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Re: Fiting squares under a curve.

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On 17/09/2021 14:22, Ben Bacarisse wrote:
>    *Main> error_of_i int_id [0, 1/3, 1]
>    5.5555555555555663e-2
>    *Main> error_of_i int_id [0, 1/9, 1/3, 1]
>    6.17283950617295e-3
[...]
> But the progression above immediately suggests we should try:
>    *Main> error_of_i int_id [0, 1/27, 1/9, 1/3, 1]
>    6.858710562415376e-4
> which is indeed a better fit, [...]

	I'm still confused by the problem, by Malcolm's description
of it, and by Ben's Haskell!  But in the case f(x) == x, perhaps
worth noting that the situation with N tiles with the last starting
at X consists of the problem with N-1 tiles scaled by a factor X,
together with the last tile.  So the error with N tiles is

   E[N] == X^2 E[N-1] + { error with tile size 1-X between X and 1 } ,

which is more-or-less quadratic in X [somewhat confused by the mod
in the error term], and so can be solved by "easy" calculus/algebra.
If indeed X == 1/3 is correct for all N, then it follows immediately
that the general solution is of form

   [0, ..., 1/81, 1/27, 1/9, 1/3, 1] ,

and each extra tile is taking away 8/9 of the error.  Formal proof
and details left as an exercise ....

-- 
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Bach,CPE

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Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-15 06:59 -0700
  Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-15 15:17 +0100
    Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-15 08:06 -0700
      Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-15 22:33 +0100
        Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-15 15:01 -0700
          Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-16 00:33 +0100
            Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-17 03:25 -0700
              Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-17 14:22 +0100
                Re: Fiting squares under a curve. Andy Walker <anw@cuboid.co.uk> - 2021-09-17 16:25 +0100
                Re: Fiting squares under a curve. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2021-09-17 20:59 +0100
                Re: Fiting squares under a curve. Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2021-09-23 02:14 -0700

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