#19112: Add a function "isometry" to the quadratic forms package.
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Reporter: tgaona | Owner:
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-6.10
Component: quadratic forms | Resolution:
Keywords: isometry | Merged in:
Authors: Tyler Gaona | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/tgaona/ticket/19112 | 582ae4500eb3652446c6f829310298953f8dbd02
Dependencies: | Stopgaps:
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Comment (by tgaona):
Replying to [comment:23 jdemeyer]:
> Why does the final term need to be a special case? I don't like special
cases unless they are justified.
It was originally necessary because the `short_vector_list_up_to_length()`
function didn't work for 1-dimensional quadratic forms, and the final
iteration of the algorithm operates on two 1-dimensional forms. However,
since that's been replaced by the PARI method, it's no longer necessary to
have it as a special case, so thanks for pointing that out.
> Are you sure it's worth to check `is_diagonal`? Is there significant
gain?
I suppose not, and simplifies the code to remove it, so I will.
> Are you sure it's worth to check `is_rationally_isometric`?
I believe this is necessary. The loop that looks for two vectors such that
the modified bases including them will be non-singular will try vectors
indefinitely until it finds a satisfactory pair. If the forms aren't
equivalent there's no guarantee such a pair will be found.
> Do you really need the `copy` in the function `isometry`?
Nope. That's a relic from an early version when `isometry` and
`diagonal_isometry` were one function. I'll remove it, thanks for catching
that.
> Can you explain this block of code:
{{{
+ # Find a vector w such that Q(v) = F(w) where v = [1, ..., 0]
+ # v, w = vectors_of_common_length_dev(Q, F, q_basis, f_basis,
i)
+ v = vector([0] * (n - i))
+ index = 0;
+ while True:
+ v[index] = v[index] + 1
+ index = (index + 1) % (n - i)
+ c = Q(v)
+ try:
+ w = F.solve(c)
+ #print("Find vectors {0} and {1} such that Q(v) =
F(w)".format(v, w))
+ if not zero_row(f_basis, w, i) and not
zero_row(q_basis, v, i):
+ break
+ except ArithmeticError:
+ # No solution found, try another vector.
+ pass
}}}
This block finds a pair of vectors `v` and `w` such that `Q(v) == F(v)`.
These vectors will represent a linear combination of the vectors in the
basis for each quadratic form. It's necessary that modifying the bases to
include these vectors not produce a basis whose matrix is singular. So
essentially this loop just looks for a pair of vectors satsifying these
properties. It starts with `v = [1, 0, 0]` (for a 3-dimensional form) and
finds `w` by calling `F.solve(Q(v))`. If this pair doesn't work, it finds
a new `v` and starts over. I get new `v`'s by incrementing each term in
the vector so the first few vectors that are generated are: `[1, 0, 0],
[1, 1, 0], [1, 1, 1], [2, 1, 1]...`
Also, I realized that matrices have an `is_singular` function, so I'm
getting rid of the `zero_row` function.
----
New commits:
||[http://git.sagemath.org/sage.git/commit/?id=582ae4500eb3652446c6f829310298953f8dbd02
582ae45]||{{{Minor bugfixes/refactoring on isometry}}}||
--
Ticket URL: <http://trac.sagemath.org/ticket/19112#comment:31>
Sage <http://www.sagemath.org>
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