On Sun, 30 Jun 2013 13:23:50 -0700 (PDT)
Eric Gourgoulhon <egourgoul...@gmail.com> wrote:
> Meanwhile, I've written the following workaround in python:
> basically, it scans all the sqrt's in a given symbolic expression,
> send them to radcan, takes the absolute value of the output and then
> calls simplify(). It is not optimal but it works:
>
> sage: assume(x<0)
> sage: simplify_sqrt_real( sqrt(x^2) )
> -x
> sage: simplify_sqrt_real( sqrt(x^2-2*x+1) )
> -x + 1
> sage: simplify_sqrt_real( sqrt(x^2) + sqrt(x^2-2*x+1) )
> -2*x + 1
>
> Here is the source code:
>
> def simplify_sqrt_real(expr):
> r"""
> Simplify sqrt in symbolic expressions in the real domain.
>
> EXAMPLES:
>
> Simplifications of basic expressions::
>
> sage: assume(x<0)
> sage: simplify_sqrt_real( sqrt(x^2) )
> -x
> sage: simplify_sqrt_real( sqrt(x^2-2*x+1) )
> -x + 1
> sage: simplify_sqrt_real( sqrt(x^2) + sqrt(x^2-2*x+1) )
> -2*x + 1
>
> """
> from sage.symbolic.ring import SR
> from sage.calculus.calculus import maxima
> # 1/ Search for the sqrt's in expr
> sexpr = str(expr)
> if 'sqrt(' not in sexpr: # no sqrt to simplify
> return expr
> pos_sqrts = [] # positions of the sqrt's in sexpr
> the_sqrts = [] # the sqrt sub-expressions in sexpr
> for pos in range(len(sexpr)):
> if sexpr[pos:pos+5] == 'sqrt(':
> pos_sqrts.append(pos)
> parenth = 1
> scan = pos+5
> while parenth != 0:
> if sexpr[scan] == '(': parenth += 1
> if sexpr[scan] == ')': parenth -= 1
> scan += 1
> the_sqrts.append( sexpr[pos:scan] )
Instead of going through strings, you might want to try pattern
matching:
sage: t = sin(x)*sqrt(x^2)*e^(sqrt(x+1))
sage: w = SR.wild()
sage: t.find(w^(1/2))
[sqrt(x + 1), sqrt(x^2)]
Note that this won't catch sqrt() in the denominator. You will need to
specify a negative exponent for that:
sage: u = 1/sqrt(x+1)
sage: u.find(w^(-1/2))
[1/sqrt(x + 1)]
> # 2/ Simplifications of the sqrt's
> new_expr = "" # will contain the result
> pos0 = 0
> for i, pos in enumerate(pos_sqrts):
> # radcan is called on each sqrt:
> x = SR(the_sqrts[i])
> simpl = SR(x._maxima_().radcan())
This is x.maxima_methods().radcan().
> # the absolute value of radcan's output is taken, the call to
> simplify()
> # taking into account possible assumptions regarding the sign
> of simpl:
> new_expr += sexpr[pos0:pos] + '(' +
> str(abs(simpl).simplify()) + ')' pos0 = pos + len(the_sqrts[i])
> new_expr += sexpr[pos0:]
> return SR(new_expr)
Putting the new values back can be done with subs().
sage: t
sqrt(x^2)*e^(sqrt(x + 1))*sin(x)
sage: t.subs({sqrt(x^2): x})
x*e^(sqrt(x + 1))*sin(x)
Cheers,
Burcin
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