1. The Hong Kong Polytechnic University Industrial Centre 1 MatLAB Lesson 4 : Polynomial Edward Cheung email: icec@polyu.edu.hk Room W311g 2008

2. 2 Polynomial A polynomial is a function and can be written as follows:- Where f(x) is a function of x The degree or order of a polynomial is n, the highest power of x. a i = 1, 2,…,n, n+1 are coefficients of the polynomial

3. 3 Operations On Polynomials A polynomial can be described with a row vector in MatLab The coefficients become the elements of the vector starting from the highest power of x Interested operations are:-  Get roots from polynomial  Get polynomial from root  Perform arithmetic operations  Addition, subtraction, multiplication, division  Visualisation

4. 4 Root Finding Functions >>help polyfun  Functions for Interpolation and polynomials. roots() – roots of polynomial function fzero() – find root for continous function of single variable  Usage :- x = fzero(fun,x0,options)  fzero looks for a point where fun change sign and defines a zero as a point where the function crosses the x-axis. and not applicable for zeros of even multiplicity

5. 5 Example  Calculate zero of cosine between 0 & -  >> x=fzero(@cos,[0,-pi]) x = -1.5708 % -pi/2 >> fzero('(x-1)^2',0) % root of even multiplicity search may fail % returns NaN = not a number % a value or symbol produced as a result of invalid input operands. E.g. 0/0, inf/inf

6. 6 Polynomial Roots Consider the following cubic equation >>p=[1 0 -7 6]; % i/p coefficient row vector r=roots(p) % find roots for p(x) r = % o/p root as a column vector -3.0000 2.0000 1.0000 % r are the values of x that make p(x)=0

7. 7 Find Imaginary Roots >> r2=roots([1 -6 18 -30 25]) r2 = 1.0000 + 2.0000i 1.0000 - 2.0000i 2.0000 + 1.0000i 2.0000 - 1.0000i Solve:-

8. 8 Coefficient of Polynomial from Roots p=Poly(r)  Return a row vector p such that the elements of p are the coefficients of a polynomial with roots given by r >> r=[-4 -2 -1 1]; p=poly(r) p = 1 6 7 -6 -8

9. 9 Characteristic Polynomial For any square matrix A, we can write:-  is called the eigenvalue of A (eigenvert in German means own value or characteristic value)  x is called the eigenvector (x≠0) p( ) Is called the characteristic polynomial of A Eigenvalues of matrix A can be computed by solving the equation

10. 10 Example on Eigenvalues Suppose we want to find the eigenvalues of a matrix X >> X=[8 -4;2 2]; px=poly(X) %Find characteristic polynomial of X px = 1 -10 24 >> roots(px) % solve for px give eigenvalues of X ans = 6.0000 4.0000 >> eig(X) % use eigenvalue function on X ans = % same result as above 6 4

11. 11 Polynomial Evaluation >> p=[1 1 -2 3]; polyval(p,3) ans = 33 >> x=-5:0.1:5; y=polyval(p,x); plot(x,y) grid Example:- Plot & evaluate the following polynomial at x=3.

12. 12 Polynomial Arithmetic Addition & subtraction of polynomial follows the addition and subtraction of vectors  You should check the order of matrix and patch zero on missing terms to avoid error caused by unequal length Multiplication of polynomial by a scalar is just scalar multiplication Multiplication of two polynomials can be done by the convolve function; conv() Division of two polynomials can be done by the deconvolve function; deconv()

13. 13 Convolution & Deconvolution >> u=[1 2 3]; % u(s) defined v=[4 5 6]; % v(s) defined w=conv(u,v) % multiple u(s) & v(s) w = 4 13 28 27 18 >> [q,r]=deconv(w,v) % q=quotient, r=remainder q = 1 2 3 r = 0 0 0 0 0 Consider the following polynomials:-

14. 14 Partial Fraction Expansion [r,p,k] = residue(b,a) finds the residues, poles, and direct term of a partial fraction expansion of the ratio of two polynomials, b(s) and a(s), of the form: [b,a] = residue(r,p,k) converts the partial fraction expansion back to the polynomials with coefficients in b and a. If there are no multiple roots, then: If p j =…=p (j+m-1) is a pole of multiplicity m, then the expansion include terms of the form

15. 15 Example on Partial Fraction Expansion >> b=[4 -40 98]; >> a=[1 -15 74 -120]; >> [r,p,k]=residue(b,a) r = 1.0000 2.0000 1.0000 p = 6.0000 5.0000 4.0000 k = [] The direct term coefficient vector k is empty because length(a) > length(b)

16. 16 Example on Partial Fraction Expansion (Cont.) Given r,p & k, convert the partial fraction expression to factional polynomial. >> [b2,a2]=residue(r,p,k) b2 = 4 -40 98 a2 = 1 -15 74 -120

17. 17 Data Analysis & Curve Fitting Curve fitting is a technique to model data with an equation Better than interpolation In curve fitting, one need to find a smooth curve to best fit the data. Best fit is interpreted as minimizing the sums of squared errors at the data points. (least squares method) >> eqn=polyfit(X,Y,N)  Fits a polynomial of degree N to data described by the vector X & Y, where X is the independent variable. Returns a row vector of length N+1 as the coefficients in the order of descending power

18. 18 Example on Linear Regression >> x=0.1:0.1:10; % create data x y=5+1.5*x; % create data y y1=y+randn(1,100); % perturbation y1 plot(x,y1,'*')

19. 19 Example on Linear Regression (cont.) eq=polyfit(x,y1,1) % eq = 1.4868 5.1147 >> y2=polyval(eq,x); plot(x,y1,'x',x,y2,'m') legend('data','fit') The data can be Described by the polynomial

20. 20 Example on Non-linear Regression Fit non-linear data with higher order polynomial >> x=0:0.2:1.4; y=cos(x); plot(x,y,'x')

21. 21 Example on Non-linear Regression (cont.) >> eqn1=polyfit(x,y,2) % eqn1 = % -0.3664 -0.0924 1.0081 >> x1=0:0.01:1.4; >> y1=polyval(eqn1,x1); >> hold on >> line1=plot(x1,y1,'g');

22. 22 Example on Non-linear Regression (cont.) >> delete(line1) >> eqn2=polyfit(x,y,3) % eqn2 = % 0.1040 -0.5848 0.0220 0.9994 >> y2=polyval(eqn2,x1); >> line2=plot(x1,y2,'g');

23. 23 3-D Line Plots >> t=-3*pi:0.1:3*pi; >> plot3(t,sin(t),cos(t)) >> grid on

24. 24 3-D Line Plots (cont.) Different Views >> view([1,0,0]) >> view([0,1,0]) >> view([0,0,1])

25. 25 Surface Mesh Plots If z=f(x,y) represents a surface on xyz axes First use [X,Y] = meshgrid(x,y) to generate a grid of points in the xy plane and then evaluate the function z If x=[xmin:spacing:xmax] & y=[ymin:spacing:ymax], meshgrid will generate the coordinates of a rectangular grid with one corner at (xmin, ymin) and the opposite corner at (xmax, ymax). [X,Y] = meshgrid(x) is the same as [X,Y] = meshgrid(x,x)  Usage>> [X,Y]=meshgrid(min:spacing:max); Calculate z & use mesh(X,Y,Z) to plot the surface Not desirable to use too small spacing because of difficulty for visualize and X,Y matrices can become too large

26. 26 Example – meshgrid generation >> [X,Y]=meshgrid(-2:2) X = -2 -1 0 1 2 Y = -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 plot(X,Y,'rx'),axis([-3 3 -3 3])

27. 27 Mesh Plot Example >>Z=sin(sqrt(X.^2+Y.^2)+eps)./(sqrt(X.^2+Y.^2)+eps ); mesh(X,Y,Z)

28. 28 Mesh Plot Examples >> [X,Y]=meshgrid(-2:0.2:2); Z=sin(sqrt(X.^2+Y.^2)+0.0001)./(sqrt(X.^2+Y.^2)+0.0001); mesh(X,Y,Z) >> [X,Y]=meshgrid(-2:0.1:2); >> Z=X.*exp(-((X-Y.^2).^2+Y.^2)); >> mesh(X,Y,Z),xlabel('x'),ylabel('y'),zlabel('z')

29. 29 Contour, Mesh & Surface Plots >> [X,Y]=meshgrid(-2:0.2:2); Z=sin(X).*sin(Y).*exp(-X.^2-Y.^2); contour(X,Y,Z,10)

30. 30 Other 3-D Plots contour(x,y,z) – creates a contour plot mesh(x,y,z) – creates a 3D mesh surface plot meshc(x,y,z) – same as mesh but draws a contour plot under the surface meshz(x,y,z) – same as mesh but draws a series of vertical reference lines under the surface surf(x,y,z) – creates a shaded 3D mesh surface plot surfc(x,y,z) –same as surf but draws a contour plot under the surface waterfall(x,y,z) – same as mesh but draw mesh lines in one direction only

31. 31 Waterfall Plot Example