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Polynomials, Curve Fitting, and Interpolation

Polynomials, Curve Fitting, and Interpolation. Numerical Computing with . MATLAB for Scientists and Engineers. You will be able to. Use MATLAB to manipulate polynomials for calculating values, derivatives and for multiplications as well as divisions,

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Polynomials, Curve Fitting, and Interpolation

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  1. Polynomials, Curve Fitting, and Interpolation Numerical Computing with . MATLAB for Scientists and Engineers

  2. You will be able to • Use MATLAB to manipulate polynomials for calculating values, derivatives and for multiplications as well as divisions, • Findthe polynomials for curve fitting in a least square sense, • Interpolate intermediate values between sampled points.

  3. Polynomials • Functions of the form: • n is the degree, or the order of the polynomial. • Examples polynomial of degree 5 polynomial of degree 2 polynomial of degree 1

  4. Representing Polynomials • MATLAB represents a polynomial as a vector of coefficients of the x with the highest power first. p = [ 5 0 0 6 4 2]; d = [ 2 -4 20]; p = [ -4 12];

  5. Value of a Polynomial • Value of a polynomial p at a value of x • Examples y = polyval(p, x); 18 p = [ 5 0 0 6 4 2]; x = 0; y = polyval(p,x) x = [0 1]; y = polyval(p,x) x = -1:0.01:1; figure, plot(x, polyval(p,x)); 16 14 12 10 8 6 4 2 0 -2 -1 -0.5 0 0.5 1

  6. Roots of a Polynomial • Roots: a set of values where the polynomial has the value of zero.

  7. Example 1: Polynomial and Its Roots • Plot the following polynomial for the range of x=-2.1~2.1together with its real roots. p = [1 0 -5 0 4 0]; x = -2.1:0.01:2.1; % Plot the polynomial figure(1), plot(x, polyval(p, x), 'LineWidth',2); hold on; plot([-2.5 2.5], [0 0]); % Find its roots and plot them using markers. r = roots(p); ry = zeros(size(r)); h = plot(r,ry,'ro', 'MarkerSize', 10, 'LineWidth', 2); % Display the polynomial in mathematical terms. t = text(-0.5, 2, '{\itf}({\itx})\it=x^5-5x^3+4x'); set(t,'FontName', 'Times', 'FontSize', 16);

  8. Addition, Multiplication and Division • Addition: sum of two equal length vectors • Multiplication: convolution of two vectors • Division: deconvolution of two vectors p1 = [1 0 -5 0 4 0]; p2 = [0 0 0 1 2 0]; pa = p1 + p2 pm = conv(p1, p2) [q r] = deconv(p1, p2)

  9. Derivatives of Polynomials 1/2 • Derivative of a single polynomial • Derivative of a product of two polynomials p1 = [1 0 -5 0 4 0]; p1d = polyder(p1) p2 = [1 2 0]; pd = polyder(p1, p2)

  10. Derivatives of Polynomials 2/2 • Derivative of a quotient of two polynomials [n d ] = polyder(p1, p2)

  11. Exercise 1: Mass of Glass Cup • There is a glass cup of the following shape. When the density of the glass is given as , plot the mass of the cup as a function of the thickness x. What is the thickness x, when the mass of the cup is 400g. 8cm x cm 20cm 5x cm

  12. Solution 1 • Script and Plot • Thickness x when the mass is 400g. glass_cup.m

  13. Polynomials Passing through Points • Find a polynomial passing through the red points. • Substitute the points to the polynomial and get the unknown coefficients. (1,1) (2,0) (-2,-1)

  14. Script • The matrix equation after the substitution. poly_3points.m px = [-2 1 2]; py = [-1 1 0]; A = [4 -2 1; 1 1 1; 4 2 1]; c = inv(A) * py'; x = -3:0.01:3; y = polyval(c,x); figure(1); h = plot(x,y,'b-',px,py,'ro','LineWidth',2); grid; set(h(2),'MarkerSize',10);

  15. Least Square Polynomial Fitting • Find a1 and a0s.t. square errors become min. 2 1.5 Solution 1 0.5 0 -0.5 p = polyfit(x,y,n) -1 -1.5 -2 -3 -2 -1 0 1 2 3

  16. Script • 1st order polynomial approximation using the least square method polyfit_3points.m px = [-2 1 2]; py = [-1 1 0]; p = polyfit(px, py, 1); x = -3:0.01:3; y = polyval(p,x); figure(1); h = plot(x,y,'b-',px,py,'ro','LineWidth',2); grid; set(h(2),'MarkerSize',10); hold on; plot([-3 3],[0 0],'r-',[0 0], [-2 2],'r-'); hold off;

  17. Polynomial Fitting with Diff. Degrees • Compare several polynomial curve fitting schemes with different degrees.

  18. Script polyfit_6points.m %% Experimental Data x = [1 3 4 6 8 10]; y = 2.5*x+2+4*randn(size(x)); figure(1); plot(x,y,'ro','LineWidth',2,'MarkerSize',10); grid; hold on; axis([0 11 -5 30]); %% Polyfit using polynomials x1 = 0:0.01:11; linespec = ['b-','r-','k-']; forn=1:2:5 p = polyfit(x,y,n); y1 = polyval(p,x1); plot(x1,y1,linespec(n), 'LineWidth', 2); end hold off; legend('Experiment','1st order poly', '3rd order poly',... '5th order poly', 'Location', 'SouthEast');

  19. Curve Fitting with Other Functions

  20. Example • Find the best-fit for the following data set. x = 0:0.5:5; y = [3.50 2.69 2.02 1.74 1.17 1.05 0.89 0.63 0.50 0.35 0.15];

  21. Script unknown_fit.m x = 0:0.5:5; y = [3.50 2.69 2.02 1.74 1.17 1.05 0.89 0.63 0.50 0.35 0.15]; figure(1); plot(x,y,'ro','LineWidth',2,'MarkerSize',10); grid; hold on; axis([0 5 0 3.6]); ...

  22. Exercise 2 • Find the best-fit for the following data set using the functions on page 19.

  23. Solution 2 • Script and the screenshot best_fit.m x = [1.0 1.8 2.9 4.1 4.6 5.4 7.1 7.6 8.2 10.0]; y = [2.00 1.35 0.93 0.70 0.63 0.55 0.43 0.40 0.38 0.31]; figure(1); plot(x,y,'ro','LineWidth',2,'MarkerSize',10); grid; hold on; xlabel('x'); ylabel('y'); ...

  24. Interpolation • One dimensional interpolation yi = interp1( x, y, xi, 'method' )

  25. Script interp1_demo.m x = [0.1 0.2 0.4 0.5 0.9 1.0]; y = x .* sin(pi*x); figure(1); plot(x,y,'ro','LineWidth',2,'MarkerSize',10); grid; hold on; xlabel('x'); ylabel('y'); xi = 0:0.01:1; yi_n = interp1(x,y,xi,'nearest'); yi_l = interp1(x,y,xi,'linear'); yi_s = interp1(x,y,xi,'spline'); yi_c = interp1(x,y,xi,'pchip'); plot(xi,yi_n,xi,yi_l,xi,yi_s,xi,yi_c,'LineWidth',2); legend('Exp','Nearest','Linear','Spline','Cubic', ... 'Location','NorthWest');

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