1. Covenant College November 14, 20151 Laura Broussard, Ph.D. Professor COS 131: Computing for Engineers Ch. 3: Vectors and Arrays

2. Covenant College Ch. 3: Vectors and Arrays I. Introduction II. Concept: Using Built-in Functions III. Concept: Data Collections IV. MATLAB Vectors V. Engineering Example VI. MATLAB Arrays VII. Engineering Example November 14, 20152

3. Covenant College November 14, 20153 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

4. Covenant College Ch. 3: Vectors and Arrays I. Introduction II. Concept: Using Built-in Functions III. Concept: Data Collections IV. MATLAB Vectors V. Engineering Example VI. MATLAB Arrays VII. Engineering Example November 14, 20154

5. Covenant College November 14, 20155 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

6. Covenant College November 14, 20156 III. Concept: Data Collections 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

7. Covenant College November 14, 20157 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

8. Covenant College Ch. 3: Vectors and Arrays I. Introduction II. Concept: Using Built-in Functions III. Concept: Data Collections IV. MATLAB Vectors V. Engineering Example VI. MATLAB Arrays VII. Engineering Example November 14, 20158

9. Covenant College November 14, 20159 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

10. Covenant College IV. MATLAB Vectors A. Creating Vectors B. Size (measuring) C. Indexing D. Shortening E. Operations ArithmeticConcatenation LogicalSlicing Library November 14, 201510

11. Covenant College November 14, 201511 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)

12. Covenant College November 14, 201512 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

13. Covenant College 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 numbers in between E = zeros(1,4) Exercise 3.1 Working with Vectors, pp. 49-50 November 14, 201513

14. Covenant College November 14, 201514 A.Creating vectors Workspace window information  The name  The value Contents (small vectors) or Description - such as  The class (data type) of the vector Scalars are vectors (unit length) IV. MATLAB Vectors

15. Covenant College November 14, 201515 IV. MATLAB Vectors B.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

16. Covenant College November 14, 201516 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

17. Covenant College November 14, 201517 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

18. Covenant College November 14, 201518 IV. MATLAB Vectors C.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

19. Covenant College November 14, 201519 IV. MATLAB Vectors C.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.

20. Covenant College November 14, 201520 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

21. Covenant College November 14, 201521 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)

22. Covenant College November 14, 201522 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.

23. Covenant College November 14, 201523 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

24. Covenant College November 14, 201524 IV. MATLAB Vectors E.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

25. Covenant College November 14, 201525 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

26. Covenant College November 14, 201526 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

27. Covenant College November 14, 201527 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

28. Covenant College November 14, 201528 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

29. Covenant College November 14, 201529 IV. MATLAB Vectors E.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

30. Covenant College November 14, 201530 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

31. Covenant College November 14, 201531 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

32. Covenant College November 14, 201532 IV. MATLAB Vectors E.Operations: Slicing – Anonymous A = [2 5 7 1 3 4] –General form for generating a vector of numbers: : : Omit, default = 1 Key word end ≡ the length of the vector Operator : (alone) is short for 1:end odds = 1:2:end –Indexing vector of logical values must be same length or shorter than the vector being indexed >> A([false true false true]) ans = 5 1

33. Covenant College November 14, 201533 IV. MATLAB Vectors E.Operations - Slicing –General form of statements for slicing vectors (copy part of A to part of B) B( = A( ) –When and 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

34. Covenant College November 14, 201534 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

35. Covenant College November 14, 201535 IV. MATLAB Vectors Slicing

36. Covenant College November 14, 201536 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.

37. Covenant College Ch. 3: Vectors and Arrays I. Introduction II. Concept: Using Built-in Functions III. Concept: Data Collections IV. MATLAB Vectors V. Engineering Example VI. MATLAB Arrays VII. Engineering Example November 14, 201537

38. Covenant College November 14, 201538 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

39. Covenant College November 14, 201539 V. Engineering Example – Forces and Moments Examine code to left and work through commentary on page 64.

40. Covenant College Ch. 3: Vectors and Arrays I.Introduction II.Concept: Using Built-in Functions III.Concept: Data Collections IV.MATLAB Vectors V.Engineering Example VI.MATLAB Arrays VII. Engineering Example November 14, 201540

41. Covenant College November 14, 201541 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

42. Covenant College IV. MATLAB Arrays A. Properties B. Creating C. Accessing Elements D. Removing Elements E. Operations ArithmeticConcatenation LogicalSlicing LibraryReshaping Linearizing November 14, 201542

43. Covenant College November 14, 201543 MATLAB Arrays A.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

44. Covenant College November 14, 201544 MATLAB Arrays A.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

45. Covenant College November 14, 201545 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

46. Covenant College November 14, 201546 MATLAB Arrays Properties of an Array Original ArrayTransposed Array

47. Covenant College November 14, 201547 MATLAB Arrays Properties of an Array –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

48. Covenant College November 14, 201548 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

49. Covenant College November 14, 201549 MATLAB Arrays Creating an Array –Exercise 3.11: Creating Arrays –Smith text, page 67, top

50. Covenant College November 14, 201550 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.

51. Covenant College November 14, 201551 MATLAB Arrays Removing Elements of an Array –In arrays elements must be removed as comlete rows or columns –See page 68 for examples

52. Covenant College November 14, 201552 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

53. Covenant College November 14, 201553 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

54. Covenant College November 14, 201554 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

55. Covenant College November 14, 201555 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

56. Covenant College November 14, 201556 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

57. Covenant College November 14, 201557 MATLAB Arrays Slicing Arrays –General form of statements for moving sections of one array into sections of another array: B(, ) = A(, ) Where each is an index vector, A is an existing array, and B can be an existing array, a new array, or absent altogether (B = ans)

58. Covenant College November 14, 201558 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

59. Covenant College November 14, 201559 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

60. Covenant College November 14, 201560 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

61. Covenant College November 14, 201561 MATLAB Arrays Array Manipulation Script Work through the line by line commentary on this program. There is a lot here you should understand. See pages 75-76

62. Covenant College Ch. 3: Vectors and Arrays I. Introduction II. Concept: Using Built-in Functions III. Concept: Data Collections IV. MATLAB Vectors V. Engineering Example VI. MATLAB Arrays VII. Engineering Example November 14, 201562

63. Covenant College November 14, 201563 Engineering Example – Computing Soil Volume Problem Description – pages 76-77 Landscape Survey Calculating Soil Volume

64. Covenant College November 14, 201564 Engineering Example – Computing Soil Volume Script to compute total soil