Here are three one line versions of the quadratic roots. They include most of
the ideas which have been included so far.
f=: 13 :'(((a %~ ]), (c % ])) _0.5 * b +`-@.(0>[) %: (*:b) - 4*a*c)[''a b
c''=:y'
f
3 : '(((a %~ ]), (c % ])) _0.5 * b +`-@.(0>[) %: (*:b) - 4*a*c)[''a b c''=:y'
f 1 3 2
_2 _1
f 1 0 1
0j_1 0j1
g=: 13 : '((2*a)%~(-b)(+,-)%:(*:b)-4*a*c)[''a b c ''=:y'
g
3 : '((2*a)%~(-b)(+,-)%:(*:b)-4*a*c)[''a b c ''=:y'
g 1 3 2
_1 _2
g 1 0 1
0j1 0j_1
h=: 13 : '(2*0{y)%~(-1{y)(+,-)%:(*:1{y)-4*(0{y)*2{y'
h
(2 * 0 { ]) %~ ([: - 1 { ]) (+ , -) [: %: ([: *: 1 { ]) - 4 * (0 { ]) * 2 { ]
h 1 3 2
_1 _2
h 1 0 1
0j1 0j_1
Only the third one has a tacit version which can be studied further:
5!:4 <'h'
┌─ 2
├─ *
┌─────┤ ┌─ 0
│ └───┼─ {
│ └─ ]
├─ ~ ─── %
│ ┌─ [:
│ ├─ -
──┤ ┌───┤ ┌─ 1
│ │ └────┼─ {
│ │ └─ ]
│ │ ┌─ +
│ ├───┼─ ,
│ │ └─ -
└─────┤
│ ┌─ [:
│ ├─ %:
│ │ ┌─ [:
│ │ ├─ *:
└───┤ ┌───┤ ┌─ 1
│ │ └────┼─ {
│ │ └─ ]
│ ├─ -
└────┤ ┌─ 4
│ ├─ *
│ │ ┌─ 0
└───┤ ┌───┼─ {
│ │ └─ ]
└────┼─ *
│ ┌─ 2
└───┼─ {
└─ ]
Any ideas to improve these?
Linda
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of km
Sent: Thursday, December 13, 2012 10:57 AM
To: [email protected]
Subject: Re: [Jprogramming] Cool roots
Try
rootsHR =: 3 : 0
'a b c' =. y
((a %~ ]) , (c % ])) _0.5 * b +`-@.(0>[) %: (*:b) - 4*a*c
)
rootsHR 1 3 2
_2 _1
Inside this rootsHR, a b c are "locals" which do not interfere with the global
a b c .
Kip Murray
Sent from my iPad
On Dec 13, 2012, at 1:13 AM, "Linda Alvord" < <mailto:[email protected]>
[email protected]> wrote:
> It there some way to have these three work the same way in a script. Here's
> my jijs:
>
> a=:0 { ]
> b=:1 { ]
> c=:2 { ]
> roots2=:(2*a)%~([:-b)(+,-)[:%:([:*:b)-4*a*c
> roots2 1 3 2
> _1 _2
> roots2 1 1 1
> _0.5j0.866025 _0.5j_0.866025
> roots2 1 0 _1
> 1 _1
> 'a b c'=:1 1 1
> rootsHR=:((a %~ ]) , (c % ])) _0.5 * b +`-@.(0>[) %: (*:b) - 4*a*c
> rootsHR
> _0.5j_0.866025 _0.5j0.866025
> erase names 'a b c'
> 1 1 1
> a=:0 { ]
> b=:1 { ]
> c=:2 { ]
> roots2=:(2*a)%~([:-b)(+,-)[:%:([:*:b)-4*a*c
> roots2 1 3 2
> _1 _2
>
> Linda
>
>
> -----Original Message-----
> From: <mailto:[email protected]>
> [email protected]
> [ <mailto:[email protected]>
> mailto:[email protected]] On Behalf Of km
> Sent: Wednesday, December 12, 2012 11:09 PM
> To: <mailto:[email protected]> [email protected]
> Subject: Re: [Jprogramming] Cool roots
>
> Cool!
>
> Sent from my iPad
>
>
> On Dec 12, 2012, at 6:00 PM, Henry Rich < <mailto:[email protected]>
> [email protected]> wrote:
>
>> Oh, I see. the signum function mustn't return 0. Should be
>>
>> ((a %~ ]) , (c % ])) _0.5 * b +`-@.(0>[) %: (*:b) - 4*a*c
>>
>>
>> Henry Rich
>>
>> On 12/12/2012 10:30 AM, km wrote:
>>> Henry, the combined form below fails when b is 0. Kip
>>>
>>> r
>>> 3 : 0
>>> 'a b c' =. y
>>> ((a %~ ]) , (c % ])) _0.5 * (%: (*:b) - 4*a*c) ((* *) + ]) b
>>> )
>>> r 1 3 2
>>> _2 _1
>>> r 1 2 1
>>> _1 _1
>>> r 1 0 1
>>> 0 _
>>> r 1 0 _1
>>> 0 __
>>> roots 1 0 1
>>> 0j1 0j_1
>>> roots 1 0 _1
>>> 1 _1
>>>
>>> Sent from my iPad
>>>
>>>
>>> On Dec 12, 2012, at 6:25 AM, Henry Rich < <mailto:[email protected]>
>>> [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 <
>>>>> <mailto:[email protected]> [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 <
>>>>>> <mailto:[email protected]> [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 < <mailto:[email protected]>
>>>>>>>> [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
>>>>>>>>> < <mailto:[email protected]> [email protected]>
>>>>>>>>> wrote:
>>>>>>>>>
>>>>>>>>> Example from the Iverson and McDonnell *Phrasal
>>>>>>>>>> Forms*<http://www.jsoftware.**com/papers/fork.htm<http://www.
>>>>>>>>>> j software.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 < <mailto:[email protected]>
>>>>>>>>>> [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/**> 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/**> 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/**>
>>>>>>>>> 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%3chttp:/www.jsoftware.com/>
>>>>>>>> http://www.jsoftware.com/**forums.htm<http://www.jsoftware.com/
>>>>>>>> f
>>>>>>>> orums.htm>
>>>>>>>>
>>>>>>>>
>>>>>>>> ------------------------------**------------------------------*
>>>>>>>> *
>>>>>>> ----------
>>>>>>> For information about J forums see
>>>>>>> <http://www.jsoftware.com/**forums.htm%3chttp:/www.jsoftware.com/f>
>>>>>>> http://www.jsoftware.com/**forums.htm<http://www.jsoftware.com/f
>>>>>>> o
>>>>>>> rums.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> 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>
> http://www.jsoftware.com/forums.htm
----------------------------------------------------------------------
For information about J forums see <http://www.jsoftware.com/forums.htm>
http://www.jsoftware.com/forums.htm
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