MATLAB

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• MATLAB Matrix Laboratory
• Usage
• Math and computation Algorithm
• development Data acquisition Modeling
  simulation, and prototyping
• Data analysis exploration, and visualization
• Scientific and engineering graphics
• Application development, including graphical user
  interface building

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Development Environment
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Desktop Tools

• Command Window

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Desktop Tools

• Start easy access to tools, demos, and
  documentation.

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Current Directory Browser
file you want to run must either be in the
current directory or on the search path
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Workspace Browser
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Editor/Debugger

• M-files are programs you write to run MATLAB
  functions.

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Matrix

• To enter a matrix, simply type in the Command
  Window
• A 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
• MATLAB displays the matrix you just entered.
• A
• 16 3 2 13
• 5 10 11 8
• 9 6 7 12
• 4 15 14 1

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Transpose

• So A'
• produces
• ans
• 16 5 9 4
• 3 10 6 15
• 2 11 7 14
• 13 8 12 1

A 16 3 2 13 5 10 11
8 9 6 7 12 4 15 14
1
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Diagonal Elements

• diag(A)
• produces
• ans
• 16
• 10
• 7
• 1

A 16 3 2 13 5 10 11
8 9 6 7 12 4 15 14
1
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Assigning a new Value to an Element

• X(4,4) 17
• X
• 16 3 2 13
• 5 10 11 8
• 9 6 7 12
• 4 15 14 17

A 16 3 2 13 5 10 11
8 9 6 7 12 4 15 14
1
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Storing an Element

• gtgtt A(4,3)
• t
• 14

A 16 3 2 13 5 10 11
8 9 6 7 12 4 15 14
1
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The Colon Operator

• The expression 110 is a row vector containing
  the integers from 1 to 10
• 1 2 3 4 5 6 7 8
  9 10
• To obtain nonunit spacing, specify an increment.
  For example, 100-750 is
• 100 93 86 79 72 65 58 51

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The Colon Operator

• Subscript expressions involving colons refer to
  portions of a matrix.
• A(1k , j)
• is the first k elements of the jth column of A
• colon by itself refers to all the elements in a
  row or column of a matrix .

A( , 1) A( 2 , )
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Some Other Good Operations

• To swap the two middle columns.
• A B(,1 3 2 4)
• To Erase an entire row or column
• A(1,)
• To Add a new row or column
• BA1 2 1 5

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Variable

• MATLAB does not require any type declarations or
  dimension statements.
• the variable already exists, MATLAB changes its
  contents
• MATLAB is case sensitive
• To view the matrix assigned to any variable,
  simply enter the variable name

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Format

• All computations in MATLAB are done in double
  precision.
• FORMAT may be used to switch between
  different output
• display formats as follows
• FORMAT Default. Same as SHORT.
• FORMAT SHORT Scaled fixed point format with 5
  digits
• FORMAT LONG Scaled fixed point format
  with 15 digits
• FORMAT HEX Hexadecimal format.

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Cell Arrays

• They are multidimensional arrays whose elements
  are copies of other arrays.

If you subsequently change A, nothing happens to C
Example gtgtC A sum(A) prod(prod(A)) ans C
4x4 double 1x4 double
20922789888000
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useful constants.

• Pi 3.14159265...
• i Imaginary unit sqrt(-1)
• j Same as i
• eps Floating-point relative precision 2 -52
• realmin Smallest floating-point number, 2-1022
• realmax Largest floating-point number, 2-21023
• Inf Infinity
• NaN Not-a-number

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Operations

• Addition
• - Subtraction
• Multiplication
• / Division
• \ Left division (described in "Matrices and
  Linear Algebra" in the MATLAB
  documentation)
• Power
• Complex conjugate transpose
• ( ) Specify evaluation order

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Example
gtgt3/10 ans 0.3000 gtgt 10\3 ans
0.3000

• 1j 22j'
• ans
• 1.0000 - 1.0000i
• 2.0000 - 2.0000i

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• rho (1sqrt(5))/2
• rho
• 1.6180
• a abs(34i)
• a
• 5
• z sqrt(besselk(4/3,rho-i))
• z
• 0.3730 0.3214i

huge exp(log(realmax)) huge
1.7977e308 toobig pihuge toobig Inf
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Generating Matrices

• Z zeros(2,4)
• Z
• 0 0 0 0
• 0 0 0 0
• F 5ones(3,3)
• F
• 5 5 5
• 5 5 5
• 5 5 5

N fix(10rand(1,10)) N 4 9 4
4 8 5 2 6 8 0 R
randn(4,4) R 1.0668 0.2944 -0.6918
-1.4410 0.0593 -1.3362 0.8580 0.5711
-0.0956 0.7143 1.2540 -0.3999
-0.8323 1.6236 -1.5937 0.6900
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Some Useful functions

• A'
• d det(A)
• X inv(A)
• e eig(A)
• poly(A)
• sin(x)
• sinh(x)
• asin(x)

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Array Operators

• Addition
• - Subtraction
• . Element-by-element multiplication
• ./ Element-by-element division
• .\ Element-by-element left division
• . Element-by-element power
• .' Unconjugated array transpose

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Example

• n (09)'
• Then pows n n.2 2.n
• builds a table of squares and powers of 2.
• pows
• 0 0 1
• 1 1 2
• 2 4 4
• 3 9 8
• 4 16 16
• 5 25 32
• 6 36 64
• 7 49 128
• 8 64 256
• 9 81 512

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• gtgt a1 2 3 4
• gtgt b1 2 3 4'
• gtgt ab
• ans
• 30
• gtgt ba
• ans
• 1 2 3 4
• 2 4 6 8
• 3 6 9 12
• 4 8 12 16
• gtgt a.b'
• ans
• 1 4 9 16

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• B
• 7.5 -5.5 -6.5 4.5
• -3.5 1.5 2.5 -0.5
• 0.5 -2.5 -1.5 3.5
• -4.5 6.5 5.5 -7.5
• gtgtB(12,23) 0
• B
• 7.5 0 0 4.5
• -3.5 0 0 -0.5
• 0.5 -2.5 -1.5 3.5
• -4.5 6.5 5.5 -7.5

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Geraphics

• PLOT
• x 0pi/1002pi
• y sin(x)
• plot(x,y)
• xlabel('x 02\pi')
• ylabel('Sine of x')
• title('Plot of the Sine Function','FontSize',12)

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Multiple Data Sets in One Graph

• y2 sin(x-.25)
• y3 sin(x-.5)
• plot(x,y,x,y2,x,y3)

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Colors

• 'c cyan
• 'mmagenta
• 'y yellow
• 'r red
• 'g green
• 'b blue
• 'w white
• 'kblack

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plot(x,y,'ks')
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plot(x,y,'kgt')
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plot(x,y,'r')
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This example plots the data twice using a
different number of points for the dotted line
and marker plots.

• x1 0pi/1002pi
• x2 0pi/102pi
• plot(x1,sin(x1),'r',x2,sin(x2),'r')

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Complex Data

• When the arguments to plot are complex, the
  imaginary part is ignored
• For the special case of giving the plot a single
  complex argument, the command is a shortcut for a
  plot of the real part versus the imaginary part.
• Therefore, plot(Z) where Z is a complex vector
  or matrix, is equivalent to plot(real(Z),imag(Z))

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Adding Plots to an Existing Graph

• hold on
• Example
• x,y,z peaks
• contour(x,y,z,20,'k')
• hold on
• pcolor(x,y,z)
• shading interp
• hold off

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Multiple Plots in One Figure

• subplot(m,n,p)
• Example
• t 0pi/102pi
• X,Y,Z cylinder(4cos(t))
• subplot(2,2,1) mesh(X)
• subplot(2,2,2) mesh(Y)
• subplot(2,2,3) mesh(Z)
• subplot(2,2,4) mesh(X,Y,Z)

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Setting Axis

• axis(xmin xmax ymin ymax zmin zmax)
• axis auto
• axis on
• axis off
• grid on
• grid off

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Example

• t -pipi/100pi
• y sin(t)
• plot(t,y)
• axis(-pi pi -1 1)
• xlabel('-\pi \leq \itt \leq \pi')
• ylabel('sin(t)')
• title('Graph of the sine function')
• text(1,-1/3,'\itNote the odd symmetry.')

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Mesh and Surface Plots

• Mesh (x,y,z)
• produces wireframe surfaces that color only the
  lines connecting the defining points.
• surf (x,y,z)
• displays both the connecting lines and the faces
  of the surface in color.

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• X,Y meshgrid(-8.58)
• R sqrt(X.2 Y.2) eps
• Z sin(R)./R
• mesh(X,Y,Z,'EdgeColor','black')

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transparency

• hidden off

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Surf Example
surf(X,Y,Z) colormap hsv colorbar
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View
view(az,el) view(az,el)
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Surface Plots with Lighting

• surf(X,Y,Z,'FaceColor','red','EdgeColor','none')
• camlight left lighting phong
• view(-15,65)

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Flow Control

• if rem(n,2) 0
• .
• elseif rem(n,4) 0
• else
• .
• end

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Important!

• when A and B are matrices, A B does not test
  if they are equal, it tests where they are equal
  the result is another matrix of 0's and 1's
  showing element-by-element equality.
• Solution
• if isequal(A,B), ...

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Some Other Helpful Functions

• Isequal(A,B,)
• Determine if arrays are numerically equal
• Isempty(A)
• Determine if item is an empty array
• isequalwithequalnans(A,B,...)
• Determine if arrays are numerically equal,
  treating NaNs as equal
• ismember(A,S)
• Detect members of a specific set
• isnumeric(A)
• Returns logical true (1) if A is a numeric
  array and logical false (0) otherwise.

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Some Other Helpful Functions

• isprime(A)
• returns an array the same size as A containing
  logical true (1) for the elements of A which are
  prime.
• isreal(A)
• returns logical false (0) if any element of
  array A has an imaginary component, even if the
  value of that component is 0.
• ischar(A)
• ischar(A) returns logical true (1) if A is a
  character array and logical false (0) otherwise.

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• all(A)
• If A is a vector, all(A) returns logical true
  (1) if all of the elements are nonzero, and
  returns logical false (0) if one or more elements
  are zero.
• If A is a matrix, all(A) treats the columns of A
  as vectors, returning a row vector of 1s and 0s.

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• B any(A)
• If A is a vector, any(A) returns logical true
  (1) if any of the elements of A are nonzero, and
  returns logical false (0)
• if all the elements are zero.
• If A is a matrix, any(A) treats the columns of A
  as vectors, returning a row vector of 1s and 0s.

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Example

• gtgtA 0.53 0.67 0.01 0.38 0.07 0.42 0.69
• B (A lt 0.5)
• Ans
• 0 0 1 1 1 1 0
• gtgtall(B)
• Ans 0
• gtgtany(B)
• Ans 1

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Relational Operators
EXAMPLE gtgtX 5ones(3,3) gtgtX gt 1 2 3 4 5 6
7 8 10 ans 1 1 1 1 1 0
0 0 0

• A lt B
• A gt B
• A lt B
• A gt B
• A B
• A B

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Logical Operators
The precedence for the logical operators
The truth table
The second operand is evaluated only when the
result is not fully determined by the first
operand.
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Example
Logical Operation on Elements
Short Circuit and

• gtgtu 0 0 1 1 0 1
• gtgtv 0 1 1 0 0 1
• gtgtu v
• ans
• 0 1 1 1 0 1

to avoid generating a warning when the divisor,
b, is zero. x (b 0) (a/b gt 18.5)
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Flow Control

• switch (rem(n,4)0) (rem(n,2)0)
• case 0
• .
• case 1
• .
• case 2
• otherwise
• error('This is impossible')
• end

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Important!

• Unlike the C language switch statement, MATLAB
  switch does not fall through. If the first case
  statement is true, the other case statements do
  not execute. So, break statements are not
  required.

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Flow Control

• for variable scalar1 step scalar2
• statement 1
• ...
• statement n
• end

Example a zeros(k,k) Preallocate
matrix for m 1k for n 1k
a(m,n) 1/(mn -1) end end
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• while expression
• statements
• End

The statements are executed while the real part
of expression has all nonzero elements.
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Two Useful Functions for Loops

• Continue
• Passes control to the next iteration of for or
  while loop
• Break
• statement lets you exit early from a for or
  while loop.

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Characters and Numbers

• To define a string
• gtgts 'Hello
• characters are stored as numbers, but not in
  floating-point format.
• To see the characters as numbers
• gtgta double(s)
• a
• 72 101 108 108 111
• To reverses the conversion
• gtgts char(a)

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M-files

• Scripts, which do not accept input arguments or
  return output arguments. They operate on data in
  the workspace.
• Functions, which can accept input arguments and
  return output arguments. Internal variables are
  local to the function.

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Functions

• Checks to see if the name is a variable.
• Checks to see if the name is an internal function
  (eig, sin) that was not overloaded.
• Checks to see if the name is a local function
  (local in sense of multifunction file).
• Checks to see if the name is a function in a
  private directory.
• Locates any and all occurrences of function in
  method directories and on the path. Order is of
  no importance.
• At execution, MATLAB
• Checks to see if the name is wired to a specific
  function (2, 3, 4 above)
• Uses precedence rules to determine which instance
  from 5 above to call (we may default to an
  internal MATLAB function). Constructors have
  higher precedence than anything else.

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Functions

• nargin and nargout indicate how many input or
  output arguments, respectively, a user has
  supplied.

if nargout 0 plot(x,y) else x0 x
y0 y end
if nargin lt 5, subdiv 20 end if nargin lt 4,
angl 10 end if nargin lt 3, npts 25 end
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• function x0,y0 myplot(fname,lims,npts,angl,sub
  div)
• MYPLOT Plot a function.
• MYPLOT(fname,lims,npts,angl,subdiv)
• The first two input arguments are
• required the other three have default
  values.
• ...
• if nargin lt 5, subdiv 20 end
• if nargin lt 4, angl 10 end
• if nargin lt 3, npts 25 end
• ...
• if nargout 0
• plot(x,y)
• else
• x0 x
• y0 y
• end

function h falling(t) global GRAVITY h
1/2GRAVITYt.2
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A review on Mathematical Functions

• Binary addition AB plus(A,B)
• Unary plus A uplus(A) Binary
• Subtraction A-B minus(A,B)
• Unary minus -A uminus(A) Matrix
• Multiplication AB mtimes(A,B) Array-wise
• Multiplication A.B times(A,B) Matrix right
• Division A/B mrdivide(A,B) Array-wise right
• Division A./B rdivide(A,B) Matrix left
• Division A\B mldivide(A,B) Array-wise left
• Division A.\B ldivide(A,B) Matrix
• Power AB mpower(A,B) Array-wise
• Power A.B power(A,B) Complex
• Transpose A ctranspose(A) Matrix
• Transpose A. transpose(A)

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