Hello,
> 1) Taylor series of a rational function.
>
> This works:
> sage: cos(x).taylor(x,0,2)
>
> This doesn't:
> sage: x/(1+x).taylor(x,0,2)
>
> This is very confusing:
> sage: var('x')
> sage: x/(1+x).taylor(x,0,2)
This is due to the fact that '.' binds tighter than '/'. For example,
sage: x/(1+x).taylor(x,0,2)
x/(x + 1)
sage: x/((1+x).taylor(x,0,2))
x/(x + 1)
sage: (x/(1+x)).taylor(x,0,2)
x - x^2
In Sage, "(x/(1+x))" creates an object and the you call the taylor()
method on that object.
>
> 2) Map for matrices. This works:
>
> sage: x=polygen(QQ)
> sage: m=(1-x*matrix([[1,1],[1,0]]))^-1; m
> sage: matrix([[m[i,j].subs(x=1) for j in range(2)] for i in range(2)])
>
> But, surely there is a direct way substitute x=1 for all entries?
If you just need to substitute, you can do:
sage: m.subs(x=1)
[-1 -1]
[-1 0]
If you want to apply a more general map to the coefficients, then you can do:
sage: m.apply_map(lambda e: e.subs(x=1))
[-1 -1]
[-1 0]
> Another thing I found confusing was that this slight variation gave
> division by 0.
>
> sage: var('x')
> sage: m=(1-x*matrix([[1,1],[1,0]]))^-1; m
> sage: matrix([[m[i,j].subs(x=1) for j in range(2)] for i in range(2)])
> The problem is that sage hasn't simplified the entries of m. It
> thinks
Yep, maybe Sage should try and do some simplification of the maxima
expression before trying the simplification. There is a ticket for
implementing a symbolic matrix class which should take care of that
issue.
> 3) How do I get access to maxima's ratsimp?
sage: n = m.apply_map(lambda e: e.rational_simplify()); n
[ -1/(x^2 + x - 1) -x/(x^2 + x - 1)]
[ -x/(x^2 + x - 1) (x - 1)/(x^2 + x - 1)]
sage: n.subs(x=1)
[-1 -1]
[-1 0]
In the new symbolic matrix class ( when it gets written ;-] ), you
should be able to do m.rational_simplify().
Hope that helps,
--Mike
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