  Parker and L\^e introduced random butterfly transforms (RBTs) as a
preprocessing technique to replace pivoting in dense LU factorization.
Unfortunately, their FFT-like recursive structure restricts the dimensions of
the matrix. Furthermore, on multi-node systems, efficient management of the
communication overheads restricts the matrix's distribution even more. To
remove these limitations, we have generalized the RBT to arbitrary matrix sizes
by truncating the dimensions of each layer in the transform. We expanded
Parker's theoretical analysis to generalized RBT, specifically that in exact
arithmetic, Gaussian elimination with no pivoting will succeed with probability
1 after transforming a matrix with full-depth RBTs. Furthermore, we
experimentally show that these generalized transforms improve performance over
Parker's formulation by up to 62\% while retaining the ability to replace
pivoting. This generalized RBT is available in the SLATE numerical software
library.
