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MATLAB

MATLAB. Lecture Two (Part II) Thursday/Friday, 3 - 4 July 2003. Chapter 5. Applications. Linear Algebra. Solving a linear system 5x = 3y - 2z + 10 8y + 4z = 3x +20 2x + 4y - 9z = 9 Cast in standard matrix form A x = b. Linear Equations. A = [5 -3 2; -3 8 4; 2 4 -9];

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MATLAB

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  1. MATLAB Lecture Two (Part II) Thursday/Friday, 3 - 4 July 2003

  2. Chapter 5 Applications

  3. Linear Algebra • Solving a linear system 5x = 3y - 2z + 10 8y + 4z = 3x +20 2x + 4y - 9z = 9 • Cast in standard matrix form A x = b

  4. Linear Equations A = [5 -3 2; -3 8 4; 2 4 -9]; b = [10; 20; 9]; x = A \ b • Check solution c = A*x • c should be equal to b.

  5. Gaussian Elimination C = [ A b]; % augmented matrix • row reduced echelon form Cr = rref(C);

  6. Eigenvalues and Eigenvectors • The problem A v =  v • With pencil-paper, we must solve for det (A - ) = 0 then solve the linear equation • MATLAB way [V, D] = eig(A)

  7. Matrix Factorizations • LU decomposition [L, U] = lu(A); such that L*U = A, L is a lower triangular matrix, U is an upper triangular matrix.

  8. Matrix Factorization • QR factorization [Q, R] = qr(A); such that Q*R = A; Q is orthogonal matrix and R is upper triangular matrix.

  9. Matrix factorization • Singular Value Decomposition [U, D, V] = svd(A); such that UDV = A, where U and V are orthogonal and D is diagonal matrix.

  10. Sparse Matrix • See help sparfun for detail

  11. Curve Fitting • Polynomial curve fitting a=polyfit(x,y,n) do a least-square fit with polynomial of degree n. x and y are data vectors, and a is coefficients of the polynomial, a = [an, an-1, …, a1, a0]

  12. Curve Fitting • y = polyval(a, x) • compute the value of polynomial at value x y = a(1) xn + a(2) xn-1 + … + a(n) x + a(n+1) • x can be a vector

  13. Example-1: Straight-line fit: • Problem: Fit the set of data, and make a plot of it. • Step-1: find the coefficients • Step-2: Evaluate y at finer scale • Step-3: Plot and see

  14. Interpolation • Find a curve that pass through all the points ynew = interp1(x,y,xnew,method) • method is a string. Possible choices are 'nearest', 'linear', 'cubic', or 'spline'.

  15. Data Analysis and Statistics • mean average value • median half way • std standard deviation • max largest value • min smallest value • sum sum total • prod product

  16. Numerical Integration • Quad Adaptive Simpson's rule • Quad8 Adaptive Newton-Cotes • Use quad('your_func', a, b); or quad('your_func',a, b, opt…)

  17. Ordinary Differential Equations • General first-order ODE dx/dt = f(x, t) where x and f are vectors • MATLAB ODE solvers ode23 2nd/3rd Runge-Kutta ode45 4/5th Runge-Kutta

  18. Examples of ODE • Solve the first order differential equation dx/dt = x + t with the initial condition x(0)=0.

  19. dx/dt = x + t, Example function xdot = simpode(t,x) % SIMPODE: computes % xdot = x + t. xdot = x + t;

  20. dx/dt = x + t example tspan = [0 2]; x0; [t, x] = ode23('simpode', tspan, x0); plot(t, x) xlabel('t'), ylabel('x')

  21. Second Order Equations • Nonlinear pendulum d2 /dt2 + 2 sin = 0 • Convert into set of first-order ODE with z1 = , z2 = d/dt, dz1/dt = z2, dz2/dt = - 2 sin(z1)

  22. Pendulum Function File Function zdot = pend(t, z) %Inputs : t = time % z = [z(1); z(2)] %Outputs: zdot = [z(2); -w^2 sin (z1)] wsq = 1.56; % omega value zdot = [z(2); - wsq*sin(z(1))];

  23. Nonlinear Algebraic Equations • Finding zeros of equation f(x) = 0 • MATLAB function x_sol = fzero('your_func', x0, tol, trace); where tol and trace are optional arguments

  24. Examples of Algebraic Equations • Solve transcendental equation sin x = exp(x) - 5 • Define function f(x) = sin(x) - exp(x) + 5

  25. Chapter 6 Graphics

  26. Simple Plotting • plot(x,y,'r') • plot(x,y,':', x2,y2, '+') • plot(x,y,'b--') • See Table on Chapter 6 for style options.

  27. Other Plotting Commands • xlabel, ylabel labels on axis • title title text • text text in graphics • legend legend • axis axis limits/scale • gtext text with local • located by mouse

  28. Specialized 2D Plots • semilogx log x vs y • semilogy x vs log y • loglog log x vs log y • polar polar plot • bar bar chart • hist histogram • pie pie plot

  29. 3D Plots • plot3(x, y, z, 'style-option') • Example: t=linspace(0, 6*pi, 100); x=cos(t); y=sin(t); z = t; plot3(x,y,z)

  30. Advanced Features • "Handle graphics" gives a better control of graphic output. It is a lower level graphics.

  31. Chapter 8, What Else is There? • Symbolic Math and other Toolboxes • External Interface: Mex-files • Graphics User Interface

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