matlab polynomials
DESCRIPTION
MATLAB Polynomials. Nafees Ahmed Asstt. Professor, EE Deptt DIT, DehraDun. Introduction. Polynomials x 2 +2x-7 x 4 +3x 3 -15x 2 -2x+9 In MATLAB polynomials are created by row vector i.e. s 4 +3s 3 -15s 2 -2s+9 >>p=[ 1 3 -15 -2 9]; 3x 3 -9 - PowerPoint PPT PresentationTRANSCRIPT
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