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1.3 Arrays, Files, and Plots + Fourier series

1.3 Arrays, Files, and Plots + Fourier series. By Mr. Q. Arrays. A set of numbers in a specific order or pattern 1 x n dimensional matrix In Matlab , you must always define arrays: To input y=f(x) you must Define x

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1.3 Arrays, Files, and Plots + Fourier series

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  1. 1.3 Arrays, Files, and Plots + Fourier series By Mr. Q

  2. Arrays • A set of numbers in a specific order or pattern • 1 x n dimensional matrix • In Matlab, you must always define arrays: • To input y=f(x) you must • Define x • Define y as a function of x using the hundreds of functions available in Matlab

  3. Arrays • Define y=sin(x) on the interval [0, 2π] • Is it continuous? • How accurate should you have it? • How ‘big’ is the array(s)? • What is the 10th value?

  4. Files • 6x3 + 5x2-20x+40=0 • e2 • ln(20) • log 9 • cos () • cos(48o) • Find zeros of a polynomial • ex • ln(x) • cos x (radians) • cos x (degrees) • (radians) • (degrees)

  5. Plot • Plot the function w=e-0.3x+3x on [0,5] y= Graph displays Force x is in meters w,y is in newtons Label line ‘w’ and ‘y’ Approximate their intersection Add a grid • Must define independent and dependent variables • Plot(independent, dependent,…, ‘+’) • title (‘ ‘) • xlabel (‘ ‘) • ylabel(‘ ‘) • gtext(‘ ‘) • [x,y] ginput( n ) • grid

  6. Matrices • Solve using Matlab • 2x+2y+3z=10 • 4x-y+z=-5 • 5x-2y+6z=1 • Combination of arrays • n x m dimensions • Constructed using a semi colon between rows • Ax=b • x=?

  7. FOURIER sERIES • Breaks down repeating, step, or periodic functions into a sum of sine and cosine • Used: • Originally to solve heat equation • Differential Equations-Eigensolutions • Electrical Engineering • Vibrational Analysis • Signal Processing, etc.

  8. Fourier series • If • Find: • a0, a1, a2, a3 • b1, b2, b3, b4 • to approximate f(x) in a Fourier Series • Given f(x) where xє(-π,π) • Then f(x) can be approximated a.e. by: • f(x) ≈ • Where

  9. P1.23 FT approximation • Consider the Step Function • The Fourier Transform for the above Function • Taking the First FOUR Terms of the Infinite Sum

  10. Graphing the function • OR Solution 2 • x0=[-pi,-1e-6,1e-6,pi] • f0=[-1,-1,1,1] • How can we plot f(x)? • Solution 1 • x1=[-pi,0] • f1= [-1,-1] • x2=[0,pi] • f2=[1,1]

  11. Plot the function • Solution 2 • plot(x,f);grid;title(‘f(x)’);xlabel(‘x’); • Solution 1 • plot(x1,f1,x2,f2);grid; • title(‘f(x)’);xlablel(‘x’)

  12. Graphing the approximation • Solution 2: • ftot=zeros(1,length(x) • for k=1:2:7; • fc=sin(k*x)/k; • ftot=fc+ftot; • end • How can we graph? • Solution 1 • x=-pi:0.01:pi; • f=4/pi(sin(x)/1+sin(3*x)/3+sin(5*x)/5+sin(7*x)/7);

  13. Can we get a better approximation? • function fourier(n) • x=-pi:0.01:pi; • f1=[-1,-1,1,1]; • x1=[-pi,-1e-6,1e-6,pi]; • ftot=zeros(1,length(x)); • for k=1:2:n; • fc=sin(k*x)/k; • ftot=ftot+fc; • end • f=4/pi*ftot; • plot(x,f,x1,f1) • Let’s make a program for solution 2 and plot all on same axis!

  14. The heat equation • Q=m c Δt • Ut=αUxx (Diff. Eq) • U(x,t)= • Where • And f(x)=initial temperature distribution at t=0

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