Re: Change definition of complex::norm

[Brad Lucier <lucier at math dot purdue dot edu>]

> 
> Brad Lucier <lucier@math.purdue.edu> writes:
> 
> | > |  is
> | > | that the formula
> | > | 
> | > | norm(z) = real(z)^2 + imag(z)^2
> | > | 
> | > | does give good relative error for normal results, and good absolute
> | > | error for denormal results.
> | > 
> | > This isn't the case in the test you posted.  Did I misread it?
> | 
> | This *is* the case for the test I posted.  The difference between
> | 0 and 4.whatever\times e-324 is within the absolute error of
> | denormal results.
> 
> I recall the part of your assertion I agreed with
> 
>   # The relevant error tolerance when calculating a quantity x is roughly
>   # 
>   # \max (|x| * 2^{-53}, 2^{-53-1024})
> 
>   # The first quantity is the relative tolerance, the second is the absolute
>   # tolerance for numbers in the denormal range.  One would like the
>   # computation of any quantity x to be within a small multiple of this
>   # tolerance.  There can also be qualitative issues---returning a finite
>   # answer as often as possible, etc.
> 
> 2^{-53-1024} is about 0.6175820573e-324. I guess one of the point
> of our disagreement is whether a factor of about 7 or 8 is considered
> "small". 

Sorry, I forgot that the exponent range is not symmetric, the formula I
gave for the absolute error tolerance is incorrect.  The smallest
nonzero denormal in double precision IEEE is 4.940656e-324.  And in the
example I gave, the answer in exact arithmetic is exactly halfway between
4.940656e-324 and 0.0, so there is little to choose between them.

Brad