Re: MidRad to/from InfSup (was: the "set paradigm" is harmful)
On Fri, 13 Feb 2009 11:28:35 -0100, Arnold Neumaier
<Arnold.Neumaier@xxxxxxxxxxxx> wrote:
Siegfried M. Rump schrieb:
On Thu, 12 Feb 2009 12:15:10 -0100, Arnold Neumaier
<Arnold.Neumaier@xxxxxxxxxxxx> wrote:
Shin'ichi Oishi proposed
set round up
m=0.5*(l+u)
r=m-l
I think the formula that should be in the standard is
set round up
m=l+0.5*(u-l);r=m-l
In binary arithmetic, or when intermediate results are kept to higher
precision, the two formulas appear to be equivalent.
Given this last remark, I suggest to use Oishi's proposal at least
for binary arithmetic:
- it uses only 3 rather than 4 operations,
- the midpoint is always inside [l,u], see Higham: ASNA.
But in Intlab (version 5.2), you use my new proposal, not Oishi's!
Arnold Neumaier
I use what you call your proposal since INTLAB Version 1, that
is in 1998. I think Oishi's proposal is even older than that.
My proposal is used within mtimes for matrix multiplication,
so there is no performance issue. The reason is that u-l works good
for narrow intervals, also near overflow, but l+u does not. However,
u-l is not safe for extremely wide intervals like [-realmax,realmax],
but this seems less important. But the result is always correct. For
l+u care has to be taken for [-inf,inf]. It is a design decision.
I am concerned with large full and even much larger sparse matrices.
As I said, for matrix multiplication there is no time penalty, so
it does not matter which method to use.
For scalar intervals, the design decision may be different. Then
25 % less computing time as in Oishi's method may be important.
Best wishes
Siegfried
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