"extended sin and cos" meant "complex sin and cos"
Kip Murray wrote:
And
0j1 exp i. 4
1 0j1 _1 0j_1
but you still can't duplicate with exp your J 6.03 feat with 0j1 ^ i. 5 4 .
Thank you for working on this.
I am uneasy about replacing zero crossings \ and / with - in the graphs
of real cos and sin, but I want to have apparent periodicities 0 = (exp
z) - exp z + 0j2p1 and
0 = (cos z) - cos z + 2p1 and 0 = (sin z) - sin z + 2p1 as in (now using
my sin and exp)
sin 0j1 + 2p1 * i. 4 1
0j1.1752
0j1.1752
0j1.1752
0j1.1752
exp 1 + 0j2p1 * i. 4 1
2.71828
2.71828
2.71828
2.71828
NB. compare
1 o. 0j1 + 2p1 * i. 4 1
0j1.1752
_3.77946e_16j1.1752
_7.55892e_16j1.1752
_1.13384e_15j1.1752
^ 1 + 0j2p1 * i. 4 1
2.71828
2.71828j_6.65787e_16
2.71828j_1.33157e_15
2.71828j_1.99736e_15
Below if you replace 0.5p1 1.5p1 by (0.5p1 + 2p1 * i.10) and do
something similar for
sin, that might do for practical purposes. Later an extended sin and
cos could
be defined in terms of exp.
Roger Hui wrote:
cos should check for 1.5p1 as well as 0.5p1 :
cos=: 2&o. * 0.5p1 1.5p1 -.@(e.!.0)~ |
With this change, we get:
mexp (i.4) * ^.0j1
1 0j1 _1 0j_1
----- Original Message -----
From: Roger Hui <[email protected]>
Date: Monday, May 25, 2009 10:10
Subject: Re: [Jchat] Clean elementary functions?
To: Chat forum <[email protected]>
I believe that if you want a "clean" result at all,
it's better to give it for sin and cos only if the argument is the
best possible 64-bit representation for pi (or pi%2). Thus:
sin =: 1&o. * 1p1 ~:!.0 |
cos =: 2&o. * 0.5p1 ~:!.0 |
re =: 9&o.
im =: 11&o.
mexp=: ^...@re * (cos j. sin)@im
exp =: mexp : (mexp@(^...@[ * ]))
2 exp 3
8
1j1 exp 2
0j2
0j1 exp 2
_1
0 = 1 + exp 0j1 * 1p1
1
exp _24
3.77513e_11
hex =: {:@(2&(3!:3))"0
unhex=: 3!:2@((}:2(3!:3) 1p1)&,)"1
sin 1p1 _1p1
0 0
hex 1p1
400921fb54442d18
sin unhex (}:hex 1p1),"1 0 '79'
5.6655e_16 _3.21629e_16
cos 0.5p1 _0.5p1
0 0
hex 0.5p1
3ff921fb54442d18
cos unhex (}:hex 0.5p1),"1 0 '79'
2.83275e_16 _1.60814e_16
----- Original Message -----
From: Kip Murray <[email protected]>
Date: Friday, May 22, 2009 18:54
Subject: Re: [Jchat] Clean elementary functions?
To: Chat forum <[email protected]>
NB. Here is a model for ^ that has 0 = 1 + ^ 0j1*1p1
load 'numeric'
cos =: clean@(2&o.)
sin =: clean@(1&o.)
mexp =: 3 : 0
'u v' =. +. y
(^u)*(cos v)+0j1*sin v
)
exp =: 3 : 0
mexp y
:
mexp y * ^. x
)
i =: 0j1
pi =: 1p1
2 exp 3
8
i exp 2
_1
0 = 1 + exp i*pi
1
exp _24
3.77513e_11
NB. That is, exp x remains positive for real x
Kip Murray
Kip Murray wrote:
This is great! /Kip
Roger Hui wrote:
In J6.03:
0j1 ^ i.5 4
1 0j1 _1 0j_1
1 0j1 _1 0j_1
1 0j1 _1 0j_1
1 0j1 _1 0j_1
1 0j1 _1 0j_1
----- Original Message -----
From: Roger Hui <[email protected]>
Date: Thursday, May 21, 2009 17:46
Subject: Re: [Jchat] Clean elementary functions?
To: Chat forum <[email protected]>
I am in sympathy with your concern.
A solution in the particular case of z^n is to do
repeated squaring instead of ^n*^.z .
The interpreter already does that for real z .
----- Original Message -----
From: Kip Murray <[email protected]>
Date: Thursday, May 21, 2009 15:48
Subject: [Jchat] Clean elementary functions?
To: Chat forum <[email protected]>
Caution, rant follows. /Kip Murray
NB. It is too bad that whereas
0j1*0j1
_1
NB. and
*:0j1
_1
NB. we get
0j1^2
_1j1.22465e_16
NB. The culprit appears to be
1 o. o. 1
1.22465e_16
NB. because 0j1^2 is calculated as
^ 2 * ^. 0j1
_1j1.22465e_16
NB. which has real part
2 o. o. 1
_1
NB. and imaginary part
1 o. o. 1
1.22465e_16
NB. Sometimes I think verb "clean"
should be incorporated NB.
in the elementary functions! Maybe just in 1&o. and 2&o.
NB. The TI-83 calculator gives
"clean" results for i^2 and e^(i
pi).
load 'numeric'
clean 0j1^2
_1
clean ^ 0j1 * 1p1
_1
0j1^2
_1j1.22465e_16
^ 0j1 * 1p1
_1j1.22465e_16
clean
1e_10&$: :(4 : 0)
if. L. y do.
x clean each y
else.
if. (3!:0 y) e. 16 16384 do.
j./"1 y * x <: | y=. +.y
else.
y * x <: |y
end.
end.
)
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