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SandBoxSymPy
  
last edited 15 years ago by page

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Editor: page
Time: 2007/09/19 00:53:07 GMT-7
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Running SymPy in a SageBlock

SymPy initialization:

\begin{sageblock}
#sys.path.append("/home/page/sympy")
import sympy
# the follow command is only needed because we are running SymPy inside Sage
Integer = int
\end{sageblock}

We are running SymPy version: $\sage{sympy.__version__}$

First simple confidence test:

\begin{sageblock}
from sympy import Symbol
a=Symbol('a')
b=Symbol('b')
c=Symbol('c')
e=( a*b*b+2*b*a*b )**c
\end{sageblock}

The resulting SymPy expression is:

\begin{verbatim}
print e
\end{verbatim}

\begin{equation}
\sage{e}
\end{equation}

Limits

  Here is a simple limit in SymPy

\begin{sageblock}
from sympy import limit
x = Symbol("x")
e=limit((3**(1/x)+5**(1/x))**x, x, 0)
\end{sageblock}

and the result is:

\begin{equation}
\sage{e}
\end{equation}

Unfortunately for this limit Axiom gives:
$\sage{axiom('limit((3**(1/x)+5**(1/x))**x, x=0)::OutputForm')}$.

And Maxima gives: 
$\sage{maxima('limit((3**(1/x)+5**(1/x))**x, x, 0)')}$.

So the Axiom and Maxima developers have some more work to do!

But worse, Reduce actually gets it wrong...
\begin{reduce}
limit((3**(1/x)+5**(1/x))**x, x,0);
\end{reduce}


From robert.dodier Fri Apr 20 15:22:26 -0500 2007
From: robert.dodier
Date: Fri, 20 Apr 2007 15:22:26 -0500
Subject: question about limit
Message-ID: <20070420152226-0500@wiki.axiom-developer.org>

Hello, about this limit problem, limit((3^(1/x) + 5^(1/x))^x, x, 0), I seem to find that the limit is different depending on whether 0 is approached from above or below. (I get 5 as the limit from above, and 3 as the limit from below.) So either "failed" or "und" (undetermined) seems like an acceptable response, and 5 is OK only with qualification; it doesn't seem right to return 5 unqualified.


From kratt6 Fri Apr 20 15:56:42 -0500 2007
From: kratt6
Date: Fri, 20 Apr 2007 15:56:42 -0500
Subject: see Gruntz
Message-ID: <20070420155642-0500@wiki.axiom-developer.org>

Very likely, the implementation computes by default the limit from above.  I guess that Gruntz' algorithm is restricted to the real case, but I do not know.

Martin

From billpage Fri Apr 20 17:59:01 -0500 2007
From: billpage
Date: Fri, 20 Apr 2007 17:59:01 -0500
Subject: left and right limits can be different (not two-sided) but ...
Message-ID: <20070420175901-0500@wiki.axiom-developer.org>

For this limit, approaching from the right, Axiom gives:
$\sage{axiom('limit((3**(1/x)+5**(1/x))**x, x=0,"right")::OutputForm')}$
and from the left:
$\sage{axiom('limit((3**(1/x)+5**(1/x))**x, x=0,"left")::OutputForm')}$;

while Maxima gives: 
left (from below): $\sage{maxima('limit((3**(1/x)+5**(1/x))**x, x, 0,minus)')}$,
right (from above): $\sage{maxima('limit((3**(1/x)+5**(1/x))**x, x, 0,plus)')}$.



From billpage Fri Apr 20 18:26:50 -0500 2007
From: billpage
Date: Fri, 20 Apr 2007 18:26:50 -0500
Subject: Dominik Gruntz
Message-ID: <20070420182650-0500@wiki.axiom-developer.org>

"Axiom Reference":http://portal.axiom-developer.org/refs/articles/gruntz-limits-th11432.pdf/file_view

**On Computing Limits in a Symbolic Manipulation System;**
Dominik Gruntz.
"ETH Diss 11432":http://www.cs.fh-aargau.ch/~gruntz/publications2.html
"abstract":ftp://ftp.inf.ethz.ch/pub/publications/dissertations/th11432.abstract
"postscript":ftp://ftp.inf.ethz.ch/pub/publications/dissertations/th11432.ps.gz
, 1996.



Running SymPy in a SageBlock


SymPy initialization:


\begin{sageblock}
#sys.path.append("/home/page/sympy")
import sympy
# the follow command is only needed because we are running SymPy inside Sage
Integer = int
\end{sageblock}


We are running SymPy version: $\sage{sympy.__version__}$


First simple confidence test:


\begin{sageblock}
from sympy import Symbol
a=Symbol(a)
b=Symbol(b)
c=Symbol(c)
e=( abb+2bab )*c
\end{sageblock}


The resulting SymPy expression is:


\begin{verbatim}
print e
\end{verbatim}


\begin{equation}
\label{eq1}
\sage{e}
\end{equation}


Limits


  Here is a simple limit in SymPy


\begin{sageblock}
from sympy import limit
x = Symbol("x")
e=limit((3(1/x)+5(1/x))**x, x, 0)
\end{sageblock}


and the result is:


\begin{equation}
\label{eq2}
\sage{e}
\end{equation}


Unfortunately for this limit Axiom gives:
$\sage{axiom(limit((3**(1/x)+5**(1/x))**x, x=0)::OutputForm)}$.


And Maxima gives: 
$\sage{maxima(limit((3**(1/x)+5**(1/x))**x, x, 0))}$.


So the Axiom and Maxima developers have some more work to do!


But worse, Reduce actually gets it wrong...


                 
limit((3**(1/x)+5**(1/x))**x, x,0);
reduce
                 

$$\displaylines{\qdd
1
\cr}
$$






question about limit --robert.dodier,  Fri, 20 Apr 2007 15:22:26 -0500 reply
Hello, about this limit problem, limit((3^(1/x) + 5^(1/x))^x, x, 0), I seem to find that the limit is different depending on whether 0 is approached from above or below. (I get 5 as the limit from above, and 3 as the limit from below.) So either "failed" or "und" (undetermined) seems like an acceptable response, and 5 is OK only with qualification; it doesn't seem right to return 5 unqualified.


see Gruntz --kratt6,  Fri, 20 Apr 2007 15:56:42 -0500 reply
Very likely, the implementation computes by default the limit from above.  I guess that Gruntz' algorithm is restricted to the real case, but I do not know.


Martin


left and right limits can be different (not two-sided) but ... --billpage,  Fri, 20 Apr 2007 17:59:01 -0500 reply
For this limit, approaching from the right, Axiom gives:
$\sage{axiom(limit((3**(1/x)+5**(1/x))**x, x=0,"right")::OutputForm)}$
and from the left:
$\sage{axiom(limit((3**(1/x)+5**(1/x))**x, x=0,"left")::OutputForm)}$;


while Maxima gives: 
left (from below): $\sage{maxima(limit((3**(1/x)+5**(1/x))**x, x, 0,minus))}$,
right (from above): $\sage{maxima(limit((3**(1/x)+5**(1/x))**x, x, 0,plus))}$.


Dominik Gruntz --billpage,  Fri, 20 Apr 2007 18:26:50 -0500 reply
Axiom Reference


On Computing Limits in a Symbolic Manipulation System;
Dominik Gruntz.
ETH Diss 11432
abstract
postscript
, 1996.
Some or all expressions may not have rendered properly, because Latex returned the following error:
sage: unset PYTHONPATH; PATH=/usr/local/bin:$PATH HOME=/var/zope/var/LatexWiki sage 4110322647062224839-18px.sage
Traceback (most recent call last):
  File "4110322647062224839-18px.py", line 26, in <module>
    e=limit((Integer(3)(Integer(1)/x)+Integer(5)(Integer(1)/x))**x, x, Integer(0))
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, *kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 138, in __new__
    return InfLimit(expr.subs(x, xlim+1/xoo), xoo)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 223, in __new__
    result = S.Exp(expr.exp  S.Log(expr.base)).inflimit(x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic.py", line 995, in inflimit
    return Basic.InfLimit(self, x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 228, in __new__
    result = expr.func([a.inflimit(x) for a in expr.args])
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic.py", line 995, in inflimit
    return Basic.InfLimit(self, x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, *kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 231, in __new__
    result = mrv_inflimit(expr, x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, *kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 257, in mrv_inflimit
    r = c.inflimit(x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic.py", line 995, in inflimit
    return Basic.InfLimit(self, x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, *kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 231, in __new__
    result = mrv_inflimit(expr, x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, *kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 257, in mrv_inflimit
    r = c.inflimit(x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic.py", line 995, in inflimit
    return Basic.InfLimit(self, x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, *kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 231, in __new__
    result = mrv_inflimit(expr, x)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/core/basic_methods.py", line 122, in wrapper
    func_cache_it_cache[k] = r = func(args, *kw_args)
  File "/usr/local/sage-2.6/local/lib/python2.5/site-packages/sympy/series/limits_newcore.py", line 238, in mrv_inflimit
    raise RuntimeError(Detected recursion while computing mrv_inflimit(%s, %s) % (expr, x))
RuntimeError: Detected recursion while computing mrv_inflimit(1/_xoolog(exp(_xoolog(3)) + exp(_xoo*log(5))), _xoo)