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 :)

[nbruin]

comment:9

Burcin is spot-on:

sage: S=(I(10).operands()[0].operator()); S
sum
sage: type(S)
<class 'sage.symbolic.function_factory.NewSymbolicFunction'>

(Note the "New", not "BuiltIn" or similar. It's a completely generic placeholder)
we just need a symbolic function hooked up that can do some mildly intelligent evaluation when asked for it.

Incidentally, we can just map back to maxima and do the right thing there:

sage: maxima_calculus(I(10))
('sum(abs(4*_SAGE_VAR_k^2+56*_SAGE_VAR_k-329),_SAGE_VAR_k,1,10))/500
sage: SR(maxima_calculus(I(10)).simplify_sum())
1301/250

(I haven't checked if it's correct). You can see why the "simplify_sum" is required: the newly created "sum" function in SR is linked to the inert "'sum".

[kcrisman]

comment:12

A relevant ask.sagemath question.

Another one.

Maybe it's time we fixed this.

[rwst]

comment:13

Replying to @kcrisman:

Maybe it's time we fixed this.

It is not clear if forcing people to use N() and getting a float, even if there is an integer simplification of the sum, is the right thing to do. Granted, the error thrown on N(I(10)) is a bug, and this ticket is about it. Here is a minimal example:

sage: (k,n) = var('k,n')
sage: f(n)=sum(abs(-k*k+n),k,1,n)
sage: f(n=8)
sum(abs(-k^2 + 8), k, 1, 8)
sage: N(f(8))

However, I would expect f(n=8).simplify() or .expand() to give me the result 162, and this is #17422

[rwst]

comment:26

Fixing these doctests will need the solutions found for fixing #17849 (or vice versa).

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

5a1ac7e	Merge branch 'develop' into t/9424/provide_symbolic_sum_function_with_evalf
9375510	Merge branch 'develop' into t/17759/public/17759
16aa81d	17759: handle hold=True and hypergeometric
1f4edf2	Merge branch 'public/17759' of trac.sagemath.org:sage into t/9424/provide_symbolic_sum_function_with_evalf
27a1692	9424: delegate to superclass; fix doctest

[sagetrac-git]

Changed commit from 32bca6a to 27a1692

[rwst]

Changed branch from u/rws/provide_symbolic_sum_function_with_evalf to u/rws/9424

[rwst]

New commits:

fa73285	17759: convenience class symbolic ExpressionTreeWalker(Converter)
b213d69	Merge branch 'u/rws/provide_symbolic_sum_function_with_evalf' of trac.sagemath.org:sage into tmp2

[rwst]

Changed commit from 27a1692 to b213d69

[mezzarobba]

comment:31

TestsFailed (says the patchbot)

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

985fdb4	Merge branch 'develop' into t/9424/9424
4f7b161	9424: make ceil/floor accept kwds; fix doctests

[sagetrac-git]

Changed commit from b213d69 to 4f7b161

[dkrenn]

Reviewer: Daniel Krenn

[dkrenn]

comment:34

LGTM

[vbraun]

Changed branch from u/rws/9424 to 4f7b161