Theorem (Graeffe's Method).  Given the polynomial   of degree n with real distinct roots  .  Define the sequence    as follows:
    
        

is a polynomial of degree n with roots    for  .  Furthermore, the roots of   are approximated by

           for   .  

Where the appropriate sign can be determined by evaluating  .  

Proof.

    Using the root squaring theorem, the first iteration produces

          

which is a polynomial of degree n with roots  .  

    The second iteration produces

          

which is a polynomial of degree n with roots  .  

    The third iteration produces

          

which is a polynomial of degree n with roots  .  

    Assume that we have    then the k-th iteration produces

          

which is a polynomial of degree n with roots  .  

Therefore, by mathematical induction the process can be carried out indefinitely, but for practical purposes we use a finite number of terms of the sequence  .

References

1. Introduction to Numerical Analysis, Section 10-17, Graeffe's Root-Squaring Technique   
   F. B. Hildebrand   
   McGraw-Hill Book Co., Inc., (1974), New York, pp. 602-608.  
2. A First Course in Numerical Analysis, Section 8.10-2, Graeffe's Root-Squaring Method   
   Anthony Ralston    
   McGraw-Hill Book Co., Inc., (1965), New York, pp. 359-364.  
3. Principles of Numerical Analysis, Section 3.1, The Graeffe Process.
   Alston S. Householder  
   McGraw-Hill Book Co., Inc., (1953), New York, pp. 106-114.
4. Notes on the Graeffe Method of Root Squaring (in Mathematical Notes)  
   G. C. Best  
   American Mathematical Monthly, Vol. 56, No. 2. (Feb., 1949), pp. 91-94, Jstor.  
5. On the Graeffe Method of Solution of Equations  
   L. L. Cronvich  
   American Mathematical Monthly, Vol. 46, No. 4. (Apr., 1939), pp. 185-190. Jstor.  
6. Graeffe's Method and Complex Roots (in Questions, Discussions, and Notes)  
   B. A. Hausmann  
   American Mathematical Monthly, Vol. 43, No. 4. (Apr., 1936), pp. 225-229. Jstor.  
7. On Graeffe's Method for the Numerical Solution of Algebraic Equations  
   C. A. Hutchinson  
   American Mathematical Monthly, Vol. 42, No. 3. (Mar., 1935), pp. 149-161, Jstor.  

(c) John H. Mathews 2005