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

[Eduardo O. <edu...@gm...>]

Hi all, especially Stavros...
First: thanks for all the hints!
Second: this is a simple demo of ratpow.lisp:

  a : taylor(exp(2*x),x,0,5);
  b : ratdisrep(a);
  :lisp #$b$
  :lisp #$a$
  load("ratpow.lisp")$
  ratp_hipow(a, x);
  ratp_lopow(a, x);
  ratp_coeffs(a, x);
  ratp_dense_coeffs(a, x);
  ratp_dense_coeffs_lo(a, x);

Third: this _almost_ gives me a very nice way to implement a method
that I always teach to my students!!! Suppose that E is exp(%i*x);
this code

  f  : sin(x)^4 * cos(x)^2;
  f1 : expand(exponentialize(f));
  f2 : subst([x=1/%i, %e=E], f1);
  f3 : rat(f2, E);

converts f to a rational function on E. What do I need to use instead
of the "rat" in the last step to convert f to a _Laurent polynomial_
on E instead of a _rational function_ on E?

  Thanks in advance!
  Cheers,
    Eduardo Ochs

P.S.: (more on the method later...)

On Mon, 17 Jul 2023 at 15:16, Stavros Macrakis <mac...@gm...> wrote:

> 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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