These seemed to be what I expected: p. 1 1 _1 ┌──┬─────────────────┐ │_1│1.61803 _0.618034│ └──┴─────────────────┘ p. _1 _1 1 ┌─┬─────────────────┐ │1│1.61803 _0.618034│ └─┴─────────────────┘ Linda
-----Original Message----- From: [email protected] [mailto:[email protected]] On Behalf Of Linda A Alvord Sent: Thursday, January 7, 2016 5:59 AM To: [email protected] Subject: Re: [Jprogramming] the p. equation solver Roger, If I type in fresh jqt: p. 1 2 3 4 5 ┌─┬───────────────────────────────────────────────────────────────────────────┐ │5│0.137832j0.678154 0.137832j_0.678154 _0.537832j0.358285 _0.537832j_0.358285│ └─┴───────────────────────────────────────────────────────────────────────────┘ p.. 1 2 3 4 5 2 6 12 20 However I get the same answer when I copied the example in the vocabulary in the second example. What explains this? Linda -----Original Message----- From: [email protected] [mailto:[email protected]] On Behalf Of Roger Hui Sent: Thursday, January 7, 2016 2:20 AM To: Programming forum Subject: Re: [Jprogramming] the p. equation solver p. uses Laguerre's Method with polishing on the original polynomial using Newton iteration. It can no doubt be simplified and/or improved. A couple of examples of using p. : c=: p. <1+i.20 $c 21 3 7$c 2432902008176640000 _8752948036761600000 13803759753640704000 _12870931245150988800 8037811822645051776 _3599979517947607200 1206647803780373360 _311333643161390640 63030812099294896 _10142299865511450 1307535010540395 _135585182899530 11310276995381 _756111184500 40171771630 _1672280820 53327946 _1256850 20615 _210 1 c are the coefficient's of Wilkinson's polynomial <https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial>, whose roots are p. c ┌─┬──────────────────────────────────────────────────┐ │1│20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1│ └─┴──────────────────────────────────────────────────┘ On Wed, Jan 6, 2016 at 10:53 PM, 'Bo Jacoby' via Programming < [email protected]> wrote: > Dear experts. > The equation solver (p.) is marvelous! However I do not know how it is > implemented. > Computing the complex roots of algebraic equations should be > considered basic. If you can do that, then you need not to learn about > division or square roots. > I made this elementary equation solver in J. (Durand-Kerner method). > > poly =.+/@([*]^~/~i.@#@[) > > step=.]-poly*[:*/-/~@]^<:@=/~@i.@<:@#@[ > > iter =.](step^:_)_0.4j0.9^i.@<:@# norm =.[:(% {:)]}.~[:-[:+/[:*/\0=|. > solv =.iter@norm > A polynomial is represented by an array of coefficients, the constant > term first. > poly computes values of a polynomial. > 0 40 0 _10 0 0 poly i:4 > 480 150 0 _30 0 30 0 _150 _480 > norm normalizes a polynomial such that the leading coefficient is one, > without changing the roots. > norm 0 40 0 _10 0 0 > 0 _4 0 1 > step improves a set of approximate roots to a normalized polynomial. > 0 _4 0 1 step _0.4j0.9^i.3 > 1.98849j0.159402 _0.860465j_0.735105 _1.12803j0.575703 iter makes a > sufficient number of steps. > iter 0 _4 0 1 > 2 0 _2 > solv solves even non-normalized polynomials > solv 0 40 0 _10 0 0 > 2 0 _2 > My questions: > 1. Is this how (p.) is implemented? > 2. Can it be simplified or improved? > Thanks! Bo. > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
