On 20.04.2012 13:55, gsagrawal wrote:
i was evaluating this function.Few points which i noticed are below
1. in current TrigonometricFunction we dont have "csc" and "sec "
which are kind of must in trigonometry simplification ( for now
may bwe can have empty classes ..just to use theorems)
Indeed we have no csc and sec. In any case, the algorithm cannot really
deal with them (because csc and 1/sin look "equally good" to it).
2. After 4 or 5 loops this is taking too much time and the final
expression is in terms of sin only (converts all cos to sin )
What do you mean loops. Passing hint=[5] or so? Then yes, this is going
to get slow, for various reasons. It should never produce worse results
though, at least in terms of (1) the total degree of the result, and (2)
the number of terms. It will eventually turn things like sin(x)**3 into
linear expressions of sin(3x) etc, though.
3. before going to apply ratsimpmodprime function we can call some
basic identity substitution (sample code is given in the end)
That's what the curent trigsimp does. There really isn't much point in
applying identities this algorithm knows about. What the code should do
is replace occurrences of e.g. cot(x) by 1/tan(x), and then perhaps in
the end undo such things if the result gets somehow "nicer"
4. Also , identity like 1-sin(x)**2 = cos(x)**2 are not applied (try
trigsimp_groebner((1+sin(x))*(1-sin(x)) . this can be handled if
we apply all identity first as mentioned in 3rd point)
Yes. Anything beyond reducing the degree is somewhat fiddly. Basically
the algorithm excludes certain terms (in this case cos(x)**2) if it can
be rewritten in terms of smaller or equal degree. One could try to do a
final optimization step where *all* terms of said degree are used, but
this is going to get slow even faster...
In any case, thank you for pointing this issue out, I'm taking note of it.
5. Perhaps in place of passing groebner basis like :
sin(x)**2*tan(x) + sin(x)*cos(x) - tan(x) (i dont know how this is
generated at first place) ,we should pass only basic formulas
(here i think you mean 1+tan(x)**2=1/cos(x)**2 )
No this does not work. The formulas have to be polynomials. In fact, the
formulas we "pass" are those listed under "ideal". The groebner basis
computation is essentially some magic. It generates many more formulae
which are needed for complicated reasons.
6. And yes sometime it gives very funny expressions
Examples?
I
Thank you :-).
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