I was rereading this thread and I remember that in APL there is a function
that can be applied to B to get A. Is there an equivalent function in J?
f=: 13 :'^@o.j.0.5*i.y'
]A=: f 3 4
1 0j1 _1 0j_1
1 0j1 _1 0j_1
1 0j1 _1 0j_1
g=: 13 :'([:^o.)j.0.5*i.y'
B=:g 3 4
1 6.12323e_17j1 _1j1.22465e_16 _1.83697e_16j_1
1j_2.44929e_16 3.06162e_16j1 _1j3.67394e_16 _4.28626e_16j_1
1j_4.89859e_16 5.51091e_16j1 _1j6.12323e_16 _2.44991e_15j_1
((f 3 4)-:g 3 4)
1
Linda
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Roger Hui
Sent: Thursday, January 17, 2013 12:11 PM
To: Programming forum
Subject: Re: [Jprogramming] Hermitian from triangular
> You may not know that ^ x j. y by definition is (^ x) * (cos + 0j1
> *
sin) y
This is too complex a definition for my taste. I prefer one where (for
example) ^z is defined to be a function which is equal to its derivative,
and derive the equation you cited as a theorem. And ^. is the inverse of ^
.
Regarding 0 = 1 + ^ 1p1 * 0j1, see
http://www.jsoftware.com/jwiki/Essays/Euler's_Identity
In J7.01, you can do this:
^@o. j. 0.5 * i. 3 4
1 0j1 _1 0j_1
1 0j1 _1 0j_1
1 0j1 _1 0j_1
^@o. j. 2e9 + 0.5 * i. 3 4
1 0j1 _1 0j_1
1 0j1 _1 0j_1
1 0j1 _1 0j_1
On Thu, Jan 17, 2013 at 6:30 AM, km <[email protected]> wrote:
> Linda, about logarithms of negative numbers
>
> First of all, you know the number e =: ^ 1 and you know ^ y is e^y .
> You may not know that ^ x j. y by definition is (^ x) * (cos + 0j1
> * sin) y where cos =: 2&o. and sin =: 1&o. . I first learned this
> in a college math class called Complex Analysis. A good reference is E.
B. Saff and A.
> D. Snider, Fundamentals of Complex Analysis, Pearson Education, Inc. 2003.
>
> Anyway, a famous identity in higher math is
>
> _1 = ^ 0j1 * o. 1
> 1
>
> which should tell you that
>
> (0j1 * o.1) = ^. _1
> 1
>
> i.e., negative numbers can have logarithms to the base e . For more
> on this, please see Saff and Snider's Chapter 3.
>
> Kip Murray
>
> Sent from my iPad
>
>
> On Jan 17, 2013, at 4:22 AM, "Linda Alvord" <[email protected]>
> wrote:
>
> > Isn't the log of negative numbers indefined?
> >
> > This is a problem:
> >
> > %1&o.+0
> > _
> > %1&o.-0
> > _
> >
> > This is nice!
> >
> > %1&o.%_
> > _
> > %1&o.%__
> > __
> >
> >
> > The csc is very small for negative numbers close to zero and very
> > large
> for
> > very small positive numbers.
> >
> > Linda
> >
> > -----Original Message-----
> > From: [email protected]
> > [mailto:[email protected]] On Behalf Of Bo
> > Jacoby
> > Sennt: Thursday, January 17, 2013 3:37 AM
> > To: [email protected]
> > Subject: Re: [Jprogramming] Hermitian from triangular
> >
> > Henry, How is negative zero different from positive zero when taking
> > the log?
> > ^.%__ NB. log -0
> > __
> > ^.%_ NB. log +0
> > __
> >
> >
> > - Bo
> >
> >
> >> ________________________________
> >> Fra: Henry Rich <[email protected]>
> >> Til: [email protected]
> >> Sendt: 0:38 torsdag den 17. januar 2013
> >> Emne: Re: [Jprogramming] Hermitian from triangular
> >>
> >> Negative zero makes sense as a last vestige of gradual underflow;
> >> and
> > anyway, it's well-behaved: it looks like 0 except when you take the
> > log, reciprocal, or square root. In any normal computation, it goes
> > away. In contrast, NaN messes up anything it touches.
> >>
> >> I think we've had negative 0 in J forever. If NaN is a data virus,
> >> -0
> is a
> > virus that has been inserted into our DNA.
> >>
> >> Henry Rich
> >>
> >> On 1/16/2013 4:45 PM, Raul Miller wrote:
> >>> On Wed, Jan 16, 2013 at 4:35 PM, Henry Rich <[email protected]>
> wrote:
> >>>> Negative zero isn't a bug, it's a feature that numerical types,
> >>>> especially William Kahan, wanted to get into IEEE-754 to help out
> >>>> some things. I'm not expert enough to explain.
> >>>
> >>> Something similar could be said about NaN.
> >> -------------------------------------------------------------------
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