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MATLAB Tutorials

MATLAB Tutorials. Session IV Mathematical Applications using MATLAB. Rajeev Madazhy Email: rmadaz1@lsu.edu Dept of Mechanical Engineering LSU. Department of Mechanical Engineering, LSU Session IV. Last Session…. Plotting in MATLAB 2-D Graphics 3-D Graphics

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MATLAB Tutorials

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  1. MATLAB Tutorials Session IV Mathematical Applications using MATLAB Rajeev Madazhy Email: rmadaz1@lsu.edu Dept of Mechanical Engineering LSU Department of Mechanical Engineering, LSU Session IV

  2. Last Session…. • Plotting in MATLAB • 2-D Graphics • 3-D Graphics • Mesh, surface and contour plots in 3-D plots • Exercises in Graphics

  3. Session IV Outline…. • Using fplot function • Minimization • Zero finding • Curve fitting • Interpolation • Integration

  4. Using fplot() function…. fplot(fun,[xmin, xmax]) plots the function specified by the string fun between the x-axis limits specified by [xmin xmax]. For example fplot(‘sin(x)’,[0 2*pi]) will plot sine of x with x ranging from 0 to 2. Try it.

  5. Using fplot() function…. You can also use fplot for the function defined by yourself. >>fplot(‘function1’,[0 8]) where function1 is a M-file function: fun=‘function1’; fplot(fun,[0 8]) function y=function1(x) y=2*exp(-x).*sin(x);

  6. Output….

  7. Minimization…. • In many situations, we want to find the function extremes, the local maximum (peaks) and minimum (valleys). • fmin(fun,x1,x2) can be used to find a local minimum of function f(x) defined by string fun in the interval x1 < x < x2. • In order to find the local maximum, function f(x) should be replaced by -f(x).

  8. Example….

  9. Zero Finding…. • We have learned to use roots to find the zeros of a polynomial. • fzero(fun,x0) can be used to find zero of a general function f(x) near x0. • If you want to solve the equation f(x)=c, just define a function g(x)=f(x)-c. Then find the zero of g(x).

  10. Example….

  11. Curve Fitting by Polynomial Perform curve fit of N point of data (xi, yi) (i=1,…N) by a n-th order polynomial For example, n=4, f(x)=a0+a1*x+a2*x^2+a3*x^3+a4*x^4 Find a0, …, a4 which minimize the sum of the squared error at the data points Solve linear equations to find a0, …, a

  12. Using polyfit function…. • In Matlab, the function polyfit(x, y, n) finds the coefficients of a polynomial p(x) of degree n that fits the data provided by array x and y in a least-squares sense. • The program on next slide illustrates the use of this function

  13. Poly_fit M-File….

  14. Output….

  15. More on Curve Fitting…. • If x and y can be modeled by a power function or exponential function in the form y=a*x^b • y=a*exp(b*x) Taking natural logarithms of both side will yield ln(y)=ln(a)+b*ln(x) • ln(y)=ln(a)+b*x Then we can use the first order polynomial to fit the data and therefore find a and b

  16. Interpolation…. • Interpolation is defined as a way of estimating values of a function between those given by some set of data points. • The simplest way of interpolation is called linear interpolation in which the intermediate values are calculated by the straight line between the entered points. • Matlab provides function interp1 for one dimensional interpolation.

  17. Using interp1 function…. • yi = interp1(x,y,xi,'method'), where array x and y provide the data points that are known, yi is the result of interpolation at xi. • Available methods are: • 'nearest' - nearest neighbor interpolation • 'linear' - linear interpolation • 'spline' - cubic spline interpolation • 'cubic' - cubic interpolation The default is linear interpolation.

  18. Example…. Known temperature every hour for twelve hours. Find the temperature at 9.3 and 11.7 using interp1

  19. M-File….

  20. Output in command window….

  21. Example 2….

  22. Output….

  23. Integration…. Many engineering problems require the calculation of the definite integral of a function bounded in the finite interval [a, b]. Numerically, we can approximate the definite integral by the weighted sum of a number of function values

  24. Using trapz, quad, quad8 functions…. • Matlab provides three function , trapz, quad, quad8, for calculation of integrals. • The function trapz(x, y) calculates the integral by summing the area of trapezoids formed from the data points defined by array x and y • The function quad(fun, a, b) or quadl(fun, a, b) calculate the integral based on the concept of quadrature.

  25. Matlab M-File….

  26. Output….

  27. Recap…. • Using fplot function • Minimization • Zero finding • Curve fitting • Interpolation • Integration

  28. Next Session…. Ordinary Differential Equations…. Examples using MATLAB to solve ODE’s…. Mention of DDE’s….

  29. Thank You

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