Generating Extreme-Scale Matrices With Specified Singular Values or Condition Number

A widely used form of test matrix is the randsvd matrix constructed as the product A = U V * , where U and V are random orthogonal or unitary matrices from the Haar distribution and is a diagonal matrix of singular values. Such matrices are random but have a specified singular value distribution. The cost of forming an m n randsvd matrix is m 3 + n 3 flops, which is prohibitively expensive at extreme scale; moreover, the randsvd construction requires a significant amount of communication, making it unsuitable for distributed memory environments. By dropping the requirement that U and V be Haar distributed and that both be random, we derive new algorithms for forming A that have cost linear in the number of matrix elements and require a low amount of communication and synchronization. We specialize these algorithms to generating matrices with a specified 2-norm condition number. Numerical experiments show that the algorithms have excellent efficiency and scalability.