8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



If B n and B n -\ are of the same sign p 2 was taken too large and must be dimin- 

 ished. Then take 



fr _ B n 



100 Bn-l 



and continue the operation. The required root will be: 



1.262 GraefiVs Method. This method determines approximate values of all 

 the roots of a numerical equation, complex as well as real. Write the equation 

 of the nth degree 



f(x) = aox n aix n ~ l + (hx n ~ 2 . . . . a n = o. 

 The product 



~* - . . . . A n = o 



contains only even powers of x. It is an equation of the nth degree in x 2 . The 

 coefficients are determined by 



A = 2 



AI = 0i 2 20Q02 



A% = 2 2 20103 + 20 04 



AS = 03 2 20204 + 20105 ~ 20Q06 



^4 4 = 04 2 20305 + 20206 20107 + 20 08 



The roots of the equation 



A y n - Ay"- 1 + A 2 y n ~ 2 - . . . . A n = o 



are the squares of the roots of the given equation. Continuing this process we 

 get an equation 



R u n - Riu n ~ l + R 2 u n ~ 2 - . . . . R n = o 



whose roots are the 2 r th powers of the roots of the given equation. Put X = 2 r . 

 Let the roots of the given equation be Ci, c 2 , . . . . , c n . Suppose first that 



Then for large values of X, 



"D Z) 



x _ 2 x _ 7 



If the roots are real they may be determined by extracting the Xth roots of 

 these quantities. Whether they are is determined by taking the sign which 

 approximately satisfies the equation f(x) = o. 



Suppose next that complex roots enter so that there are equalities among 

 the absolute values of the roots. Suppose that 



| ci \ 2 \ c 2 \ > I c 3 I . . . . ^ c p ; c p | > 



