Provide symbolic sum function with evalf

[burcin]

Symbolics sums returned from maxima cannot be numerically evaluated, since they don't define an _evalf_() method.

This was reported by dirkd on sage-support:

Why is evaluating this expression problematical?

y1(x)=x^2;y2(x)=5-x;
a0=1;an=3;Delta=(an-a0)/n;p(k)=a0+(k-1/2)*Delta;
I(n)=sum(abs(y2(p(k))-y1(p(k)))*Delta,k,1,n);
N(I(10))

SAGE respons:
<snipped traceback>
  File "expression.pyx", line 3797, in
sage.symbolic.expression.Expression.n (sage/symbolic/expression.cpp:
17022)
TypeError: cannot evaluate symbolic expression numerically

Here is the thread:

http://groups.google.com/group/sage-support/t/615b15ca638c9652

See also #15346

Depends on #17759

CC: @sagetrac-whuss @kcrisman @eviatarbach

Component: symbolics

Author: Ralf Stephan

Branch/Commit: 4f7b161

Reviewer: Daniel Krenn

Issue created by migration from https://trac.sagemath.org/ticket/9424

[rwst]

comment:7

Your problem is threefold: 1. you use n both as sum endpoint and variable, 2. Sage sum is only intended with symbolic endpoints, and 3. the need to use Python summation may require defining a Python function instead of a Sage symbolic function:

sage: sum?
...
Warning: This function only works with symbolic expressions. To sum any
     other objects like list elements or function return values,
     please use python summation...

sage: I(n)=sum(abs(y2(p(k))-y1(p(k)))*Delta,k,1,n);
sage: I
n |--> 2*sum(abs(-4*k^2 - 3*(2*k - 1)*n + 3*n^2 + 4*k - 1), k, 1, n)/n^3

sage: def I(n):
....:     return (2*sum(abs(-4*k^2 - 3*(2*k - 1)*n + 3*n^2 + 4*k - 1) for k in range(1,n+1))/n^3)
....: 
sage: I(10)
1301/250
sage: [I(i) for i in range(1,11)]
[2, 5, 46/9, 5, 26/5, 277/54, 254/49, 665/128, 418/81, 1301/250]

[kcrisman]

comment:8

I don't think any of these invalidate the ticket; the point is to extend the behavior. Why is 1. a problem? This seems like it should be a nice function to me. See Burcin's reply in the thread:

> If I leave out the N( )-operator on the last line the block evaluates
> to
> 
> 
> 1/500*sum(abs(-4*k^2 - 56*k + 329), k, 1, 10)
> 
> which can be evaluated in a new inputbox. Why not in one step?
The result returned from maxima uses a symbolic function object created
on the fly. This is quite different from the sum() function
available on the command line, and unfortunately, it doesn't define a
numerical evaluation function, _evalf_().

Burcin knows this code very well, so I would be surprised if he misdiagnosed this. But I figure maybe changing to enhancement will appease everyone :)