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Functions

Functions. ผศ.ดร.อนันต์ ผลเพิ่ม Anan Phonphoem http://www.cpe.ku.ac.th/~anan anan@cpe.ku.ac.th. Overview. Polynomial review Functions Built-in functions User-defined functions. Order = 2. Order = 3. Polynomial. f(x) = a 1 x n + a 2 x n-1 + a 3 x n-2 + … + a n x + a n+1. n.

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Functions

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  1. Functions ผศ.ดร.อนันต์ ผลเพิ่ม Anan Phonphoem http://www.cpe.ku.ac.th/~anan anan@cpe.ku.ac.th

  2. Overview • Polynomial review • Functions • Built-in functions • User-defined functions

  3. Order = 2 Order = 3 Polynomial f(x) = a1xn + a2xn-1 + a3xn-2 + …+ anx + an+1 n Degree = Order = Example: y = 3x2 + 4 y = 12x3 + 2x2 + 1

  4. 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 ]

  5. y = 5x4 + 6x3 + 3x2 + 2 b = 4a3– 6a2 T = x3 + x2 + x + 1 Polynomial Coefficient [ 5 6 3 0 2 ] [ 4 -6 0 0 ] [ 1 1 1 1 ]

  6. Roots of y 1 , 2 Roots of polynomial a = [ 1 -3 2] y = x2– 3x+ 2 = (x – 1) (x– 2) c = roots(a) = [ 1 2 ] T = roots( [ 1 -3 2 ] )

  7. Polynomial of roots Roots of y = 1 , 2 >>r = [ 1 2 ]; >>poly(r) ans = 1 -3 2 (x – 1) (x– 2) >>P = poly([1 2]) y = x2– 3x+ 2

  8. roots ( [1 –3 2 ] ) poly ( [1 2 ] ) Poly ( ) roots ( ) y = x2– 3x+ 2 [ 1 2 ] [ 1 –3 2 ] (x – 1)(x – 2)

  9. conv( [1 -3 2], [1 3 -10] ) = Polynomial multiplication f(x) = x2– 3x + 2 g(x) = x2 + 3x – 10 f(x) g(x) = x4– 17x2 + 36x – 20 [ 1 0 -17 36 -20 ] >>a=[1 -3 2]; >>b=[1 3 -10]; >>conv(a,b) ans = [1 0 -17 36 -20]

  10. quotient remainder 17 2 = 5 + 3 3 Polymonial division

  11. f(x) = x3– 4x2 + 2 g(x) = x2 + 3x – 10 x3– 4x2 + 2 f(x) g(x) = x2 + 3x – 10 31x – 68 = (x – 7) + x2 + 3x – 10 Polynomial division >>a=[1 -4 0 2]; >>b=[1 3 -10]; >>deconv(a,b) ans = [1 -7] >>[S,T] = deconv(a,b) S = 1 -7 T = 0 0 31 -68

  12. Plotting -2 ≤ x ≤ 2 >>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)’)

  13. Black Box ? Output Input Some Mechanics Functions

  14. 100 16 x 10 4 y What function is it? ? y = f(x) y = sqrt(x) Square root

  15. x = 4 y = 3 x = 1 y = 2 x , y z = 2.2 z = 5 z z = √(x2+y2) What function is it? ? z = f(x,y) • Multiple inputs • Multiple outputs • Complicate

  16. Functions • Type of functions • Built-in functions (predefined in MATLAB) • User-defined functions(create your own function) Built-in functions

  17. 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)

  18. Imaginary Axis x Real Axis 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

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

  20. 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

  21. Functions • Type of functions • Built-in functions (predefined in MATLAB) • User-defined functions(create your own function) User-defined functions

  22. x x User-defined functions Write a function to find area of the field Area = x * x If we want to install a fence around the field Write a function to find the length of the fence L = 4 * x

  23. >>a a = [ 1 10 ] >>b b = [ 2 5 ] >>a+b ans = [ 3 15 ] %Example of program %Program Test1.m a = [ 1 10]; b = [ 2 5]; c = a + b >>Test1 c = [ 3 15 ] Interactive mode (Calculator) Running a script file (Program) Operation modes in MATLAB

  24. Program file • Program file (M-files) “.m” • Two types of M-Files • Script file • Function file

  25. Function file format • First line must begin with a function definition function [outputs] = function_name(inputs) • Function name should be the same as .m file function [outputs] = Test1(inputs) Should be save as “Test1.m” • MATLAB is case sensitive !

  26. x x User-defined function Area.m %This is function to find area of a field function y = Area(x) %Area of square box is x*x y = x * x >>Area(3) y = 9 ans = 9 >>result = Area(2) y = 4 result = 4

  27. x x User-defined function length.m %This is function to find the length of the fence function L = length(x) %Length of the fence is 4*x L = 4 * x >>length(4) L = 16 ans = 16 >>result = length(5) L = 20 result = 20

  28. User-defined function x y length2.m %This is function to find the length of the fence function L2 = length2(x,y) %Length of the fence is 2x+2y L2 = 2.*x + 2.*y >>x=[4 10];>>y=[15 20]; >>length2(x,y) L2 = 38 60 ans = 38 60 >>result = length(4,5) L2 = 18 result = 18

  29. User-defined function r h volume.m %This is function to find the volume function v = volume(r,h) %Length of the fence is pi*r*r*h area = pi .* (r.^2); v = area .* h >>r = 3; >>h = 5; >>z = volume(r,h) z = 141.3717 >>r = 3; >>h = 5; >>z = volume(h,r) z = 235.6194 >>h = 3; >>r = 5; >>z = volume(r,h) z = 235.6194

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