Whereas matrix rank is additive under direct sum, in 1981 Sch\"onhage showed that one of its generalizations to the tensor setting, border rank, can be strictly subadditive for tensors order three. Whether higher has remained open. In this work, we settle problem by providing analogs Sch\"onhage's construction four and higher. work was motivated study computational complexity multiplication; di...

The SPQR RANK package contains routines that calculate the numerical rank of large, sparse, numerically rank-deficient matrices. The routines can also calculate orthonormal bases for numerical null spaces, approximate pseudoinverse solutions to least squares problems involving rankdeficient matrices, and basic solutions to these problems. The algorithms are based on SPQR from SuiteSparseQR (ACM...