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Re: Quick n-th Square of BigInteger

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From Eric Sosman <esosman@ieee-dot-org.invalid>
Newsgroups comp.lang.java.programmer
Subject Re: Quick n-th Square of BigInteger
Date Fri, 08 Jun 2012 17:04:28 -0400
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On 6/8/2012 3:03 PM, Jan Burse wrote:
> Dear All,
>
> What is your favorite algorithm to compute the n-th Square of
> a BigInteger, i.e.
>
> Given: x, n
> Compute: y = max { z | z^n =< x }

     (I'm assuming you mean n'th root, which is what your final
line seems to imply.)

     It's not something I've found a need to do, so I have no
"favorite" algorithm.  For positive `x' I think I'd start
by observing that

	x.bitCount()-1 <= lg(x) < x.bitCount()

hence the lg() of the desired root is between

	floor((x.bitCount()-1)/n) <= lg(root) < ceil(x.bitCount()/n)

 From this you can get lower and upper bounds on the root itself,
something like (just typed in free-hand)

	BigInteger lo = BigInteger.ONE.shiftLeft(
	    Math.floor((x.bitCount()-1)/n));
	BitInteger hi = BigInteger.ONE.shiftLeft(
	    Math.ceil(x.bitCount()/n));

... and then do a binary search between those extrema.  (Looks like
they'd differ by a factor of two usually, maybe a factor of four
occasionally -- but I might be wrong about that.)

     Simplifications might be possible if `n' is known to be an
integer.

     For negative `x' and odd integer `n' you could find the bounds
for -x, negate and interchange them, and then do the same binary
search.

     For negative `x' and `n' even or non-integer, I think you're
out of luck.

-- 
Eric Sosman
esosman@ieee-dot-org.invalid

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Thread

Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 21:03 +0200
  Re: Quick n-th Square of BigInteger Gene Wirchenko <genew@ocis.net> - 2012-06-08 13:34 -0700
    Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 22:36 +0200
      Re: Quick n-th Root of BigInteger markspace <-@.> - 2012-06-08 13:55 -0700
        Re: Quick n-th Root of BigInteger glen herrmannsfeldt <gah@ugcs.caltech.edu> - 2012-06-08 21:06 +0000
          Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 23:21 +0200
        Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 23:34 +0200
          Re: Quick n-th Root of BigInteger Lew <lewbloch@gmail.com> - 2012-06-08 14:43 -0700
            Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 23:47 +0200
              Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 23:47 +0200
              Re: Quick n-th Root of BigInteger Lew <lewbloch@gmail.com> - 2012-06-08 14:55 -0700
                Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 00:00 +0200
                Re: Quick n-th Root of BigInteger Lew <lewbloch@gmail.com> - 2012-06-08 15:10 -0700
                Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 00:12 +0200
                Re: Quick n-th Root of BigInteger Lew <lewbloch@gmail.com> - 2012-06-08 15:18 -0700
            Re: Quick n-th Root of BigInteger glen herrmannsfeldt <gah@ugcs.caltech.edu> - 2012-06-08 22:59 +0000
              Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 01:05 +0200
            Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 01:00 +0200
              Re: Quick n-th Root of BigInteger Lew <lewbloch@gmail.com> - 2012-06-08 16:15 -0700
                Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 01:51 +0200
                Re: Quick n-th Root of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 02:32 +0200
  Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 23:00 +0200
  Re: Quick n-th Square of BigInteger Eric Sosman <esosman@ieee-dot-org.invalid> - 2012-06-08 17:04 -0400
    Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 23:19 +0200
      Re: Quick n-th Square of BigInteger Eric Sosman <esosman@ieee-dot-org.invalid> - 2012-06-08 17:40 -0400
        Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-08 23:43 +0200
          Re: Quick n-th Square of BigInteger Lew <lewbloch@gmail.com> - 2012-06-08 14:52 -0700
            Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 00:30 +0200
              Re: Quick n-th Square of BigInteger glen herrmannsfeldt <gah@ugcs.caltech.edu> - 2012-06-08 23:05 +0000
    Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-11 14:53 +0200
  Re: Quick n-th Square of BigInteger Roedy Green <see_website@mindprod.com.invalid> - 2012-06-08 15:32 -0700
    Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 00:37 +0200
      Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 00:39 +0200
      Re: Quick n-th Square of BigInteger glen herrmannsfeldt <gah@ugcs.caltech.edu> - 2012-06-08 23:34 +0000
        Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 01:44 +0200
          Re: Quick n-th Square of BigInteger glen herrmannsfeldt <gah@ugcs.caltech.edu> - 2012-06-09 01:06 +0000
    Re: Quick n-th Square of BigInteger glen herrmannsfeldt <gah@ugcs.caltech.edu> - 2012-06-08 23:25 +0000
      Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 01:29 +0200
        Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 01:29 +0200
    Re: Quick n-th Square of BigInteger Wanja Gayk <brixomatic@yahoo.com> - 2012-06-17 15:00 +0200
  Re: Quick n-th Square of BigInteger Leif Roar Moldskred <leifm@dimnakorr.com> - 2012-06-09 08:42 -0500
    Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 16:54 +0200
      Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 17:56 +0200
        Re: Quick n-th Square of BigInteger Leif Roar Moldskred <leifm@dimnakorr.com> - 2012-06-09 12:52 -0500
      Re: Quick n-th Square of BigInteger Joshua Cranmer <Pidgeot18@verizon.invalid> - 2012-06-09 12:55 -0400
        Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 21:23 +0200
      Re: Quick n-th Square of BigInteger Leif Roar Moldskred <leifm@dimnakorr.com> - 2012-06-09 12:44 -0500
        Re: Quick n-th Square of BigInteger markspace <-@.> - 2012-06-09 11:50 -0700
          Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 21:13 +0200
            Re: Quick n-th Square of BigInteger markspace <-@.> - 2012-06-09 12:25 -0700
              Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-09 21:29 +0200
        Re: Quick n-th Square of BigInteger Joshua Cranmer <Pidgeot18@verizon.invalid> - 2012-06-09 21:27 -0400
          Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-10 12:08 +0200
            Re: Quick n-th Square of BigInteger Joshua Cranmer <Pidgeot18@verizon.invalid> - 2012-06-10 08:23 -0400
          Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-10 13:31 +0200
            Re: Quick n-th Square of BigInteger Wanja Gayk <brixomatic@yahoo.com> - 2012-06-17 15:11 +0200
        Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-10 12:04 +0200
      Re: Quick n-th Square of BigInteger Roedy Green <see_website@mindprod.com.invalid> - 2012-06-16 17:45 -0700
        Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-17 03:17 +0200
        Re: Quick n-th Square of BigInteger Jan Burse <janburse@fastmail.fm> - 2012-06-17 03:38 +0200
  Troll Parade Closing, Award Ceremony Jan Burse <janburse@fastmail.fm> - 2012-06-10 12:31 +0200
    Re: Troll Parade Closing, Award Ceremony Jan Burse <janburse@fastmail.fm> - 2012-06-10 12:35 +0200
    Re: Troll Parade Closing, Award Ceremony Leif Roar Moldskred <leifm@dimnakorr.com> - 2012-06-10 06:28 -0500
      Re: Troll Parade Closing, Award Ceremony Jan Burse <janburse@fastmail.fm> - 2012-06-10 13:30 +0200
    Re: Troll Parade Closing, Award Ceremony Joshua Cranmer <Pidgeot18@verizon.invalid> - 2012-06-10 08:25 -0400
      Re: Troll Parade Closing, Award Ceremony Jan Burse <janburse@fastmail.fm> - 2012-06-10 14:48 +0200
    Re: Troll Parade Closing, Award Ceremony Daniel Pitts <newsgroup.nospam@virtualinfinity.net> - 2012-06-10 11:49 -0700

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