Opened 2 years ago

Lattice precision for p-adics — at Version 7

Reported by: Owned by: caruso major sage-8.1 padics sd87 roed, TristanVaccon, saraedum, mmezzarobba, swewers N/A u/caruso/lattice_precision (Commits) 858492b4b0cd87dfb63fba0f3ad239801b98cb62

In several recent papers, David Roe, Tristan Vaccon and I explain that lattices allow a sharp track of precision: if f is a function we want to evaluate and x is an input given with some uncertainty modeled by a lattice H, then the uncertainty on the output f(x) is exactly df_x(H).

For much more details, I refer to my lecture notes http://xavier.toonywood.org/papers/publis/course-padic.pdf

The aim of this ticket is to propose a rough implementation of these ideas.

Below is a small demo (extracted from the doctest).

```   Below is a small demo of the features by this model of precision:

sage: R = ZpLP(3, print_mode='terse')
sage: x = R(1,10)

Of course, when we multiply by 3, we gain one digit of absolute
precision:

sage: 3*x
3 + O(3^11)

The lattice precision machinery sees this even if we decompose the
computation into several steps:

sage: y = x+x
sage: y
2 + O(3^10)
sage: x + y
3 + O(3^11)

The same works for the multiplication:

sage: z = x^2
sage: z
1 + O(3^10)
sage: x*z
1 + O(3^11)

This comes more funny when we are working with elements given at
different precisions:

sage: R = ZpLP(2, print_mode='terse')
sage: x = R(1,10)
sage: y = R(1,5)
sage: z = x+y; z
2 + O(2^5)
sage: t = x-y; t
0 + O(2^5)
sage: z+t  # observe that z+t = 2*x
2 + O(2^11)
sage: z-t  # observe that z-t = 2*y
2 + O(2^6)

sage: x = R(28888,15)
sage: y = R(204,10)
sage: z = x/y; z
242 + O(2^9)
sage: z*y  # which is x
28888 + O(2^15)

The SOMOS sequence is the sequence defined by the recurrence:

..MATH::

u_n =  rac {u_{n-1} u_{n-3} + u_{n-2}^2} {u_{n-4}}

It is known for its numerical instability. On the one hand, one can
show that if the initial values are invertible in mathbb{Z}_p and
known at precision O(p^N) then all the next terms of the SOMOS
sequence will be known at the same precision as well. On the other
hand, because of the division, when we unroll the recurrence, we
loose a lot of precision. Observe:

sage: R = Zp(2, 30, print_mode='terse')
sage: a,b,c,d = R(1,15), R(1,15), R(1,15), R(3,15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
4 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
13 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
55 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
21975 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
6639 + O(2^13)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
7186 + O(2^13)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
569 + O(2^13)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
253 + O(2^13)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
4149 + O(2^13)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
2899 + O(2^12)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
3072 + O(2^12)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
349 + O(2^12)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
619 + O(2^12)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
243 + O(2^12)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
3 + O(2^2)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
2 + O(2^2)

If instead, we use the lattice precision, everything goes well:

sage: R = ZpLP(2, 30, print_mode='terse')
sage: a,b,c,d = R(1,15), R(1,15), R(1,15), R(3,15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
4 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
13 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
55 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
21975 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
23023 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
31762 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
16953 + O(2^15)
sage: a,b,c,d = b,c,d,(b*d+c*c)/a; print d
16637 + O(2^15)

sage: for _ in range(100):
....:     a,b,c,d = b,c,d,(b*d+c*c)/a
sage: a
15519 + O(2^15)
sage: b
32042 + O(2^15)
sage: c
17769 + O(2^15)
sage: d
20949 + O(2^15)

BEHIND THE SCENE:

The precision is global. It is encoded by a lattice in a huge
vector space whose dimension is the number of elements having this
parent.

Concretely, this precision datum is an instance of the class
attached to the parent and is created at the same time as the
parent. (It is actually a bit more subtle because two different
parents may share the same instance; this happens for instance for
a p-adic ring and its field of fractions.)

This precision datum is accessible through the method
"precision()":

sage: R = ZpLP(5, print_mode='terse')
sage: prec = R.precision()
sage: prec
Precision Lattice on 0 object

This instance knows about all elements of the parent, it is
automatically updated when a new element (of this parent) is
created:

sage: x = R(3513,10)
sage: prec
Precision Lattice on 1 object
sage: y = R(176,5)
sage: prec
Precision Lattice on 2 objects
sage: z = R.random_element()
sage: prec
Precision Lattice on 3 objects

The method "tracked_elements()" provides the list of all tracked
elements:

sage: prec.tracked_elements()
[3513 + O(5^10), 176 + O(5^5), ...]

Similarly, when a variable is collected by the garbage collector,
the precision lattice is updated. Note however that the update
might be delayed. We can force it with the method "del_elements()":

sage: z = 0
sage: prec
Precision Lattice on 3 objects
sage: prec.del_elements()
sage: prec
Precision Lattice on 2 objects

The method "precision_lattice()" returns (a matrix defining) the
lattice that models the precision. Here we have:

sage: prec.precision_lattice()
[9765625       0]
[      0    3125]

Observe that 5^10 = 9765625 and 5^5 = 3125. The above matrix then
reflects the precision on x and y.

Now, observe how the precision lattice changes while performing
computations:

sage: x, y = 3*x+2*y, 2*(x-y)
sage: prec.del_elements()
sage: prec.precision_lattice()
[    3125 48825000]
[       0 48828125]

The matrix we get is no longer diagonal, meaning that some digits
of precision are diffused among the two new elements x and y. They
nevertheless show up when we compute for instance x+y:

sage: x
1516 + O(5^5)
sage: y
424 + O(5^5)
sage: x+y
17565 + O(5^11)

It is these diffused digits of precision (which are tracked but do
not appear on the printing) that allow to be always sharp on
precision.

PERFORMANCES:

Each elementary operation requires significant manipulations on the
lattice precision and then is costly. Precisely:

* The creation of a new element has a cost O(n) when n is the
number of tracked elements.

* The destruction of one element has a cost O(m^2) when m is the
distance between the destroyed element and the last one.
Fortunately, it seems that m tends to be small in general (the
dynamics of the list of tracked elements is rather close to that
of a stack).

It is nevertheless still possible to manipulate several hundred
variables (e.g. squares matrices of size 5 or polynomials of degree
20 are accessible).

The class "PrecisionLattice" provides several features for
introspection (especially concerning timings). If enables, it
maintains an history of all actions and stores the wall time of
each of them:

sage: R = ZpLP(3)
sage: prec = R.precision()
sage: prec.history_enable()
sage: M = random_matrix(R, 5)
sage: d = M.determinant()
sage: print prec.history()  # somewhat random
---
0.004212s  oooooooooooooooooooooooooooooooooooo
0.000003s  oooooooooooooooooooooooooooooooooo~~
0.000010s  oooooooooooooooooooooooooooooooooo
0.001560s  ooooooooooooooooooooooooooooooooooooooooo
0.000004s  ooooooooooooooooooooooooooooo~oooo~oooo~o
0.002168s  oooooooooooooooooooooooooooooooooooooo
0.001787s  ooooooooooooooooooooooooooooooooooooooooo
0.000004s  oooooooooooooooooooooooooooooooooooooo~~o
0.000198s  ooooooooooooooooooooooooooooooooooooooo
0.001152s  ooooooooooooooooooooooooooooooooooooooooo
0.000005s  ooooooooooooooooooooooooooooooooo~oooo~~o
0.000853s  oooooooooooooooooooooooooooooooooooooo
0.000610s  ooooooooooooooooooooooooooooooooooooooo
...
0.003879s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.000006s  oooooooooooooooooooooooooooooooooooooooooooooooooooo~~~~~
0.000036s  oooooooooooooooooooooooooooooooooooooooooooooooooooo
0.006737s  oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.000005s  oooooooooooooooooooooooooooooooooooooooooooooooooooo~~~~~ooooo
0.002637s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.007118s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.000008s  oooooooooooooooooooooooooooooooooooooooooooooooooooo~~~~o~~~~oooo
0.003504s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.005371s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.000006s  ooooooooooooooooooooooooooooooooooooooooooooooooooooo~~~o~~~ooo
0.001858s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.003584s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.000004s  oooooooooooooooooooooooooooooooooooooooooooooooooooooo~~o~~oo
0.000801s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.001916s  ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
0.000022s  ooooooooooooooooooooooooooooo~~~~~~~~~~~~~~~~~~~~~~oooo~o~o
0.014705s  ooooooooooooooooooooooooooooooooooo
0.001292s  ooooooooooooooooooooooooooooooooooooo
0.000002s  ooooooooooooooooooooooooooooooooooo~o

The symbol o symbolized a tracked element. The symbol ~ means that
the element is marked for deletion.

The global timings are also accessible as follows:

sage: prec.timings()   # somewhat random
'del': 0.11911273002624512,
'mark': 0.0004909038543701172,
'partial reduce': 0.0917658805847168}

```

comment:1 Changed 2 years ago by caruso

• Branch set to u/caruso/lattice_precision

comment:2 Changed 2 years ago by caruso

• Description modified (diff)

New commits:

 ​b46b474 `Pseudo-code for lattice precision`

comment:3 Changed 2 years ago by git

• Commit changed from b46b474adaa8d0f54a2bfa039939e9185ee79b4e to 4b2af06c55734697998f25742eea3659334144ae

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

 ​4b2af06 `First more-or-less working implementation`

comment:5 Changed 23 months ago by caruso

• Type changed from task to enhancement

I've worked more on my implementation and, at least, it seems now to be usable.

It still requires a lot of work (convert to Cython, write templates, write doctests, rewrite completely the class `pRational` which is a hack, etc). I however post it because I think that some people might have fun playing with it and discovering what lattice precision can do. Enjoy :-)

@mmezzarobba: I add your name in Cc because I think that you could be interested. If you're not, feel free to remove it.

Last edited 23 months ago by caruso (previous) (diff)

comment:6 Changed 23 months ago by git

• Commit changed from 4b2af06c55734697998f25742eea3659334144ae to 858492b4b0cd87dfb63fba0f3ad239801b98cb62

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

 ​858492b `Second rough implementation of lattice precision`

comment:7 Changed 23 months ago by caruso

• Description modified (diff)
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