Lecture 3

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1. Convergence of Newton-Raphson Method Suppose differs from the root by a small quantity such that and Then = = This shows that the subsequent error at each step, is proportional to the square of the previous error and as such the convergence is quadratic. Thus Newton Raphson method has second order convergence.  

2. Rule-I Descarte's Rule of sign (for real roots) In any algebra equation The number of positive roots cannot exceed the number of changes of signs (or variations) from positive to negative and negative to positive in The number of negative roots cannot exceed the number of variations in If we consider the equation Rule-II Every equation of an odd degree has at least one real root opposite to that of ( ), whereas every equation of an even degree whose last term is negative has at least two real roots, one positive and the other negative. Rule-III Relation between roots and coefficients If 1, 2, are the roots of the equation n  + 1  n-1  + 2  n-2  + .  .  . +  n-1 + n  = 0   Some rules to Locate Roots of  

3. In all the previous methods we require prior information about the root, but in this method we do not required any previous information and it is capable of giving all the roots of algebraic equation at a time. This is a direct method to find the roots of any polynomial equation with real coefficients. The basic idea behind this method is to separate the roots of the equations by squaring the roots. This can be done by separating even and odd powers of   in    n  + 1  n-1  + 2  n-2  + .  .  . +  n-1 + n  = 0 Now separate odd and even degree powers of x and squaring on both sides. Thus we get,   ( n  + 2  n-2  + 4  n-4  + .  .  . ) 2  = (  n  + 1  n-1  + 3  n-3  + .  .  . ) 2 2n n  - ( 1 2   - 2 2 ) 2n-2  + ( 2 2   - 2 1 3  + 2 4 ) 2n-4  +.  .  . +(-1) n n 2  = 0 Substituting y for  2  we have  n  + 1 n-1  + . . . +  n-1 + n  = 0 Where 1  = 1 2   - 2 2  2  = 2 2   - 2 1 3  + 2 4 n  = n 2   Graeffe's Root Squaring Method

4. If 1, 2, are the roots of original equation then 1 2 , 2 2 , 3 2 … 2 are the roots of the above equation. This procedure can be repeated many times so that the final equation n  + 1  n-1  +  .  .  . + n-1 + n = 0 has the roots  1 , 2  , .  .  ., n  such that i  = i  (2^m)   where = 1, 2, . . ., Type equation here. if we repeat the process m times. If we assume | 1 | > | 2 | > . . . | n |     then    | 1 | >> | 2 | >> . . .>> |  n | that is the roots  i  are very widely separated for large  . Then = Now since i  = i  (2^m)    

5. Apply Graeffe's root squaring method to solve the equation Given Rearranged given equation by separating even and odd degree powers of x, we get Squaring both side we get Now, let we get Again squaring both sides, we get   Example-1

6. Now, let to get Again squaring both side, we get Putting we get Comparing the above equation with , we get Let the roots of equation (2)are = = 2 = = Thus, roots of given equation are 5 , 2 and 1.  

7. Find all roots of the equation -2 -5 +6=0by Graeffe’s method. Given -5 Squaring on both sides 2 -5 ) 2 Put 2 = to get -5) 2 = (2 -6) 2 2 +49) = 14 2 +36 Again squaring on both sides and putting 2 = we get +49) 2 = (1 +36) 2 Or 2 +1393) = 9 +1296 Again squaring and putting 2 = we get +1393) 2 =( 9 +1296) Or -6818 -1679616=0 If the roots of this equation are R 1 ,R 2 ,R 3   Example-2

8. = = 2 = = By Descartes rule of sign, there is one negative root which is found by substituting the values in the given equation, we find Hence the roots are 3,-2,1.  

9. Complex Roots If and form a complex pair ,then the coefficients of in successive squarings would fluctuate both in magnitude and sign by an amount For sufficiently large and can be determined by , = If the equation has only one pair of complex roots say then we can find all real roots . is given by And is given by  

10. Example-1

11. Squaring again and putting , we obtain Or Squaring once again and putting we get Or If be the roots of the original equation, then roots of the above equation are . Thus, we have  

12. From above equations, we observe that the magnitude of the coefficient and have become constant. This indicates that and are the real roots where as and are a pair of complex roots. the real roots and Now let us find the complex roots  

13. Hence, we have Where Also, from original equation, This gives Hence the complex roots are