  Integer relation algorithms can convert numerical results for Feynman
integrals to exact evaluations, when one has reason to suspect the existence of
reductions to linear combinations of a basis, with rational or algebraic
coefficients. Once a tentative reduction is obtained, confidence in its
validity is greatly increased by computing more decimal digits of the terms and
verifying the stability of the result. Here we give examples of how the PSLQ
and LLL algorithms have yielded remarkable reductions of Feynman integrals to
multiple polylogarithms and to the periods and quasi-periods of modular forms.
Moreover, these algorithms have revealed quadratic relations between Feynman
integrals. A recent application concerning black holes involves quadratic
relations between combinations of Feynman integrals with algebraic
coefficients.
