az, el);
view(-37.5, 60);