Difference between revisions of "/Advanced Level/Polynomials/"

By using Scilab you can create polynomials, find their roots and perform operations on them such as addition, subtraction, multiplication, division, simplification, etc.

-->x = poly(0, 'x')
 x  =
   x

-->p = 1+x+2*x^2
  p  =
             2  
   1 + x + 2x   

-->p1 = poly([3 2], 'x')
or

type

-->p1 = poly([3 2], 'x', 'r') 

 p1  =  
             2  

6 - 5x + x

-->p2 = poly([6 -5 1], 'x', 'c') 
 p2  =
             2  

6 - 5x + x

In the above polynomial p2 which is same as p1, it is observed that, the third parameter, when supplied, may be 'c' or 'coeff' in which case the first parameter is a vector containing the coefficients of the polynomial, starting from the constant as the first element and power of the symbolic variable increasing by one for each element in the vector.

When the third parameter is not supplied, it defaults to 'r' or 'roots'.

-->p1 
 p1  =
             2  

6 - 5x + x

Recall p1 by typing

 ans  =

   2.  

3.

-->p2
 p2  =
             2  
   6 - 5x + x  

-->p3 = p1 + p2  
 p3  =
                2  

12 - 10x + 2x

Subtraction can be performed in a similar way.

-->p4 = p1 * p2 
p4  =
                2     3   4  

36 - 60x + 37x - 10x + x

-->q1 = 1+x
  q1  =

1 + x

-->q2 = 1+2*x+x^2 
  q2  = 
              2  
   1 + 2x + x 

-->q = q1 / q2 
  q  =

     1     
   -----   

1 + x

-->q3 = 1+3*x+2*x^2
  q3  =
              2  

1 + 3x + 2x

-->q1/q3
  ans  =

     0.5     
   -------   

0.5 + x

-->derivat(p1)
 ans  =

- 5 + 2x

-->derivat(p1*p2)
  ans  =

  − 60 + 74x − 30x^2 + 4x^3

-->coeff(p1)
 ans  =

6. - 5. 1.

-->horner(p1,0)
 ans  =

   6.  

type

-->horner(p1,[0 1 2])
 ans  =

6. 2. 0.

-->p3
p3  =
                2  

12 - 10x + 2x

For this purpose the factors() command is used.

This command performs numeric real factorization.

Let us see an example to illustrate this concept.

Let us Recall p3 which was p1+p2 and

-->factors(p3)
ans  =
      ans(1)
 - 2 + x   
      ans(2)
 - 3 + x   

-->q3
  q3  =
              2  

1 + 3x + 2x

-->factors(q3)
  ans  =
      ans(1)
   0.5 + x   
      ans(2)
   1 + x   

--> s = %s;

Let us declare 's' as symbolic variable with the command

But the command 'poly' used at the start of the tutorial to define a polynomial in x, works for any variable.

-->num =1;
-->den =s^2+3*s+2;

-->tf=num/den

 tf  =
       1        
   ---------    
             2  
   2 + 3s + s  

and

-->pfe=pfss(tf)
 pfe  =
      pfe(1)
     1     
   -----   
   1 + s   
pfe(2)
   - 1     
   -----   
   2 + s   

-->orig = pfe(1)+pfe(2)
  orig  =

       1        
   ---------    
             2  
   2 + 3s + s   

which is exactly same as the transfer function we defined using the num and den variables.

Consider a Second Order Transfer Function with no zeros of the form

<math> g(s)=\frac{n_0}{d_2s^2 + d_1s + d_0} </math>

dividing numerator and denominator by d2 gives

<math> g(s)=\frac{\frac{n_0}{d_2}}{s^2 + \frac{d_1}{d_2}s + \frac{d_0}{d_2}} </math>

Equating the RHS of the above equation to the standard Second Order System

<math> g(s)=\frac{k}{s^2 + 2\zeta\omega_n s + \omega_n^2} </math>

<math> \omega_n^2 = \frac{d_0}{d_2}</math>

and

<math> 2\zeta\omega_n = \frac{d_1}{d_2}</math>

which help to solve for <math> \omega_n </math> and <math> \zeta </math>

In Scilab 'trfmod()' with option 'f' gives

Numerator <math> k=\frac{n_0}{d_2} </math>

and Denominator <math> \omega_n </math> and <math> \zeta </math>

<math> g(s) = \frac{5}{2s^2+3s+8} </math>

we get this equal to

<math> g(s) = \frac{\frac{5}{2}}{2s^2+\frac{3}{2}s+4} </math>

This gives the natural frequency

<math> \omega_n = 2 </math>

and damping ratio

<math> \zeta = \frac{3}{8}= 0.375 </math>

-->g = 5/(2*s^2+3*s+8)

In this tutorial we have learnt how to create polynomials in scilab and how to perform various operations on those polynomials. There are many other functions in Scilab which will be covered in other spoken tutorials. Keep watching the Scilab links.