how to best simplify product of square roots

asked 2020-01-12 11:17:34 +0200

rue82
113 ●4 ●13

updated 2020-01-13 11:25:29 +0200

I'd like to simplify expressions like

p1,p2,p3 = var('p1 p2 p3')
assume(p1>0,p2>0,p3>0)
R = p1*p2*sqrt(p3)*sqrt(p3/p1)*sqrt(p3/p2)
R

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')
assume(p1>0,p2>0,p3>0)
# 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
simplify_chain_real(R)
#R

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Comments

what is simplify_chain_real ? Not in Sagemath 9.1.beta0...

Emmanuel Charpentier ( 2020-01-12 11:56:27 +0200 )edit

One has to import it:

sage: from sage.manifolds.utilities import simplify_chain_real

It is documented here

eric_g ( 2020-01-12 18:05:43 +0200 )edit

Indeed, simplify_chain_real does the job here:

sage: simplify_chain_real(R)
sqrt(p1)*sqrt(p2)*p3^(3/2)

eric_g ( 2020-01-12 18:12:09 +0200 )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 ( 2020-01-12 18:15:33 +0200 )edit

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

rue82 ( 2020-01-13 11:24:02 +0200 )edit

see more comments

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

Comments

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 ( 2020-01-13 19:38:49 +0200 )edit

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