MATLAB – Ch 2 - N umeric, Cell, & Structure Arrays

1. MATLAB – Ch 2 - Numeric, Cell, & Structure Arrays EGR1302

2. Outline • Introduction • Arrays • Multidimensional arrays • Element-by-element operations • Polynomial operations using arrays

3. Introduction • Array capabilities in Matlab • Serves as basic building block in Matlab • Allows for complex operations using one command or function • Means that Matlab programs can be very short

4. Introduction • Arrays in Chapter 1 • Array assignment • “[]” contains numbers being assigned • Commas or spaces separate elements in row • Semi-colons separate rows • Special format for assigning row array with regularly spaced numbers

5. Introduction • Arrays in Chapter 1 • Use just variable name in expressions • Operation is performed on every element in the array • Use variable name & array index in expressions • Use length function to determine number of elements in an array

6. Section 2.1 Arrays

8. Cartesian coordinates review • Represent a point p in 3D space • x, y, & z • Represent unit vectors

9. Cartesian coordinates review • Represent a vector p from origin to point p • In Matlab • Row vector • Column vector

10. Creating vectors in Matlab • To create a row vector • Type numbers within square brackets • Separate numbers with a space or a comma >>p = [3,7,9] p = 3 7 9

11. Creating vectors in Matlab • To create a column vector • Type numbers within square brackets • Separate numbers with a semi-colon • OR transpose a row vector >>p = [3;7;9] p = 3 7 9 >>p = [3,7,9]’

12. Appending one vector to another • Append 2 row vectors >> r = [2,4,20]; >> w = [9,-6,3]; >> u = [r,w] u = 2 4 20 9 -6 3

13. Generating vectors of regularly spaced elements • Use colon “:” operator • x1 = first value in the series • x2 = last value in the series • d = increment between numbers in series (default is 1 if d is omitted) • Use linspace function • n =number of points (default is 1 if n is omitted)

14. Generating vectors of regularly spaced elements • Use logspace function • a = exponent for first value (i.e., 10a) • b = exponent for last value (i.e., 10b) • n =number of points (default is 50 if n is omitted)

15. 2D Arrays • Array • Collection of scalars arranged in logical manner • Row vector • Array with a single row of scalars • Column vector • Array with a single column of scalars • Matrix • An array with multiple rows and columns

16. 2D Arrays • Square brackets denote matrices • Recall  Parallel lines denote a determinant

17. Creating Matrices • Type row by row • Semi-colon separating rows • Comma or space separating elements in a row >> A = [2,4,10;16,3,7] A = 2 4 10 16 3 7

18. Array addressing • v(5)  5th element in vector v • A(2,3)  Element in 2nd row and 3rd column in array A • Row number is always first! • D(1,3) = 4 • Replaces the element in 1st row, 3rd column of array D with 4

19. Array addressing • Colon operator • Selects submatrices • v(:)  all row or column elements in vector v • v(2:4)  2nd through 4th elements in v • A(3,:)  all elements in the 3rd row of matrix A • A( :,2:4)  all elements in the 2nd through 4th columns of A • B(2:4,1:3)  all elements in the 2nd through 4th rows and 1st through 3rd columns of B

20. Useful array functions • See Table 2.1-1, p. 77 • Max • For a vector x, returns algebraically greatest element • For a matrix B, returns • Row vector x containing greatest element in each column of B • Row vector k containing indices of greatest elements in each column of B >> y = max(x) >> [x,k] = max(B)

21. Vector terms • Length • Number of elements in a vector • Magnitude • Vector’s geometric length • Absolute value • Absolute values of each elements in vector

22. Array editor • Graphical interface for working with arrays • View and edit workspace variables • Clear workspace variables • Plotting workspace variables

23. Section 2.2 Multidimensional arrays

24. 3D & 4D arrays • 1st dimension is row • 2nd dimension is column • Higher dimensions are referred to as pages

25. Section 2.3 Element-by-element operations

26. Scalar multiplication • Increase magnitude of a vector by multiplying it by scalar r=[3,5,2]; >> v=2*r v = 6 10 4

27. Array addition & subtraction • 2 arrays with same size • Sum or difference has same size • Add or subtract corresponding elements >> A=[6,-2;10,3]; >> B=[9,8;-12,14]; >> C=A+B C = 15 6 -2 17

28. Array addition & subtraction • Associative • Commutative

29. Multiplication of two arrays • Two definitions of multiplication of two arrays • Array multiplication • Element-by-element operation • Matrix multiplication • Division and exponentiation also must be carefully defined

30. Table 2.3-1, p.85 Symbol + - + - .* ./ .\ .^ Operation Scalar-array addition Scalar-array subtraction Array addition Array subtraction Array multiplication Array right division Array left division Array exponentiation Form A + b A – b A + B A – B A.*B A./B A.\B A.^B Examples [6,3]+2=[8,5] [8,3]-5=[3,-2] [6,5]+[4,8]=[10,13] [6,5]-[4,8]=[2,-3] [3,5].*[4,8]=[12,40] [2,5]./[4,8]=[2/4,5/8] [2,5].\[4,8]=[2\4,5\8] [3,5].^2=[3^2,5^2] 2.^[3,5]=[2^3,2^5] [3,5].^[2,4]=[3^2,5^4]

31. Array multiplication • Vectors must be of the same size • Matrices must be of the same size • Dot (.) and asterisk (*) form one symbol

32. Built-in Matlab functions • sqrt(x) and exp(x) • Automatically operate on array arguments to produce an array result the same size as the array argument. • Thus these functions are said to be vectorizedfunctions

33. Built-in Matlab functions • When multiplying, dividing, or exponentiating these functions, you must use element-by-element operations if the arguments are arrays. • To compute z = (ey sin x) cos2x, you must type

34. Section 2.4 Matrix operations

35. Addition & Subtraction • Matrix addition and subtraction are identical to element-by-element addition and subtracted

36. Vector Multiplication • Vector dot product • Recall  result is a scalar

37. Vector-Matrix Multiplication • Matrix multiplied by column vector • Result is a column vector • Number of columns in matrix must equal number of rows in vector

38. Matrix multiplication • To multiply two matrices A & B • Number of columns in A must equal number of rows in • The resulting product AB has • Same number of rows as A • Same number of columns as B 6 –2 10 3 4 7 (6)(9) + (– 2)(– 5) (6)(8) + (– 2)(12) (10)(9) + (3)(– 5) (10)(8) + (3)(12) (4)(9) + (7)(– 5) (4)(8) + (7)(12) 9 8 –5 12 = 64 24 75 116 1 116 =

39. Matrix multiplication • Matrix multiplication is NOT commutative • The order of the matrices in the equation is important • Exceptions • Null matrix 0 • Identity matrix I

40. Section 2.5 Polynomial operations using arrays

41. Polynomials in Matlab • Defined as row vector • Containing coefficients • Starting with coefficient of highest power of x • Addition & subtraction • Add row vectors • BUT if polynomials are of different degrees, add zeros to coefficient array of lower degree polynomial • Fool Matlab into thinking that the lower degree polynomial has the same degree

42. Polynomial functions • Roots(a)  calculate roots • Poly(a)  computes coefficients of polynomial whose roots are contained in a • See Table 2.5-1 on p.108 for additional functions

43. Plotting a polynomial • Function polyval(a,x) • Evaluates a polynomial at specified values of its independent variable x, which can be a matrix or a vector. • The polynomial’s coefficients of descending powers are stored in the array a. • The result is the same size as x.