# Changes between Version 2 and Version 7 of Ticket #23505

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Timestamp:
08/02/17 00:00:07 (16 months ago)
Comment:

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Unmodified
• Property Commit changed from b46b474adaa8d0f54a2bfa039939e9185ee79b4e to 858492b4b0cd87dfb63fba0f3ad239801b98cb62
• Property Type changed from task to enhancement
 v2 {{{ sage: R = ZpLP(3) 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) 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 = \frac {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, 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) 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) 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} }}} PS: For now, the code is far from being ready for review. Any help will be of course quite appreciated.