  Many fields of science and engineering require finding eigenvalues and
eigenvectors of large matrices. The solutions can represent oscillatory modes
of a bridge, a violin, the disposition of electrons around an atom or molecule,
the acoustic modes of a concert hall, or hundreds of other physical quantities.
Often only the few eigenpairs with the lowest or highest frequency (extremal
solutions) are needed. Methods that have been developed over the past 60 years
to solve such problems include the Lanczos [1,2] algorithm, Jacobi-Davidson
techniques [3], and the conjugate gradient method [4]. Here we present a way to
solve the extremal eigenvalue/eigenvector problem, turning it into a nonlinear
classical mechanical system with a modified Lagrangian constraint. The
constraint induces exponential inflationary growth of the desired extremal
solutions.
