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authorRelease Manager <release@sagemath.org>2018-03-07 00:24:21 +0100
committerVolker Braun <vbraun.name@gmail.com>2018-03-07 00:24:21 +0100
commit30d778252cb5774929460c6a62de8b18e5714384 (patch)
tree5338bea158194cadd7ea3d80e4f307313e30531c
parentTrac #21524: configure.ac: write build/make/Makefile within an AC_CONFIG_FILE... (diff)
parentFix doctest (diff)
Trac #23505: Lattice precision for p-adics
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. You can play with the latest version of this by clicking on launch binder [https://github.com/saraedum/sage-binder-env/tree/t-23505 -lattice-precision here]. 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 "sage.rings.padic.lattice_precision.PrecisionLattice". It is 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 {'add': 0.25049376487731934, 'del': 0.11911273002624512, 'mark': 0.0004909038543701172, 'partial reduce': 0.0917658805847168} }}} URL: https://trac.sagemath.org/23505 Reported by: caruso Ticket author(s): Xavier Caruso Reviewer(s): David Roe, Julian RĂ¼th
-rw-r--r--src/doc/en/reference/references/index.rst7
-rw-r--r--src/sage/categories/pushout.py7
-rw-r--r--src/sage/rings/padics/all.py4
-rw-r--r--src/sage/rings/padics/factory.py651
-rw-r--r--src/sage/rings/padics/generic_nodes.py465
-rw-r--r--src/sage/rings/padics/lattice_precision.py2835
-rw-r--r--src/sage/rings/padics/local_generic.py79
-rw-r--r--src/sage/rings/padics/local_generic_element.pyx6
-rw-r--r--src/sage/rings/padics/misc.py4
-rw-r--r--src/sage/rings/padics/padic_base_generic.py40
-rw-r--r--src/sage/rings/padics/padic_base_leaves.py274
-rw-r--r--src/sage/rings/padics/padic_generic.py50
-rw-r--r--src/sage/rings/padics/padic_lattice_element.py1311
-rw-r--r--src/sage/rings/padics/padic_printing.pyx7
14 files changed, 5581 insertions, 159 deletions
diff --git a/src/doc/en/reference/references/index.rst b/src/doc/en/reference/references/index.rst
index a78c2ae..4a9098b 100644
--- a/src/doc/en/reference/references/index.rst
+++ b/src/doc/en/reference/references/index.rst
@@ -673,6 +673,13 @@ REFERENCES:
.. [Crossproduct] Algebraic Properties of the Cross Product
:wikipedia:`Cross_product`
+.. [CRV2018] Xavier Caruso, David Roe and Tristan Vaccon.
+ *ZpL: a p-adic precision package*, (2018) :arxiv:`1802.08532`.
+
+.. [CRV2014] Xavier Caruso, David Roe and Tristan Vaccon.
+ *Tracking p-adic precision*,
+ LMS J. Comput. Math. **17** (2014), 274-294.
+
.. [CS1986] \J. Conway and N. Sloane. *Lexicographic codes:
error-correcting codes from game theory*, IEEE
Trans. Infor. Theory **32** (1986) 337-348.
diff --git a/src/sage/categories/pushout.py b/src/sage/categories/pushout.py
index 602117f..bccd0b5 100644
--- a/src/sage/categories/pushout.py
+++ b/src/sage/categories/pushout.py
@@ -2318,7 +2318,7 @@ class CompletionFunctor(ConstructionFunctor):
"""
rank = 4
_real_types = ['Interval', 'Ball', 'MPFR', 'RDF', 'RLF', 'RR']
- _dvr_types = [None, 'fixed-mod','floating-point','capped-abs','capped-rel','lazy']
+ _dvr_types = [None, 'fixed-mod', 'floating-point', 'capped-abs', 'capped-rel', 'lattice-cap', 'lattice-float']
def __init__(self, p, prec, extras=None):
"""
@@ -2329,14 +2329,15 @@ class CompletionFunctor(ConstructionFunctor):
- ``prec``: an integer, yielding the precision in bits. Note that
if ``p`` is prime then the ``prec`` is the *capped* precision,
while it is the *set* precision if ``p`` is ``+Infinity``.
+ In the ``lattice-cap`` precision case, ``prec`` will be a tuple instead.
- ``extras`` (optional dictionary): Information on how to print elements, etc.
If 'type' is given as a key, the corresponding value should be a string among the following:
- 'RDF', 'Interval', 'RLF', or 'RR' for completions at infinity
- - 'capped-rel', 'capped-abs', 'fixed-mod' or 'lazy' for completions at a finite place
- or ideal of a DVR.
+ - 'capped-rel', 'capped-abs', 'fixed-mod', 'lattice-cap' or 'lattice-float'
+ for completions at a finite place or ideal of a DVR.
TESTS::
diff --git a/src/sage/rings/padics/all.py b/src/sage/rings/padics/all.py
index 3e361a9..0c6d31d 100644
--- a/src