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Continued Fraction Trouble
*To*: mathgroup at smc.vnet.net
*Subject*: [mg111031] Continued Fraction Trouble
*From*: Nicholas <physnick at gmail.com>
*Date*: Sat, 17 Jul 2010 08:17:51 -0400 (EDT)
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?
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