Class Prep

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Class Prep

• Bring to Class
• In-Class Exercise
• Paper for Classroom Printer
• Check HannayD folder under My Documents(or
  H/csc070)
• Run MATLAB
• Set Current Directory

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Start up MATLAB

• Run MATLAB
• Change MATLAB Current Directory
• h/csc070/70week02

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Week 02-b(3.1-3.4)

• MATLAB Vectors
• Finding Values within an Array
• Elementary Math Functions

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Main Collection Type Array

• A Collection is a data structure into which you
  put stuff, and from which you get stuff.
• MATLABs basic type of collection is called an
  Array.
• Arrays are homogeneous collection of things.
• All "things" are the same "type"
• Arrays are indexable
• c.f. Arrays in Machine/Assembly Language
• Example a text string is an array of characters
• If ch 'ABC' then ch1 is 'A', ch2 is 'B', ch3
  is 'C'
• Recall that ch1 is stored as 65 (see Appendix B)

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Array Processing

• Big part of MATLAB's power is operation on entire
  Arrays, both on a per element basis and in a
  summary basis.
• Per Element y sin(x) 2x
• Summary p prod(x) sum(x)
• Each element of an array has a value and a
  position (sometimes called index ).

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Vectors

• The simplest array type is called a vector.
• Simple example (using MATLAB syntax)
• X 82 7 42 16 8 127 33
• To index into a vector needs to know the position
  within the vector
• First element has position 1
• e.g. X(1) which happens to equal 82
• Last element has position either the length of
  the array or the special label end
• e.g. X(length(X)), or X(end) equals 33

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Row Column Vectors

• A row vector in MATLAB can be created by an
  explicit list, starting with a left bracket,
  entering the values separated by spaces (or
  commas) and closing the vector with a right
  bracket.
• gtgt x 0 0.25pi 0.5pi 0.75pi pi
• x
• 0 0.7854 1.5708 2.3562
  3.1416
• A column vector can be created in a similar way,
  with the rows are separated by semicolons.
• gtgt y 0 pi 0.5pi 2pi
• y
• 0
• 3.1416
• 1.5708
• 6.2832

x is a row vector. y is a column vector.
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Simple Vector Commands
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Hands-On DEMO Creating Vectors

• Leave off semi-colon to see what you get each
  time
• gtgt a 110
• gtgt b 00.11
• gtgt c 7 8 9
• gtgt d 10 11 12
• gtgt length(b) NOTE length is 11 not 10
• gtgt linspace(0,100,21)

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Hands-On DEMO "Average" of 5 numbers

• Store list of 5 values in variable x
• e.g. x 56 57 33 98 45
• Display arithmetic mean and median
• mean(x)
• median(x)
• NOTE No "average" function (ambiguous)
• median() gives you the MIDDLE value (keep this in
  mind for class exercise)

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Hands-On DEMO Basic Operations on Vectors

• gtgt v 028 v 0 2 4 6 8
• gtgt u 0-1-4
• u 0 -1 -2 -3 -4
• gtgt uv
• ans 0 1 2 3 4
• Additionally, scalar multiplication is defined in
  the standard way. Also note that scalar division
  is defined in a way that is consistent with
  scalar multiplication
• gtgt -2u
• ans 0 2 4 6 8
• gtgt v/3
• ans 0 0.6667 1.3333 2.0000 2.6667

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Partial Vectors

• a 210
• a(46) 18
• a(46) 8 6 4
• a(46) 4 5 6 7 - error
• b 12 3 4 5 6 7
• b(13) a(57) "called slicing"

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Hands-On DEMO Extracting Partial Vector

• gtgt x linspace(31,40,10)
• gtgt y x(37)
• y 33 34 35 36 37
• gtgt y(3)
• ans 35
• The expression, y x(37), copies the third
  through seventh elements of x into the first
  through fourth elements of y. If y did not
  already exist it is created by the assignment.
• gtgt v 028 v 0 2 4 6 8
• gtgt v(13)-v(24)
• ans -2 -2 -2

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Hands-On DEMO Modifying Vectors

• x -2 0 9 1 4
• x(2) 5
• x -2 5 9 1 4
• x(4) x(1)
• x -2 5 9 -2 4
• x(8) -1
• x -2 5 9 -2 4 0 0
  -1

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Arrays as Indexes

• fred 25100
• fred(56)
• Remove the odd indexed values
• fred(12end)
• The important use of arrays as indices has to do
  with picking out the elements of a vector (or
  array) that have a certain property or
  characteristic.
• To do this requires two things
• definition of the characteristic
• "Finding" the indices of the vector with that
  property

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find Function

• dates 16 17 18 19 20
• rain 0 0.2 0.1 0 1.7
• Save indices where array has a certain property
• index find(rain gt 0)
• Now use index twice
• goodRain rain(index)
• goodDates dates(index)
• We could plot only days with rainfall
• plot(goodDates, goodRain)

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Hands-On DEMO Speed Demon

• Work in command window
• Assign 65 to limit
• Enter vector of speeds 60 70 65 80 66
• Use find to list those over the speed limit
• Indices (subscripts)
• Speeds at those indices
• Calculate of cars going over the speed limit
• limit 65
• indicesfind(speedsgtlimit)
• speeds(indices)
• percent_over_limit ... length(indices)/length(sp
  eeds)100

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Predefined (Built-In) Functions

• To call a built-in function, include its name
  followed by any arguments enclosed in parentheses
  following the name
• Arguments may be constants, variables, or
  expressions (scalars, vectors, or matrices)
• ysin(x/2)
• Use a function in an expression to calculate a
  value
• w2sin(a)

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Hands-On DEMO Using Built-In Functions
gtgt p sin (pi/4) p 0.7071 gtgt q
log (4) q 1.3863

• Trigonometric functions
• sin ( ) asin ( ) sind (
  )
• cos ( ) acos ( ) cosd ( )
• tan ( ) atan ( ) tand ( )
• Angle is in radians Angle in
  degrees

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Degrees vs. Radians for Measuring Angles

• http//www.teacherschoice.com.au/Maths_Library/Ang
  les/Angles.htm

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Help on Predefined Functions

• help elfun ? categorized list of elementary math
  functions
• Trigonometric
• Exponential
• Complex
• Rounding and remainder
• Follow link for sin (same as typing "help sin")
• Follow link to doc sin (same as typing "doc sin")
• Can find through "Help" menu also
• What do we need to know from documentation?
• List of functions available types of arguments
  values

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Hands-On DEMO Functions of Vectors

• Most MATLAB functions will work equally well with
  both scalars and arrays (of any dimension)

gtgt A1 2 3 4 5 gtgt sin(A) ans 0.8415
0.9093 0.1411 -0.7568 -0.9589 gtgt
sqrt(A) ans 1.0000 1.4142 1.7321
2.0000 2.2361
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Hands-On DEMO Trig Functions(remember that sin,
cos, tan use radians sind, cosd, tand use
degrees)

• Calculate twice the sine of 45 degrees
• 2sind(45) should equal sqrt(2)
• Generate a vector of angles from 0 to 360 in
  steps of 10 degrees
• Generate a vector of corresponding sine values
• plot(angles,sine) more on plot later
• Display a 2-column table with angle and sine
  values
• angles010360
• sinesind(angles)
• angles' sine'

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  Selected Exponential Functions

• exp Exponential ex.
• log Natural logarithm (often called ln(...)).
• log10 Common (base 10) logarithm.
• log2 Base 2 logarithm.
• pow2 Base 2 power.
• sqrt Square root.

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Hands-On DEMO Exponential Functions

• Create a single MATLAB command to display a
  2-column table listing the first 10 powers of 2.
  
• The table should show values of "p" from 0 to 9
  in the first column an 2p in the second.
• 09' pow2(09)'

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Rounding and Remainder Functions

• fix Round towards zero.
• floor Round towards minus infinity.
• ceil Round towards plus infinity.
• round Round towards nearest integer.
• mod Modulus (signed remainder after division).
• rem Remainder after division.
• sign -1, 0, or 1 based on sign of number.

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Hands-On DEMO Rounding Functions

• Create a 5-column table that illustrates the
  differences between the rounding functions round,
  fix, floor, and ceil for positive and negative
  values.
• Apply functions to a vector of numbers (saved as
  variable x) ranging from -2 to 2 in increments
  of 0.1
• x-2.12'
• x round(x) fix(x) floor(x) ceil(x)

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Additional format Specifications(MATLAB performs
all computations in double precision)

•   The format command switches among different
  display formats.
•  Command      Result                   Example
  format short      5 digit scaled fixed
  point      3.1416 format long       15 digit
  scaled fixed point   3.14159265358979 format
  short e   5 digit floating-point          3.1416e
  00 format long e    15 digit floating-point  
  3.141592653589793e00
• format short g   general purpose           
  5 or 1.25 or 3.0e-12
• format bank       Fixed dollars and
  cents         7.95 format rat        Ratio of
  small integers         355/113   format
  compact Suppresses excess line feeds. format
  loose     Add line feeds.

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Hands-On DEMO Formatting

• p020
• format short e exponential
• p' pow2(p)' pow2(-p)'
• format short g general purpose
• p' pow2(p)' pow2(-p)'

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Hands-On DEMO Operations on Transposes
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Hands-On DEMO Element by Element Operations
Note the . operator for "Element by Element"
Multiplication
Plotting a function using vector math x
linspace(0, 20, 100) define 100 x values (from
0 to 20) y 5exp(-0.3x).sin(x) compute y
vector plot(x,y), xlabel('X'), ylabel('Y'),
title('Vector calc')

• linspace( ) function can be very effective for
  creating the x vector

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Question

• Given x 1 2 3
• Why does
• gtgt x2
• Yield an ERROR,
• but
• gtgt x.2
• is fine?
• We are asking for xx but dimensions are wrong.
• Squaring individual elements of x is fine

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Hands-On DEMO "Dot" Division by Vector

• gtgt x1 2 3 4 6 12
• gtgt 12./x
• What about x./12?
• How about x/12?
• Finally 12/x?
• "Dot" division ( multiplication) operate element
  by element

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In-Class Exercise 2b Height of a Building

• One day, the barometric pressure at ground level
  was measured at 30.28 inches of mercury, and on
  the roof of a building it was measured at 30.16
  inches of mercury.
• You are given the formula
• height in feet 25,000 ln(ground pressure /
  roof pressure)
• Calculate heights of building based on pressure
  measurements over 5 days. Then display the
  "middle" value to nearest foot.
• Write a MATLAB script to solve the problem. Use
  the appropriate predefined functions for natural
  log (ln in problem description, but not in
  MATLAB) and rounding
• Comment all Units of Measure.
• Add your name, comments and print script output

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END