Continued fraction/Arithmetic: Difference between revisions

which will perform basic mathmatical operations on continued fractions.

 

:Gosper has invented an algorithm for performing analytic addition, subtraction, multiplication, and division using continued fractions. It requires keeping track of eight integers which are conceptually arranged at the polyhedron vertices of a cube. Although this algorithm has not appeared in print, similar algorithms have been constructed by Vuillemin (1987) and Liardet and Stambul (1998).

 

:Gosper's algorithm for computing the continued fraction for (ax+b)/(cx+d) from the continued fraction for x is described by Gosper (1972), Knuth (1998, Exercise 4.5.3.15, pp. 360 and 601), and Fowler (1999). (In line 9 of Knuth's solution, X_k<-|_A/C_| should be replaced by X_k<-min(|_A/C_|,|_(A+B)/(C+D)_|).) Gosper (1972) and Knuth (1981) also mention the bivariate case (axy+bx+cy+d)/(Axy+Bx+Cy+D).

My description follows [http://perl.plover.com/yak/cftalk/INFO/gosper.txt part of Gosper reproduced on perl.plover.com]. This document is text and unnumbered, you may wish to start by searching for "Addition, Multiplication, etc. of Two Continued Fractions" prior to reading the whole thing.

 

[http://perl.plover.com/classes/cftalk/TALK/slide001.html perl.plover.com] also includes a series of slides as a class on continued fractions. The example [1;5,2] + 1/2 in [[Continued fraction arithmetic/G(matrix NG, Contined Fraction N) | G(matrix NG, Contined Fraction_N)]] is worked through in this class.

 

For these tasks continued fractions will be of the form: