[fricas-devel] Integration

[Waldek Hebisch]

I looked a bit at FriCAS failures in Rubi testsutite. More than
8000 positions in testsutite contains elliptic integrals in
answer. FriCAS currently can not generate ellipic integrals
in answers, so unless the integral really is elementary
(it happens sometimes, but is quite rare) FriCAS can not
do it. This is more than 10% of the testsuite and single
biggest reason for failures.

More about this: it seems that most elliptic cases is
very simple, easily reducing to defining formulas
by few substitutions. It seem relatively easy to add
ad-hoc handling for such cases. Main problem is that
we do not want to loose completeness for elementary cases,
so we can generate elliptics only after we decided that
integral is nonelementary.

Probably next biggest problem is polylogaritms (of order 5000
positions). We can handle one case, when argument of
polylogarithm is an exponential. But Rubi testsuite seem
to contain mostly different case.

There are also integrals expressed in terms of hypergeometric
functions (few thousends). They are used to integrate
algebraic functions and some mixed cases involving
algebraics and exponentials. I need to look closer,
but at least some of them we should be able to handle
like existing code for incomplete gamma.

Together the cases above seem to cover vast majority of
failures on Rubi testsuite. There are also failures
which current FriCAS methods in principle should handle,
but are not handled due to incomplete or buggy implementation.

ATM better handling of the above is still in planning stage.
I have made nice theoretical progress, both for elliptic
functions and for polylogarithms. But in both cases
theory is still too weak to give complete algorithm.
So we probably should try adding ad-hoc extentions.
There is reasonable chance that such extentions will
be part of complete implementation in the future.

[oldk1331]

Do you think there are some easier sub-tasks that we can
help? For example, writing these tests in FriCAS format,
adding rules for some elliptic cases, etc.

Post by Waldek Hebisch
I looked a bit at FriCAS failures in Rubi testsutite. More than
8000 positions in testsutite contains elliptic integrals in
answer. FriCAS currently can not generate ellipic integrals
in answers, so unless the integral really is elementary
(it happens sometimes, but is quite rare) FriCAS can not
do it. This is more than 10% of the testsuite and single
biggest reason for failures.
More about this: it seems that most elliptic cases is
very simple, easily reducing to defining formulas
by few substitutions. It seem relatively easy to add
ad-hoc handling for such cases. Main problem is that
we do not want to loose completeness for elementary cases,
so we can generate elliptics only after we decided that
integral is nonelementary.
Probably next biggest problem is polylogaritms (of order 5000
positions). We can handle one case, when argument of
polylogarithm is an exponential. But Rubi testsuite seem
to contain mostly different case.
There are also integrals expressed in terms of hypergeometric
functions (few thousends). They are used to integrate
algebraic functions and some mixed cases involving
algebraics and exponentials. I need to look closer,
but at least some of them we should be able to handle
like existing code for incomplete gamma.
Together the cases above seem to cover vast majority of
failures on Rubi testsuite. There are also failures
which current FriCAS methods in principle should handle,
but are not handled due to incomplete or buggy implementation.
ATM better handling of the above is still in planning stage.
I have made nice theoretical progress, both for elliptic
functions and for polylogarithms. But in both cases
theory is still too weak to give complete algorithm.
So we probably should try adding ad-hoc extentions.
There is reasonable chance that such extentions will
be part of complete implementation in the future.