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Christian Stump
Sep 13, 2012, 5:39:22 AM9/13/12
to sage-s...@googlegroups.com
Hello,
can anyone tell me how I can use sage to check that the following two (fairly simple) expressions coincide. Some unneeded background: both come from identities in character theory for complex reflection groups, Sage was able to solve similar expressions, see below, and this is the smallest example I have were it doesn't work ?
sage: f
-(e^(2/3*I*pi) + 1)*e^(2/15*I*pi - 32*XX) + 3*(e^(2/3*I*pi) + 1)*e^(2/15*I*pi - 12*XX) - 5*(e^(2/3*I*pi) + 1)*e^(2/15*I*pi + 4*XX) + 3*(e^(2/3*I*pi) + 1)*e^(2/15*I*pi + 8*XX) - 5*(e^(2/3*I*pi) + 1)*e^(4*XX) + 6*(e^(2/3*I*pi) + 1)*e^(8*XX) - (e^(2/3*I*pi) + 1)*e^(28*XX) + 3*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-12*XX) - 5*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-8*XX) + 3*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(8*XX) - (((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(28*XX) - 5*(((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(4*XX) + 5*(((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4*XX) + 3*(((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(8*XX) - 3*(((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(8*XX) + 3*(((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(-12*XX) - 3*(((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-12*XX) - (((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(-32*XX) + (((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-32*XX) - ((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-32*XX) + 3*((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-12*XX) - 5*((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4*XX) + 3*((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(8*XX) + 5*e^(4/15*I*pi + 4*XX) + 5*e^(14/15*I*pi + 4*XX) + 5*e^(8/15*I*pi + 4*XX) + 5*e^(2/15*I*pi + 4*XX) + 5*e^(2/3*I*pi + 4*XX) + 5*e^(6/5*I*pi - 8*XX) + 5*e^(4/5*I*pi - 8*XX) + 5*e^(2/5*I*pi - 8*XX) - 3*e^(4/15*I*pi + 8*XX) - 3*e^(14/15*I*pi + 8*XX) - 3*e^(8/15*I*pi + 8*XX) - 3*e^(2/15*I*pi + 8*XX) - 3*e^(4/15*I*pi - 12*XX) - 3*e^(14/15*I*pi - 12*XX) - 3*e^(8/15*I*pi - 12*XX) - 3*e^(2/15*I*pi - 12*XX) - 6*e^(2/3*I*pi + 8*XX) - 3*e^(6/5*I*pi + 8*XX) - 3*e^(4/5*I*pi + 8*XX) - 3*e^(2/5*I*pi + 8*XX) - 3*e^(6/5*I*pi - 12*XX) - 3*e^(4/5*I*pi - 12*XX) - 3*e^(2/5*I*pi - 12*XX) + e^(4/15*I*pi - 32*XX) + e^(14/15*I*pi - 32*XX) + e^(8/15*I*pi - 32*XX) + e^(2/15*I*pi - 32*XX) + e^(6/5*I*pi + 28*XX) + e^(4/5*I*pi + 28*XX) + e^(2/5*I*pi + 28*XX) + e^(2/3*I*pi + 28*XX) + 5*e^(-8*XX) - 6*e^(8*XX) + e^(88*XX)
sage: g
e^(-32*XX) - 2*e^(28*XX) + e^(88*XX)
Other similar examples were property simplified:
sage: h
(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-42*XX) - 5*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-6*XX) + 5*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(6*XX) - (((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(18*XX) + 5*e^(6/5*I*pi - 6*XX) + 5*e^(4/5*I*pi - 6*XX) + 5*e^(2/5*I*pi - 6*XX) - 5*e^(6/5*I*pi + 6*XX) - 5*e^(4/5*I*pi + 6*XX) - 5*e^(2/5*I*pi + 6*XX) - 4*e^(1/10*I*pi + 3*XX) + 4*e^(7/10*I*pi + 3*XX) + 4*e^(3/10*I*pi + 3*XX) + 4*e^(1/10*I*pi + 3*XX) - 4*e^(7/10*I*pi + 3*XX) - 4*e^(3/10*I*pi + 3*XX) - 4*e^(6/5*I*pi + 3*XX) - 4*e^(4/5*I*pi + 3*XX) - 4*e^(2/5*I*pi + 3*XX) + 4*e^(6/5*I*pi + 3*XX) + 4*e^(4/5*I*pi + 3*XX) + 4*e^(2/5*I*pi + 3*XX) + 2*e^(1/10*I*pi + 18*XX) - 2*e^(7/10*I*pi + 18*XX) - 2*e^(3/10*I*pi + 18*XX) - 2*e^(1/10*I*pi + 18*XX) + 2*e^(7/10*I*pi + 18*XX) + 2*e^(3/10*I*pi + 18*XX) + 2*e^(1/10*I*pi - 12*XX) - 2*e^(7/10*I*pi - 12*XX) - 2*e^(3/10*I*pi - 12*XX) - 2*e^(1/10*I*pi - 12*XX) + 2*e^(7/10*I*pi - 12*XX) + 2*e^(3/10*I*pi - 12*XX) + e^(6/5*I*pi + 18*XX) + e^(4/5*I*pi + 18*XX) + e^(2/5*I*pi + 18*XX) - e^(6/5*I*pi - 42*XX) - e^(4/5*I*pi - 42*XX) - e^(2/5*I*pi - 42*XX) + 5*e^(-6*XX) - 5*e^(6*XX) - e^(18*XX) + e^(78*XX)
sage: h.factor().expand()
e^(-42*XX) - 2*e^(18*XX) + e^(78*XX)
I now tested the first equality by taking the first terms of the Taylor expansion of f and numerically evaluated the coefficients which turn out to coincide with the coefficients of the Taylor expansion of g.
Thanks for pointer or further ideas! Christian
can anyone tell me how I can use sage to check that the following two (fairly simple) expressions coincide. Some unneeded background: both come from identities in character theory for complex reflection groups, Sage was able to solve similar expressions, see below, and this is the smallest example I have were it doesn't work ?
sage: f
-(e^(2/3*I*pi) + 1)*e^(2/15*I*pi - 32*XX) + 3*(e^(2/3*I*pi) + 1)*e^(2/15*I*pi - 12*XX) - 5*(e^(2/3*I*pi) + 1)*e^(2/15*I*pi + 4*XX) + 3*(e^(2/3*I*pi) + 1)*e^(2/15*I*pi + 8*XX) - 5*(e^(2/3*I*pi) + 1)*e^(4*XX) + 6*(e^(2/3*I*pi) + 1)*e^(8*XX) - (e^(2/3*I*pi) + 1)*e^(28*XX) + 3*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-12*XX) - 5*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-8*XX) + 3*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(8*XX) - (((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(28*XX) - 5*(((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(4*XX) + 5*(((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4*XX) + 3*(((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(8*XX) - 3*(((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(8*XX) + 3*(((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(-12*XX) - 3*(((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-12*XX) - (((((e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(4/15*I*pi) - 1)*e^(-32*XX) + (((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4/15*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-32*XX) - ((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-32*XX) + 3*((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(-12*XX) - 5*((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(4*XX) + 3*((((e^(4/15*I*pi) - 1)*e^(2/15*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/15*I*pi) - 1)*e^(8*XX) + 5*e^(4/15*I*pi + 4*XX) + 5*e^(14/15*I*pi + 4*XX) + 5*e^(8/15*I*pi + 4*XX) + 5*e^(2/15*I*pi + 4*XX) + 5*e^(2/3*I*pi + 4*XX) + 5*e^(6/5*I*pi - 8*XX) + 5*e^(4/5*I*pi - 8*XX) + 5*e^(2/5*I*pi - 8*XX) - 3*e^(4/15*I*pi + 8*XX) - 3*e^(14/15*I*pi + 8*XX) - 3*e^(8/15*I*pi + 8*XX) - 3*e^(2/15*I*pi + 8*XX) - 3*e^(4/15*I*pi - 12*XX) - 3*e^(14/15*I*pi - 12*XX) - 3*e^(8/15*I*pi - 12*XX) - 3*e^(2/15*I*pi - 12*XX) - 6*e^(2/3*I*pi + 8*XX) - 3*e^(6/5*I*pi + 8*XX) - 3*e^(4/5*I*pi + 8*XX) - 3*e^(2/5*I*pi + 8*XX) - 3*e^(6/5*I*pi - 12*XX) - 3*e^(4/5*I*pi - 12*XX) - 3*e^(2/5*I*pi - 12*XX) + e^(4/15*I*pi - 32*XX) + e^(14/15*I*pi - 32*XX) + e^(8/15*I*pi - 32*XX) + e^(2/15*I*pi - 32*XX) + e^(6/5*I*pi + 28*XX) + e^(4/5*I*pi + 28*XX) + e^(2/5*I*pi + 28*XX) + e^(2/3*I*pi + 28*XX) + 5*e^(-8*XX) - 6*e^(8*XX) + e^(88*XX)
sage: g
e^(-32*XX) - 2*e^(28*XX) + e^(88*XX)
Other similar examples were property simplified:
sage: h
(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-42*XX) - 5*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(-6*XX) + 5*(((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(6*XX) - (((e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(2/5*I*pi) + 1)*e^(18*XX) + 5*e^(6/5*I*pi - 6*XX) + 5*e^(4/5*I*pi - 6*XX) + 5*e^(2/5*I*pi - 6*XX) - 5*e^(6/5*I*pi + 6*XX) - 5*e^(4/5*I*pi + 6*XX) - 5*e^(2/5*I*pi + 6*XX) - 4*e^(1/10*I*pi + 3*XX) + 4*e^(7/10*I*pi + 3*XX) + 4*e^(3/10*I*pi + 3*XX) + 4*e^(1/10*I*pi + 3*XX) - 4*e^(7/10*I*pi + 3*XX) - 4*e^(3/10*I*pi + 3*XX) - 4*e^(6/5*I*pi + 3*XX) - 4*e^(4/5*I*pi + 3*XX) - 4*e^(2/5*I*pi + 3*XX) + 4*e^(6/5*I*pi + 3*XX) + 4*e^(4/5*I*pi + 3*XX) + 4*e^(2/5*I*pi + 3*XX) + 2*e^(1/10*I*pi + 18*XX) - 2*e^(7/10*I*pi + 18*XX) - 2*e^(3/10*I*pi + 18*XX) - 2*e^(1/10*I*pi + 18*XX) + 2*e^(7/10*I*pi + 18*XX) + 2*e^(3/10*I*pi + 18*XX) + 2*e^(1/10*I*pi - 12*XX) - 2*e^(7/10*I*pi - 12*XX) - 2*e^(3/10*I*pi - 12*XX) - 2*e^(1/10*I*pi - 12*XX) + 2*e^(7/10*I*pi - 12*XX) + 2*e^(3/10*I*pi - 12*XX) + e^(6/5*I*pi + 18*XX) + e^(4/5*I*pi + 18*XX) + e^(2/5*I*pi + 18*XX) - e^(6/5*I*pi - 42*XX) - e^(4/5*I*pi - 42*XX) - e^(2/5*I*pi - 42*XX) + 5*e^(-6*XX) - 5*e^(6*XX) - e^(18*XX) + e^(78*XX)
sage: h.factor().expand()
e^(-42*XX) - 2*e^(18*XX) + e^(78*XX)
I now tested the first equality by taking the first terms of the Taylor expansion of f and numerically evaluated the coefficients which turn out to coincide with the coefficients of the Taylor expansion of g.
Thanks for pointer or further ideas! Christian
Christian Stump
Sep 13, 2012, 5:53:48 AM9/13/12
to sage-s...@googlegroups.com
Okay, I got the simplification by doing
sage: f.expand().simplify()
while
sage: f.simplify()
or
sage: f.simplify_full()
did actually not work...
sage: f.expand().simplify()
while
sage: f.simplify()
or
sage: f.simplify_full()
did actually not work...
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