Re: [pure-lang-users] math.pure

[Albert G. <Dr....@t-...>]

Albert Graef wrote:
> John Cowan wrote:
>> Here is the Common Lisp discussion of where the branch cuts of the
>> transcendental functions are: http://tinyurl.com/5rbsng .
> 
> Thanks a bunch for that link, that should help me to revise the complex 
> stuff in math.pure.

Ok, I did some minor cosmetic surgery to the complex trigs and hyps, 
actually most of my definitions (originally pilfered from Bronstein) 
seemed to be pretty much in line with that document already. I also 
fixed up my completely nonsensical complex sqrt definition, wonder how 
that slipped in. ;-)

What still remains to be done is to add some type guards on the mixed 
real/complex rules, so that we can later (when Eddie's GSL module 
materializes) overload the common operators for scalar+vector/matrix 
kind of stuff.

One thing that keeps me wondering is Kahan's suggestion for the acosh:

acosh z@(x+:y)	|
acosh z@(r<:t)	= 2*ln (sqrt ((z+1)/2)+sqrt ((z-1)/2));

This is the definition I currently use (from Mathworld):

acosh z@(x+:y)	|
acosh z@(r<:t)	= ln (z+sqrt (z-1)*sqrt (z+1));

This has the same branch cut (-inf,1), and seems to yield the same 
results AFAICT (up to rounding of course). Kahan's definition looks more 
complicated and requires more operations, but given that Kahan wrote it, 
I guess that there must be some reason he wrote it that way. ;-)

So can anyone think of a reason why one might prefer Kahan's definition? 
Numerical stability maybe?

Cheers,
Albert

-- 
Dr. Albert Gr"af
Dept. of Music-Informatics, University of Mainz, Germany
Email:  Dr....@t-..., ag...@mu...
WWW:    http://www.musikinformatik.uni-mainz.de/ag