MATLAB Polynomials

Nafees AhmedNafees AhmedAsstt. Professor, EE DepttDIT, DehraDun

Introduction Polynomials

x2+2x-7x4+3x3-15x2-2x+9

In MATLAB polynomials are created by row vector i.e.s4+3s3-15s2-2s+9>>p=[ 1 3 -15 -2 9];3x3-9>>q=[3 0 0 -9]%write the coefficients of every term

Polynomial evaluation : polyval(c,s)Exp: Evaluate the value of polynomial y=2s2+3s+4 at s=1, -3>>y=[2 3 4];>>s=1;>>value=polyval(y, s)>>value =

9

Polynomials Evaluation

Similarly>>s=-3;>>value=polyval(y, s)>>value =

13OR

>>s=[1 -3];>> value=polyval(y, s)value =

9 13

OR>> value=polyval(y,[1 -3])

Roots of Polynomials

Roots of polynomials: roots(p)

>>p=[1 3 2]; % p=s2+3s+2

>>r=roots(p)

r = -2

-1

Try this: find the roots of s4+3s3-15s2-2s+9=0

Polynomials mathematics

• Addition

>>a=[0 1 2 1]; %s2+2s+1

>>b=[1 0 1 5]; % s3+s+1

>>c=a+b %s3+s2+3s+6• Subtraction

>>a=[3 0 0 2]; %s3+2

>>b=[0 0 1 7]; %s+7

>>c=b-a %-s3+s+5

c=

-3 0 1 5

Polynomials mathematics • Multiplication : Multiplication is done by convolution operation .

Sytnax z= conv(x, y)

>>a=[1 2 ]; %s+2

>>b=[1 4 8 ]; % s2+4s+8

>>c=conv(a, b) % s3+6s2+16s+16

c=

1 6 16 16

Try this: find the product of (s+3),(s+6) & (s+2). Hint: two at a time

• Division : Division is done by deconvolution operation.

Syntax is [z, r]=deconv(x, y)

Where

x=divident vector y=divisor vector

z=Quotients vector r=remainders vector

Polynomials mathematics

>>a=[1 6 16 16]; %a=s3+6s2+16s+16

>>b=[1 4 8]; %b=s2+4s+8

>>[c, r]=deconv(a, b)

c=

1 2

r=

0 0 0 0

Try this: divide s2-1 by s+1

Formulation of Polynomials

• Making polynomial from given roots:

>>r=[-1 -2]; %Roots of polynomial are -1 & -2

>>p=poly(r); %p=s2+3s+2

p=

1 3 2

• Characteristic Polynomial/Equation of matrix ‘A”: =det(sI-A)

>>A=[0 1; 2 3];

>>p=poly(A) %p= determinant (sI-A)

p= %p=s2-3s-2

1 -3 -2

Polynomials Differentiation & Integration

Polynomial differentiation : syntax is

dydx=polyder(y)

>>y=[1 4 8 0 16]; %y=s4+4s3+8s2+16

>>dydx=polyder(y) %dydx=4s3+12s2+16s

dydx=

4 12 16 0

Polynomial integration : syntax is

x=polyint (y, k) %k=constant of integration

OR

x=polyint(y) %k=0

>>y=[4 12 16 1]; %y=4s3+12s2+16s+1

>>x=polyint(y,3) %x=s4+4s3+8s2+s+3

x=

1 4 8 1 3(this is k)

Polynomials Curve fitting

In case a set of points are known in terms of vectors x & y, then a polynomial can be formed that fits the given points. Syntax is

c=polyfit(x, y, k) %k is degree of polynomial

Ex: Find a polynomial of degree 2 to fit the following data

Sol:

>>x=[0 1 2 4];

>>y=[1 6 20 100];

>>c=polyfit(x, y, 2) %2nd degree polynomial

c=

7.3409 -4.8409 1.6818

>>c=polyfit(x, y, 3) %3rd degree polynomial

c =

1.0417 1.3750 2.5833 1.0000

X 0 1 2 4

Y 1 6 20 100

Polynomials Curve fitting

Ex: Find a polynomial of degree 1 to fit the following data

Sol:

>>current=[10 15 20 25 30];

>>voltage=[100 150 200 250 300];

>>resistance=polyfit(current, voltage, 1)

resistance=

10.0000 -0.0000

i.e. Voltage = 10x Current

Current 10 15 20 25 30

voltage 100 150 200 250 300

Polynomials Evaluation with matrix arguments

Ex: Evaluate the matrix polynomial X2+X+2, given that the square matrix

X= 2 3

4 5

Sol:

>>A=[1 1 2]; %A= X2+X+2I

>>X=[2 3; 4 5];

>>Z=polyvalm(A,X) %poly+val(evaluate)+m(matix)

Z=

20 24

32 44