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[email protected] • ENGR-25_Arrays-2.ppt1
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engr/Math/Physics 25
Chp2 MATLABArrays: Part-2
[email protected] • ENGR-25_Arrays-2.ppt2
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Learning Goals
Learn to Construct 1D Row and Column Vectors
Create MULTI-Dimensional ARRAYS and MATRICES
Perform Arithmetic Operations on Vectors and Arrays/Matrices
Analyze Polynomial Functions
[email protected] • ENGR-25_Arrays-2.ppt3
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Last Time Vector/Array Scalar
Multiplication by Term-by-Term Scaling
Vector/Array Addition & Subtraction by Term-by-Term Operations• “Tip-toTail” Geometry
>> r = [ 7 11 19];>> v = 2*rv = 14 22 38 vrw
[email protected] • ENGR-25_Arrays-2.ppt4
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Scalar-Array Multiplication
Multiplying an Array B by a scalar w produces an Array whose elements are the elements of B multiplied by w.
8.514.44
6.293.33
1412
897.3
>> B = [9,8;-12,14];>> 3.7*Bans = 33.3000 29.6000 -44.4000 51.8000
Using MATLAB
[email protected] • ENGR-25_Arrays-2.ppt5
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array Multiplication
Multiplication of TWO ARRAYS is not nearly as straightforward as Scalar-Array Mult.
MATLAB uses TWO definitions for Array-Array multiplication:1. ARRAY Multiplication
– also called element-by-element multiplication
2. MATRIX Multiplication (see also MTH6)
DIVISION and EXPONENTIATION must also be CAREFULLY defined when dealing with operations between two arrays.
[email protected] • ENGR-25_Arrays-2.ppt6
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Element-by-Element OperationsSymbol Operation Form Example
+ Scalar-array addition A + b [6,3]+2=[8,5]
- Scalar-array subtraction A – b [8,3]-5=[3,-2]
+ 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]= [0.5,0.625]
.\ Array left division A.\B [2,5].\[4,8]=[2\4,5\8]= [2,1.6]
.^ Array exponentiation A.^B[3,5].^2=[3^2,5^2]2.^[3,5]=[2^3,2^5]
[3,5].^[2,4]=[3^2,5^4]
[email protected] • ENGR-25_Arrays-2.ppt7
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array Operations
Array or Element-by-Element multiplication is defined ONLY for arrays having the SAME size. The definition of the product x.*y, where x and y each have n×n elements:
if x and y are row vectors. For example, if
x.*y = [x(1)y(1), x(2)y(2), ... , x(n)y(n)]
x = [2, 4, – 5], y = [– 7, 3, – 8]
then z = x.*y gives
40,12,1485,34,72 z
[email protected] • ENGR-25_Arrays-2.ppt8
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array Operations cont
If u and v are column vectors, the result of u.*v is a column vector. The Transpose operation z = (x’).*(y’) yields
Note that x’ is a column vector with size 3 × 1 and thus does not have the same size as y, whose size is 1 × 3
40
12
14
85
34
72
z
Thus for the vectors x and y the operations x’.*y and y.*x’ are NOT DEFINED in MATLAB and will generate an error message.
[email protected] • ENGR-25_Arrays-2.ppt9
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array Operations cont
The array operations are performed between the elements in corresponding locations in the arrays. For example, the array multiplication operation A.*B results in an array C that has the same size as A and B and has the elements cij = aij bij . For example, if
Then C = A.*B Yields
26
87
49
511BA
854
4077
2469
85711C
[email protected] • ENGR-25_Arrays-2.ppt10
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array Operations cont
The built-in MATLAB functions such as sqrt(x) and exp(x) automatically operate on array arguments to produce an array result of the same size as the array argument x• Thus these functions are said to be
VECTORIZED
Some Examples
>> r = [7 11 19];>> h = sqrt(r)h = 2.6458 3.3166 4.3589
>> u = [1,2,3];>> f = exp(u)f = 2.7183 7.3891 20.0855
[email protected] • ENGR-25_Arrays-2.ppt11
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array Operations cont
However, when multiplying or dividing these functions, or when raising them to a power, we must use element-by-element dot (.) operations if the arguments are arrays.
To Calc: z = (eu sinr)•cos2r, enter command
>> z = exp(u).*sin(r).*(cos(r)).^2z = 1.0150 -0.0001 2.9427
MATLAB returns an error message if the size of r is not the same as the size of u. The result z will have the same size as r and u.
[email protected] • ENGR-25_Arrays-2.ppt12
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array DIVISION
The definition of array division is similar to the definition of array multiplication except that the elements of one array are divided by the elements of the other array. Both arrays must have the same size. The symbol for array right division is ./
Recall r = [ 7 11 19] u = [1,2,3] then z = r./u gives
333.65.57
31921117
z
>> z = r./uz = 7.0000 5.5000 6.3333
[email protected] • ENGR-25_Arrays-2.ppt13
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array-Array DIVISION cont.
Consider
>> A./Bans = -6 4 -3 2
Taking C = A./B yields
23
54
49
2024BA
23
46
2439
520424C
A = 24 20 -9 4B = -4 5 3 2
[email protected] • ENGR-25_Arrays-2.ppt14
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array EXPONENTIATION
MATLAB enables us not only to raise arrays to powers but also to raise scalars and arrays to ARRAY powers.
Use the .^ symbol to perform exponentiation on an element-by-element basis• if x = [3, 5, 8], then typing x.^3 produces
the array [33, 53, 83] = [27, 125, 512]
We can also raise a scalar to an array power. For example, if p = [2, 4, 5], then typing 3.^p produces the array [32, 34, 35] = [9, 81, 243].
[email protected] • ENGR-25_Arrays-2.ppt15
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Array to Array Power>> A = [5 6 7; 8 9 8; 7 6 5]A = 5 6 7 8 9 8 7 6 5
>> B = [-4 -3 -2; -1 0 1; 2 3 4]B = -4 -3 -2 -1 0 1 2 3 4
>> C = A.^BC = 0.0016 0.0046 0.0204 0.1250 1.0000 8.0000 49.0000 216.0000 625.0000
[email protected] • ENGR-25_Arrays-2.ppt16
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Matrix-Matrix Multiplication
Multiplication of MATRICES requires meeting the CONFORMABILITY condition
The conformability condition for multiplication is that the COLUMN dimensions (k x m) of the LEAD matrix A must be EQUAL to the ROW dimension of the LAG matrix B (m x n)
If
125
89
74
310
26
BA Then
Error BACCAB 3x2 is
[email protected] • ENGR-25_Arrays-2.ppt17
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Matrix-Mult. Mechanics
Multiplication of A (k x m) and B (m x n) CONFORMABLE Matrices produces a Product Matrix C with Dimensions (k x n)
The elements of C are the sum of the products of like-index Row Elements from A, and Column Elements from B; to whit
1161
11675
2464
127845794
12381053910
122865296
125
89
74
310
26
C
[email protected] • ENGR-25_Arrays-2.ppt18
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Matrix-Vector Multiplication
A Vector and Matrix May be Multiplied if they meet the Conformability Critera: (1xm)*(mxp) or (kxm)*(mx1)• Given Vector-a, Matrix-B, and aB; Find
Dims for all
c
bbb
bbbaaaB
232221
131211
1211
231213112212121121121111131211 babababababaccc
Then the Dims: a(1x2), B(2x3), c(1x3)
[email protected] • ENGR-25_Arrays-2.ppt19
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Summation Notation Digression
Greek letter sigma (Σ, for sum) is another convenient way of handling several terms or variables – The Definition
For the previous example
qqq
q
pp xxxxxxxsum
1232
11
231213112212121121121111131211 babababababaccc
2
111
pppba
2
121
pppba
2
131
pppba
[email protected] • ENGR-25_Arrays-2.ppt20
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Matrix Mult by Σ-Notation
In General the product of Conformable Matrices A & B when
Then Any Element, cij, of Matrix C for• i = 1 to k (no. Rows) j = 1 to n (no. Cols)
knkk
n
n
ccc
ccc
ccc
21
22221
11211
CAB
mp
ppjipij bac
1
e.g.;
7
13553
p
pppbac
nmmk BA
[email protected] • ENGR-25_Arrays-2.ppt21
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Matrix Mult Example
>> A = [3 1.7 -7; 8.1 -0.31 4.6; -1.2 2.3 0.73; 4 -.32 8; 7.7 9.9 -0.17]
A =
3.0000 1.7000 -7.0000 8.1000 -0.3100 4.6000 -1.2000 2.3000 0.7300 4.0000 -0.3200 8.0000 7.7000 9.9000 -0.1700
A is Then 5x3
[email protected] • ENGR-25_Arrays-2.ppt22
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Matrix Mult Example cont
>> B = [0.67 -7.6; 4.4 .11; -7 -13]
B =
0.6700 -7.6000 4.4000 0.1100 -7.0000 -13.0000
B is Then 3x2
>> C = A*B
C =
58.4900 68.3870 -28.1370 -121.3941 4.2060 -0.1170 -54.7280 -134.4352 49.9090 -55.2210
Result, C, is Then 5x2
[email protected] • ENGR-25_Arrays-2.ppt23
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Matrix-Mult NOT Commutative
Matrix multiplication is generally not commutative. That is, AB ≠ BA even if BA is conformable• Consider an
Illustrative ExampleBA
76
10&
43
21
2524
1312
74136403
72116201AB
4027
43
47263716
41203110BA
[email protected] • ENGR-25_Arrays-2.ppt24
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Commutation Exceptions
Two EXCEPTIONS to the NONcommutative property are the NULL or ZERO matrix, denoted by 0 and the IDENTITY, or UNITY, matrix, denoted by I.• The NULL matrix contains all ZEROS and is
NOT the same as the EMPTY matrix [ ], which has NO elements.
Commutation of the Null & Identity Matrices
AAIIA0A00A Strictly speaking 0 & I are always SQUARE
[email protected] • ENGR-25_Arrays-2.ppt25
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Identity and Null Matrices
Identity Matrix is a square matrix and also it is a diagonal matrix with 1’s along the diagonal• similar to scalar “1” Null Matrix is one in
which all elements are 0• similar to scalar “0”
etc.
100
010
001
.or 10
01
000
000
000
Both are “idempotent” Matrices: for A = 0 or I → 32
and
AAA
AA
T
[email protected] • ENGR-25_Arrays-2.ppt26
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
eye and zeros
Use the eye(n) command to Form an nxn Identity Matrix
>> I = eye(5)I = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
Use the zeros(mxn) to Form an mxn 0-Filled Matrix• Strictly Speaking a
NULL Matrix is SQUARE
>> Z24 = zeros(2,4)
Z24 =
0 0 0 0 0 0 0 0
[email protected] • ENGR-25_Arrays-2.ppt27
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
PolyNomial Mult & Div Function conv(a,b) computes the
product of the two polynomials described by the coefficient arrays a and b. The two polynomials need not be the same degree. The result is the coefficient array of the product polynomial.
function [q,r] = deconv(num,den) produces the result of dividing a numerator polynomial, whose coefficient array is num, by a denominator polynomial represented by the coefficient array den. The quotient polynomial is given by the coefficient array q, and the remainder polynomial is given by the coefficient array r.
[email protected] • ENGR-25_Arrays-2.ppt28
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
PolyNomial Mult Example
Find the PRODUCT for
35136972 223 xxxgxxxxf
>> f = [2 -7 9 -6];>> g = [13,-5,3];>> prod = conv(f,g)prod = 26 -101 158 -144 57 -18
185714415810126 2345 xxxxxxprod
[email protected] • ENGR-25_Arrays-2.ppt29
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
PolyNomial Quotient Example
Find the QUOTIENT 3513
69722
23
xx
xxxquot
>> f = [2 -7 9 -6];>> g = [13,-5,3];>> [quot1,rem1] = deconv(f,g)quot1 = 0.1538 -0.4793rem1 = 0 0.0000 6.1420 -4.5621
5621.4142.6 rem4973.01538.0 xxxquot
[email protected] • ENGR-25_Arrays-2.ppt30
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
PolyNomial Roots
The function roots(h) computes the roots of a polynomial specified by the coefficient array h. The result is a column vector that contains the polynomial’s roots.>> r = roots([2, 14, 20])r = -5 -2>> rf = roots(f)rf = 2.0000 0.7500 + 0.9682i 0.7500 - 0.9682i
25020142 2 xxxx
>> rg = roots(g)rg = 0.1923 + 0.4402i 0.1923 - 0.4402i
[email protected] • ENGR-25_Arrays-2.ppt31
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Plotting PolyNomials
The 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.
Plot over (−4 ≤ x ≤ 7) the function 6972 23 xxxxf
[email protected] • ENGR-25_Arrays-2.ppt32
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
The Demo Plot
[email protected] • ENGR-25_Arrays-2.ppt33
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
All Done for Today
Not Covered in Chapter 2• §2.6 = Cell Arrays• §2.7 = Structure
Arrays
Cell-Arrays&
StructureArrays
[email protected] • ENGR-25_Arrays-2.ppt34
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engr/Math/Physics 25
Appendix
[email protected] • ENGR-25_Arrays-2.ppt35
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Chp2 Demox = linspace(-4,7,20);p3 = [2 -7 9 -6];y = polyval(p3,x);plot(x,y, x,y, 'o', 'LineWidth', 2), grid, xlabel('x'),...ylabel('y = f(x)'), title('f(x) = 2x^3 - 7x^2 + 9x - 6')
-4 -2 0 2 4 6 8-300
-200
-100
0
100
200
300
400
500
x
y =
f(x
)
f(x) = 2x3 - 7x2 + 9x - 6
[email protected] • ENGR-25_Arrays-2.ppt36
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Chp2 Demo
>> f = [2 -7 9 -6];>> x = [-4:0.02:7];>> fx = polyval(f,x);>> plot(x,fx),xlabel('x'),ylabel('f(x)'), title('chp2 Demo'), grid