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Random butterfly matrices were introduced by Parker in 1995 to remove theneed for pivoting when using Gaussian elimination. The growing applications ofbutterfly matrices have often eclipsed the mathematical understanding of how orwhy butterfly matrices are able to accomplish these given tasks. To help beginto close this gap using theoretical and numerical approaches, we explore theimpact on the growth factor of preconditioning a linear system by butterflymatrices. These results are compared to other common methods found inrandomized numerical linear algebra. In these experiments, we showpreconditioning using butterfly matrices has a more significant dampeningimpact on large growth factors than other common preconditioners and a smallerincrease to minimal growth factor systems. Moreover, we are able to determinethe full distribution of the growth factors for a subclass of random butterflymatrices. Previous results by Trefethen and Schreiber relating to thedistribution of random growth factors were limited to empirical estimates ofthe first moment for Ginibre matrices.

Growth factors of random butterfly matrices and the stability of avoiding pivoting