| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 2035.1 | | HANNAH::OSMAN | see HANNAH::IGLOO$:[OSMAN]ERIC.VT240 | Tue Apr 02 1996 20:06 | 6 |
|
What does "e .. 1" mean in this case ?
It almost looks like something like "e is in the range a..b".
/Eric
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| 2035.2 | Makes sense as a range | WIBBIN::NOYCE | EV5 issues 4 instructions per meter | Tue Apr 02 1996 20:07 | 3 |
| I read it to mean
e <= exponent_of_a_number <= 1
where e < -m
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| 2035.3 | yes, it was a range | FLOYD::YODER | MFY | Tue Apr 02 1996 20:17 | 1 |
| Yes, it was meant as a range. Sorry, I was mixing notations.
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| 2035.5 | solution and generalization | FLOYD::YODER | MFY | Thu Apr 11 1996 18:43 | 17 |
| I have found a proof that is easier than the first one I found. Unfortunately
there doesn't seem to be anything special about the unit circle here. Also, the
generalization seems less elegant than the original result somehow. Sigh.
A generalization of the given theorem is:
If u, v are nonnegative reals (not necessarily in D) such that u + v = 1, then
round(u) # round(v) <= 1.
For bases b > 2 the proof is easy (including when b is odd). The maximum errors
in round(u) and round(v) are (1/2)b^-m; but the next number in D above 1 is N =
1+b^(1-m), so round(u)+round(v) is closer to 1 than N.
This gives a 16-ton clue as to how to prove the b = 2 case: here, unless u = v =
1/2, in which case the theorem is obvious, one of u and v must be > 1/2 and the
other < 1/2. The rounding error for the smaller one, then, is at most
(1/4)2^-m, and reasoning similar to the above gives us the desired result.
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| 2035.6 | Only halfway there? | WIBBIN::NOYCE | EV5 issues 4 instructions per meter | Fri Apr 12 1996 15:03 | 2 |
| .5 helps if you can show that u*u isn't too much bigger than u^2.
That isn't obvious to me yet.
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| 2035.7 | reply to .6 | FLOYD::YODER | MFY | Mon Apr 15 1996 14:31 | 2 |
| The new u and v, as it were, that you apply .5 to are u^2 and v^2, that is, the
*exact* (not rounded) squares of the given u and v.
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| 2035.8 | | WIBBIN::NOYCE | EV5 issues 4 instructions per meter | Mon Apr 15 1996 15:54 | 5 |
| Got it. I read too slowly before.
Sort of a "meet-in-the-middle" analysis for error propagation across
multiple operations: Add in the error for the early operations, and
use the axioms of rounding for the last operation. Cute!
|