#13213: Comparisons in real number field
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Reporter: vdelecroix | Owner: vdelecroix
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-5.3
Component: number fields | Resolution:
Keywords: real number field, comparison | Work issues:
Report Upstream: N/A | Reviewers: Burcin Erocal
Authors: Vincent Delecroix | Merged in:
Dependencies: | Stopgaps:
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Changes (by vdelecroix):
* status: needs_review => needs_work
Old description:
> Provide the order from RR for real quadratic number field. The current
> implementation does
> {{{
> sage: K.<sqrt2> = NumberField(x^2 - 2, 'sqrt2', embedding=1.4142)
> sage: sqrt2 < 100
> False
> }}}
>
> There is a patch for quadratic field. Another one is comming for general
> number fields. Note that this patch is partly a duplicate because of
> #7160.
>
> The lost of speed is about x5 for positive discriminant and almost
> nothing for negative ones
> {{{
> sage: K.<sqrt2> = QuadraticField(2,'sqrt2',embedding=1.4142)
> sage: a = (3*sqrt2 + 18)/7
> sage: b = (5*sqrt2 + 14)/5
> sage: %timeit a < b
> 625 loops, best of 3: 2.36 µs per loop
>
> sage: K.<s> = QuadraticField(-2)
> sage: a=3*s+2/4
> sage: b=5/7*s+1/3
> sage: %timeit a < b
> 625 loops, best of 3: 600 ns per loop
> }}}
> Timings without the patch
> {{{
> sage: sage: K.<sqrt2> = QuadraticField(2,'sqrt2',embedding=1.4142)
> sage: a = (3*sqrt2 + 18)/7
> sage: b = (5*sqrt2 + 14)/5
> sage: %timeit a < b
> 625 loops, best of 3: 491 ns per loop
>
> sage: K.<s> = QuadraticField(-2)
> sage: a=3*s+2/4
> sage: b=5/7*s+1/3
> sage: %timeit a < b
> 625 loops, best of 3: 488 ns per loop
> }}}
New description:
The order of quadratic field is not induced from the order of RR and CC
(when there is an embedding). More precisely we have
{{{
sage: K.<sqrt2> = NumberField(x^2 - 2, 'sqrt2', embedding=1.4142)
sage: sqrt2 < 100
False
}}}
which is not compatible with the order of RR and
{{{
sage: K.<i> = QuadraticField(-1)
sage: i > 1
True
sage: 1 > i
True
}}}
which is not compatible (!) with the order of CC.
There is a patch for quadratic field. Another will come for general number
fields. Note that this patch is partly a duplicate because of #7160. The
modifications for the order on complex number field modify the behavior of
many commands (output order).
The lost of speed is about x5 for positive discriminant and almost nothing
for negative ones
{{{
sage: K.<sqrt2> = QuadraticField(2,'sqrt2',embedding=1.4142)
sage: a = (3*sqrt2 + 18)/7
sage: b = (5*sqrt2 + 14)/5
sage: %timeit a < b
625 loops, best of 3: 2.36 µs per loop
sage: K.<s> = QuadraticField(-2)
sage: a=3*s+2/4
sage: b=5/7*s+1/3
sage: %timeit a < b
625 loops, best of 3: 600 ns per loop
}}}
Timings without the patch
{{{
sage: sage: K.<sqrt2> = QuadraticField(2,'sqrt2',embedding=1.4142)
sage: a = (3*sqrt2 + 18)/7
sage: b = (5*sqrt2 + 14)/5
sage: %timeit a < b
625 loops, best of 3: 491 ns per loop
sage: K.<s> = QuadraticField(-2)
sage: a=3*s+2/4
sage: b=5/7*s+1/3
sage: %timeit a < b
625 loops, best of 3: 488 ns per loop
}}}
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13213#comment:9>
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