Kurt Pagani wrote:
>
> When integrating sin(x)*cos(2*n*x), n=1,10,20,30, there is a unusual growth
> in run-time. I guess that the cause might be found in trigonometric
> transformations. Is that so?
Yes. Compare:
(6) -> complexElementary(normalize(sin(x)*cos(2*20*x)))
(6)
+---+ 164 +---+ 160 +---+ 4
x\|- 1 x\|- 1 x\|- 1
------- ------- -------
+---+ 2 +---+ 2 +---+ 2 +---+
- \|- 1 (%e ) + \|- 1 (%e ) - \|- 1 (%e ) + \|- 1
---------------------------------------------------------------------------
+---+ 82
x\|- 1
-------
2
4(%e )
Type: Expression(Integer)
Time: 14.47 (EV) + 0.01 (OT) = 14.48 sec
(7) -> complexIntegrate(sin(x)*cos(2*20*x),x)
+---+ 82 +---+ 80 +---+ 2
x\|- 1 x\|- 1 x\|- 1
- 39(%e ) + 41(%e ) + 41(%e ) - 39
(7) ---------------------------------------------------------
+---+ 41
x\|- 1
6396(%e )
Type: Expression(Integer)
Time: 0.01 (EV) = 0.01 sec
'complexElementary' is doing slightly harder job than needed for
integration, but may give some idea how expensive is such transformation.
> Of course we can integrate using the parameter "n" followed by evaluation,
> however, I wonder whether this behavior is normal.
In a sense this normal behavior: FriCAS assumes that it can
expand/transform expressions to more conveniet forms and
such transformations may be costly. In the case above there
are some additional factors:
- integrator first writes function in terms of tangent and
later converts it to exponential form. This is more expensive
than direct convertion
- apparently during computations FriCAS uses subresultant GCD.
More precisely, sbcl profiler indicates that most of
execution time is spent in subresultant GCD. Subresultant GCD
is much slower than modular version which should be applicable
here.
--
Waldek Hebisch
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