Two variants of HJLS-PSLQ with applications

Abstract

The HJLS and PSLQ algorithms are the most popular algorithms for finding nontrivial integer relations for several real numbers. It has been already shown that PSLQ is essentially equivalent to HJLS under certain settings. We here call them HJLS-PSLQ. In the present work, we provide two variants of HJLS-PSLQ. The first one is a new modification of Bailey and Broadhurst's multi-pair version. We prove the termination of our modification, while the original multi-pair version may not terminate. The second one is an incremental version of HJLS-PSLQ. For those applications requiring to call HJLS-PSLQ many times, such as finding the minimal polynomial of an algebraic number without knowing the degree, we show the incremental version is more efficient than HJLS-PSLQ, both theoretically and practically.