Blade: A package for block-triangular form improved Feynman integrals decomposition

In this article, we present the package {\tt Blade} as the first implementation of the block-triangular form improved Feynman integral reduction method. The block-triangular form has orders of magnitude fewer equations compared to the plain integration-by-parts system, allowing for strictly block-by-block solutions. This results in faster evaluations and reduced resource consumption. We elucidate the algorithms involved in obtaining the block-triangular form along with their implementations. Additionally, we introduce novel algorithms for finding the canonical form and symmetry relations of Feynman integrals, as well as for performing spanning-sector reduction. Our benchmarks for various state-of-the-art problems demonstrate that {\tt Blade} is remarkably competitive among existing reduction tools. Furthermore, the {\tt Blade} package offers several distinctive features, including support for complex kinematic variables or masses, user-defined Feynman prescriptions for each propagator, and general integrands.