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
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