Dear list,
Setup : sage 5.11 on Debian amd64 self-compiled (Debian's compilers and
tuned Atlas library)
trying to put on paper the elementary proof that the convolution of two
normals is a normal, I stumbled on what I think to be a disastrous bug.
What follows is shown in text mode, but running it in emacs with sage-view
or in the notebook with the "typeset" option is much clearer...
var("x,t,mu,sigma")
assume(x,"real",t,"real",mu,"real",sigma,"real",sigma>0)
phi(x,mu,sigma)=e^-(((x-mu)^2)/(2*sigma^2))/sqrt(2*pi*sigma^2)
var("m1,m2,s1,s2")
assume(m1,"real",m2,"real",s1,"real",s2,"real",s1>0,s2>0)
g(x,m1,m2,s1,s2)=integrate(phi(t,m1,s1)*phi(x-t,m2,s2),t,-oo,oo)
prints :
(x, m1, m2, s1, s2) |--> sqrt(1/2)*sqrt(pi)*e^(-1/2*(m1^2 + 2*m1*m2 + m2^2
- 2*(m1 + m2)*x + x^2)/(s1^2 +
s2^2))/(sqrt(pi*s1^2)*sqrt(pi*s2^2)*sqrt((s1^2 + s2^2)/(s1^2*s2^2)))
Ugly (in text mode), but exact.
h=g(x,m1,m2,s1,s2) # to get a symbolic expression
print h.denominator()
This prints
sqrt(pi*s1^2)*sqrt(pi*s2^2)
which is simple, but obviously *FALSE*. And, by the way,
maxima.denom(h).sage()
gives the much more reasonable :
sqrt(2)*sqrt(pi*s1^2)*sqrt(pi*s2^2)*sqrt((s1^2 + s2^2)/(s1^2*s2^2))
which, simplify_radical()ized (legitimate under our assumptions), gives the
expected :
sqrt(2)*pi*sqrt(s1^2 + s2^2)
I think that this is a bug (what is used for this kind of expression ?
Sympy ?). Has it been already reported ? Should I report it ?
HTH,
Emmanuel Charpentier
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