source file: mills2.txt Date: Fri, 13 Oct 1995 09:46:36 -0700 Subject: Re: 12-tet in China From: "David R. Canright" Paul Hahn's adaptation of the square root algorithm to the Nth root certainly works: Answer := Of_Number; loop Delta := (Of_Number / (Answer ^ (N-1))) - Answer; if abs Delta <= Answer * Epsilon then return Answer; end if; Answer := Answer + Delta / N; end loop; But as he points out: > Powers of two reduce the amount of calculation > required, which is nice, but not necessary. For doing by hand, the simplest way is find the square root of two, then the square root of that, then the cube root of that. This converges to 10 significant digits in a total of 12 iterations (4 for each root). The direct 12th root approach also converges in 12 iterations, but the latter requires calculating 11th powers 12 times, while the former only requires finding 2nd powers 4 times. Either way, the algorithm is not inherently difficult. David C. -- David Canright (408) 656-2782 (or -2206) Math. Dept., Code MA/Ca (408) 656-2355 (FAX) Naval Postgraduate School DCanright@NPS.Navy.mil Monterey, CA 93943 USA http://math.nps.navy.mil/~dcanrig/ Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 13 Oct 1995 19:04 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id KAA08088; Fri, 13 Oct 1995 10:04:17 -0700 Date: Fri, 13 Oct 1995 10:04:17 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu