[Jprogramming] Cool roots

Wed Dec 12 12:25:24 UTC 2012

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 <rogerhui.canada at gmail.com>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>
>>     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 <HenryHRich at nc.rr.com> 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 <km at math.uh.edu> 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 <rogerhui.canada at gmail.com>
>>>>> wrote:
>>>>>
>>>>>   Example from the Iverson and McDonnell *Phrasal
>>>>>> Forms*<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 <km at math.uh.edu> 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
>>>>>>>
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>>>
>>
>>
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