View source for /Advanced Level/Polynomials/

'''Title of script''': Polynomials in Scilab

'''Author: Anuradha Amrutkar'''

'''Keywords: scilab, polynomials'''



{| style="border-spacing:0;"
! <center>Visual Clue</center>
! <center>Narration</center>

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| 1<sup>st</sup> Slide
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Welcome to the spoken tutorial on Polynomials in Scilab.

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

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Scilab Console
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Please open Scilab Console window to practise this tutorial

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Scilab Console and type

 -->x = poly(0, 'x')
  x  <nowiki>=</nowiki>
    x
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Let us create a polynomial in x with one root at zero. It can be done using the following command

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Type

 -->p = 1+x+2*x^2
   p  <nowiki>=</nowiki>
              2  
    1 + x + 2x   
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| This variable x can now be used to define another polynomial in x using

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Now type

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

type

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

  p1  <nowiki>= </nowiki> 
              2  

6 - 5x + x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| We can directly define a polynomial by specifying all its roots. For example Let us create a polynomial p1 having roots 3 and 2, and x being the symbolic variable. The polynomial is p1 = 6âˆ’5x+x^2.In the above polynomial p1 it is observed that the third parameter, when supplied, may be 'r' or 'roots' in which case the first parameter is a vector containing the roots of the polynomial.

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Now type

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

6 - 5x + x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Now we will create a polynomial p2 having coefficients 6,âˆ’5,1 assuming the symbolic variable as x again. The polynomial is p2 = 6âˆ’5x+x^2.

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.

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Switch Back to 2<sup>nd</sup> Slide
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Thus, the polynomial with two roots is a polynomial of order two. Similarly, a polynomial with three coefficients is also a polynomial of order two.

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

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Type

 -->p1 
  p1  <nowiki>=</nowiki>
              2  

6 - 5x + x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| It is possible to perform a number of operations on polynomials, such as, find its roots, add, subtract, multiply, divide and simplify.

Recall p1 by typing

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"|  -->roots(p1)
  ans  <nowiki>=</nowiki>

    2.  

3. 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| the roots of p1 can be obtained as:

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->p2
  p2  <nowiki>=</nowiki>
              2  
    6 - 5x + x  
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Also recall p2

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->p3 = p1 + p2  
  p3  <nowiki>=</nowiki>
                 2  

12 - 10x + 2x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Now add the two polynomials p1 and p2 and store the result in the polynomial p3.

Subtraction can be performed in a similar way.

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->p4 = p1 * p2 
 p4  <nowiki>=</nowiki>
                 2     3   4  

36 - 60x + 37x - 10x + x
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Product of two polynomials is also a polynomial, and is calculated using the multiplication operator (*).




|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"|  type   
 -->q1 = 1+x
   q1  <nowiki>=</nowiki>

1 + x
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| We will define another polynomial say q1 as

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->q2 = 1+2*x+x^2 
   q2  <nowiki>= </nowiki>
               2  
    1 + 2x + x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| and q2 as

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->q = q1 / q2 
   q  <nowiki>=</nowiki>

      1     
    <nowiki>----- </nowiki>  

1 + x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Dividing a polynomial with another polynomial is done as shown




|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->q3 = 1+3*x+2*x^2
   q3  <nowiki>=</nowiki>
               2  

1 + 3x + 2x 



| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| also let us define another polynomial say q3 as

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->q1/q3
   ans  <nowiki>=</nowiki>

      0.5     
    <nowiki>------- </nowiki>  

0.5 + x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| and divide q1 by q3 as




|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->derivat(p1)
  ans  <nowiki>=</nowiki>

- 5 + 2x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| The derivative of the polynomial p1 can be obtained using the derivat command




|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->derivat(p1*p2)
   ans  <nowiki>=</nowiki>

   âˆ’ 60 + 74x âˆ’ 30x^2 + 4x^3



| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Also we can find derivative of p1*p2




|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->coeff(p1)
  ans  <nowiki>=</nowiki>

6. - 5. 1. 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| The coefficients of the polynomial p1 can be found out using the coeff command as

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->horner(p1,0)
  ans  <nowiki>=</nowiki>

    6.  

type

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

6. 2. 0.
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| <nowiki>To evaluate a polynomial p1 at ' 0 ' or at set of values ' [0 1 2] ' the horner command is used.</nowiki>




|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->p3
 p3  <nowiki>=</nowiki>
                 2  

12 - 10x + 2x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| We will now see how to find factors of a polynomial

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 

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->factors(p3)
 ans  <nowiki>=</nowiki>
       ans(1)
  - 2 + x   
       ans(2)
  - 3 + x   
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| find its factors using the factors() command

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"|  Type
 -->q3
   q3  <nowiki>=</nowiki>
               2  

1 + 3x + 2x 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Also let us recall q3 and find its factors

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->factors(q3)
   ans  <nowiki>=</nowiki>
       ans(1)
    0.5 + x   
       ans(2)
    1 + x   
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| 

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

--> s =&nbsp;%s;
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Another important feature that scilab provides is partial fraction decomposition of the linear system. Let us see an example


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

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Slide 3
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Please note that defining symbolic variable using the above command works only for variable s and z.

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

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Scilab Console and type


 -->num =1;
 -->den =s^2+3*s+2;
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Now we will define a transfer function with numerator and denominator abbreviated as 'num' and 'den' respectively.

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Type

<tt>-->tf=num/den</tt>

  tf  <nowiki>=</nowiki>
        1        
    <nowiki>--------- </nowiki>   
              2  
    2 + 3s + s  
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Let us now define a variable say 'tf' in the form of fraction

and 

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->pfe=pfss(tf)
  pfe  <nowiki>=</nowiki>
       pfe(1)
      1     
    <nowiki>----- </nowiki>  
    1 + s   
 pfe(2)
    - 1     
    <nowiki>----- </nowiki>  
    2 + s   
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| perform the partial fraction decomposition operation on 'tf' using the 'pfss() command as

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| type

 -->orig = pfe(1)+pfe(2)
   orig  <nowiki>=</nowiki>

        1        
    <nowiki>--------- </nowiki>   
              2  
    2 + 3s + s   
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Let us retrieve the original transfer function by adding the output obtained from the pfss() command


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

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Slide 4
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Let us now discuss how to find poles and zeros of a transfer function.

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>

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Slide 5
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| We obtain

<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> 

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Slide 6
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| Consider an example

<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>

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Scilab Console type 
-->g = 5/(2*s^2+3*s+8)
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| 
 
   

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| 
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| This brings us to the end of spoken tutorial on Polynomials in Scilab.

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.

|-
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:none;padding:0.0382in;"| Slide 7
| style="border-top:none;border-bottom:0.0139in solid #000000;border-left:0.0139in solid #000000;border-right:0.0139in solid #000000;padding:0.0382in;"| The Spoken Tutorials are part of the Talk to a Teacher project, supported by the National Mission on Education through ICT abbreviated as NMEICT given by MHRD government of India. More information on the same is available at this website[http://spoken-tutorial.org/NMEICT-Intro http://spoken-tutorial.org/NMEICT-Intro]. Thanks for joining us. This is Anuradha Amrutkar signing off. Goodbye. 

|}