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Topic: cube root of a given number
Replies: 112   Last Post: Jan 10, 2013 1:39 PM

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Iain Davidson

Posts: 1,173
Registered: 12/12/04
Re: cube root of a given number
Posted: Aug 6, 2007 1:35 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 6 Aug, 15:18, arithmonic <djes...@gmail.com> wrote:
> > > On 26 jul, 04:37, "sttscitr...@tesco.net" <sttscitr...@tesco.net>

> All the readers can find information on this topic on GCF at (page
> 3-4, Section 1.1.1):
>
> http://assets.cambridge.org/052181/8052/sample/0521818052ws.pdf


Dumbingo

What is it that the extract from Finch's book is supposed to
demonstrate ?

"1.1.1 Generalized Continued Fractions

It is well known that any quadratic irrational possesses a periodic
regular continued fraction expansion and vice versa"

This is a false statement. It is not the case that every quadratic
irrational has a periodic RCF expansion. Every quadratic irrational
has a RCF
expansion that is either periodic or eventually periodic.

"Comparatively few people have examined the generalized continued
fraction"

It is not clear which of the "two types" of generalized continued
fraction
Finch is referring to. If he is referring to the "bifurcating" CF then
what he says is true. Basically, all these "bifurcating CFs" are is
pretty patterns on
a page. To generalize simple CFs you have to generalize their
mathematical properties not how they look when written down.

The basic properties of a SCF expansion of a real number t are
1) All best rational approximations to t are produced
2) If t is rational, the SCF expansion terminates
3) if t is a quadratic irrational, the SCF expansion is purely
periodic
if t is a reduced root of an irreducible quadratic equation and
eventually
periodic otherwise.
4) A SCF expansion will indicate if t', a decimal , is an
approximation to a quadratic irrational t

A generalization of the SCF should have at least one of these
properties - say the ability to distinguish between rationals,
quadratic irrationals and cubic irrationals.
If you think of the SCF of t as a sequence of unimodular matrix
transformations applied to a point written using homogeneous
coordinates (t,1), giving a sequence of homogeneous points (t(1),1)
(t(2),1) ...,(t(n),1) periodicity will occur when t(m), t(m+k) are in
the same ratio. In terms of linear algebra t(m), t(m+k), ....are fixed
points (or eigenvectors) of powers of a unimodular matrix.
If you want periodicity to occur when considering say cubic
irrationals , you must consider a point of the form (s,t,1).
(s and t must ,of course, belong to the same cubic field). In other
words
you must have a "2-dimenesional" CF.

The Gupta-Mittal CF referred to by Finch is in fact of this form, but
no general law of formation for convergents is stated.

The Gupta-Mittal CF for two reals a, b is based on these equations

A(i) = int(a(i))
B(i) = int)b(i))
a(i+1) = 1/(b(i) -B(i))
b(i+1) = (a(i) -A(i))/(b(i) -B(i))

But this is essentially, a Jacobi-Perron -type continued fraction/

Gupta and Mittal seem to think that the SCF is "defective" in some way
as the SCF expansions of higher irrationals are not periodic.

The situation is actually the reverse. The SCF has optimal properties.
The difficulty is finding generalizations that have at least one of
the SCF's
desired properties.

Given a real number or a set of real numbers, the Dumbingo CF
cannot discover to what degree of irrationality they belong unless
their
algebraic equations are known beforehand. It doesn't produce best
approximations
Another intellectual triumph for Dumbingo Gormless Moron



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