1 Functions ผศ. ดร. อนันต์ ผลเพิ่ม Anan Phonphoem anan [email protected].

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Functions

ผศ.ดร.อนั�นัต์ ผลเพิ่��มAnan Phonphoem

http://www.cpe.ku.ac.th/[email protected]

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Overview

Polynomial review Functions

Built-in functions User-defined functions

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Polynomial

Example:

f(x) = a1xn + a2xn-1 + a3xn-2 + …+ anx + an+1

Degree = Order =

n

y = 3x2 + 4 Order = 2

y = 12x3 + 2x2 + 1 Order = 3

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Polynomial Coefficient

f(x) = a1xn + a2xn-1 + a3xn-2 + …+ anx + an+1

[ a1 a2 a3 … an-1 an an+1 ]

y = 12x3 + 2x2 + 1

[ 12 2 0 1 ]

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Polynomial Coefficient

[ 5 6 3 0 2 ]

[ 4 -6 0 0 ]

[ 1 1 1 1 ]

y = 5x4 + 6x3 + 3x2 + 2

b = 4a3 – 6a2

T = x3 + x2 + x + 1

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Roots of polynomial

y = x2 – 3x + 2

= (x – 1) (x – 2)

Roots of y 1 , 2

a = [ 1 -3 2]

c = roots(a)

= [ 1 2 ]

T = roots( [ 1 -3 2 ] )

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Polynomial of roots

Roots of y = 1 , 2

y = x2 – 3x + 2

(x – 1) (x – 2)

>>r = [ 1 2 ];

>>poly(r)ans = 1 -3 2

>>P = poly([1 2])

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Poly ( ) roots ( )

y = x2 – 3x + 2 roots ( [1 –3 2 ] ) [ 1 2 ]

[ 1 –3 2 ] (x – 1)(x – 2)poly ( [1 2 ] )

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Polynomial multiplicationf(x) = x2 – 3x + 2

g(x) = x2 + 3x – 10

f(x) g(x) = x4 – 17x2 + 36x – 20

conv( [1 -3 2], [1 3 -10] ) = [ 1 0 -17 36 -20 ]

>>a=[1 -3 2];>>b=[1 3 -10];>>conv(a,b)ans = [1 0 -17 36 -20]

>>a=[1 -3 2];>>b=[1 3 -10];>>conv(a,b)ans = [1 0 -17 36 -20]

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Polymonial division

173 = 5 + 2

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remainderquotient

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Polynomial division

f(x) = x3 – 4x2 + 2 g(x) = x2 + 3x – 10

f(x) g(x)

= x3 – 4x2 + 2

x2 + 3x – 10

= (x – 7) + 31x – 68

x2 + 3x – 10

>>a=[1 -4 0 2];>>b=[1 3 -10];>>deconv(a,b)ans = [1 -7]

>>a=[1 -4 0 2];>>b=[1 3 -10];>>deconv(a,b)ans = [1 -7]

>>[S,T] = deconv(a,b)S = 1 -7T = 0 0 31 -68

>>[S,T] = deconv(a,b)S = 1 -7T = 0 0 31 -68

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Plotting

24 16)( xxxr

>>ar=[1 0 -16 0 0];>>x=[-2:0.2:2];>>R=polyval(ar,x);>>plot(x,R);>>xlable(‘x’)>>ylabel(‘r(x)’)>>title(‘Plotting r(x)’)

>>ar=[1 0 -16 0 0];>>x=[-2:0.2:2];>>R=polyval(ar,x);>>plot(x,R);>>xlable(‘x’)>>ylabel(‘r(x)’)>>title(‘Plotting r(x)’)

-2 ≤ x ≤ 2

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Functions

Input ?

Black Box

Some Mechanics

Output

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x

What function is it?

?Square root

y = f(x)y = sqrt(x) y100 1016 4

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x , y

What function is it?

?z = f(x,y) zx = 4y = 3

z = 5x = 1y = 2

z = 2.2z = √(x2+y2)

-Multiple inputs-Multiple outputs-Complicate

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Functions

Type of functions Built-in functions (predefined in MATLAB) User-defined functions (create your own

function)

Built-in functions

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Built-in functions (I)

Exponential functions exp(x) = Exponential = ex

sqrt(x) = Squart root = √x Logarithmic functions

log(x) = Natural log = ln x log10(x) = Based 10 log = log10(x)

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Built-in functions (II) Complex number functions

abs(x) = Absolute x = |x| imag(x) = imaginary part of x real(x) = real part of x angle(x) =angle of x x = a + ib

Imaginary Axis

Real Axis

x

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Built-in functions (III) Numeric functions

ceil(x) = Round toward fix(x) = Round toward 0 round(x) = Round toward nearest integer

Floor(x) = Round toward nearest –

>>x = [ 3.45 1.98 -2.16 ] >>ceil(x)ans = 4 2 -2 >>fix(x)ans = 3 1 -2

>>x = [ 3.45 1.98 -2.16 ] >>ceil(x)ans = 4 2 -2 >>fix(x)ans = 3 1 -2

>>round(x)ans = 3 2 -2>>floor(x)ans = 3 1 -3

>>round(x)ans = 3 2 -2>>floor(x)ans = 3 1 -3

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Built-in functions (IV)

Trigonometric functions sin(x) cos(x)

Inverse trigonometric acos(x) = arccos x = cos –1 x atan(x) = arctan x = tan –1 x

Hyperbolic functions cosh(x)= Hyperbolic cosine = cosh x =

(ex+e–x)/2

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Functions

Type of functions Built-in functions (predefined in MATLAB) User-defined functions (create your own

function)User-defined functions

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User-defined functions

x

x

Write a function to find area of the fieldArea = x * x

If we want to install a fence around the fieldWrite a function to find the length of the fenceL = 4 * x

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Operation modes in MATLAB

>>aa = [ 1 10 ]>>bb = [ 2 5 ]>>a+bans = [ 3 15 ]

>>aa = [ 1 10 ]>>bb = [ 2 5 ]>>a+bans = [ 3 15 ]

Interactive mode (Calculator)

%Example of program%Program Test1.ma = [ 1 10];b = [ 2 5];c = a + b

%Example of program%Program Test1.ma = [ 1 10];b = [ 2 5];c = a + b

>>Test1c = [ 3 15 ]

>>Test1c = [ 3 15 ]

Running a script file (Program)

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Program file

Program file (M-files) “ .m ” Two types of M-Files

Script file Function file

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Function file format

First line must begin with a function definition

function [outputs] = function_name(inputs) Function name should be the same as .m

filefunction [outputs] = Test1(inputs)

Should be save as “Test1.m” MATLAB is case sensitive !

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User-defined function

x

x

%This is function to find area of a fieldfunction y = Area(x)%Area of square box is x*xy = x * x

%This is function to find area of a fieldfunction y = Area(x)%Area of square box is x*xy = x * x

Area.m

>>Area(3)y = 9ans = 9

>>Area(3)y = 9ans = 9

>>result = Area(2)y = 4result = 4

>>result = Area(2)y = 4result = 4

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User-defined function

x

x

%This is function to find the length of the fencefunction L = length(x)%Length of the fence is 4*xL = 4 * x

%This is function to find the length of the fencefunction L = length(x)%Length of the fence is 4*xL = 4 * x

length.m

>>length(4)L = 16ans = 16

>>length(4)L = 16ans = 16

>>result = length(5)L = 20result = 20

>>result = length(5)L = 20result = 20

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User-defined function

y

x

%This is function to find the length of the fencefunction L2 = length2(x,y)%Length of the fence is 2x+2yL2 = 2.*x + 2.*y

%This is function to find the length of the fencefunction L2 = length2(x,y)%Length of the fence is 2x+2yL2 = 2.*x + 2.*y

length2.m

>>x=[4 10];>>y=[15 20];>>length2(x,y)L2 = 38 60ans = 38 60

>>x=[4 10];>>y=[15 20];>>length2(x,y)L2 = 38 60ans = 38 60

>>result = length(4,5)L2 = 18result = 18

>>result = length(4,5)L2 = 18result = 18

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User-defined function

%This is function to find the volumefunction v = volume(r,h)%Length of the fence is pi*r*r*harea = pi .* (r.^2);v = area .* h

%This is function to find the volumefunction v = volume(r,h)%Length of the fence is pi*r*r*harea = pi .* (r.^2);v = area .* h

volume.m

>>r = 3;>>h = 5;>>z = volume(r,h)z = 141.3717

>>r = 3;>>h = 5;>>z = volume(r,h)z = 141.3717

r

h

>>r = 3;>>h = 5;>>z = volume(h,r)z = 235.6194

>>r = 3;>>h = 5;>>z = volume(h,r)z = 235.6194

>>h = 3;>>r = 5;>>z = volume(r,h)z = 235.6194

>>h = 3;>>r = 5;>>z = volume(r,h)z = 235.6194