1.3 ARRAYS, FILES, AND PLOTS + FOURIER SERIES BY MR. Q.

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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  Define y as a function of x using the hundreds of functions available in Matlab

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 10 th value?

FILES

Must define independent and dependent variables Plot(independent, dependent,…, ‘+’) title (‘ ‘) xlabel (‘ ‘) ylabel(‘ ‘) gtext(‘ ‘) [x,y] ginput( n ) grid PLOT

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

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

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

How can we plot f(x)? Solution 1 x1=[-pi,0] f1= [-1,-1] x2=[0,pi] f2=[1,1] OR Solution 2 x0=[-pi,-1e-6,1e-6,pi] f0=[-1,-1,1,1] GRAPHING THE FUNCTION

Solution 1 plot(x1,f1,x2,f2);grid; title(‘f(x)’);xlablel(‘x’) Solution 2 plot(x,f);grid;title(‘f(x )’);xlabel(‘x’); PLOT THE FUNCTION

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); Solution 2: ftot=zeros(1,length(x) for k=1:2:7; fc=sin(k*x)/k; ftot=fc+ftot; end GRAPHING THE APPROXIMATION

Let’s make a program for solution 2 and plot all on same axis! 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) CAN WE GET A BETTER APPROXIMATION?

THE HEAT EQUATION

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