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

Empirical determinations of Feynman integrals using integer relation algorithms