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Newton's iteration for square roots

Let g(a)=s-f(a) be the function whose roots we seek to find, where f(a) is an arbitrary function and s is a constant. If we find the roots of g(a)=0, then we also have found the roots of f(a)=s. Newton's iteration for g(a)=0 can be written as g(at)=-g'(at)(at+1-at), or

 
s-f(at)=f'(at)(at+1-at) (223)

If we now consider f(a)=a2, then we can write ([*]) as:

 
s-at2=2at(at+1-at) (224)

which gives Newton's iterative procedure for finding square roots  
  (225)

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Stanford Exploration Project
7/5/1998