Hi Ted,
On 11-12-13 04:52 PM, (Ted Harding) wrote:
[...]
Now, computer programs for numerical computation can broadly
be divided into two types.
In one, "arbitrary precision" is available: you can tell
the program how many decimal digits you want it to work to.
An example of this is 'bc':
http://en.wikipedia.org/wiki/Bc_programming_language
You can set as many decimal ditgits as you like, *provided*
they fall within the storage capacity of your computer, for
which an upper bound is the storage capacity of the Universe
(see above). For integers and results which surpass the
decimal places you have set, the result will be an approximation.
Inevitably.
AFAICT, with bc and other tools doing arithmetic on arbitrary large
integers, operations like +, -, *, ^ etc either give the exact answer
or they fail. That's the beauty of those tools. Otherwise you could
call them "pointless" or "broken".
Arbitrary-precision vs fixed-precision is slightly off topics though.
In particular I didn't suggest that doing sum() on an integer vector
should use arbitrary large integers internally to do the computation.
Cheers,
H.
In the other type, the program is written so as to embody
integers to a fixed maximum number of decimal (or binary)
digits. An example of this is R (and most other numerical
programs). This may be 32 bits or 64 bits. Any result ot
computation which involve smore than this numer of bits
is inevitably an approximation.
Provided the user is aware of this, there is no need for
your "It should always return the correct value or fail."
It will return the correct value if the integers are not
too large; otherwise it will retuirn the best approximation
that it can cope with in the fixed finite storage space
for which it has been programmed.
There is an implcit element of the arbitrary in this. You
can install 32-bit R on a 64-bit-capable machine, or a
64-bit version. You could re-program R so that it can
work to, say, 128 bits or 256 bits even on a 32-bit machine
(using techniques like those that underlie 'bc'), but
that would be an arbitrary choice. However, the essential
point is that some choice is unavoidable, since if you push
it too far the Universe will run out of particles -- and the
computer industry will run out of transistors long before
you hit the Universe limit!
So you just have to accept the limits. Provided you are aware
of the approximations which may set in at some point, you can
cope with the consequences, so long as you take account of
some concept of "adequacy" in the inevitable approximations.
Simply to "fail" is far too unsophisticated a result!
Hoping this is useful,
Ted.
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E-Mail: (Ted Harding)<ted.hard...@wlandres.net>
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Date: 14-Dec-11 Time: 00:52:49
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