Karim Belabas on Sun, 17 Aug 2014 09:20:56 +0200

 Re: Vec() and leading zeros of generating function

```* Kevin Ryde [2014-08-16 07:47]:
> Vec() can give terms from a polynomial generating function but it
> doesn't include leading zeros.  Eg.
>
>     Vec(x^2/(1-2*x) + O(x^5))
>     =>
>     [1, 2, 4]
>
> Is there a good way to start from the zero'th term so I would get
>
>     [0, 0, 1, 2, 4]
>
> I have to confess to only doing this by cut-and-paste programming, so
> perhaps there's something completely different I don't know which is
> better. :-) My main use has been to compare against expected or desired
> values, so
>
>     want = [1, -1, 1, -1]
>     Vec(1/(1+x) + O(x^length(want))) == want || error("oops")

You can use Vec's optional "length" parameter (making it negative
prepends 0s instead of appending them):

want = [0,0,1,2,4];
s = x^2/(1-2*x) + O(x^5);

? Vec(s,5)
%3 = [1, 2, 4, 0, 0]   \\ fix length = 5, append 0s
? Vec(s,-5)
%4 = [0, 0, 1, 2, 4]   \\ prepend

? Vec(s, -#want) == want    \\ note the '-' #want
%5 = 1

The is often marginally easier than concatenating 0s yourself, as others
already suggested. (As Loic's suggestion, this requires care if the series
has negative valuation: set the reference value accordingly...)

It might be simpler to just convert the desired value to generating function

? want = Ser(want)
%5 = x^2 + 2*x^3 + 4*x^4 + O(x^5)
? s == want
%6 = 1

(use === instead  of == if you insist on having the same accuracy, i.e.
"identical" series instead of merely "equal to given accyracy" ones)

Cheers,

K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 69 50
351, cours de la Liberation    http://www.math.u-bordeaux1.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux1.fr/  [PARI/GP]
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