Improve interface to SymPy solvers

[rwst]

At the moment with solve(...,algorithm='sympy') the SymPy function solveset is called for single expressions, and solve for systems. However, it turns out that solve can handle some single-expression tasks where solveset fails. This ticket should add code that analyzes solveset output and calls solve in case no solution was found. This probably includes simplifying the returned ConditionSets.

References:

• https://groups.google.com/g/sympy/c/v_YLkX4QuRY/m/50uHEGbHBAAJ (Mar 2022)

Depends on #24171

CC: @mforets

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/24142

[rwst]

Dependencies: #22322

[rwst]

comment:2

These sets can be returned by solveset: FiniteSet, EmptySet, Interval, Intersection, Union, ConditionSet, ImageSet, Complement, plus the algebraic sets of numbers like ZZ, RR, CC. We can directly translate FiniteSet, EmptySet, Interval. ConditionSet is a AND-combination of a relation and a set, so this can be translated into list format too if the set is finite/interval. Often RR/CC are given as set, and they may be redundant (CC is default with no domain= option given, and RR is implicated with any inequality as condition).

[rwst]

comment:3

Should we translate to sage.set as optional output?

[rwst]

comment:4

Let's use sage.set when we can't express the solveset result as list, and later support an option to always use sage.set. As to calling solve do this when EmptySet or the original expression is returned.

[rwst]

comment:5

Possibilities:

• translate SymPy's result if possible to list or dict notation; if not possible leave it as is. Disadvantages are that new classes are introduced from SymPy.
• translate SymPy's result to a sage.set and then, if possible, to list or dict notation; if not possible leave it as sage.set. This would make it necessary to improve sage.set with at least image sets and complex regions.
• always translate SymPy's result to a sage.set like above, making set notation the default even for the Maxima solver.

A further problem is that output is different between SymPy's solve and solveset. The relation/boolean notation of solve can be translated to set notation for the reals however (see #24156).

[rwst]

comment:6

Replying to @rwst:

ConditionSet is a AND-combination of a relation and a set, so this can be translated into list format too if the set is finite/interval.

Not so, it is a AND-combination of a relation and an element statement, i.e. symbol in set. This means it cannot be expressed in list format.

[rwst]

Changed dependencies from #22322 to #22322, #24171

[rwst]

comment:8

As an example this result set from SymPy:

In [9]: solveset(abs(x) - n, x, S.Reals)
Out[9]: ([0, ∞) ∩ {n}) ∪ ((-∞, 0] ∩ {-n})

is with #24156 and #24171 constructible in Sage:

sage: from sage.sets.set import Set_object_union, Set_object_intersection
sage: _ = var('n')
sage: Set_object_union(Set_object_intersection(RealSet(x>=0),Set([n])), Set_obje
....: ct_intersection(RealSet(x<=0),Set([-n])))
Set-theoretic union of Set-theoretic intersection of [0, +oo) and {n} and Set-theoretic intersection of (-oo, 0] and {-n}

[kcrisman]

comment:10

See possibly also #31926 or some of its constituent tickets.

[mkoeppe]

Changed dependencies from #22322, #24171 to #24171

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1 +1,4 @@
 At the moment with `solve(...,algorithm='sympy')` the SymPy function `solveset` is called for single expressions, and `solve` for systems. However, it turns out that `solve` can handle some single-expression tasks where `solveset` fails. This ticket should add code that analyzes `solveset` output and calls `solve` in case no solution was found. This probably includes simplifying the returned `ConditionSet`s.
+
+References:
+- https://groups.google.com/g/sympy/c/v_YLkX4QuRY/m/50uHEGbHBAAJ (Mar 2022)