Update points() in projective_homset.py and affine_homset.py to work over CC and CDF

[rlmiller]

Updated numerical point finding in the projective_homset.py and affine homset.py files to work over the complex numbers and the complex double field.

[rlmiller]

New commits:

​35cf9f1

23627 update points() to work over CC

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c8fb086

23693 updated points() to work over CC in affine_homset.py

[bhutz]

This ticket and #23693 have the same branch and contains both the changes for affine_homset and projective_homset. What did you mean to do with these two tickets?

Otherwise, just a few minor things:

• CDF should work in addition to CC

• a couple of the comments in the affine_homset reference projective. Fix those.

• the sorting line is commented out in affine_homset and should be restored

• You need to update the docmentation of .rational_points() to describe the changes and add an example there.

[raghukul01]

Rebase from current master

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​1c5dba9

Added Support for CDF

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​e5e83ed	Added Support for CDF
​fa9418f	#23627 Added support to handle duplicates

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​e4f49f5

23627 Added support to handle duplicates

[bhutz]

Just a couple simple comments before I test functionality.

Instead of importing, its better to use members function whenever possible (i.e., .floor())

The two for loops

for i in range(len(dupl_points)):
    for j in range(len(dupl_points)):
        if j > i:
            u = dupl_points[i]

can be simplified to

for i in range(len(dupl_points)):
    u = dupl_points[i]
    for j in range(i+1, len(dupl_points)):

[raghukul01]

Actually after product with tolerance, the result becomes float, and hence floor(), member function doesn't exist. So there are 2 options

1) Convert tolerance to RealNumber? and then use .floor()

2) use floor from math

currently I implemented using 2), should I change to 1)? or do you have any other idea?

Thanks

Replying to bhutz:

Just a couple simple comments before I test functionality.

Instead of importing, its better to use members function whenever possible (i.e., .floor())

The two for loops

for i in range(len(dupl_points)):
    for j in range(len(dupl_points)):
        if j > i:
            u = dupl_points[i]

can be simplified to

for i in range(len(dupl_points)):
    u = dupl_points[i]
    for j in range(i+1, len(dupl_points)):

[bhutz]

I would do 1) since you should do basic input checking on tolerance anyway, so you can convert to RealField? and check that it's positive.

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​dc753b8

23627 Added support to handle duplicates

[bhutz]

in testing, I found that the method does very poorly when the base ring is numerical such as CC or RR. For simple subschemes, it was ok, but for even mildly complicated schemes (such as projective closures of intersections of affine curves) you could easily get wildly wrong answers. So, I think the verbose warning is warranted in this case.

This also led me to separate out the case where the subscheme is defined over a number field and the user wants points numerically approximated which is much faster. This seems to work well in my testing (up to precision/tolerance).

Additionally, the documenation was significantly updated as well as normalizing all the inputs for the various points/rational_points methods.

So I'm marking this as needs-review, but it should undergo fairly extensive testing as I'm sure there are cases I missed.

---

New commits:

​3a02dd0

23627: add numerical point support

[raghukul01]

There are couple of documentation updated

• Value of dimension need to be corrected (line no := for projective_homset 74, 188, 383, 453; for affine_homset 129, 138).

• Calculation of dimension in projective homset is wrong (needs to be one less than I.dimension(), line no :- 185, 380).

• Subscheme should be replaced by scheme (line no: for projective homset 75, 293; for affine homset 129, 317).

• Projective word should be replaced by affine in affine homset (line no := 283, 436).

• Example in docstring of projective_homset for CC, and CDF are similar. different example need to be added.

• For NumberFields embeddings should be specified during formation, this needs to be added in documentation.

• In affine homset example below is better to illustrate working in CDF, compared to current example

  sage: A.<x1, x2> = AffineSpace(CDF, 2)
  sage: E = A.subscheme([x1^2 + x2^2 + x1*x2 + 1, x1 + x2])
  sage: E(A.base_ring()).points() 
  verbose 0 (124: affine_homset.py, points) Warning: computations in the
  numerical fields are inexact;points may be computed partially or
  incorrectly.
  [(-2.77555756156e-17 + 1.0*I, 2.77555756156e-17 - 1.0*I), (-1.0*I,
  1.0*I)]
  

• There should be an example in projective and affine homset to illustrate use of tolerance, some thing like this

  sage: A.<x1, x2> = AffineSpace(QQ, 2)
  sage: E = A.subscheme([x1^1000 + x2^2 + x1*x2 + 1, x1 + x2])
  sage: len(E(A.base_ring()).numerical_points(F=CDF, zero_tolerance =1e-11))
  987
  sage: len(E(A.base_ring()).numerical_points(F=CDF, zero_tolerance =1e-10))
  1000
  

Following gives wrong result

sage: P.<x1, x2> = ProjectiveSpace(QQ, 1)
sage: E = A.subscheme([x1^2 -3*x1 + 2, x2])
sage: E(A.base_ring()).numerical_points(F=CDF, zero_tolerance =1e-10)
[(1.0, 0.0), (2.0, 0.0)]
sage: E(A.base_ring()).points()
[(1, 0), (2, 0)]

They are same projective point, and hence only 1 should be returned.

This also

sage: P.<x1> = ProjectiveSpace(QQ, 0)
sage: E = A.subscheme([x1^2 -3*x1 + 2])
sage: E(A.base_ring()).points(bound=5)
[(1, -5), (1, -4), (1, -3), (1, -2), (1, -1), (1, 0), (1, 1), (1, 2),
(1, 3), (1, 4), (1, 5), (2, -5), (2, -4), (2, -3), (2, -2), (2, -1), (2,
0), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5)]

---

New commits:

​dbb9326

Reviewier's patch #23627

[bhutz]

Changes made. A few comment below

• The dimension checks are actually correct, that is why the comments are there

• made doc changes where appropriate (can't really have a 0 dim scheme in sage)

• I modified the examples as specified

• However, the two examples above that are 'failing' are actually problems with the examples. Notice that you are defining a project space P and using subschemes of A.

---

New commits:

​79b5e0e	Merge branch 'u/raghukul01/stoll' of git://trac.sagemath.org/sage into 23627
​7c09986	23627: changes from review

[raghukul01]

Those 2 examples were so silly of me :P

[vbraun]

Test failures... Always run 'make ptestlong' before setting stuff to positive review.

[raghukul01]

I guess most of the doc test are failing due to following line in sage/coding/source-coding/hamming_code.py PFn = [list(p) for p in X.point_set(F).points(F)] (line no 147)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​537ecbd

23627: fix doctest

[bhutz]

looks like hamming_codes was calling .points() and trying to pass in the field. This was never possible in the first place (only could pass in bound or precision) it was only coming up since the keyword parameters were moved to *kwds and not listed explicitly.

This one change fixed all doctests for me.

[raghukul01]

Passed on my system as well.

[vbraun]

merge conflict

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d076ca9

Merge branch 8.3.beta6 into 23627

[bhutz]

merge conflict resolved

[raghukul01]

Following doctest are failing however they are not due to this ticket, they are failing on the develop(8.3.beta7) branch as well.

sage -t --long --warn-long 91.1 src/sage/tests/cmdline.py  # 1 doctest failed
sage -t --long --warn-long 91.1 src/sage/functions/other.py  # 1 doctest failed
sage -t --long --warn-long 91.1 src/sage/rings/complex_arb.pyx  # 24 doctests failed
sage -t --long --warn-long 91.1 src/sage/misc/sphinxify.py  # 1 doctest failed
sage -t --long --warn-long 91.1 src/sage/rings/real_arb.pyx  # 7 doctests failed
sage -t --long --warn-long 91.1 src/sage/matrix/matrix_complex_ball_dense.pyx  # 1 doctest failed
sage -t --long --warn-long 91.1 src/sage/rings/polynomial/polynomial_complex_arb.pyx  # 2 doctests failed

[vbraun]

On 32-bit

**********************************************************************
File "src/sage/schemes/projective/projective_homset.py", line 160, in sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field.points
Failed example:
    E(P.base_ring()).points()
Expected:
    verbose 0 (70: projective_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
    [(-1.0000000000000004*I : 1.0 : 0.0),
     (0.0 : 0.9999999999999997*I : 1.0),
     (0.0 : 2.7755575615628914e-17 - 1.0*I : 1.0),
     (0.9999999999999997*I : 1.0 : 0.0),
     (2.7755575615628914e-17 - 1.0*I : 0.0 : 1.0),
     (2.7755575615628914e-17 + 1.0*I : 0.0 : 1.0)]
Got:
    verbose 0 (70: projective_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
    [(-1.3091741232762466e-17 - 1.0*I : 0.0 : 1.0),
     (-1.3091741232762466e-17 + 1.0*I : 0.0 : 1.0),
     (-1.0000000000000002*I : 1.0 : 0.0),
     (0.0 : -1.0000000000000002*I : 1.0),
     (0.0 : 0.9999999999999998*I : 1.0),
     (0.9999999999999998*I : 1.0 : 0.0)]
**********************************************************************
File "src/sage/schemes/projective/projective_homset.py", line 335, in sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field.numerical_points
Failed example:
    X(K).numerical_points(F=CDF)
Expected:
    [(-1.475773161594552 : 1.475773161594552 : 1.0),
     (1.475773161594551 : 1.4757731615945517 : 1.0)]
Got:
    [(-1.475773161594552 : 1.475773161594552 : 1.0),
     (1.4757731615945517 : 1.4757731615945517 : 1.0)]
**********************************************************************
2 items had failures:
   1 of  22 in sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field.numerical_points
   1 of  20 in sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field.points
    [75 tests, 2 failures, 1.65 s]
----------------------------------------------------------------------
sage -t --long src/sage/schemes/projective/projective_homset.py  # 2 doctests failed
----------------------------------------------------------------------

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​84b290a

23627: fix doctests for 32bit

[bhutz]

corrected a minor whitespace typo as well.

[raghukul01]

How can I run doctest for a 32-bit machine? Mine is 64.

[raghukul01]

tested, LGTM

[vbraun]

sage -t --long src/sage/schemes/projective/projective_homset.py
**********************************************************************
File "src/sage/schemes/projective/projective_homset.py", line 146, in sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field.points
Failed example:
    L=E(P.base_ring()).points();L
Expected:
    verbose 0 (70: projective_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
    [(-0.500000000000000 + 0.866025403784439*I : 1.00000000000000 : 0.000000000000000),
    (-0.500000000000000 - 0.866025403784439*I : 1.00000000000000 : 0.000000000000000),
    (-1.00000000000000*I : 0.000000000000000 : 1.00000000000000),
    (0.000000000000000 : 0.000000000000000 : 1.00000000000000),
    (1.00000000000000*I : 0.000000000000000 : 1.00000000000000),
    (1.00000000000000 : 1.00000000000000 : 0.000000000000000)]
Got:
    verbose 0 (71: projective_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
    [(-0.500000000000000 + 0.866025403784439*I : 1.00000000000000 : 0.000000000000000),
     (-0.500000000000000 - 0.866025403784439*I : 1.00000000000000 : 0.000000000000000),
     (-1.00000000000000*I : 0.000000000000000 : 1.00000000000000),
     (0.000000000000000 : 0.000000000000000 : 1.00000000000000),
     (1.00000000000000*I : 0.000000000000000 : 1.00000000000000),
     (1.00000000000000 : 1.00000000000000 : 0.000000000000000)]
**********************************************************************
File "src/sage/schemes/projective/projective_homset.py", line 161, in sage.schemes.projective.projective_homset.SchemeHomset_points_projective_field.points

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​f4c1aff	Merge branch 8.3.rc1 into t/23627/23627_numerical_points
​c6f8d4f	23627: update doctest

[bhutz]

merged in and updated doctests for latest beta