Alex Raichev <[EMAIL PROTECTED]> writes:
> Hi all:
>
> Is there Sage function that computes Taylor expansions for
> multivariate functions?
If you are willing to install the optional fricas package:
sage: reset()
sage: X=axiom('x::TS FRAC INT')
sage: Y=axiom('y::TS FRAC INT')
sage: axiom._eval_line(')set str cal 3')
sage: sin(X+Y)
1 3 1 2 1 2 1 3
(y + x) + (- - y - - x y - - x y - - x ) + O(4)
6 2 2 6
sage: axiom.coefficient(sin(X+Y), 5)
1 5 1 4 1 2 3 1 3 2 1 4 1 5
--- y + -- x y + -- x y + -- x y + -- x y + --- x
120 24 12 12 24 120
-----------------------------------------------------------------------------
I guess I should provide some details. x and y are at first symbols, then via
"::" coerced to TaylorSeries Fraction Integer (short: TS FRAC INT, both works).
The third line instructs FriCAS to calculate series only up to order 3, note
that FriCAS's series really are lazy formal power series. Note that these are
*not* just iterated power series, as eg. in Mathematica :-)
The next line applies sin to the TaylorSeries X+Y.
Unfortunately, the interface to FriCAS is currently rather weak - there is a
problem with pexpect, there is not sage wrapper for the most important function
"::" of FriCAS, and there is still a lot of work to do to convert FriCAS
objects to Sage objects. (although a lot has been done under the hood,
already)
An alternative route would be to extend the Mike Hansen / Ralf Hemmecke lazy
power series.
Hope this helps.
Martin
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