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Re: Continued Fraction Trouble

  • To: mathgroup at smc.vnet.net
  • Subject: [mg111040] Re: Continued Fraction Trouble
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sun, 18 Jul 2010 01:03:20 -0400 (EDT)

On my system, I do not get the behavior that you showed

$Version

7.0 for Mac OS X x86 (64-bit) (February 19, 2009)

quadcon[a_, c_, n_] :==
 FromContinuedFraction[Take[Flatten@
    Table[{a c, a}, {1/8 (6 + 4 n)}], n]]

data == Table[quadcon[1, -5, i], {i, 1, 200}];

N[data, 2]

{-5.0,-4.0,-3.8,-3.7,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6,-3.6
 ,!
-3.6,-3.6}

However, asking for low precision calculations is not conducive to good results. Recommend that you use NumberForm to control what is displayed

NumberForm[data // N, {4, 1}]


Bob Hanlon

---- Nicholas <physnick at gmail.com> wrote:

==========================
I ran across an unexpected behavior of FromContinuedFraction that I
hope someone can help me with.  I originally thought is was a
representation problem, but now I am not sure.

When I was playing with the continued fraction solutions of a
quadratic

x^2 -a c x -c == 0,

The continued fraction solution

x == a c + 1/( a + 1/ ( a c + ...

converges if the discriminant (a^2 c^2 + 4 c) is nonnegative.  I tried
to do this in Mathematica, using the function "quadcon" defined by

quadcon[a_, c_, n_] :==
 FromContinuedFraction[
  Take[Flatten@Table[{a c, a}, {1/8 (6 + 4 n)}], n]]

which makes a list like {ac, a, ac, a, ac, a} that is then entered
into FromContinuedFraction, but I got some weird results.  For
example, when I tried it with a==1, c ==-5, (which should converge), I
get that it converges to the right value for the first 90 values of n,
but then starts to wander around after that in fits of convergence:

N[Table[quadcon[1, -5, i], {i, 1, 200}],2]

{-5.0, -4.0, -3.8, -3.7, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -0.077, 1.3, 5.5, 30., -62., -30., -25., -23., -1.2*10^2,
 2.0*10^2, 99., -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6,
\
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, -3.6, \
-3.6, -3.6, -3.6, -3.6, 2.1, 2.1, -0.16, 0.57, 5.9, 1.9, 17., 26., \
23., 17., -29., -29., -34., -34., -23., -23., -22., -24., 1.9*10^2,
 2.0*10^2}

However, when I call quadcon with real numbers instead, I get the
desired result for as long as I was patient enough to check.  For
example,

N@Table[quadcon[1., -5., i], {i, 1, 20}]

{-5., -4., -3.75, -3.66667, -3.63636, -3.625, -3.62069, -3.61905, \
-3.61842, -3.61818, -3.61809, -3.61806, -3.61804, -3.61804, -3.61804,
\
-3.61803, -3.61803, -3.61803, -3.61803, -3.61803} ...

This seems to happen when c is negative and the discriminant is
positive.  If the discriminant is zero (i.e. c==-1 or -4),
Mathematica's continued fraction converges, and when a>0 and c>0, it
also converges to the correct result.

There are some theorems about convergence of quadratic continued
fractions, but I have not tested these, and the problem persists if I
use rational numbers as well, but I have not looked into it deeply.

The problem seems to be averted if I plug in undetermined values into
quadcon, then replace with numbers after, but I have not tested this
extensively, and it is not reassuring.

eg

N@quadcon[-9., -5., 20]

44.8886

N@quadcon[-9, -5, 20]

1.26434

N@Simplify[quadcon[a, c, 20]] /. {a -> -9, c -> -5}

44.8886


What am I missing?


--

Bob Hanlon


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