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Re: Working with the set of real numbers

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From Oscar Benjamin <oscar.j.benjamin@gmail.com>
Date Wed, 5 Mar 2014 11:59:47 +0000
Subject Re: Working with the set of real numbers
To Dave Angel <davea@davea.name>
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On 4 March 2014 23:20, Dave Angel <davea@davea.name> wrote:
>
> One problem with complexity claims is that it's easy to miss some
>  contributing time eaters. I haven't done any measuring on modern
>  machines nor in python, but I'd assume that multiplies take
>  *much* longer for large integers, and that divides are much
>  worse. So counting iterations isn't the whole story.

Agreed but there's a big difference between log(N) iterations and N iterations!

> On the assumption that division by 2 is very fast,  and that a
>  general multiply isn't too bad, you could improve on Newton by
>  observing that the slope is 2.
>
>    err = n - guess * guess
>     guess += err/2

I gues you mean like this:

def sqrt(n):
    err = guess = 1
    while err > 1e-10:
        err = n - guess * guess
        guess += err/2
    return guess

This requires log2(10)*N iterations to get N digits. So the penalty
for using division would have to be extreme in order for this to
better. Using Decimal to get many digits we can write that as:

def sqrt2(n, prec=1000):
    '''Solve x**2 = y'''
    eps = D(10) ** -(prec + 5)
    err = guess = D(1)
    with localcontext() as ctx:
        ctx.prec = prec + 10
        while abs(err) > eps:
            err = n - guess*guess
            guess += err/2
    return guess

This method works out much slower than Newton with division at 10000
digits: 40s (based on a single trial) vs 80ms (timeit result).

> Some 37 years ago I microcoded  a math package which included
>  square root.  All the math was decimal, and there was no hardware
>  multiply or divide. The algorithm I came up with generated the
>  answer one digit at a time, with no subsequent rounding needed.
>  And it took just a little less time than two divides. For that
>  architecture,  Newton's method would've been too
>  slow.

If you're working with a fixed small precision then it might be.

> Incidentally, the algorithm did no divides, not even by 2. No
>  multiplies either.  Just repeated subtraction,  sorta like divide
>  was done.
>
> If anyone is curious,  I'll be glad to describe the algorithm;
>  I've never seen it published, before or since.  I got my
>  inspiration from a method used in mechanical,  non-motorized,
>  adding machines.  My father had shown me that approach in the
>  50's.

I'm curious.


Oscar

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Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 03:28 -0800
  Re: Finding size of Variable Peter Otten <__peter__@web.de> - 2014-02-04 12:40 +0100
    Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 04:43 -0800
      Re: Finding size of Variable Asaf Las <roegltd@gmail.com> - 2014-02-04 04:53 -0800
        Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 05:18 -0800
      Re: Finding size of Variable Dave Angel <davea@davea.name> - 2014-02-04 08:09 -0500
        Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 05:19 -0800
          Re: Finding size of Variable Dennis Lee Bieber <wlfraed@ix.netcom.com> - 2014-02-04 09:06 -0500
            Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 21:00 -0800
  Re:Finding size of Variable Dave Angel <davea@davea.name> - 2014-02-04 14:21 -0500
    Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 21:15 -0800
      Re: Finding size of Variable Peter Otten <__peter__@web.de> - 2014-02-05 09:27 +0100
  Re: Finding size of Variable Tim Golden <mail@timgolden.me.uk> - 2014-02-04 19:28 +0000
  Re: Finding size of Variable Tim Chase <python.list@tim.thechases.com> - 2014-02-04 13:29 -0600
    Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 21:35 -0800
      Re: Finding size of Variable Rustom Mody <rustompmody@gmail.com> - 2014-02-04 21:45 -0800
        Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-04 22:00 -0800
      Re: Finding size of Variable Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2014-02-05 11:00 +0000
        Re: Finding size of Variable Chris Angelico <rosuav@gmail.com> - 2014-02-05 22:44 +1100
          Re: Finding size of Variable wxjmfauth@gmail.com - 2014-02-06 02:15 -0800
            Re: Finding size of Variable Ned Batchelder <ned@nedbatchelder.com> - 2014-02-06 06:10 -0500
              Re: Finding size of Variable wxjmfauth@gmail.com - 2014-02-06 05:51 -0800
                Re: Finding size of Variable wxjmfauth@gmail.com - 2014-02-06 06:15 -0800
                Re: Finding size of Variable Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2014-02-08 02:48 +0000
                Re: Finding size of Variable Ethan Furman <ethan@stoneleaf.us> - 2014-02-07 19:02 -0800
                Re: Finding size of Variable Mark Lawrence <breamoreboy@yahoo.co.uk> - 2014-02-08 13:17 +0000
                Re: Finding size of Variable David Hutto <dwightdhutto@gmail.com> - 2014-02-08 17:45 -0500
                Re: Finding size of Variable Rustom Mody <rustompmody@gmail.com> - 2014-02-08 17:25 -0800
                Re: Finding size of Variable David Hutto <dwightdhutto@gmail.com> - 2014-02-08 21:56 -0500
                Re: Finding size of Variable Chris Angelico <rosuav@gmail.com> - 2014-02-09 13:59 +1100
                Re: Finding size of Variable David Hutto <dwightdhutto@gmail.com> - 2014-02-08 22:07 -0500
                Re: Finding size of Variable Ned Batchelder <ned@nedbatchelder.com> - 2014-02-08 22:09 -0500
                Re: Finding size of Variable David Hutto <dwightdhutto@gmail.com> - 2014-02-08 22:09 -0500
                Re: Finding size of Variable Ned Batchelder <ned@nedbatchelder.com> - 2014-02-08 22:16 -0500
                Re: Finding size of Variable Rustom Mody <rustompmody@gmail.com> - 2014-02-08 19:30 -0800
                Re: Finding size of Variable wxjmfauth@gmail.com - 2014-02-10 06:07 -0800
                Re: Finding size of Variable Asaf Las <roegltd@gmail.com> - 2014-02-10 06:25 -0800
                Re: Finding size of Variable Mark Lawrence <breamoreboy@yahoo.co.uk> - 2014-02-10 14:39 +0000
                Re: Finding size of Variable Tim Chase <python.list@tim.thechases.com> - 2014-02-10 08:43 -0600
                Re: Finding size of Variable wxjmfauth@gmail.com - 2014-02-11 10:53 -0800
                Re: Finding size of Variable Mark Lawrence <breamoreboy@yahoo.co.uk> - 2014-02-11 19:04 +0000
                Re: Finding size of Variable wxjmfauth@gmail.com - 2014-02-11 23:49 -0800
                Re: Finding size of Variable Chris Angelico <rosuav@gmail.com> - 2014-02-12 19:06 +1100
                Re: Finding size of Variable Jussi Piitulainen <jpiitula@ling.helsinki.fi> - 2014-02-12 10:57 +0200
                Re: Finding size of Variable Chris Angelico <rosuav@gmail.com> - 2014-02-12 20:24 +1100
                Re: Finding size of Variable Jussi Piitulainen <jpiitula@ling.helsinki.fi> - 2014-02-12 11:35 +0200
                Working with the set of real numbers (was: Finding size of Variable) Ben Finney <ben+python@benfinney.id.au> - 2014-02-12 19:17 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) wxjmfauth@gmail.com - 2014-02-12 00:35 -0800
                Re: Working with the set of real numbers (was: Finding size of Variable) wxjmfauth@gmail.com - 2014-02-12 00:46 -0800
                Re: Working with the set of real numbers Ben Finney <ben+python@benfinney.id.au> - 2014-02-12 19:52 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) Grant Edwards <invalid@invalid.invalid> - 2014-02-12 15:24 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) "Gisle Vanem" <gvanem@yahoo.no> - 2014-02-12 17:23 +0100
                Re: Working with the set of real numbers (was: Finding size of Variable) Chris Angelico <rosuav@gmail.com> - 2014-02-12 19:47 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) Jussi Piitulainen <jpiitula@ling.helsinki.fi> - 2014-02-12 11:23 +0200
                Re: Working with the set of real numbers (was: Finding size of Variable) albert@spenarnc.xs4all.nl (Albert van der Horst) - 2014-03-04 02:45 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Chris Angelico <rosuav@gmail.com> - 2014-03-04 14:02 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) Rustom Mody <rustompmody@gmail.com> - 2014-03-03 19:13 -0800
                Re: Working with the set of real numbers (was: Finding size of Variable) Chris Angelico <rosuav@gmail.com> - 2014-03-04 14:46 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) Rustom Mody <rustompmody@gmail.com> - 2014-03-03 21:19 -0800
                Re: Working with the set of real numbers (was: Finding size of Variable) Steven D'Aprano <steve@pearwood.info> - 2014-03-04 05:53 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Chris Angelico <rosuav@gmail.com> - 2014-03-04 17:35 +1100
                Re: Working with the set of real numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-03-05 00:05 +1300
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-03-04 23:43 +1100
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-03-04 21:49 +0200
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-03-05 06:58 +1100
                Re: Working with the set of real numbers Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-04 20:55 +0000
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-03-04 23:05 +0200
                Re: Working with the set of real numbers Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-04 22:08 +0000
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-03-05 08:18 +1100
                Re: Working with the set of real numbers Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-04 22:02 +0000
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-03-05 09:18 +1100
                Re: Working with the set of real numbers Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-04 22:54 +0000
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-03-05 10:01 +1100
                Re: Working with the set of real numbers Dave Angel <davea@davea.name> - 2014-03-04 18:20 -0500
                Re: Working with the set of real numbers Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-05 11:59 +0000
                Re: Working with the set of real numbers Dave Angel <davea@davea.name> - 2014-03-05 07:57 -0500
                Re: Working with the set of real numbers Dave Angel <davea@davea.name> - 2014-03-05 08:32 -0500
                Re: Working with the set of real numbers Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-06 12:27 +0000
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-03-07 00:16 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) Ian Kelly <ian.g.kelly@gmail.com> - 2014-03-04 04:19 -0700
                Re: Working with the set of real numbers (was: Finding size of Variable) albert@spenarnc.xs4all.nl (Albert van der Horst) - 2014-03-05 02:27 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Ian Kelly <ian.g.kelly@gmail.com> - 2014-03-04 04:23 -0700
                Re: Working with the set of real numbers (was: Finding size of Variable) albert@spenarnc.xs4all.nl (Albert van der Horst) - 2014-03-05 02:15 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Steven D'Aprano <steve@pearwood.info> - 2014-03-05 03:41 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Rustom Mody <rustompmody@gmail.com> - 2014-03-04 20:15 -0800
                Re: Working with the set of real numbers (was: Finding size of Variable) Roy Smith <roy@panix.com> - 2014-03-04 23:25 -0500
                Re: Working with the set of real numbers Ben Finney <ben+python@benfinney.id.au> - 2014-03-05 15:37 +1100
                Re: Working with the set of real numbers Rustom Mody <rustompmody@gmail.com> - 2014-03-04 20:57 -0800
                Re: Working with the set of real numbers Roy Smith <roy@panix.com> - 2014-03-05 00:29 -0500
                Re: Working with the set of real numbers (was: Finding size of Variable) Steven D'Aprano <steve@pearwood.info> - 2014-03-05 07:52 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Steven D'Aprano <steve@pearwood.info> - 2014-03-05 08:38 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) wxjmfauth@gmail.com - 2014-03-05 01:00 -0800
                Re: Working with the set of real numbers Ned Batchelder <ned@nedbatchelder.com> - 2014-03-05 06:23 -0500
                Re: Working with the set of real numbers (was: Finding size of Variable) Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-05 12:21 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2014-03-05 17:43 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Chris Angelico <rosuav@gmail.com> - 2014-03-06 05:01 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) Chris Kaynor <ckaynor@zindagigames.com> - 2014-03-05 10:03 -0800
                Re: Working with the set of real numbers (was: Finding size of Variable) Grant Edwards <invalid@invalid.invalid> - 2014-03-05 19:13 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-03-05 21:22 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Roy Smith <roy@panix.com> - 2014-03-05 21:31 -0500
                Re: Working with the set of real numbers (was: Finding size of Variable) Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2014-03-06 03:06 +0000
                Re: Working with the set of real numbers (was: Finding size of Variable) Chris Angelico <rosuav@gmail.com> - 2014-03-06 14:14 +1100
                Re: Working with the set of real numbers (was: Finding size of Variable) Roy Smith <roy@panix.com> - 2014-03-05 23:05 -0500
                Re: Working with the set of real numbers (was: Finding size of Variable) Grant Edwards <invalid@invalid.invalid> - 2014-03-06 03:34 +0000
                Re: Working with the set of real numbers Mark Lawrence <breamoreboy@yahoo.co.uk> - 2014-03-05 12:50 +0000
                Re: Working with the set of real numbers Steven D'Aprano <steve+comp.lang.python@pearwood.info> - 2014-03-05 17:49 +0000
                Re: Working with the set of real numbers Ben Finney <ben+python@benfinney.id.au> - 2014-02-12 19:56 +1100
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-12 20:16 +1100
                Re: Working with the set of real numbers Ben Finney <ben+python@benfinney.id.au> - 2014-02-12 21:07 +1100
                Re: Working with the set of real numbers Rustom Mody <rustompmody@gmail.com> - 2014-02-12 06:11 -0800
                Re: Working with the set of real numbers Ian Kelly <ian.g.kelly@gmail.com> - 2014-02-12 13:45 -0700
                Re: Working with the set of real numbers Rustom Mody <rustompmody@gmail.com> - 2014-02-12 17:47 -0800
                Re: Working with the set of real numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-13 11:09 +1300
                Re: Working with the set of real numbers Steven D'Aprano <steve@pearwood.info> - 2014-02-13 03:31 +0000
                Re: Working with the set of real numbers Ben Finney <ben+python@benfinney.id.au> - 2014-02-13 14:45 +1100
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-13 15:17 +1100
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-12 21:20 +1100
                Re: Working with the set of real numbers wxjmfauth@gmail.com - 2014-02-12 02:55 -0800
                Re: Working with the set of real numbers Ned Batchelder <ned@nedbatchelder.com> - 2014-02-12 06:55 -0500
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-02-12 14:48 +0200
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-13 00:20 +1100
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-02-12 16:13 +0200
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-13 04:52 +1100
                Re: Working with the set of real numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-13 11:24 +1300
                Re: Working with the set of real numbers Dave Angel <davea@davea.name> - 2014-02-12 17:56 -0500
                Re: Working with the set of real numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-14 18:26 +1300
                Re: Working with the set of real numbers Ben Finney <ben+python@benfinney.id.au> - 2014-02-12 22:44 +1100
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-12 22:58 +1100
                Re: Working with the set of real numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-13 11:32 +1300
                Re: Working with the set of real numbers Grant Edwards <invalid@invalid.invalid> - 2014-02-12 23:23 +0000
                Re: Finding size of Variable Mark Lawrence <breamoreboy@yahoo.co.uk> - 2014-02-12 14:04 +0000
                Re: Finding size of Variable Rustom Mody <rustompmody@gmail.com> - 2014-02-12 06:14 -0800
                Re: Finding size of Variable Mark Lawrence <breamoreboy@yahoo.co.uk> - 2014-02-12 14:25 +0000
                Re: Finding size of Variable Rustom Mody <rustompmody@gmail.com> - 2014-02-12 06:32 -0800
                Re: Working with the set of real numbers Oscar Benjamin <oscar.j.benjamin@gmail.com> - 2014-02-13 12:48 +0000
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-02-13 16:00 +0200
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-14 06:25 +1100
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-02-13 21:47 +0200
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-14 07:08 +1100
                Re: Working with the set of real numbers Devin Jeanpierre <jeanpierreda@gmail.com> - 2014-02-13 22:05 -0800
                Re: Working with the set of real numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-15 00:30 +1300
                Re: Working with the set of real numbers Devin Jeanpierre <jeanpierreda@gmail.com> - 2014-02-14 16:26 -0800
                Re: Working with the set of real numbers albert@spenarnc.xs4all.nl (Albert van der Horst) - 2014-03-05 02:38 +0000
                Re: Working with the set of real numbers Gregory Ewing <greg.ewing@canterbury.ac.nz> - 2014-02-14 19:37 +1300
                Re: Working with the set of real numbers Chris Angelico <rosuav@gmail.com> - 2014-02-14 17:44 +1100
                Re: Working with the set of real numbers Rustom Mody <rustompmody@gmail.com> - 2014-02-14 07:13 -0800
                Re: Working with the set of real numbers Dave Angel <davea@davea.name> - 2014-02-14 07:30 -0500
                Re: Working with the set of real numbers Grant Edwards <invalid@invalid.invalid> - 2014-02-14 15:09 +0000
                Re: Working with the set of real numbers Rotwang <sg552@hotmail.co.uk> - 2014-02-13 21:29 +0000
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-02-14 00:00 +0200
                Re: Working with the set of real numbers Rotwang <sg552@hotmail.co.uk> - 2014-02-13 22:21 +0000
                Re: Working with the set of real numbers Marko Rauhamaa <marko@pacujo.net> - 2014-02-14 01:16 +0200
                Re: Working with the set of real numbers Ben Finney <ben+python@benfinney.id.au> - 2014-02-14 03:57 +1100
                Re: Finding size of Variable Ned Batchelder <ned@nedbatchelder.com> - 2014-02-10 10:02 -0500
                Re: Finding size of Variable Neil Cerutti <neilc@norwich.edu> - 2014-02-11 14:29 +0000
        Re: Finding size of Variable Dennis Lee Bieber <wlfraed@ix.netcom.com> - 2014-02-05 22:14 -0500
      Re: Finding size of Variable Dave Angel <davea@davea.name> - 2014-02-05 08:43 -0500
        Re: Finding size of Variable Ayushi Dalmia <ayushidalmia2604@gmail.com> - 2014-02-05 06:33 -0800
          Re: Finding size of Variable Mark Lawrence <breamoreboy@yahoo.co.uk> - 2014-02-05 15:22 +0000

csiph-web