The Hong Kong Polytechnic University Industrial Centre 1 MatLAB Lesson 4 : Polynomial Edward Cheung email: [email protected] Room W311g 2008.

The Hong Kong Polytechnic UniversityIndustrial Centre

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MatLABLesson 4 : Polynomial

Edward Cheung

email: [email protected]

Room W311g

2008

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Polynomial

12

12

31

21 ...)( nnnnnn axaxaxaxaxaxf

•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.

ai = 1, 2,…,n, n+1 are coefficients of the polynomial

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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

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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

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Example

• 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

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Polynomial Roots

• Consider the following cubic equation

67)( 3 xxxp

>>p=[1 0 -7 6]; % i/p coefficient row vectorr=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

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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

2530186)( 234 xxxxxp

Solve:-

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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

8676)( 234 xxxxxp

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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

xAx

11 ...)()(

)(

nnn cccp

AI

0xAI

0)( p

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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 Xpx = 1 -10 24>> roots(px) % solve for px give eigenvalues of Xans = 6.0000 4.0000>> eig(X) % use eigenvalue function on X ans = % same result as above 6 4

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Polynomial Evaluation

>> p=[1 1 -2 3];polyval(p,3)ans =

33

32)( 23 xxxxp

3333233)3( 23 p

>> x=-5:0.1:5;y=polyval(p,x);plot(x,y)grid

Example:- Plot & evaluate the following polynomial at x=3.

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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()

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Convolution & Deconvolution

>> u=[1 2 3]; % u(s) definedv=[4 5 6]; % v(s) definedw=conv(u,v) % multiple u(s) & v(s)w = 4 13 28 27 18>> [q,r]=deconv(w,v) % q=quotient, r=remainderq = 1 2 3r = 0 0 0 0 0

654)(

32)(2

2

sssv

sssu

Consider the following polynomials:-

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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 pj=…=p(j+m-1) is a pole of multiplicity m, then the expansion include terms of the form

nn

mm

sasasaa

sbsbsbb

sa

sb

1

23

121

12

31

21

...

...

)(

)(

sn

n kps

r

ps

r

ps

r

sa

sb

...

)(

)(

2

2

1

1

mj

mj

j

j

j

j

ps

r

ps

r

ps

r

)(...

)(1

2

1

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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 1.0000p = 6.0000 5.0000 4.0000k = []

1207415

9840423

2

sss

ss

4

1

5

2

6

1

sss

The direct term coefficient vector k is empty because length(a) > length(b)

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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 98a2 = 1 -15 74 -120

1207415

98404

)(

)(23

2

2

2

sss

ss

sa

sb

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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

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Example on Linear Regression

>> x=0.1:0.1:10; % create data xy=5+1.5*x; % create data yy1=y+randn(1,100); % perturbation y1plot(x,y1,'*')

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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 beDescribed by the polynomial

1147.54868.11 xy

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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')

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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');

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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');

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3-D Line Plots

>> t=-3*pi:0.1:3*pi;>> plot3(t,sin(t),cos(t))>> grid on

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3-D Line Plots (cont.)

Different Views>> view([1,0,0])>> view([0,1,0])>> view([0,0,1])

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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

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Example – meshgrid generation

>> [X,Y]=meshgrid(-2:2)X = -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2Y = -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])

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Mesh Plot Example

>>Z=sin(sqrt(X.^2+Y.^2)+eps)./(sqrt(X.^2+Y.^2)+eps);

mesh(X,Y,Z)

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22 )sin(),(

YX

YXYXf

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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')

])[( 222 yyxxez

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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)

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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

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Waterfall Plot Example