Krylov

Krylov is a Russian surname that appears in science, mathematics, and culture, and is associated with several figures as well as mathematical concepts named in honor of those figures. In the field of numerical linear algebra, the name Krylov is most closely linked to Krylov subspace methods, a class of iterative algorithms for solving large sparse linear systems and eigenvalue problems. These methods operate by constructing projections onto successive Krylov subspaces generated from a matrix A and a starting vector b, defined as K_m(A, b) = span{b, Ab, A^2 b, ..., A^{m-1} b}. Prominent examples include the Conjugate Gradient method (for symmetric positive-definite matrices), the Lanczos process, the Arnoldi method, GMRES, MINRES, and BiCGSTAB. They are valued for their ability to handle very large problems typical of discretized differential equations and other applications in scientific computing.

Beyond numerical linear algebra, Krylov’s name appears in dynamical systems and probability through the Krylov–Bogoliubov averaging method, used to study nonlinear oscillations and long-term behavior of systems. The surname is encountered across Russian-language academia and culture, and transliteration variants appear in other languages.

As a surname, Krylov can refer to multiple individuals across fields. In mathematics and applied sciences, it is most often encountered in connection with Krylov subspace methods and related theory. The term Krylov and its variants are transliterations used in Russian, Ukrainian, and other languages.