On Mar 30, 10:14 am, Eric Gourgoulhon <egourgoul...@gmail.com> wrote:
> Hello,
>
> I've noticed this strange behavior (bug ?) on the following elementary
> computation:
>
> sage: a = matrix([[sqrt(x),0,0,0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0,
> 1]])
> sage: det(a)
> ...
> TypeError: no conversion of this rational to integer
Good catch!
The problem seems to be that `det` punts to `charpoly` for 4x4
matrices and larger, and that has some issues:
sage: a=matrix([[x,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: charpoly(a)
0
this is due to the use of "x" as a variable:
sage: var('y')
y
sage: a=matrix([[y,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: charpoly(a)
x^4 + (-y - 3)*x^3 + (3*y + 3)*x^2 + (-3*y - 1)*x + y
which is correct.
That is not the (only) issue you are running into, however. Unwrapping
the code for a.charpoly:
sage: a=matrix([[y^(1/2),0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: self=a
sage: var='x'
sage: cp = self._maxima_(maxima).charpoly(var)._sage_()
Up to here everything is fine:
sage: cp
(x - 1)^3*(x - sqrt(y))
sage: cp.expand()
x^4 - x^3*sqrt(y) - 3*x^3 + 3*x^2*sqrt(y) + 3*x^2 - 3*x*sqrt(y) - x +
sqrt(y)
It's in the following step that things go wrong:
sage: cp.expand().polynomial(None, ring=SR[var])
TypeError
so the problem is sage.symbolic.expression_conversions.polynomial
It's silly that such a generic, dodgy routine is used here. You know
which variable you want to isolate (and we should NOT make that 'x'
but some 'custom_variable_you_never_use_somewhere_else') and you also
know the degree. So the conversion should be something like
V=SR('crazy_varname')
cp=self._maxima_(maxima).charpoly('crazy_varname')._sage_().expand()
SR[var](list(cp.coefficient(V,i) for i in range(self.nrows()+1)))
Of course, even in this you're at the mercy of maxima's expand doing
what you'd hope it does, but given that the result really should just
be a polynomial in 'crazy_varname', so we can have good hopes 'expand'
does the proper (expensive) thing here.
(You really don't want to let 'crazy_varname' leak out of charpoly,
otherwise you'll have trouble if someone decides to make a matrix of
characteristic polynomials and determine the charpoly of that. You
also don't want to generate a fresh variable every time, since the
management of variables in maxima/our interface would create a memory
leak)
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