Re: [Maxima-discuss] Internal representation of "Taylor sums"

[Stavros M. <mac...@gm...>]

The ratpow package handles this.

On Mon, Jul 17, 2023, 13:15 Eduardo Ochs <edu...@gm...> wrote:

> Hi list,
>
> a few hours ago I tried to use my luatree thing -
> http://anggtwu.net/eev-maxima.html#luatree - to understand the
> low-level representation of Taylor series, and I discovered that
>
>   1) it doesn't recognize the difference between "Taylor sums" - this
>      is an informal term; the correct term is canonical rational
>      expressions, a.k.a. CREs - and normal sums,
>
>   2) op and args also don't recognize the difference,
>
>   3) the internal representation of "Taylor sums" and "normal sums" is
>      very different, and ratp distinguishes them.
>
> We can see all that in this example:
>
>   a : taylor(sin(x),x,0,5);
>   b : x - x^3/6 + x^5/120;
>   op(a);
>   op(b);
>   args(a);
>   args(b);
>   :lisp #$a$
>   :lisp #$b$
>   ratp(a);
>   ratp(b);
>
> it yields:
>
>   (%i1) a : taylor(sin(x),x,0,5);
>                                     3    5
>                                    x    x
>   (%o1)/T/                     x - -- + --- + . . .
>                                    6    120
>   (%i2) b : x - x^3/6 + x^5/120;
>                                     5     3
>                                    x     x
>   (%o2)                            --- - -- + x
>                                    120   6
>   (%i3) op(a);
>   (%o3)                                  +
>   (%i4) op(b);
>   (%o4)                                  +
>   (%i5) args(a);
>                                          3   5
>                                         x   x
>   (%o5)                           [x, - --, ---]
>                                         6   120
>   (%i6) args(b);
>                                     5      3
>                                    x      x
>   (%o6)                           [---, - --, x]
>                                    120    6
>   (%i7) :lisp #$a$
>
>   ((MRAT SIMP (((%SIN SIMP) $X) $X) (sin(x)490 X491)
>     (($X ((5 . 1)) 0 NIL X491 . 2)) TRUNC)
>    PS (X491 . 2) ((5 . 1)) ((1 . 1) 1 . 1) ((3 . 1) -1 . 6) ((5 . 1) 1 .
> 120))
>   (%i7) :lisp #$b$
>
>   ((MPLUS SIMP) $X ((MTIMES SIMP) ((RAT SIMP) -1 6) ((MEXPT SIMP) $X 3))
>    ((MTIMES SIMP) ((RAT SIMP) 1 120) ((MEXPT SIMP) $X 5)))
>   (%i7) ratp(a);
>   (%o7)                                true
>   (%i8) ratp(b);
>   (%o8)                                false
>   (%i9)
>
> In the example above a is a MRAT and b is a MPLUS. Is there a standard
> way to access the components of an MRAT from Maxima? What do people
> use when they need to inspect the innards of MRATs from Maxima?
>
> Thanks in advance!
>   Eduardo Ochs
>   http://anggtwu.net/eev-maxima.html
>
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