I’ve been trying to write a conjunction that will find the zeros to a function using the Newton-Raphson method. The simplest way to do this is probably:
English: x_n+1 = x_n - f ( x_n ) / f ‘ ( x_n ) J: eznewt=: 2 : ‘ ( - u % u D.1 ) ^: n ‘ This works fine for scalar -> scalar functions, but won’t work if the rank of the result is AND the rank of the arguments are above 1. Probably the most evident situation where this would be a problem is if one were searching for a minimum instead of a zero, in which case the algorithm would be applied to the derivative of the function: +/@:*: D.1 newt 30 ] 3 4 As I understand it, this would give a result shape at each iteration (with a function u) of: ( $y ) , $ u eznewt 1 y where A is the original argument, or the result of the previous iteration. What we would want is a result with shape $y . First, let’s get some rules clear: - the syntax should be: (verb) newt (# of iterations) (argument) - the argument can be of any rank - the shape of the argument matches the shape of the result at any iteration - this implies that $ u y matches i. 0 or $ y The hard part is getting u % u D.1 to have the shape of the argument. If y is a vector, and u y is a scalar, then u D.1 y will be a vector, and u D.1 D.1 y will be a $y by $y matrix. All that is needed then is to take its diagonal with (<0 1)&|: . But what if y is a matrix? Since $ u D.1 y -: ( $ y ) , $ u y , I though maybe running y's axis together with |: might work newt=: 2 : '(- u diag_and_divide u D.1)^:n’ diag_and_divide=: [ % ] |:~ -@#@$@[ <@}. i.@#@$@] but something about it doesn’t. My head is $pinning now, and I figured I’d send this and then take a break. Thanks in advance! Louis ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
