1. Chapter 3 Arrays and Vectors
CSE 1010
Chapter 3
Arrays and Vectors

3. Objectives  Learn about built-in MATLAB functions
 Learn about data collections and how to:
Create them
Manipulate them
Access their elements
Perform mathematical and logical operations on them

4. Functions
A function is a named collection of instructions that operates on the data provided to produce a result according to the use of that function
For example,
>> A = cos(angle)
For help on a specific function, you can type
>> help <function name>
>> help sqrt
SQRT Square root.
SQRT(X) is the square root of the elements of X. Complex results are produced if X is not positive.

5. Data Collections Groups of data in specific structures
arrays, vectors
Using data collections allows us to:
Refer to all the data together
e.g., “all the temperature readings for May”
Move data items around as a whole group
Perform math or logical operations on the groups
e.g., compute the average, maximum, or minimum temperature for the month

6. Data Collections, cont’d
A homogeneous collection of data is constrained to accept only items of the same data type
e.g., all numeric, all character

7. Arrays and Vectors grades 5 3 7 9 11 2
An array is a named collection of like data arranged into rows and columns
A one-dimensional array (one row or one column) is called a vector (row vector, column vector)
Values in the array are called elements
An index identifies the position of an element (a value) in a vector
Example: the vector called grades has 6 elements (is of size 6)
grades 5 3 7 9 11 2 1 4 5 6 2 3

8. Creating a Vector – Constant Values (Declaring a Vector)
1 - Enter the values directly: e.g.,
A = [2, 5, 7, 1, 3]
2 - Enter the values as a range of numbers: e.g.,
B = 1:3:20
3 - Use the linspace(...) function: e.g.,
C = linspace (0, 20, 11)
ans =
 More on linspace on the next slide
4 - Use 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 2 5 7 1 3

9. linspace >> help linspace LINSPACE Linearly spaced vector.
    LINSPACE(X1, X2) generates a row vector of 100 linearly equally spaced points between X1 and X2.
    LINSPACE(X1, X2, N) generates N points between X1 and X2.
    For N < 2, LINSPACE returns X2.
    Class support for inputs X1,X2:
       float: double, single

10. Creating a Vector – Constant Values
Entering the values directly, e.g.
A = [2, 5, 7, 1, 3]
Entering the values as a range of numbers e.g.,
B = 1:3:20
>> B =
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
>> vecZeros = zeros(1,5)
vecZeros =
Ending number
Starting number
Increment
1-10

11. Array indexing
Indexing means accessing or changing specific indexed values of an array
Array index starts at 1
 xLet k be the index of vector x, then
k must be a positive integer
1 < = k < = length(x)
length (x) = 6
To access the kth element: x(k)
>> x(2) 8
>> x(4)=55 x 5 8 2 55 9 5 11

12. Array indexing Accessing beyond array length is illegal
??? … index out of bounds because…
>> x(0)
??? … index must be a positive integer or logical.
What does “logical” mean? Read your textbook, p.52.
Ready for the next Quiz?
Writing beyond array length will extend the vector automatically beyond its current end
>> x(8)= 21 x 5 8 2 9 5 11 x 5 8 2 9 5 11 21 8

13. Add and Remove
To add more than one element, create a new vector by concatenating elements
>> x = [vecDirect 12, 14]
x =
Removing elements
>> x(6) = [] % an empty vector
Not advisable because it can lead to logic problems when changing length of a vector x 5 8 2 9 5 12 14 x 5 8 2 9 5 12 14

14. Operations on vectors Basic operations are :
Arithmetic operations
Logical operations
Applying library functions
Concatenation
Slicing (generalized indexing)
First three extend from scalar operations, the rest are unique to vectors

15. Element-by-element Operations on Arrays+ - .* ./ .^
You must use the dot (.) with these!

16. Element-by-element Operations on Arrays and Scalars+ + - - * * / ./ .^ .^
You must use the dot (.) with these!
You may use dot (.) with these, but not necessary

17. Multiplication … and Multiplication
Mathematics
Matlab

18. Arithmetic Operations
Algebraic Syntax
MATLAB syntax
Addition
a + b
Subtraction
a - b
a – b
Multiplication
a x b
a * b
Division
a ÷ b
a /b
Exponentiation ab
a^b
>> A = [ ];
>> A + 5
ans =
>> A .* 3 % elt-by-elt multiplication
>> B = -1:1:3
B =

19. >> A .* B % Elt-by-Elt multiplication ans = -2 0 7 2 9
Operation
MATLAB syntax
Elt-by-elt Multiplication
a .* b
Elt-by-elt Division
a ./b
Elt-by-elt Exponentiation
a.^b
>> A = [ ]; %B =
>> A .* B % Elt-by-Elt multiplication
ans =
>> A * B % matrix multiplication!!
??? Error using ==> mtimes
Inner matrix dimensions must agree.
>> C = [1 2 3];
>> A .* C % A and C must have the same length
??? Error using ==> times
Matrix dimensions must agree.

20. Logical Operations >> A = [2 5 7 1 3]; >> B = [0 6 5 3 2];
>> A > = 5 % check which elements of A > = 5
ans =
>> A > = B % check which elements of A > = B
>> C = [1 2 3]
>> A > C
??? Error using ==> g t
Matrix dimensions must agree.
Note: “g t” means “greater than”.

21. Slicing Slicing means extracting values using a vector of indices
>> B = A(3:5) % From 3 to 5, default increment is 1
ans = 7 1 3
>> odds = 1:2:length(A)
ans =
>> A(odds) % odd values of A using predefined indices ans =
>> A(1:2:end) % odd values of A using anonymous indices
ans =
>> small = A < 4;
>> A(small)
ans =
>> A(small) = A(small) + 10
What’s the result?
What’s the result?

22. Useful Functions for Vectors
All MATLAB functions accept vectors rather than single values and return a vector of the same length
Special Functions:
>> v = [ ]
>> sum(v)
ans = 21
>> mean(v)
ans = 3.5
>> min(v)
ans = 1
>> max(v)
ans = 6
>> [value where] = min(v)
value = 1
where = 1

23. Example 1 Write a script to evaluate the function
f(t) = 5 cos(10πt) – 2 cos(20πt – π/2)
for t = -1.0 to 1.0 seconds in increments of 0.01 seconds.
Solution:
1) Input/Output:
input: values of t from -1.0 to 1.0
output : f(t)
2) Hand example:
f(-1.0) =
f(-0.99) =
And so on..

24. Example 1(MATLAB script)
Example 1 (Algorithm)
Define t = -1.0 to 1.0 with increments of -0.01
Calculate f(t) = 5cos(10πt) – 2 cos(20πt – π/2)
Display f(x)
Example 1(MATLAB script)
% Script to evaluate
% f(t) = 5cos(10πt) – cos(20πt – π/2)
% for t = -1.0 to 1.0 sec in increments of 0.01
% Author: Thomas Murphy
t = -1.0:0.01:1.0;
f = 5*cos(10*pi*t) - 2*cos(20*pi*t – pi/2);

25. Example 2 1) Write a script to evaluate the piecewise linear function
2) Input/Output:
input: 3 ranges of t, 0 to 0.9, 1 to 2 and 2 to 4
output : f(t)
3) Hand example:
f(0) = 0
f(0.1) = 0.2
And so on..

26. Example 2 (cont’d) 4) Algorithm: Define ranges of t (t1, t2 and t3)
Calculate f(t1), f(t2) and f(t3)
Concatenate the result
Display result

27. Example 2 (cont’d) 5) MATLAB script
% Script to evaluate the piecewise function f(t)
% Author: Thomas Murphy
% Date: July 20, 2006
%time regions for piecewise function
t1 = 0.0 : 0.1 : 0.9;
t2 = 1.0 : 0.1 : 1.9;
t3 = 2.0 : 0.1 : 4.0;
% evaluate piecewise function for the appropriate
f1 = 2*t1;
f2 = 4*t2 – 1; % regions
f3 = 4 – 3*t3;
t = [t1 t2 t3]; % concatenation of three results
f = [f1 f2 f3];

28. Vector = 1-dimensional Array
Row vector: v is an n × 1 array of values v1 to vn
v =
Column vector: w is a 1 x n array of values w1 to wn
w =

29. A Matrix (multidimensional array)
A matrix is an m x n array
It has m rows and n columns
Has m x n elements
Is called an Array Matrix (1-d array is referred to as a vector)
A(m x n) =
a a … a1n
a21 a22 … a2n .
am1 am2 … amn

30. Creating an Array Matrix
Entering the values directly
A Semicolon identifies the next row
>> v = [0,1,2,3,4,5,6] %create a 1 by 7 row matrix
>> w = [0;1;2;3;4;5;6] %create a 7 by 1 column matrix
>> A = [2, 5, 7; 1, 3, 42] %create a 2 by 3 matrix
>> v1 = [1,2,3]
>> v2 = [3,4,5]
>> v3 = [6,7,8]
>> B = [v1; v2; v3]
Using the functions to create vectors filled with 0, 1, or random values between 0 and 1
zeros(rows, cols)
Ones (rows, cols)
Rand (rows, cols)
>> B = zeros(2,5)%creates 2 by 5 matrix of zeros

31. Indexing an Array Matrix
Syntax:
A(row, col) returns the element(s) at the location(s) specified by the array row and column indices.
A(row, col) = value replaces the elements at the location(s) specified by the array row and column indices.
Examples:
>> A = [0,1,2;3,4,5;6,7,8]
>> A(1,1)
>> A(2,3)
>> A(2,:)
>> A(2,3) = 2
>> A(2,:)=[1, 1, 1]
%returns the (1,1) element = 0
%returns the (2,3) element = 5
%returns the second row
%change element (2,3) to 2
%change row 2 to all 1s

32. Size & Dimensions of Array Matrix
size() returns the dimensions of the matrix
length() returns the larger of the dimensions (the larger number of the number of rows and columns)
Example:
>> E = [1:1:10; 11:1:20; 21:1:30]
>> [nrows, ncol] = size(E)% size is a reserved word
nrows =3 % nrows is a variable name (user defined)
ncol = 10 % ncol is a variable name (user defined)
>> length(E);
ans = 10

33. Operating on an Array Matrix
Four techniques extend directly from operations on vectors:
Arithmetic operations
Logical operations
Applying library functions
Slicing

34. Array Matrix Elt-by-elt Operations
* / ^
Between arrays of the same size or array and scalar
Two arrays must be of the same size
>> A = [3, 2, 1 ; 8, 7, 6];
>> B = [1, 2, 3 ; 4, 5, 6];
>> A + 2
ans = 5 4 3
10 9 8
>> A / 2
ans =
>> A.*B
ans = 3 4 3 34 34

35. Array Matrix Logical Operations
Are done on an element-by-element basis
Result is an array of Boolean values, 0 and 1
>> A = [3, 2, 1 ; 8, 7, 6];
>> B = [1, 2, 3 ; 4, 5, 6];
>> A > = B
ans =
>> A > = 5

36. Displaying a Table of Data
% script to evaluate % f(t) = 5cos(10πt) – cos(20πt – π/2) % for t = -1.0 to 1.0 sec in increments of 0.01 % Author: Thomas Murphy t = -1.0:0.01:1.0; f = 5*cos(10*pi*t) - 2*cos(20*pi*t – pi/2); disp ([t f]); % disp is used to display any % text or array t = t’; f = f’; disp ([t’ f’]) % NOTE: Matlab uses the quote symbol ‘ to access the “fields of a structure”
What are these?

37. disp Disp(‘t f’); Displays the label of a table
Displ([t’ f’]); Displays the content

38. Example - Computing Soil Volume
 Given:
1 - The depth of soil from a survey in the form of a rectangular array of soil depth.
2 - The footprint of the foundations of a building to be built on that site and the depth of the foundation.
 Compute the volume of soil to be removed.

39. Survey Data

40. Building Footprint

41. Solution Script What units? clear clc
% soil depth data by square, produced by survey
depth = [
. . . ];
% estimated proportion of each square that should
% be excavated
area = [ ];
square_volume = depth .* area;
total_soil = sum(sum(square_volume))
What units?

42. clear and clc
The Matlab built-in function “clear” clear variables from memory.
 Use the Matlab help to better understand what “clear” does
clc clears the screen