Karim BELABAS on Sun, 19 Jan 2003 12:31:53 +0100 (MET) |
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Re: gp: contfracpnqn(x) documentation |
On Mon, 4 Nov 2002, Michael Somos wrote: > I have a question about the help documentation : > > ? ??contfracpnqn > contfracpnqn(x): > > when x is a vector or a one-row matrix, x is considered as the list of >partial quotients [a_0,a_1,...,a_n] of a rational number, and the result is the >2 by 2 matrix [p_n,p_{n-1};q_n,q_{n-1}] in the standard notation of continued >fractions, so p_n/q_n = a_0+1/(a_1+...+1/a_n)...). If x is a matrix with two >rows [b_0,b_1,...,b_n] and [a_0,a_1,...,a_n], this is then considered as a >generalized continued fraction and we have similarly p_n/q_n = >1/b_0(a_0+b_1/(a_1+...+b_n/a_n)...). Note that in this case one usually has b_0 > = 1. > > The library syntax is pnqn(x). > > I wondered why the subscripting started at 0 instead of 1. For example : > > ? print(contfracpnqn([])) > [1, 0; 0, 1] > ? print(contfracpnqn([a1])) > [a1, 1; 1, 0] > ? print(contfracpnqn([a1,a2])) > [a2*a1 + 1, a1; a2, 1] > ? print(contfracpnqn([a1,a2,a3])) > [(a3*a2 + 1)*a1 + a3, a2*a1 + 1; a3*a2 + 1, a2] > > seems more natural when subscripting begins at 1. Was there a particular > reason for using 0 instead in this situation? I can see reasons why 0 may > be used, but not strong ones. [a_0, ... , a_n] is understood to be the output of contfrac(), i.e the continued fraction of some real number x. Some authors indeed start the indexing at 1, but I believe the above is more widespread ( in which a_0 is the integer part of x, hence not part of the continued fraction proper ). Karim. -- Karim Belabas Tel: (+33) (0)1 69 15 57 48 Dép. de Mathématiques, Bât. 425 Fax: (+33) (0)1 69 15 60 19 Université Paris-Sud Email: Karim.Belabas@math.u-psud.fr F-91405 Orsay (France) http://www.math.u-psud.fr/~belabas/ -- PARI/GP Home Page: http://www.parigp-home.de/