source file: mills2.txt
Date: Fri, 13 Oct 1995 08:07:19 -0700

Subject: Re: 12-tet in China

From: Paul Hahn <Paul-Hahn@library.wustl.edu>

On Fri, 13 Oct 1995, Michael Wathen 556-9565 wrote:
> >This calculates a square root:
> >
> >    Answer := Of_Number;
> >    loop
> >       Delta := (Of_Number / Answer) - Answer;
> >       if abs Delta <= Answer * Epsilon then
> >          return Answer;
> >       end if;
> >       Answer := Answer + 0.5 * Delta;
> >    end loop;
> 
> Is this adaptable to the 12th root of two?  My immediate hunch is no.  It is
> probably only good for roots that can be expressed as powers of 2.  I'll
> have to think about it.  Maybe Mr. Canright would have something to say about
> this.

It's easily adaptable.  Powers of two reduce the amount of calculation
required, which is nice, but not necessary.  Here's the new algorithm:

    Answer := Of_Number;
    loop
       Delta := (Of_Number / (Answer ^ 11)) - Answer;
       if abs Delta <= Answer * Epsilon then
          return Answer;
       end if;
       Answer := Answer + Delta / 12;
    end loop;

The annoying part when doing it by hand is all the multiplication
involved in taking Answer to the 11th power with each iteration.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
       O
      /\             "A three-cushion player doesn't need to be married.
     -\-\-- o               He already has enough aggravation."

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