> I assumed you used Newton's method to shine up the final values of the roots, but I never knew you tried rational results too.
I don't remember that it is as simple as you described. Since the J source is now available you can look it up if you are interested. FYI: it's not rational results that make the difference. On Wed, Dec 12, 2012 at 4:25 AM, Henry Rich <[email protected]> wrote: > I have to disagree. That's way cool. > > I assumed you used Newton's method to shine up the final values of the > roots, but I never knew you tried rational results too. > > It is possible to produce accurate results for quadratics without such > heroic measures. You just can't use the quadratic formula like you learned > it in highschool. > > There's the usual quadratic formula: > > (2*a) %~ (-b) (+,-) %: (b^2) - 4*a*c > > and there's a less-well-known alternative form: > > (2*c) % (-b) (-,+) %: (b^2) - 4*a*c > > Each version produces two roots. The roots created are more accurate when > +-(-b) is positive, so you can get one accurate root from each formula. > > To calculate, use the combined form > > ((a %~ ]) , (c % ])) _0.5 * (%: (*:b) - 4*a*c) ((* *) + ]) b > > Accuracy has a coolness all its own. > > Henry Rich > > On 12/12/2012 3:13 AM, Roger Hui wrote: > >> ... produce more accurate results in some difficult cases. >>> >> >> c=: p. <20$1.5 >> c >> 3325.26 _44336.8 280799 _1.1232e6 3.18239e6 _6.78911e6 1.13152e7 >> _1.50869e7 >> 1.63441e7 _1.45281e7 1.0654e7 _6.45695e6 3.22847e6 _1.3245e6 441501 >> _117734 >> 24527.8 _3847.5 427.5 _30 1 >> p. c >> ┌─┬───────────────────────────**──────────────────────────────** >> ──────────────────────┐ >> │1│3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 3r2 >> 3r2 3r2│ >> └─┴───────────────────────────**──────────────────────────────** >> ──────────────────────┘ >> >> >> >> On Tue, Dec 11, 2012 at 11:29 PM, Roger Hui <[email protected]>** >> wrote: >> >> You may or may not know that p. employs some extraordinary measures which >>> produce more accurate results in some difficult cases. But those >>> extraordinary measures are not "cool". For example: >>> >>> w=: p. <1+i.20 NB. Wilkinson's polynomial<http://en.** >>> wikipedia.org/wiki/Wilkinson_**polynomial<http://en.wikipedia.org/wiki/Wilkinson_polynomial> >>> > >>> w >>> 2432902008176640000 _8752948036761600000 13803759753640704000 >>> _12870931245150988800 8037811822645051776 _3599979517947607200 >>> 1206647803780373360 _311333643161390640 63030812099294896 >>> _10142299865511450 1307535010540395 _135585182899530 11310276995381 >>> _7561... >>> >>> p. w >>> ┌─┬───────────────────────────**───────────────────────┐ >>> │1│20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1│ >>> └─┴───────────────────────────**───────────────────────┘ >>> >>> >>> >>> >>> >>> On Tue, Dec 11, 2012 at 3:40 PM, Henry Rich <[email protected]> >>> wrote: >>> >>> None of these cute ways is very accurate in tough cases: >>>> >>>> 0j15 ": (2*a) %~ (-b) (+,-) %: (b^2) - 4*a*c [ 'a b c' =. 1e_6 1e6 1e_6 >>>> 0.000000000000000 _1000000000000.000000000000000 >>>> >>>> But p. does better: >>>> >>>> 0j15 ": 1 {:: p. c,b,a >>>> _1000000000000.000000000000000 _0.000000000001000 >>>> >>>> Henry Rich >>>> >>>> >>>> >>>> On 12/11/2012 6:29 PM, Roger Hui wrote: >>>> >>>> There are some cheeky (or is it cheesy?) versions: >>>>> >>>>> (2*a) %~ (-b) (+,-) %: (b^2) - 4*a*c NB. Kip Murray >>>>> (2*a) %~ - b (+,-) %: (b^2) - 4*a*c >>>>> (+:a) %~ - b (+,-) %: (*:b) - 4*a*c >>>>> -: a %~ - b (+,-) %: (*:b) - 4*a*c >>>>> >>>>> >>>>> >>>>> On Tue, Dec 11, 2012 at 11:37 AM, km <[email protected]> wrote: >>>>> >>>>> It appears this could be translated into J as the rather cool >>>>> >>>>>> >>>>>> (2*a) %~ (-b) (+,-) %: (b^2) - 4*a*c >>>>>> >>>>>> Sent from my iPad >>>>>> >>>>>> >>>>>> On Dec 11, 2012, at 12:59 PM, Roger Hui <[email protected]> >>>>>> wrote: >>>>>> >>>>>> Example from the Iverson and McDonnell *Phrasal >>>>>> >>>>>>> Forms*<http://www.jsoftware.****com/papers/fork.htm<http://** >>>>>>> www.jsoftware.com/papers/fork.**htm<http://www.jsoftware.com/papers/fork.htm> >>>>>>> >>paper >>>>>>> (which >>>>>>> introduced fork): >>>>>>> >>>>>>> (-b)(+,-)√((b*2)-4×a×c)÷2×a >>>>>>> >>>>>>> √ is a postulated APL primitive, spelled %: in J. >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Tue, Dec 11, 2012 at 10:49 AM, km <[email protected]> wrote: >>>>>>> >>>>>>> What is the coolest way of programming the quadratic formula in J? >>>>>>> We >>>>>>> >>>>>>>> >>>>>>>> are >>>>>>> >>>>>> >>>>>> finding the roots of polynomial c + x*(b + x*a) without using p. . I >>>>>>> >>>>>>>> >>>>>>>> offer >>>>>>> >>>>>> >>>>>> >>>>>>> roots >>>>>>>> 3 : 0 >>>>>>>> 'a b c' =. y >>>>>>>> q =. %: (b^2) - 4*a*c >>>>>>>> (2*a) %~ (-b) + q,-q >>>>>>>> ) >>>>>>>> roots 1 3 2 >>>>>>>> _1 _2 >>>>>>>> roots 1 0 1 >>>>>>>> 0j1 0j_1 >>>>>>>> roots 1 _2 1 >>>>>>>> 1 1 >>>>>>>> >>>>>>>> partly as problem definition. I am looking for cool roots verbs! >>>>>>>> >>>>>>>> Kip Murray >>>>>>>> >>>>>>>> Sent from my iPad >>>>>>>> >>>>>>>> ------------------------------****----------------------------** >>>>>>>> --** >>>>>>>> ---------- >>>>>>>> For information about J forums see http://www.jsoftware.com/** >>>>>>>> forums.htm >>>>>>>> <http://www.jsoftware.com/**forums.htm<http://www.jsoftware.com/forums.htm> >>>>>>>> > >>>>>>>> >>>>>>>> ------------------------------****----------------------------** >>>>>>> --** >>>>>>> ---------- >>>>>>> For information about J forums see http://www.jsoftware.com/** >>>>>>> forums.htm >>>>>>> <http://www.jsoftware.com/**forums.htm<http://www.jsoftware.com/forums.htm> >>>>>>> > >>>>>>> >>>>>>> ------------------------------****----------------------------** >>>>>> --** >>>>>> ---------- >>>>>> For information about J forums see http://www.jsoftware.com/** >>>>>> forums.htm >>>>>> <http://www.jsoftware.com/**forums.htm<http://www.jsoftware.com/forums.htm> >>>>>> > >>>>>> >>>>>> ------------------------------****----------------------------** >>>>>> --** >>>>>> >>>>> ---------- >>>>> For information about J forums see http://www.jsoftware.com/**** >>>>> forums.htm <http://www.jsoftware.com/**forums.htm><http://www.** >>>>> jsoftware.com/forums.htm <http://www.jsoftware.com/forums.htm>> >>>>> >>>>> ------------------------------****----------------------------**--** >>>>> >>>> ---------- >>>> For information about J forums see http://www.jsoftware.com/**** >>>> forums.htm <http://www.jsoftware.com/**forums.htm><http://www.** >>>> jsoftware.com/forums.htm <http://www.jsoftware.com/forums.htm>> >>>> >>>> >>> >>> ------------------------------**------------------------------** >> ---------- >> For information about J forums see >> http://www.jsoftware.com/**forums.htm<http://www.jsoftware.com/forums.htm> >> >> ------------------------------**------------------------------** > ---------- > For information about J forums see > http://www.jsoftware.com/**forums.htm<http://www.jsoftware.com/forums.htm> ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
