1. MATLAB Polynomials

2. Introduction Polynomials x2+2x-7 x4+3x3-15x2-2x+9
In MATLAB polynomials are created by row vector i.e.
s4+3s3-15s2-2s+9
>>p=[ ];
3x3-9
>>q=[ ] %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=[ ];
>>s=1;
>>value=polyval(y, s)
>>value = 9

3. Polynomials Evaluation
Similarly
>>s=-3;
>>value=polyval(y, s)
>>value = 13 OR
>>s=[1 -3];
>> value=polyval(y, s)
value =
>> value=polyval(y,[1 -3])

4. Roots of Polynomials
Roots of polynomials: roots(p) >>p=[1 3 2]; % p=s2+3s+2 >>r=roots(p) r = Try this: find the roots of s4+3s3-15s2-2s+9=0

5. Polynomials mathematics
Addition
>>a=[ ]; %s2+2s+1
>>b=[ ]; % s3+s+1
>>c=a+b %s3+s2+3s+6
Subtraction
>>a=[ ]; %s3+2
>>b=[ ]; %s+7
>>c=b-a %-s3+s+5 c=

6. Polynomials mathematics
Multiplication : Multiplication is done by convolution operation .
Sytnax z= conv(x, y)
>>a=[1 2 ]; %s+2
>>b=[ ]; % s2+4s+8
>>c=conv(a, b) % s3+6s2+16s+16 c=
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

7. Polynomials mathematics
>>a=[ ]; %a=s3+6s2+16s+16
>>b=[ ]; %b=s2+4s+8
>>[c, r]=deconv(a, b) c= 2 r=
Try this: divide s2-1 by s+1

8. 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=
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

9. Polynomials Differentiation & Integration
Polynomial differentiation : syntax is
dydx=polyder(y)
>>y=[ ]; %y=s4+4s3+8s2+16
>>dydx=polyder(y) %dydx=4s3+12s2+16s
dydx=
Polynomial integration : syntax is
x=polyint (y, k) %k=constant of integration OR
x=polyint(y) %k=0
>>y=[ ]; %y=4s3+12s2+16s+1
>>x=polyint(y,3) %x=s4+4s3+8s2+s+3 x=
(this is k)

10. 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=[ ]; >>y=[ ]; >>c=polyfit(x, y, 2) %2nd degree polynomial c= >>c=polyfit(x, y, 3) %3rd degree polynomial c = X 1 2 4 Y 6 20
100

11. Polynomials Curve fitting
Ex: Find a polynomial of degree 1 to fit the following data Sol: >>current=[ ]; >>voltage=[ ]; >>resistance=polyfit(current, voltage, 1) resistance= i.e. Voltage = 10x Current
Current 10 15 20 25 30
voltage
100
150
200
250
300

12. Polynomials Evaluation with matrix arguments
Ex: Evaluate the matrix polynomial X2+X+2, given that the square matrix X= Sol: >>A=[1 1 2]; %A= X2+X+2I >>X=[2 3; 4 5]; >>Z=polyvalm(A,X) %poly+val(evaluate)+m(matix) Z=

13. Hello