Re: back to the roots
Michel,
I agree that in situations, such as your example, when you are
evaluating a monotonic function over an interval and therefore can use
endpoint evaluation, extra precision can be useful. However, surely,
this is not the general case. It can be a huge mistake to use input
numbers that are only good to 4 or 5 decimal digits of accuracy and then
interpret them as infinitely precise, as Ulrich himself did in his note
about the steam turbine that blew up and killed 6 people.
Cheers,
Bill
P. S. Sorry for the delayed response. I had to recover from a failed
hard drive.
On 6/28/13 2:09 PM, Michel Hack wrote:
George Corliss, replying to Bill Walster's comments on Exact Dot Prouct:
I interpret Bill's point as asking, "If your input is good to three
figures, then your answer is good to about three figures. How does
a infinitely precise arithmetic provide useful information, when you
should not pay any attention to digits 4 and beyond anyway?"
For linear problems, yes. The situation is different for non-linear
cases. If I take a square root, my relative precision actually improves
and I might get twice as many significant digits as in my input. Now
consider an algorithm that iteratively takes square roots followed by
an unwinding loop that effectively squares the results until it delivers
my answer. If I don't have VERY high precision at the deeper levels, I
will end up with ENTIRE as the best-known enclosure.
The EDP may be capable of dealing with these situations.
Michel.
---Sent: 2013-06-28 21:25:11 UTC
On 6/28/13 2:09 PM, Michel Hack wrote:
George Corliss, replying to Bill Walster's comments on Exact Dot Prouct:
I interpret Bill's point as asking, "If your input is good to three
figures, then your answer is good to about three figures. How does
a infinitely precise arithmetic provide useful information, when you
should not pay any attention to digits 4 and beyond anyway?"
For linear problems, yes. The situation is different for non-linear
cases. If I take a square root, my relative precision actually improves
and I might get twice as many significant digits as in my input. Now
consider an algorithm that iteratively takes square roots followed by
an unwinding loop that effectively squares the results until it delivers
my answer. If I don't have VERY high precision at the deeper levels, I
will end up with ENTIRE as the best-known enclosure.
The EDP may be capable of dealing with these situations.
Michel.
---Sent: 2013-06-28 21:25:11 UTC