cos 131: computing for engineers ch. 3: vectors and arrays

April 21, 2023 1

Laura Broussard, Ph.D.

Professor

COS 131: Computing for Engineers

Ch. 3: Vectors and Arrays

Ch. 3: Vectors and ArraysI. Introduction

II. Concept: Using Built-in Functions

III. Concept: Data Collections

IV. MATLAB Vectors

V. Engineering Example

VI. MATLAB Arrays

VII. Engineering Example

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I. Introduction• Emphasizes the basic calculations involving

rectangular collections of numbers in the form of vectors and arrays.

• You will learn how to:– Create them– Manipulate them– Access them– Perform mathematical/logical operations on them

• Shows MATLAB’s ability to deal with sets of numbers as aggregates as opposed to individually

• Introduces the notion of functions built into the language


I. Introduction



IV. MATLAB Vectors


VI. MATLAB Arrays


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II. Concept: Using Built-in Functions• Function – a named collection of

instructions that operates on the data provided to produce a result according to the specifications of that function

• To get help on a particular function, type the following at the Command window prompt:

• >> help <function name>

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• MATLAB groups data together in arrays and in vectors (subsets of arrays)

• Data Abstraction– Convenient to refer to groups of data as

opposed to individual data items– Allows the movement of these data as a group– Allows the use of mathematical and/or logical

operations on these groups

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III. Concept: Data Collections• Homogeneous Collection

– Arrays and vectors constrain data to only be accepted for the same data type

– These are referred to as homogeneous collections; i.e., all data items within a given array or vector are of the same data type

• We will now consider the basic operations related to vectors and arrays in MATLAB


I. Introduction



IV. MATLAB Vectors


VI. MATLAB Arrays


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IV. MATLAB Vectors• Vector – a list of like data items

(simplest means of grouping)• Elements – individual items in a vector• Elements have two attributes:

– Numerical value– Position in the given vector

• Data type: numbers or logical values• Vectors also called linear arrays or linear

matrices

IV. MATLAB Vectors

A. Creating Vectors

B. Size (measuring)

C. Indexing

D. Shortening

E. Operations Arithmetic Concatenation Logical Slicing Library

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IV. MATLAB Vectors

A. Creating Vectors – 2 methods

1. Creating vectors as a series of constant values

2. Producing new vectors by operating on existing vectors (later)

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A. Creating Vectors - from constant values:

• Enter each value directly A = [1 2 3]

• Enter a range of values with colon operator B = 1:3:20

• Entering a range of values with linspace(…) function C = linspace (0, 20, 11)

IV. MATLAB Vectors

IV. MATLAB Vectors

A. Creating from constant values (cont’d.)

• Using functions: zeros(1,n) rand(1,n) ones(1,n) randn(1,n)for vectors with 0’s, 1’s, or random numbersin between E = zeros(1,4)

Exercise 3.1 Working with Vectors, pp. 49-50

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A. Creating vectors• Workspace window information

The name The value

Contents (small vectors) or

Description - such as <1 x 11 double> The class (data type) of the vector

• Scalars are vectors (unit length)

IV. MATLAB Vectors

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IV. MATLAB VectorsB. Size of a Vector

– Specific attribute – the length of the vector; how many elements are in the vector

– Can increase or decrease the size of a vector by inserting and deleting elements

– Function size(V) when applied to vector V, returns another vector [r, c]: r = number of rows (for vectors, 1) c = number of columns (length of vector)

– Function length(V) returns the length of the vector

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IV. MATLAB Vectors

C. Indexing a Vector– Definition: accessing elements of a vector to

assess its value and/or change the value– Two ways of indexing:

• With a numerical vector

• With a logical vector

– Numerical indexing• Elements of a vector can be accessed individually or

in groups by enclosing the index of one or more required elements in parentheses

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IV. MATLAB Vectors

C. Indexing a Vector

– Numerical Indexing: A = [0 2 4 6 8]• Assessing value: >> A(1) ans = 0• Changing value: >> A(1) = 6 A = [6 2 4 6 8]

Exercise 3.2, Extending a vector, p. 51b

• MATLAB will automatically extend a vector if you write beyond its current end.

Missing elements zero-filled Exercise 3.3, Extending a vector, p. 51b• These examples use a single number to index

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IV. MATLAB VectorsC. Indexing a Vector

– Numerical Indexing A = [0 2 4 6 8]• Indexing can also be with multiple numbers.

>> A(1),(2),(3) ans = 0 ans = 2 ans = 4

• Use a vector of index values to index another vector

>> B = [ 1 3 5 ]

>> A(B) ans = 0 4 8

(Note: argument of A is a vector)• Index vector: - doesn’t need to match size of vector being indexed - values must be positive and smaller than length of

vector being indexed

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IV. MATLAB VectorsC. Indexing a Vector

– Logical Indexing A = [0 2 4 6 8]• A new (for us) data type. • Two values: true and false • Called Boolean or logical values• Can be assembled into vectors and arrays by typing in true or

false values. MATLAB responds with 0’s and 1’s >> mask = [true false false true] mask = 1 0 0 1

>> A(mask) ans = 0 6• Logical index vectors can be shorter, not longer.

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IV. MATLAB Vectors

D. Shortening a Vector A = [0 2 4 6 8]– May need to remove elements from a vector

– Use the empty vector, [ ] A vector with no elements in it

– Assign the empty vector to an element in another vector, that element is removed from A, and A is shortened by on element

>> A(4) = [ ] A = [0 2 6 8]

– Exercise 3.4: Shortening a vector, p. 52b

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IV. MATLAB Vectors

E. Operating on Vectors– Three techniques analogous to operations on

scalar (single) values:• Arithmetic operations

• Logical operations

• Applying library functions

– Two unique techniques for vectors and arrays:• Concatenation (joining)

• Slicing (generalized indexing)

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IV. MATLAB Vectors

E. Operations - Arithmetic– Can be performed collectively on individual

components of two vectors if • both vectors are same length or

• one of the vectors is scalar (one value)

– Element by element NOT matrix operations– Use symbols: .*, ./, and .^

Regular symbols *, /, and ^ reserved for matrix operations.

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IV. MATLAB Vectors

E. Operations – Arithmetic

– Addition and subtraction same for both matrix and element-by-element operations

Use the same symbols: + and -

– Exercise 3.5: Using vector mathematics, pp. 53-54

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IV. MATLAB VectorsE. Logical Operations

– Produce vectors of logical results– Use to index vectors in a style that makes the

logic of complex expressions very clear– Can perform element-by-element operations on

two vectors if both vectors are the same length or if one is a scalar

– A = [2 5 7 1 3] and B = [0 6 5 3 2]• Where are A’s elements ≥ 5?• >> A >= 5 ans = 0 1 1 0 0• >> A >= B ans = 1 0 1 0 1

– Ex 3.6, Working with vector logical expressions, p. 55t

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IV. MATLAB Vectors

E. Logical Operations– Can be assembled into more complex operations

using logical and (&) and or (|) operators– Operators come in two flavors

• & / | (single operators) • && / || (double operators)• Single operators are for logical arrays of matching

size• Double operators combine individual logical results,

and are associated with conditional statements (Ch. 7)– Exercise 3.7, Working with logical vectors, p. 55b

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IV. MATLAB Vectors

E. Logical Operations

– Function find(…)

Inputs an array of logical values

Outputs a vector of positions of true elements

Useful to find indices of the true elements of a

logical vector

– Ex. 3.8: Using the find(…) function, p. 56 t

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IV. MATLAB Vectors

E. Logical Operations a = [true true false true ]

– Negate the values of all elements of a logical vector (true false and false true) with⇒ ⇒

~ (not operator)

>> na = ~[true true false true]

na = 0 0 1 0– Each element of na is the logical inverse of the

corresponding original element– Table 3.1, Operator Precedence, p. 56

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IV. MATLAB Vectors

E. Operations - Library Functions– MATLAB has a rich collection of mathematical

functions covering mathematical, trigonometric, and statistical capabilities

– Partial list in App. A– Complete list – Help menu option– Accept vectors of numbers rather than single

values and return a vector of the same length

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IV. MATLAB VectorsE. Operations - Useful Library Functions

– sum(v) and mean(v) consume a vector and return the sum and mean of all the elements of the vector

– min(v) and max(v) return two quantities: the minimum and maximum value in vector the position of the value in vector

>> [value where] = max([2 7 42 9 -4]) value = 42 where = 2– round(v), ceil(v), floor(v), and fix(v) functions that round numbers in vector

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IV. MATLAB Vectors

E. Operations - Concatenation– MATLAB lets you construct a new vector by

concatenating (joining) other vectors

– A = [B C D … X Y Z]

B, C, D, … are any vector sdefined as a

constant or variable

length of A is sum of the lengths of B, C, D, …

– Resulting vector will be flat, not nested.

– Exercise 3.9: Concatenating Vectors, p. 57b

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IV. MATLAB Vectors

E. Operations: Slicing

– Indexing ≡ basic operation of extracting and replacing the elements of a vector

– Vectors of indices change multiple elements⇒

– Vectors can be: ⋄ previously-defined indices or ⋄ created anonymously as needed

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IV. MATLAB VectorsE. Operations: Slicing – Anonymous

A = [2 5 7 1 3 4]

– General form for generating a vector of numbers: <start> : < increment> : <end>

Omit <increment>, default = 1 Key word end ≡ the length of the vector Operator : (alone) is short for 1:end

odds = 1:2:end

E. Indexing vector of logical values must be same length or shorter than the vector being indexed

>> A([false true false true]) ans = 5 1

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IV. MATLAB Vectors

E. Operations - Slicing– General form of statements for slicing vectors

(copy part of A to part of B)

B(<rangeB> = A(<rangeA>)– When <rangeA> and <rangeB> are index vectors,

• A is an existing Array

• B can be an existing or new array

or absent altogether (named ans)

• Rules on next slide

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IV. MATLAB Vectors

E. Operations - Slicing– Rules for use of this template:

• Either

- size of rangeB = size of rangeA or

- size of rangeA = 1

• If B does not exist, it is zero-filled where assignments not explicitly made

• If B did not exist before this statement, values of A not directly assigned in rangeB remain unchanged

• See Listing 3.1 on the next slide

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IV. MATLAB Vectors

• Slicing

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IV. MATLAB Vectors

E. Operations – Slicing

– Ex 3.10 Running the vector indexing script, pp. 59-60

– Refer to line by line description on page 59-61.

This will become commonplace in our class discussions as we examine MATLAB code to understand what the software is doing.

Ch. 3: Vectors and ArraysI. Introduction



IV. MATLAB Vectors


VI. MATLAB Arrays


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V. Engineering Example – Forces and Moments

• MATLAB vectors are ideal representations of the concept of a vector used in physics

• Represent each of the vectors in this problem as a MATLAB vector with three components: the x, y, and z values of the vector

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V. Engineering Example – Forces and Moments

Examine code to leftand work throughcommentary on page64.


I. Introduction



IV.MATLAB Vectors


VI.MATLAB Arrays


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VI. MATLAB Arrays

•We now extend the vector ideas to include arrays of multiple dimensions•Initially two dimensions•Each row will have the same number of columns and each column will have the same number of rows•Referred to as arrays to differentiate them from matrices•Arrays and matrices differ in the way they execute their multiplication, division, and exponentiation operations

IV. MATLAB Arrays

A. Properties

B. Creating

C. Accessing Elements

D. Removing Elements

E. Operations Arithmetic Concatenation Logical Slicing Library Reshaping

Linearizing

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MATLAB ArraysA. Properties of an Array

– Individual elements in an array are also referred to as elements

– Unique attributes: value and position– In a 2-D array position will be (row, column)– In an n-D array, the element position will be a

vector of n index values

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MATLAB ArraysA. Properties of an Array

– Function size will return information 1 of 2 forms:• If called with a single return value

sz = size(A) It returns a vector of length n containing the size

of each dimension of the array

• If called with multiple return values

[rows, cols] = size(A) It returns the individual array dimension up to the number of values requested; ALWAYS provide as many variables as there are dimensions of the array

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MATLAB Arrays• Properties of an Array

– The length(…) function returns the maximum dimension of the array

– The transpose of an m x n array, indicated by the apostrophe character (‘) placed after the array identifier, returns an n x m array with the values in the rows and columns interchanged.

– See example on the next slide

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Original Array Transposed Array

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– Special Cases:• When a 2-D matrix has the same number of rows

and columns, it is called square

• When only nonzero values in an array occur when the row and column indices are the same, the array is called diagonal

• When there is only one row, the array is a row vector, or just a vector as we saw earlier

• When there is only one column, the array is a column vector, the transpose of row vector

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MATLAB Arrays• Creating an Array

– Created by entering values directly– Created by using one of a number of built-in MATLAB

functions that create arrays with specific characteristics:

• Enter data directly, using a ; to denote the end of a row• Functions zeros(m,n) and ones(m,n) creates an m x n matrix

with zeros or ones• Functions rand(m,n) and randn(m,n) fill an array with random

numbers in the range 0 .. 1.• Function diag(…) where diag(A) of an array A returns its

diagonal as a vector, and diag(V) of a vector returns a square array with the vector as the diagonal

• Function magic(m) which produces an m x m array which returns a square array where the values in rows and columns add up to the same value

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MATLAB Arrays• Creating an Array

– Exercise 3.11: Creating Arrays– Smith text, page 67, top

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MATLAB Arrays• Accessing Elements of an Array

– Elements of an array may be addressed by enclosing the indices of the required elements in parentheses, with the first index being the row index and the second index the column index

– Can also store values that are elements of an array as in A(2,3) = 0 will assign the value of 0 to the cell that is at row 2, column 3.

– MATLAB automatically extends the array if you write beyond its boundaries. Elements with no values between the boundaries of the array will be assigned values of zero, 0.

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MATLAB Arrays• Removing Elements of an Array

– In arrays elements must be removed as comlete rows or columns

– See page 68 for examples

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MATLAB Arrays• Operating on Arrays

– Discuss the following (direct extension from vectors):

• Arithmetic and logical operations• Application of functions• Concatenation and slicing

– Two array specific topics:• Reshaping• Linearizing

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MATLAB Arrays• Array Arithmetic Operations

– Can be performed collectively on the individual components of two arrays as long as both arrays have the same dimensions or one of them is scalar (vector length of 1)

– Multiplication, division, and exponentiation must use the “dot operator” symbols: .*, ./, and .^. Commas are not used.

– Exercise 3.12: Working with array mathematics– Smith text, page 69, bottom

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MATLAB Arrays• Array Logical Operations

– Logical arrays take the same dimensions as arithmetic arrays

– Result is an array of Boolean values with the same size as the original array

– Exercise 3.13: Working with array logical operations

– Smith text, page 70, top

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MATLAB Arrays• Applying Library Functions

– Most MATLAB mathematical functions can consume an array of numbers and return and array of the same dimensions

– As with vectors we have:• sum(v) and mean(v) – returns sum and mean of each

column• min(v) and max(v) – min and max values in each

column• See examples pages 70-71 for details

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MATLAB Arrays• Array Concatenation

– Can construct new arrays by concatenating other arrays in the following ways:

• Horizontally, as long as each component has the same number of rows

• Vertically, as long as each has the same number of columns; must use semi-colons to denote vertical concatenation

– The result will be an array with that number of rows and a number of columns equaling the sum of the number of columns in each individual array

– Exercise 3.14: Concatenating an array– Smith text, page 71, bottom

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MATLAB Arrays• Slicing Arrays

– General form of statements for moving sections of one array into sections of another array:

• B(<rangeBR>, <rangeBC>) = A(<rangeAR>, <rangeAC>)• Where each <range..> is an index vector, A is an existing

array, and B can be an existing array, a new array, or absent altogether (B = ans)

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MATLAB Arrays• Slicing Arrays

– Rules for using this template:• Either each dimension of each sliced array must be equal, or

the size of the slice from A must be 1 x 1.• If B did not exist before this statement was implemented, it is

zero filled where assignments were not explicitly made.• If B did exist before this statement, the values not directly

assigned remain unchanged

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MATLAB Arrays• Reshaping Arrays

– Useful to take an array with one set of dimensions and reshape it to another set

– Function reshape(…) does this– Reshape(A, rows, cols, …)– Takes an array A, whatever its dimensions, and reforms

it into an array sized:– (rows x cols x …) to as many dimensions as needed– Does not pad the data to fill any empty space– Product (rows x cols) of the dimensions of the original

array must equal the product of the new dimensions– Exercise 3.15 Reshaping an array– Smith text, page 72, bottom

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MATLAB Arrays• Linearized Arrays

– MATLAB secret: multi-dimensional arrays are NOT stored in some nice, rectangular chunk of memory

– Memory block allocated is sequential; stored in column order

– Can see this in the find(…) function– See discussion on page 73– Exercise 3.16: Linearizing an array– Smith text, page 73, bottom

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MATLAB Arrays• Array Manipulation Script

Work through the lineby line commentaryon this program. Thereis a lot here you should understand. See pages75-76


I. Introduction



IV. MATLAB Vectors


VI. MATLAB Arrays


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Engineering Example – Computing Soil Volume• Problem Description – pages 76-77

Landscape Survey

Calculating Soil Volume

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Engineering Example – Computing Soil Volume• Script to compute total soil

cos 131: computing for engineers ch. 3: vectors and arrays

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