Implement computation of Riemann period matrices etc.

[nbruin]

Include a class that supports analytic computation of period matrices and computation of endomorphism matrices.

Depends on #23140

CC: @adeines

Component: algebraic geometry

Keywords: sd86.5

Author: Nils Bruin, Alexandre Zotine

Branch/Commit: 6e689c2

Reviewer: Julian Rüth

Issue created by migration from https://trac.sagemath.org/ticket/23175

[nbruin]

Branch: u/nbruin/riemann_surface

[nbruin]

comment:2

We'll likely be rewriting history on this branch.

---

New commits:

f128723	Creation of a vectorized gauss_legendre integrator
8c82320	Change to node for degree=1, based on bugfix in mpmath:
a455f3e	some PEP8, reviewer comments; rationalized "degree" parameter for "nodes"
3ffce6a	Initial check-in of RiemannSurface

[nbruin]

Dependencies: #23140

[nbruin]

Changed keywords from none to sd86.5

[nbruin]

Commit: 3ffce6a

[sagetrac-git]

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

ce9fca4

some further imports

[sagetrac-git]

Changed commit from 3ffce6a to ce9fca4

[sagetrac-git]

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

0143571

renamed file

[sagetrac-git]

Changed commit from ce9fca4 to 0143571

[sagetrac-git]

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

b543141	Minor updates to documentation.
b21c02a	Added a few imports to documentation.
c9edca9	small changes

[sagetrac-git]

Changed commit from 0143571 to c9edca9

[sagetrac-git]

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

c26f623

avoid integer division in python code

[sagetrac-git]

Changed commit from c9edca9 to c26f623

[sagetrac-git]

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

8f8dbb1

avoid integer division in python code

[sagetrac-git]

Changed commit from c26f623 to 8f8dbb1

[sagetrac-git]

Changed commit from 8f8dbb1 to 54bdd61

[sagetrac-git]

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

bec1bf3	Fixed a doctest.
54bdd61	Merge branch 'u/nbruin/riemann_surface' of git://trac.sagemath.org/sage into t/23175/riemann_surface

[sagetrac-git]

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

5b1c2d6

Initial Check-in of RiemannSurface

[sagetrac-git]

Changed commit from 54bdd61 to 5b1c2d6

[nbruin]

comment:10

Documentation should now be more or less up to standards.

[saraedum]

Reviewer: Julian Rüth

[saraedum]

comment:11

I am making some style changes. I am still working on this, but here are already some comments.

Please fix/comment the following:

• I guess you want to make this more accessible than from sage.schemes.riemann_surfaces.riemann_surface import RiemannSurface?
• Why so much precision in the examples? Having 200 bits of precision seems like a lot. I tried a couple of the examples with less precision and found the output much easier to parse.
• TODO: have an example where self.genus < 0 to raise error
• Comment on the constant 100 in _determine_new_w and _newton_iteration (nothing fancy necessary, just say that this is a random bound that worked well in practice?)
• In period_matrix: Computes the period matrix of f given the homology basis and cohomology basis. What is "given" here?
• Why do you perform your own caching in period_matrix (self._PM) and riemann_matrix (self._RM), cohomology_basis (self._differentials) ?
• The self._differentials[1] is very confusing. Why do you need it?
• raise ValueError('Length of differentials list is not equal to genus.') should this not be a NotImplementedError?
• Does simple_vector_line_integral require a previous call to cohomology_basis to work? (The docstring makes it seem so.) If this is the case, then the method should probably not be public. In other words, the NOTE there is confusing.
• The if/else in homology_basis…could you add a comment on why you can be sure that cds[0] != cds[1]? If this could go wrong in low precision, you might want to replace the assert with something else.
• Should RiemannSurface really be an object and not at least a SageObject? You could probably add a comment about why this is not a scheme in the hierarchy of the namespace there.

There are some confusing comments in the source code, please try to make them more understandable to people who did not write this code:

# Using singular to compute the genus quickly. Later, use rank/2 of
        # gram matrix?

        # TODO: can we get rid of the comments below?
        # I'm not sure if this check will work, but could be useful?
        # I think singular's genus computation returns a negative genus
        # if it's reducible. Either way, if the genus is negative we should
        # terminate.

#TODO: Why not make a list in the first place with the edges?

        #mt = Matrix(ZZ,[[1 if i==j else 0 for j in range(4*g**2)] +
        #  [((S*w.monomial_coefficient(vars[i]).real_part()).round() for w in W] +
        #  [S*w.monomial_coefficient(vars[i]).imag_part()).round() for w in W] for i in range(len(vars))])

[saraedum]

Changed branch from u/nbruin/riemann_surface to u/saraedum/riemann_surface

[sagetrac-git]

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

f8b255b

Bugfixes and review suggestions.

[sagetrac-git]

Changed commit from 358f995 to f8b255b

[saraedum]

comment:18

Ok. Feel free to set this to positive review when the patchbot is happy.

[sagetrac-git]

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

34ac9e1

Several fixes:

[sagetrac-git]

Changed commit from f8b255b to 34ac9e1

[nbruin]

comment:20

Sorry! I ran into one case where the Voronoi cells as computed did not give rise to a homology basis (because we were discarding edges in a way that led to an non-connected diagram). I rewrote another routine to be better documented and clearer as well.

Knock-on effect of the changed voronoi cells is that a lot of doctests with arbitrary output changed, but those are not very insightful. Nonetheless, probably good form if the reviewer gives a thumbs-up to these changes before we set this to positive.

[sagetrac-git]

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

ce5097a

Two more fixes:

[sagetrac-git]

Changed commit from 34ac9e1 to ce5097a

[sagetrac-git]

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

3285684

Use sage_mantissa_exponent to get a cheaper log2

[sagetrac-git]

Changed commit from ce5097a to 3285684

[sagetrac-git]

Changed commit from 3285684 to 4eaaff0

[sagetrac-git]

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

4eaaff0

correction to implementation of voronoi_ghost

[saraedum]

comment:24

Feel free to set this back to positive review if you're confident that tests are going to pass.

[nbruin]

comment:25

OK. the bots seem busy. I've tested on 8.0beta11 and all tests pass. A previous complaint by a plugin about "EXAMPLE:" rather than "EXAMPLES:" has been fixed too, so following Julian's assessment: positive review. Preferably new issues get their own ticket.

[vbraun]

comment:27

Fails on 32-bit:

sage -t --long src/sage/schemes/riemann_surfaces/riemann_surface.py
**********************************************************************
File "src/sage/schemes/riemann_surfaces/riemann_surface.py", line 806, in sage.schemes.riemann_surfaces.riemann_surface.RiemannSurface.upstairs_edges
Failed example:
    edgeset = S.upstairs_edges(); edgeset
Expected:
    [[(0, 0), (1, 0)],
     [(0, 1), (1, 1)],
     [(0, 0), (5, 1)],
     [(0, 1), (5, 0)],
     [(1, 0), (4, 1)],
     [(1, 1), (4, 0)],
     [(2, 0), (3, 1)],
     [(2, 1), (3, 0)],
     [(2, 0), (4, 0)],
     [(2, 1), (4, 1)],
     [(3, 0), (5, 0)],
     [(3, 1), (5, 1)],
     [(4, 0), (5, 0)],
     [(4, 1), (5, 1)]]
Got:
    [[(0, 0), (1, 1)],
     [(0, 1), (1, 0)],
     [(0, 0), (5, 0)],
     [(0, 1), (5, 1)],
     [(1, 0), (4, 0)],
     [(1, 1), (4, 1)],
     [(2, 0), (3, 0)],
     [(2, 1), (3, 1)],
     [(2, 0), (4, 1)],
     [(2, 1), (4, 0)],
     [(3, 0), (5, 1)],
     [(3, 1), (5, 0)],
     [(4, 0), (5, 0)],
     [(4, 1), (5, 1)]]
**********************************************************************
File "src/sage/schemes/riemann_surfaces/riemann_surface.py", line 1423, in sage.schemes.riemann_surfaces.riemann_surface.RiemannSurface.period_matrix
Failed example:
    S.period_matrix() # abs tol 0.000001
Expected:
    [-0.77519780590366792 + 0.61819962132820207*I  -0.19145804690473785 - 1.3890819401503317*I  -2.1720559850746018 + 0.49575760460334677*I -0.23874427945779023 - 0.49575760460334677*I -0.43020232636252808 - 0.89332433554698493*I  -0.96665585280840577 - 2.0072815614785338*I]
    [  2.1720559850746018 + 0.49575760460334677*I  -3.1387118378830076 - 0.27512471421878286*I  -0.53645352644587770 - 1.1139572259315488*I    1.3968581791709339 + 1.1139572259315488*I   -1.7418536587120737 - 1.3890819401503317*I -0.96665585280840577 - 0.77088231882212961*I]
    [  0.53645352644587769 - 1.1139572259315488*I  -1.5031093792542835 - 0.89332433554698493*I  0.77519780590366793 + 0.61819962132820208*I   2.7085095115204795 - 0.61819962132820208*I   1.2054001322661960 - 0.27512471421878285*I -0.96665585280840577 + 0.22063289038456391*I]
Got:
    [     0.43020232636252809 - 1.3345901163161127*I     0.96665585280840577 - 0.77088231882212962*I      2.1720559850746018 + 0.49575760460334669*I  1.1926223897340549e-18 - 0.99151520920669346*I     0.96665585280840577 - 0.22063289038456391*I  -2.3852447794681098e-18 + 2.2279144518630976*I]
    [      1.7418536587120737 + 2.6254811828067359*I     0.96665585280840577 + 0.22063289038456391*I      0.53645352644587770 - 1.1139572259315489*I  -2.1684043449710089e-19 + 2.2279144518630977*I      0.96665585280840577 + 2.0072815614785338*I  -2.6020852139652106e-18 - 1.2363992426564042*I]
    [     -1.2054001322661960 + 1.2666399234254763*I      0.96665585280840577 - 2.0072815614785338*I    -0.77519780590366792 + 0.61819962132820217*I   3.2526065174565133e-19 - 1.2363992426564042*I     0.96665585280840577 + 0.77088231882212962*I -4.3368086899420177e-18 - 0.99151520920669343*I]
Tolerance exceeded in 33 of 36:
    -0.77519780590366792 vs 0.43020232636252809, tolerance 1e+00 > 1e-06
    + 0.61819962132820207 vs - 1.3345901163161127, tolerance 2e+00 > 1e-06
    -0.19145804690473785 vs 0.96665585280840577, tolerance 1e+00 > 1e-06
    - 1.3890819401503317 vs - 0.77088231882212962, tolerance 6e-01 > 1e-06
    -2.1720559850746018 vs 2.1720559850746018, tolerance 4e+00 > 1e-06
    -0.23874427945779023 vs 1.1926223897340549e-18, tolerance 2e-01 > 1e-06
    - 0.49575760460334677 vs - 0.99151520920669346, tolerance 5e-01 > 1e-06
    -0.43020232636252808 vs 0.96665585280840577, tolerance 1e+00 > 1e-06
    - 0.89332433554698493 vs - 0.22063289038456391, tolerance 7e-01 > 1e-06
    -0.96665585280840577 vs -2.3852447794681098e-18, tolerance 1e+00 > 1e-06
    - 2.0072815614785338 vs + 2.2279144518630976, tolerance 4e+00 > 1e-06
    2.1720559850746018 vs 1.7418536587120737, tolerance 4e-01 > 1e-06
    + 0.49575760460334677 vs + 2.6254811828067359, tolerance 2e+00 > 1e-06
    -3.1387118378830076 vs 0.96665585280840577, tolerance 4e+00 > 1e-06
    - 0.27512471421878286 vs + 0.22063289038456391, tolerance 5e-01 > 1e-06
    -0.53645352644587770 vs 0.53645352644587770, tolerance 1e+00 > 1e-06
    1.3968581791709339 vs -2.1684043449710089e-19, tolerance 1e+00 > 1e-06
    + 1.1139572259315488 vs + 2.2279144518630977, tolerance 1e+00 > 1e-06
    -1.7418536587120737 vs 0.96665585280840577, tolerance 3e+00 > 1e-06
    - 1.3890819401503317 vs + 2.0072815614785338, tolerance 3e+00 > 1e-06
    -0.96665585280840577 vs -2.6020852139652106e-18, tolerance 1e+00 > 1e-06
    - 0.77088231882212961 vs - 1.2363992426564042, tolerance 5e-01 > 1e-06
    0.53645352644587769 vs -1.2054001322661960, tolerance 2e+00 > 1e-06
    - 1.1139572259315488 vs + 1.2666399234254763, tolerance 2e+00 > 1e-06
    -1.5031093792542835 vs 0.96665585280840577, tolerance 2e+00 > 1e-06
    - 0.89332433554698493 vs - 2.0072815614785338, tolerance 1e+00 > 1e-06
    0.77519780590366793 vs -0.77519780590366792, tolerance 2e+00 > 1e-06
    2.7085095115204795 vs 3.2526065174565133e-19, tolerance 3e+00 > 1e-06
    - 0.61819962132820208 vs - 1.2363992426564042, tolerance 6e-01 > 1e-06
    1.2054001322661960 vs 0.96665585280840577, tolerance 2e-01 > 1e-06
    - 0.27512471421878285 vs + 0.77088231882212962, tolerance 1e+00 > 1e-06
    -0.96665585280840577 vs -4.3368086899420177e-18, tolerance 1e+00 > 1e-06
    + 0.22063289038456391 vs - 0.99151520920669343, tolerance 1e+00 > 1e-06
**********************************************************************
File "src/sage/schemes/riemann_surfaces/riemann_surface.py", line 1454, in sage.schemes.riemann_surfaces.riemann_surface.RiemannSurface.riemann_matrix
Failed example:
    S.riemann_matrix() #abs tol 0.0000001
Expected:
    [        0.50000000000000000 + 1.3228756555322953*I       -0.25000000000000000 + 0.66143782776614765*I  6.7220534694101275e-18 - 2.1684043449710089e-18*I]
    [      -0.25000000000000000 + 0.66143782776614764*I        0.25000000000000000 + 0.66143782776614765*I     0.50000000000000000 - 1.7347234759768071e-18*I]
    [-6.5052130349130266e-19 + 3.4694469519536142e-18*I     0.50000000000000000 + 3.4694469519536142e-18*I        0.25000000000000000 + 0.66143782776614765*I]
Got:
    [       0.37500000000000000 + 0.33071891388307383*I     0.50000000000000000 - 6.9388939039072284e-18*I    -0.50000000000000001 - 7.9146758591441824e-18*I]
    [    0.50000000000000001 - 4.3368086899420177e-18*I        0.24999999999999999 + 0.66143782776614764*I -1.1275702593849246e-17 + 1.3877787807814457e-17*I]
    [   -0.50000000000000000 + 1.5178830414797062e-17*I  2.7105054312137611e-18 - 5.4210108624275222e-18*I        0.24999999999999998 + 0.66143782776614762*I]
Tolerance exceeded in 10 of 18:
    0.50000000000000000 vs 0.37500000000000000, tolerance 1e-01 > 1e-07
    + 1.3228756555322953 vs + 0.33071891388307383, tolerance 1e+00 > 1e-07
    -0.25000000000000000 vs 0.50000000000000000, tolerance 8e-01 > 1e-07
    + 0.66143782776614765 vs - 6.9388939039072284e-18, tolerance 7e-01 > 1e-07
    6.7220534694101275e-18 vs -0.50000000000000001, tolerance 5e-01 > 1e-07
    -0.25000000000000000 vs 0.50000000000000001, tolerance 8e-01 > 1e-07
    + 0.66143782776614764 vs - 4.3368086899420177e-18, tolerance 7e-01 > 1e-07
    0.50000000000000000 vs -1.1275702593849246e-17, tolerance 5e-01 > 1e-07
    -6.5052130349130266e-19 vs -0.50000000000000000, tolerance 5e-01 > 1e-07
    0.50000000000000000 vs 2.7105054312137611e-18, tolerance 5e-01 > 1e-07
**********************************************************************
3 items had failures:
   1 of   6 in sage.schemes.riemann_surfaces.riemann_surface.RiemannSurface.period_matrix
   1 of   6 in sage.schemes.riemann_surfaces.riemann_surface.RiemannSurface.riemann_matrix
   1 of   6 in sage.schemes.riemann_surfaces.riemann_surface.RiemannSurface.upstairs_edges
    [206 tests, 3 failures, 28.84 s]

[sagetrac-git]

Changed commit from 4eaaff0 to 6e689c2

[sagetrac-git]

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

6e689c2

make doctests more robust

[nbruin]

comment:29

Sigh ... doctesting doctests with non-uniquely represented objects is a pain. It would be nice if we'd have a way of getting wider architecture exposure without Volker having to intervene. Let's try this.

[saraedum]

comment:31

Looks good. (Yes, I guess a better CI/CD infrastructure would be helpful. As I am just transferring the company I work for to gitlab CI/CD, I discussed this quite a bit with roed during the past few days actually. But I guess it would be quite some effort to swap our patchbot/buildbot out for a more standardized solution…)

[vbraun]

Changed branch from u/nbruin/riemann_surface to 6e689c2