Errors in symbolic integration
Risch-Bronstein-Trager algorithm (Risch algorithm in short) is a complete algorithom
for integration in terms of elementary functions. The algorithm either finds elementary integral
or proves that there is none. Existence of elementary integral is relatively
rare, so given random elementary function probably does not have elementary
integral. FriCAS? implementation of Risch algorithm is probably the
"most complete" existing implementation. Unfortunatly "most complete" does
not mean complete, some parts are still unimplemented. Unlike some other
systems FriCAS? will not give you unevaluated result when hitting unimplemented
part. Instead, it signals error with message indicating that given integral
requires unimplemented part. So when FriCAS? returns unevaluated result
almost surely there is on elementary integral. Almost surely, because
as all programs FriCAS? may have bugs...
FriCAS? in fact implements extension of Risch algorithm which extends class of integrands
to some Liouvillian functions and for integration in terms of Ei, Si, Ci, li, erf,
fresnelS, fresnelC, Gamma and
polylog. While there is complete extended algorithm current FriCAS? implementation
contains considerable gaps. Nevertheless, FriCAS? can handle a lot of examples
involving special functions that no other system can handle.
Additionaly to Risch integrator FriCAS? contains relativly weak pattern matching
integrator which can generate a few special function -- in addition to Ei, Si, Ci, li
it also can generate erf, fresnelC and fresnelS. However, if integral really requires
elliptic functions then the best thing which FriCAS? can do is to prove that integral
is nonelementary.
FriCAS? Examples
1)
fricas
integrate(sin(x)+sqrt(1-x^3),x)
Type: Union(Expression(Integer),...)
Unevaluated result means that FriCAS? proved that result is not elementary and can not
find nonelementary result.
int(sin(x)+sqrt(1-x^3),x); | reduce |
2)
fricas
integrate(sqrt(1-log(sin(x)^2)),x)
>> Error detected within library code:
integrate: implementation incomplete (constant residues)
In this case FriCAS? neither can compute elementry result nor can it prove that result is not elementary,
is it gives up with error message indicating that the handling this integral requires unimplemented
part of Bronstein-Trager algorithm.
int(sqrt(1-log(sin(x)^2)),x); | reduce |
3)
fricas
integrate(sqrt(sin(1/x)),x)
>> Error detected within library code:
integrate: implementation incomplete (constant residues)
Again, this integral needs unimplemented part of Bronstein-Trager algorithm.
int(sqrt(sin(1/x)),x); | reduce |
4)
fricas
integrate(sqrt(sin(x)),x)
Type: Union(Expression(Integer),...)
This time FriCAS? can prove that result is nonelementary (it needs elliptic functions).
int(sqrt(sin(x)),x); | reduce |
For this Maple 9 gives the following result:
And Mathematica 4 gives:
- Tue, 22 Mar 2005 11:48:00 -0600 reply