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MATLAB – Ch 2 - Numeric, Cell, & Structure Arrays EGR1302 Outline Introduction Arrays Multidimensional arrays Element-by-element operations Polynomial operations using arrays 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 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 u 0 : 0.1 : 10 Introduction Arrays in Chapter 1 Use just variable name in expressions Operation the array is performed on every element in Use variable name & array index in expressions u (7) Use length function to determine number of elements in an array Section 2.1 ARRAYS Cartesian coordinates review Represent a point p in 3D space x, y, & z Represent unit vectors iˆ 1i 0 j 0k ˆj 0i 1 j 0k kˆ 0i 0 j 1k Cartesian coordinates review Represent a vector p from origin to point p p xiˆ yˆj zkˆ In Matlab p [ x, y, z ] Row vector Column vector x p y z 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 Creating vectors in Matlab To create a column vector Type numbers within square brackets Separate numbers with a semi-colon >>p = [3;7;9] p = 3 7 9 OR transpose a row vector >>p = [3,7,9]’ 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 Generating vectors of regularly spaced elements Use colon “:” operator u x1 : d : x 2 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) u linspace( x1, x2, n) Generating vectors of regularly spaced elements Use logspace function u logspace(a, b, n) 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) 2D Arrays Array Row vector Array with a single row of scalars Column vector Collection of scalars arranged in logical manner Array with a single column of scalars Matrix An array with multiple rows and columns 2D Arrays Square brackets denote matrices 2 5 M 3 4 7 1 Recall Parallel lines denote a determinant 2 4 N 0 3 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 16 4 3 10 7 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 Array addressing Colon operator Selects submatrices v(:) v all row or column elements in vector 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 v(2:4) Useful array functions See Table 2.1-1, p. 77 Max >> y = max(x) For a vector x, returns algebraically greatest element >> [x,k] = max(B) For a matrix B, returns vector x containing greatest element in each column of B Row vector k containing indices of greatest elements in each column of B Row Vector terms Length Magnitude Number of elements in a vector Vector’s geometric length Absolute value Absolute values of each elements in vector Array editor Graphical interface for working with arrays View and edit workspace variables Clear workspace variables Plotting workspace variables Section 2.2 MULTIDIMENSIONAL ARRAYS 3D & 4D arrays 1st dimension is row 2nd dimension is column Higher dimensions are referred to as pages Section 2.3 ELEMENT-BY-ELEMENT OPERATIONS Scalar multiplication Increase magnitude of a vector by multiplying it by scalar r=[3,5,2]; >> v=2*r v= 6 10 4 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 c a b ij ij ij Array addition & subtraction Associative ( A B) C A ( B C ) Commutative A B C B C A AC B 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 Table 2.3-1, p.85 Symbol Operation Form Examples + Scalar-array addition A + b [6,3]+2=[8,5] - Scalar-array subtraction A – b + Array addition A + B [6,5]+[4,8]=[10,13] - Array subtraction A – B [6,5]-[4,8]=[2,-3] .* Array multiplication A.*B [3,5].*[4,8]=[12,40] ./ Array right division A./B [2,5]./[4,8]=[2/4,5/8] .\ Array left division A.\B [2,5].\[4,8]=[2\4,5\8] .^ Array exponentiation A.^B [3,5].^2=[3^2,5^2] [8,3]-5=[3,-2] 2.^[3,5]=[2^3,2^5] [3,5].^[2,4]=[3^2,5^4] Array multiplication Vectors must be of the same size x. * y [ x(1) y(1), x(2) y(2),..., x(n) y(n)] Matrices must be of the same size c a b ij ij ij Dot (.) and asterisk (*) form one symbol 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 vectorized functions 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 z exp( y). * sin( x). * (cos( x)).^2 Section 2.4 MATRIX OPERATIONS Addition & Subtraction Matrix addition and subtraction are identical to element-by-element addition and subtracted Vector Multiplication Vector dot product u w u w cos( ) u w u w u w 1 1 2 2 3 3 Recall result is a scalar iˆ iˆ ˆj ˆj kˆ kˆ 1 iˆ ˆj iˆ kˆ ˆj kˆ 0 u 1 w u w u w u w u w w 1 u 2 3 2 3 1 1 2 2 3 3 Vector-Matrix Multiplication Matrix multiplied by column vector a a 11 21 a x a x a x a x a x a x 12 1 11 1 12 2 22 2 21 1 22 2 Result is a column vector Number of columns in matrix must equal number of rows in vector Matrix multiplication To multiply two matrices A & B Number of columns in A must equal number of rows in The resulting product AB has number of rows as A Same number of columns as B Same 6 10 4 –2 3 7 9 –5 8 12 = (6)(9) + (– 2)(– 5) (10)(9) + (3)(– 5) (4)(9) + (7)(– 5) = 64 75 1 24 116 116 (6)(8) + (– 2)(12) (10)(8) + (3)(12) (4)(8) + (7)(12) Matrix multiplication Matrix multiplication is NOT commutative The order of the matrices in the equation is important A B B A Exceptions matrix 0 Identity matrix I Null 0 A A0 0 IA AI A Section 2.5 POLYNOMIAL OPERATIONS USING ARRAYS 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 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 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.