Lecture 4 Sept 8

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• Lecture 4
  Sept 8
• Complete Chapter 3 exercises
• Chapter 4

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Exercise 3.1 sum(2.(0 4)) Exercise
3.5
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Exercise 3.8. Write the MATLAB version of the
boolean statement a lt b lt c. Answer
Exercise 3.12 How many leap years are between
1780 and 2832, inclusive? Recall the definition
of a leap year.
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Solution to Exercise 3.12
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Chapter 4 arrays, collections and indexing

• This chapter discusses the basic calculations
  involving rectangular collections of numbers in
  the form of vectors and arrays. For each of these
  collections, you will learn how to
• Create them
• Manipulate them
• Access their elements
• Perform mathematical and logical operations on
  them

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Concept Data Collections

• This section considers two very common ways to
  group data in arrays and in vectors.
• Data Abstraction allows us to refer to groups of
  data collectively
• all the temperature readings for May or
• all the purchases from Wal-Mart.
• We can not only move these items around as a
  group, but also perform mathematical or logical
  operations on these groups, e.g.
• compute the average, maximum, or minimum
  temperatures for a month
• A Homogeneous Collection is constrained to accept
  only items of the same data type in this case,
  they will all be numbers

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MATLAB Vectors

• Individual items in a vector are usually referred
  to as its elements. Vector elements have two
  separate and distinct attributes that make them
  unique in a specific vector
• their numerical value and
• their position in that vector.
• For example, the individual number 66 is the
  third element in this vector. Its value is 66 and
  its index is 3. There may be other items in the
  vector with the value of 66, but no other item
  will be located in this vector at position 3.

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Vector Manipulation

• We consider the following basic operations on
  vectors
• Creating a Vector
• Determining the size of a Vector
• Extracting data from a vector by indexing
• Shortening or expanding a Vector
• Mathematical and logical operations on Vectors

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Creating a Vector

• Entering the values directly, e.g.
• A 2, 5, 7, 1, 3
• Entering the values as a range of numbers e.g.,
• B 1320
• Using the linspace(...) function e.g.
• c linspace (0, 20, 11)
• Using the functions zeros(1,n), ones(1,n),
  rand(1,n) and randn(1,n) to create vectors
  filled with 0, 1, or random values between 0 and
  1

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Exercise Write a code segment in Matlab to
generate a sequence containing the outcomes of 50
independent rolls of a fair die. Thus we want to
generate a vector of size(100) where each element
of the vector is a random member of the set 1,
2, 3, 4, 5, 6. Note that rand() gives a random
real number between 0 and 1.
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Later we will write code to count how many times
the outcome was j for different values of j 1,
2, , 6. If the random number generator is a good
one, we expect all these numbers to be close to
each other.
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Size of vectors and arrays

• MATLAB provides two functions to determine the
  size of arrays in general (a vector is an array
  with one row)
• the function size(A) when applied to the array A
  returns vector containing two quantities the
  number of rows and the number of columns
• The function length(A) returns the maximum value
  in the size of an array for a vector, this is
  its length.

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Indexing a Vector

• The process of extracting values from a vector,
  or inserting values into a vector
• Syntax
• v(index) returns the element(s) at the
  location(s) specified by the vector index.
• v(index) value replaces the elements at the
  location(s) specified by the vector index.
• The indexing vector may contain either numerical
  or logical values

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Exercise Write a one-line statement to compute
the average of a set of numbers stored in vector
x.
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Exercise Write a one-line statement to compute
the average of a set of numbers stored in vector
x. Answer gtgt sum(x) / length(x)
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Numerical Indexing

• The indexing vector may be of any length
• It should contain integer (non-fractional)
  numbers
• The values in the indexing vector are constrained
  by the following rules
• For reading elements, all index values j must be
• 1 lt j lt length (vector)

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Replacement Rules

• Either
• All dimensions of the blocks on either side of
  the replacement instruction must be equal, or
• There must be a single element on the RHS of the
  replacement
• If you replace beyond the end of the existing
  vector, the vector length is automatically
  increased.
• Any element not specifically replaced remains
  unchanged.
• Elements beyond the existing length not replaced
  are set to 0.

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Logical Indexing

• The indexing vector length must be less than or
  equal to the original vector length
• It must contain logical values (true or false)
• Access to the vector elements is by their
  relative position in the logical vector
• When reading elements, only the elements
  corresponding to true index values are returned
• When replacing elements, the elements
  corresponding to true index values are replaced
• Beware logical vectors in Matlab echo in the
  Command window as 1 or 0, but they are not the
  same thing.

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Exercise Let u be a vector. Write a Matlab code
segment that determines the number of occurrences
of 7 in it. Example gtgt u 1 3 5 7 9 7 5 3
1 gtgt lt your code heregt gtgt ans 2
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Exercise Let u be a vector. Write a Matlab code
segment that determines the number of occurrences
of 7 in it. Example gtgt u 1 3 5 7 9 7 5 3
1 gtgt lt your code heregt gtgt ans
2 Answer sum(u 7)
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Find function
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Exercise Write a program in Matlab to split a
vector u into two vectors v and w where v
contains all the numbers up to the largest number
and w contains the rest. Example gtgt u 1 8 3
12 6 9 11 gtgt lt your code heregt gtgt v ans 1
8 3 12 gtgt w ans 6 9 11
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Exercise Write a program in Matlab to split a
vector u into two vectors v and w where v
contains all the numbers up to the largest number
and w contains the rest. Example gtgt u 1 8 3
12 6 9 11 gtgt v u(1find(u max(u))) gtgt v ans
1 8 3 12
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Exercise Write a Matlab expression to determine
if there are any negative numbers in a vector
u. Answer
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Exercise Write a Matlab expression to determine
if there are any negative numbers in a vector
u. Answer sum(u lt 0) gt 0
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Operating on Vectors

• Three techniques extend directly from operations
  on scalar values
• Arithmetic operations
• Logical operations
• Applying library functions
• Two techniques are unique to arrays in general,
  and to vectors in particular
• Concatenation
• Slicing (generalized indexing)

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Arithmetic operations

• Arithmetic operations
• gtgt A 2 5 7 1 3
• gtgt A 5
• ans
• 7 10 12 6 8
• gtgt A . 2
• ans
• 4 10 14 2 6
• gtgt B -113
• B
• -1 0 1 2 3

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Arithmetic operations (continued)

• gtgt A . B element-by-element multiplication
• ans
• -2 0 7 2 9
• gtgt A B matrix multiplication!!
• ??? Error using gt mtimes
• Inner matrix dimensions must agree.
• gtgt C 1 2 3
• C
• 1 2 3
• gtgt A . C A and C must have the same length
• ??? Error using gt times
• Matrix dimensions must agree.

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Logical operations

• gtgt A 2 5 7 1 3
• gtgt B 0 6 5 3 2
• gtgt A gt 5
• ans
• 0 1 1 0 0
• gtgt A gt B
• ans
• 1 0 1 0 1
• gtgt C 1 2 3
• gtgt A gt C
• ??? Error using gt gt
• Matrix dimensions must agree.

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Logical operations (continued)

• gtgt A true true false false
• gtgt B true false true false
• gtgt A B
• ans
• 1 0 0 0
• gtgt A B
• ans
• 1 1 1 0
• gtgt C 1 0 0 NOT a logical vector
• gtgt A(C) yes, you can index logical vectors, but
  ...
• ??? Subscript indices must either be real
  positive integers or logical.

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Applying library functions

• All MATLAB functions accept vectors of numbers
  rather than single values and return a vector of
  the same length.
• Special Functions
• sum(v) and mean(v) consume a vector and return
  a number
• min(v) and max(v) return two quantities the
  minimum or maximum value in a vector, plus the
  position in that vector where that value
  occurred.

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Applying library functions

• round(v), ceil(v), floor(v), and fix(v) remove
  the fractional part of the numbers in a vector by
  conventional rounding, rounding up, rounding
  down, and rounding toward zero, respectively.
• We can also apply functions like sqrt or sine or
  cosine to each element of a vector.