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Re: Optimizing fixed point iteration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83104] Re: Optimizing fixed point iteration
  • From: mcmcclur at unca.edu
  • Date: Sun, 11 Nov 2007 02:56:40 -0500 (EST)
  • References: <fh1ckb$csj$1@smc.vnet.net>

On Nov 9, 5:26 am, Yaroslav Bulatov <yarosla... at gmail.com> wrote:
> Can anyone see the trick to speed up the following code significantly?
> depth;step=.02;
> Table[(c = 0;
>    NestWhile[-4 (# - a) (# - b) &, .51, (c++; Abs[#1 - #2] > .1) &, 2,
>      depth]; c), {a, -1., 0., step}, {b, 0., 1., step}] // ArrayPlot

Here's one solution that iterates up to 50 times
over a square with resolution 1000 by 1000.

convergenceCount = Compile[{{a, _Real}, {b, _Real}},
  Length[FixedPointList[-4 (# - a) (# - b) &, .51, 50,
    SameTest -> ((Abs[#] > 1 + Abs[a + b] ||
      Abs[#1 - #2] < 0.001) &)]]];
M = Table[convergenceCount[a, b],
  {a, -1, 0, 0.001}, {b, 0, 1, 0.001}]; // Timing
{12.5899, Null}

A few comments:
* The computation was performed by V6 on my Macbook
  Pro running at 2.16 GHz.

* Clearly Compile helps quite a lot.  Sadly,
  however, NestWhileList is not fully compilable.
  Hence the use of FixedPointList with the clever
  SameTest.

* The orbit diverges to rapidly infinity for most
  parameter values a and b.  For such parameters,
  your code computes the entire orbit up to depth.
  It's pretty easy to prove though that the orbit
  diverges if the value z ever satisfies
  |z| > max(|a*b|, 1+|a+b|).
  Thus, computation can stop once this threshhold
  is reached.  This is particularly important since
  the orbit can quickly exceed $MaxMachineNumber,
  forcing high precision arithmetic.

  Note that my code simply stops when the
  threshhold is exceeded, returning a small
  number.  You might need to fiddle with it a
  bit to return a large number, like your code.
  The picture looks pretty good, though.

Mark




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