  This survey describes probabilistic algorithms for linear algebra
computations, such as factorizing matrices and solving linear systems. It
focuses on techniques that have a proven track record for real-world problem
instances. The paper treats both the theoretical foundations of the subject and
the practical computational issues.
  Topics covered include norm estimation; matrix approximation by sampling;
structured and unstructured random embeddings; linear regression problems;
low-rank approximation; subspace iteration and Krylov methods; error estimation
and adaptivity; interpolatory and CUR factorizations; Nystr\"om approximation
of positive-semidefinite matrices; single view ("streaming") algorithms; full
rank-revealing factorizations; solvers for linear systems; and approximation of
kernel matrices that arise in machine learning and in scientific computing.
