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Dear Ulrich,
Thanks for your email.
You write: "As
far as I understand the article only measurements of the
gravitational constant are discussed." Not exactly. We
cannot "measure" the gravitational constant directly. Many other
measurements are require under extremely well but not perfectly
controlled conditions, which are also measured. Then from all
these fallible measurements, bounds on the error in them, and
mathematical models predicting the value of these measurements as
a function of the unknown gravitational constant, one attempts to
compute an estimate of the gravitational constant. The situation
is what appears to be an overdetermined system of nonlinear
equations with one unknown, the gravitational constant.
In reality, this system is not overdetermined if all the
measurements are included as interval bounds. By writing this
system of nonlinear interval equations, one is assuming that the
the true value of each measurement is one of the values in the
interval measurements.
When solving this system using an interval algorithm, one obtains
an interval bound on the gravitational constant. There are a
number of assumptions: Both the underlying theory about how the
gravitational constant influences physical phenomena that we can
observe and measure, and our mathematical model for this influence
are assumed to be correct. The bounds on measurement errors are
assumed to be valid. And the interval computing system used to
perform the computations is assumed to produce no containment
failures.
This, I believe, is close to the situation involving the
computation of resonating RPMs of a steam turbine you have used as
an example of why EDP is so valuable.
My problem is that I fail to see how this can be true in either
case. To encounter the kind of catastrophic cancellation you
describe below, the interval bound on the measurement of a
quantity whose value is on the order of 10^200 would have to be
0.5*[-1,+1]. I am unaware of any physical measurement that can be
made with this accuracy. This means that using interval bounds on
the errors in physical measurements, the kind of catastrophic
cancellation you cite below simply cannot occur and be prevented
with with EDP.
Cheers,
Bill
On 9/26/13 3:21 AM, Ulrich Kulisch wrote:
Bill,
I read the paper you mentioned in the mail below. As far as
I understand the article only measurements of the
gravitational constant are discussed. Although these
mesurements are occaionlally called calculation I don't see
any way of using arithmetic to compute the constant. I
wonder wether you are teasing me. I never claimed that the
EDP is good for solving all problems in the world.
But let us assume that the gravitational constant appears as
a datum in a computation perhaps in a matrix multiplication.
If you need a guaranteed answer you would read it into the
computer as an interval where the bounds differ perhaps in
the fifth digit. So you have to compute a dot product with
this interval in one component. You would compute the minima
and the maxima of the products of the vector components and
finally you have to accumulate all the minima and all the
maxima. Let us assume that this accumulation requires
computing the sum
10²⁰⁰
+
23456 - 10²⁰⁰. (1)
If your computer provides an EDP you get the
correct answer 23456 and if the EDP is supported by hardware
you get it very fast.
If your computer does not provide an EDP the
average user will accumulate (1) in conventional
floating-point arithmetic and he gets the wrong answer 0.
I still remember the old days of computing when
every computer had its own arithmetic which often was not
best possible. Computer users, vendors, and even
mathematicians often argued that a more accurate arithmetic
is not needed since the underlying data are imprecise, or
discretization errors are much larger than rounding errors.
This kind of justification is rather dubious.
One source of error does not justify adding another one.
Even an imprecise mathematical model or imprecise data will
suffer from the use of an imprecise or sloppy arithmetic.
The systematic development of a mathematical model requires
that the error resulting from the computation can largely be
excluded. This requires the best possible arithmetic. What
happens outside the computer is the responsibility of the
user. As soon as the data are in the computer treating them
as exact is a must. The vendor is responsible for the
arithmetic that is used in the computer. To be as accurate
as possible is also a must. External and internal error
sources must be separately identifiable. The dot product is
a very frequent arithmetic operation which easily can be
computed exactly. Not providing this operation on computers
is the source of many avoidable errors.
With best regards
Ulrich
Am 20.09.2013 01:15, schrieb G. William (Bill) Walster:
Great, Ulrich.
When you do answer, you might want to explain how it is that computing
EDPs will help computing with intervals to address the following problem
<http://www.scientificamerican.com/article.cfm?id=puzzling-measurement-of-big-g-gravitational-constant-ignites-debate-slide-show&WT.mc_id=SA_CAT_SPCPHYS_20130919>
It seems to me that this is a prime example of the kind of problem on
which the interval computational and research communities should
concentrate their efforts. We have a unique opportunity in situations
like this.
Cheers,
Bill
On 9/18/13 8:37 PM, Ulrich Kulisch wrote:
Bill,
I shall answer your question if I have more time. For the moment take
the present discussion which started with computing a dot product for
vectors with interval data.
I have to leave for a meeting in one hour.
Best regards
Ulrich
Am 18.09.2013 20:02, schrieb G. William (Bill) Walster:
Ulrich,
Lets assume everything you say about EDP is correct.
Just because EDP is a fundamental mathematical operation does not mean
that it should be required in an interval arithmetic standard.
Requiring EDP in an interval arithmetic standard is a separate issue.
As I have repeatedly written and received no answer from you or anybody
else, please provide a single example of an interval computation based
on non-degenerate interval inputs that substantially benefits from the
availability of EDP.
Thanks in advance,
Respectfully,
Bill
--
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch
Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
www.kit.edu
www.math.kit.edu/ianm2/~kulisch/
KIT - Universität des Landes Baden-Württemberg
und nationales Großforschungszentrum in der
Helmholtz-Gesellschaft
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