|Sine and Cosine Accuracy on AMD64 and Pentium 4 email@example.com (Scott Robert Ladd) (2005-05-26)|
|Re: Sine and Cosine Accuracy on AMD64 and Pentium 4 firstname.lastname@example.org (glen herrmannsfeldt) (2005-05-28)|
|Re: Sine and Cosine Accuracy on AMD64 and Pentium 4 email@example.com (Jack Crenshaw) (2005-07-17)|
|Re: Sine and Cosine Accuracy on AMD64 and Pentium 4 Juergen.Kahrs@vr-web.de (=?ISO-8859-1?Q?J=FCrgen_Kahrs?=) (2005-07-17)|
|Re: Sine and Cosine Accuracy on AMD64 and Pentium 4 firstname.lastname@example.org (glen herrmannsfeldt) (2005-07-22)|
|Re: Sine and Cosine Accuracy on AMD64 and Pentium 4 email@example.com (2005-07-26)|
|From:||firstname.lastname@example.org (Henry Spencer)|
|Date:||26 Jul 2005 13:18:20 -0400|
|Organization:||SP Systems, Toronto, Canada|
|References:||05-05-215 05-07-075 05-07-082|
|Posted-Date:||26 Jul 2005 13:18:20 EDT|
glen herrmannsfeldt <email@example.com> wrote:
>There are algorithms where maintaining identities is more important
>than accuracy, some of which go by the name symplectic.
>A dynamics problem, for example, may find that conservation of energy
>is more important than the accurate final positions of the objects.
It's *very* common for numerical algorithms to depend much more on the
accuracy with which relationships between intermediate values are
preserved than on the accuracy of the individual intermediate values.
Kahan & Darcy observed, for example, that what matters to most matrix
algorithms is not the component-by-component accuracy of matrix
multiplication, but the magnitude of the error in some matrix norm
(that is, the size of `norm(product - A*B) / (norm(A)*norm(B))')
introduced by multiplication. This is significant because re-ordering
of the operations in matrix multiply to exploit pipelines, caches,
etc. -- which can make a huge difference in speed -- can noticeably
change component values but doesn't introduce significant error in
It can be very difficult to figure out just which relationships are
crucial to a particular algorithm, and how best to preserve them.
"Think outside the box -- the box isn't our friend." | Henry Spencer
-- George Herbert | firstname.lastname@example.org
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