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how to best simplify product of square roots

asked 2020-01-12 04:17:34 -0500

rue82 gravatar image

updated 2020-01-13 04:25:29 -0500

I'd like to simplify expressions like

p1,p2,p3 = var('p1 p2 p3')
R = p1*p2*sqrt(p3)*sqrt(p3/p1)*sqrt(p3/p2)

without using R.canonicalize_radical(), which unfortunately messes up other factors. I understand there are some options using R.simplify_chain_real(), but what else can I try?

Let us see an example where also R.simplify_chain_real() messes things up:

p1,p2,p3 = var('p1 p2 p3')
# R = p1*p2*sqrt(p3)*sqrt(p3/p1)*sqrt(p3/p2)
R = p1*p2*sqrt(p3)*sqrt(p3/p1)*sqrt(p3/p2)/((p1 - p3)*(p2 - p3)*(p3 - 1))
%display latex
from sage.manifolds.utilities import simplify_chain_real
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what is simplify_chain_real ? Not in Sagemath 9.1.beta0...

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2020-01-12 04:56:27 -0500 )edit

One has to import it:

sage: from sage.manifolds.utilities import simplify_chain_real

It is documented here

eric_g gravatar imageeric_g ( 2020-01-12 11:05:43 -0500 )edit

Indeed, simplify_chain_real does the job here:

sage: simplify_chain_real(R)
eric_g gravatar imageeric_g ( 2020-01-12 11:12:09 -0500 )edit

On general grounds, for real expressions, simplify_chain_real is safer than canonicalize_radical (see the doc examples for a case where canonicalize_radical yields a wrong result).

eric_g gravatar imageeric_g ( 2020-01-12 11:15:33 -0500 )edit

@eric_g I understand, but still simplify_chain_real messes things up, look at the second example I added.

rue82 gravatar imagerue82 ( 2020-01-13 04:24:02 -0500 )edit

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answered 2020-01-13 07:30:13 -0500

Emmanuel Charpentier gravatar image

Possible workarounds:

  • Substitute what you want to be simplified in the original (larger) expression containing the (unspecified) pther factors (not shown in your exemples...), possibly helped by use of wildcard patterns.


sage: reset()
sage: from sage.manifolds.utilities import simplify_chain_real
sage: w0, w1=(SR.wild(u) for u in (0, 1))
sage: p1,p2,p3 = var('p1 p2 p3', domain="positive")
sage: ER = p1*p2*sqrt(p3)*sqrt(p3/p1)*sqrt(p3/p2)/((p1 - p3)*(p2 - p3)*(p3 - 1))
sage: E1, E2=var("E1, E2")
sage: E=E1*(E2+ER)
sage: E.subs(ER==simplify_chain_real(ER))
-(sqrt(p1)*sqrt(p2)*p3^(3/2)/((p1 + p2 + 1)*p3^2 - p3^3 + p1*p2 - ((p1 + 1)*p2 + p1)*p3) - E2)*E1
sage: E.subs(w0*(w1+ER)==w0*(w1+simplify_chain_real(ER)))
-(sqrt(p1)*sqrt(p2)*p3^(3/2)/((p1 + p2 + 1)*p3^2 - p3^3 + p1*p2 - ((p1 + 1)*p2 + p1)*p3) - E2)*E1
  • Specify exactly the subexpression(s) you want simplified via Maxima's part and subspart functions (available via the maxima_methods() objects).

@eric_g: shouldn't simplify_chain_real (and, better, a similarly named method for SR elements) be part of the standard code of SR ?

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Yes one could imagine that. Probably, the best way would to put the simplify_chain_real code into the existing method simplify_real, thereby avoiding to create a new method in SR elements.

eric_g gravatar imageeric_g ( 2020-01-13 12:38:49 -0500 )edit

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Asked: 2020-01-12 04:17:34 -0500

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Last updated: Jan 13