Taylor Polynomial possible "bug" in return

[LFS]

Hiya, Probably I am just doing something wrong ...
I have a cubic polynomial p(x) with "regular" coefficients and I wanted coefficients around e.g. (x-1). So I did p1=p.taylor(x,1,3).
I get:

x |--> 0.085*(x - 1)^3 - 0.255*(x - 1)^2 + 0.34*x + 1.23
The polynomial is correct, but look at the last two terms.

The 1-degree term is in x not in (x-1) and the difference has been added to the 0-degree term. 

I think i should get:

x |--> 0.085*(x - 1)^3 - 0.255*(x - 1)^2 + 0.34*(x-1) + 1.57

See (scroll all the way to the bottom): http://sage.math.canterbury.ac.nz/home/pub/237/

Thanks for any help. Linda

[LFS]

Little more: I see canterbury is using 4.8. Maybe this is fixed in version 5?

[Burcin Erocal]

On Thu, 10 Jan 2013 03:09:05 -0800 (PST)
LFS <lfah...@gmail.com> wrote:

> Hiya, Probably I am just doing something wrong ...
> I have a cubic polynomial p(x) with "regular" coefficients and I
> wanted coefficients around e.g. (x-1). So I did p1=p.taylor(x,1,3).
> I get:
>
> x |--> 0.085*(x - 1)^3 - 0.255*(x - 1)^2 + 0.34*x + 1.23
> The polynomial is correct, but look at the last two terms.
>
> The 1-degree term is in x not in (x-1) and the difference has been
> added to the 0-degree term.
>
> I think i should get:
>
> x |--> 0.085*(x - 1)^3 - 0.255*(x - 1)^2 + 0.34*(x-1) + 1.57

The taylor() method calls maxima. Maxima returns the expected result,
but while converting the expression back to Sage, we "normalize" it,
and expand the linear term.

sage: ps
2*x^3 - x^2 + 2*x + 1
sage: mps = maxima(ps)
sage: mps
2*x^3-x^2+2*x+1
sage: mps.taylor(x, 1, 3)
4+6*(x-1)+5*(x-1)^2+2*(x-1)^3
sage: ps.taylor(x, 1, 3)
2*(x - 1)^3 + 5*(x - 1)^2 + 6*x - 2


Note that series() does the right thing:

sage: ps.series(x==1, 3)
4 + 6*(x - 1) + 5*(x - 1)^2 + Order((x - 1)^3)


There is a ticket to deprecate taylor() or switch it to use series():

http://trac.sagemath.org/sage_trac/ticket/6119

This requires some work to identify differences in the behavior of
these two functions and figure out how to best remedy these. Any help
is much appreciated.


Cheers,
Burcin

[LFS]

Hiya Burcin,
Thanks for reply!
For my cubic polynomials p0 and p1, I did get the following to work:
p0=p.series(x==xD[0],4)
p1=q.series(x==xD[1],4)
print p0; p1

x |--> 1.0600000000000001 + 0.59499999999999997*x +
(-0.08500000000000002)*x^3
x |--> 1.5699999999999998 + 0.34000000000000008*(x - 1) +
(-0.25500000000000006)*(x - 1)^2 + 0.08500000000000002*(x - 1)^3

(I am guessing that here (a) you must add one more degree than you want and (b) if the derivative is not zero, you get the Order thing indicating a remainder??)
 I did like the taylor command. It was easy to understand and the syntax was intuitive (good for people like me). I don't mind the Order thing so much, but the x==1, I don't like.
BTW: The wolframalpha command is: series[0.085*x^3 - 0.510*x^2 + 1.105*x + 0.890,(x,1,3)]
I know next to nothing about programming so I hope someone else offers to help :)
Thanks again.
Linda