On Tue, Nov 05, 2024 at 05:01:36PM +0100, 'Ralf Hemmecke' via FriCAS - computer
algebra system wrote:
> Hi Martin, hi Waldek,
>
> I just stumpled over the behaviour below.
> The result is not wrong in the case of algebraic numbers, but I wonder why
> the resulting recurrence is not "normalized" in some way.
>
> OK, one cannot simply take out the "gcd" of the coefficients, because that
> wouldn´t work for AN without recognizing that the numbers in the recurrence
> are actually integers.
>
> Can you explain your design decision or is it just an oversight and there is
> room for improvement?
>
> Ralf
>
>
> %%% (1) -> cq := [(1/2)^k for k in 0..20]
>
> (1)
> 1 1 1 1 1 1 1 1 1 1 1 1 1 1
> [1, -, -, -, --, --, --, ---, ---, ---, ----, ----, ----, ----, -----,
> 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384
> 1 1 1 1 1 1
> -----, -----, ------, ------, ------, -------]
> 32768 65536 131072 262144 524288 1048576
> Type:
> List(Fraction(Integer))
> %%% (2) -> sq := guessRec(cq, functionName=='co).1
>
> (2) [co(n): - 2 co(n + 1) + co(n) = 0, co(0) = 1]
> Type:
> Expression(Integer)
> %%% (3) -> ca := [((1/2)::AN)^k for k in 0..20]
>
> (3)
> 1 1 1 1 1 1 1 1 1 1 1 1 1 1
> [1, -, -, -, --, --, --, ---, ---, ---, ----, ----, ----, ----, -----,
> 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384
> 1 1 1 1 1 1
> -----, -----, ------, ------, ------, -------]
> 32768 65536 131072 262144 524288 1048576
> Type:
> List(AlgebraicNumber)
> %%% (4) -> sa := guessRec(ca, functionName=='co).1
>
> (4)
> [
> co(n):
> - 738578176637708424660750 co(n + 1) + 369289088318854212330375 co(n)
> = 0
> ,
> co(0) = 1]
> Type:
> Expression(Integer)
In case of algebraic numbers we use fraction-free solver which works
over quite gerneal rings but due to this can not remove common
factors. In case of algebraic numbers one can try to remove
integer common divisor. I think that this is not done mainly due
to structure of the solver. Namely over integers solver removes
common divisors already during solving. But general solver can
not do this in general, it would need special case for algebraic
numbers to do this. In your example solver could cheat, that
is recognize that all numbers are rational and hand work to
integer solver. But such case is probably quite rare in normal
use and when testing there is advantage in passing such problems
to general solver.
--
Waldek Hebisch
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