On 26 July 2011 15:00, Cliff Reiter <[email protected]> wrote: > Consider the Euclidean model for gcd. It appears to model +. > reasonably for (positive) nice numbers. It nears comparison tolerance > for naughty numbers. > Best, Cliff > > gcd=:4 : 0 > r0=.x > r1=.y > ......................
As the use of <. is incorporated in | (residue), a somewhat simpler definition of the Euclidean gcd suffices: gcd=. 4 : 0 while. 0~:y do. 'x y'=. y,y|x end. x ) However, there are more interesting points to consider. -- What is the meaning (in GCD) of the word `greatest' when the arguments are complex numbers? (Perhaps `greatest' then is related to the magnitude of the complex number, but then there may not be a unique `greatest' number.) -- Is the above model of GCD the one implemented in +.? Is this the only reasonable model? In particular, what if the notions of `integer quotient' and `residue' -- and therefore of GCD -- are based not on <. but something else? -- What properties does +. exhibit for complex (even integer or rational) and floating-point arguments? (And through this, what are the valuable uses of GCD with such arguments?) The article of Eugene McDonnell that Roger referred to only considers part of these questions. A more detailed study of the field is: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.1389 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
