Corrections to infinities returned by mpmath

[rwst]

It seems in some cases the infinities returned from mpmath need to be changed by the function's evalf() in order to be correct:
Before:

sage: zeta(SR(1))
Infinity
sage: zeta(SR(1.))   <---------
+infinity
sage: type(zeta(SR(1)))
<type 'sage.symbolic.expression.Expression'>
sage: type(zeta(SR(1)).pyobject())
<class 'sage.rings.infinity.UnsignedInfinity'>
sage: type(zeta(SR(1.)))
<type 'sage.symbolic.expression.Expression'>
sage: type(zeta(SR(1.)).pyobject())
<type 'sage.rings.real_mpfr.RealNumber'>

The ticket should result in:

sage: zeta(SR(1))
Infinity
sage: zeta(SR(1.))
Infinity
sage: type(zeta(SR(1)))
<type 'sage.symbolic.expression.Expression'>
sage: type(zeta(SR(1)).pyobject())
<class 'sage.rings.infinity.UnsignedInfinity'>
sage: type(zeta(SR(1.)))
<type 'sage.symbolic.expression.Expression'>
sage: type(zeta(SR(1.)).pyobject())
<class 'sage.rings.infinity.UnsignedInfinity'>

Ideally, fix other such cases too in this ticket.

Depends on #22838

CC: @paulmasson

Component: symbolics

Author: Ralf Stephan

Branch/Commit: 5d2d860

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/19439

[rwst]

comment:1

Polylog uses zeta here, so it's mainly because of RR(1).zeta() not being converted in symbolic/function.pyx or the RR functions not using the InfinityRing elements.

[rwst]

Description changed:

--- 
+++ 
@@ -11,5 +11,11 @@
 +infinity
 sage: type(_)
 <type 'sage.rings.real_mpfr.RealNumber'>
+sage: polylog(SR(1.),SR(1.))
++infinity
+sage: zeta(SR(1.))
++infinity
+sage: type(_)
+<type 'sage.symbolic.expression.Expression'>
 ```
 

[rwst]

comment:2

The problematic output is rather for the last lines, ie symbolic input, which should produce symbolic output. It does, by wrapping FP infinity into SR. It would be better if that would rather produce the infinity from the ring.

[paulmasson]

Description changed:

--- 
+++ 
@@ -19,3 +19,4 @@
 <type 'sage.symbolic.expression.Expression'>
 ```
 
+Additionally, evaluating the beta function at poles returns `NaN` rather than unsigned infinity.

[rwst]

Description changed:

--- 
+++ 
@@ -19,4 +19,4 @@
 <type 'sage.symbolic.expression.Expression'>
 ```
 
-Additionally, evaluating the beta function at poles returns `NaN` rather than unsigned infinity.
+These are mpmath infinities. In #18386 the same problem with arb comes up. As an exception to the what-goes-in-goes-out rule they should be converted upon return to elements of the infinity ring. The conversion should additionally convert all FP NaNs to the symbolic constant NaN.

[rwst]

Description changed:

--- 
+++ 
@@ -19,4 +19,4 @@
 <type 'sage.symbolic.expression.Expression'>
 ```
 
-These are mpmath infinities. In #18386 the same problem with arb comes up. As an exception to the what-goes-in-goes-out rule they should be converted upon return to elements of the infinity ring. The conversion should additionally convert all FP NaNs to the symbolic constant NaN.
+These are MPFR infinities, mpmath may do this too. In #18386 the same problem with arb comes up. As an exception to the what-goes-in-goes-out rule they should be converted upon return to elements of the infinity ring. The conversion should additionally convert all FP NaNs to the symbolic constant NaN.