On the squared unsymmetric Lanczos method

The biorthogonal Lanczos and the biconjugate gradient methods have been proposed as iterative methods to approximate the solution of nonsymmetric and indefinite linear systems. Sonneveld (1989) obtained the conjugate gradient squared by squaring the matrix polynomials of the biconjugate gradient method. Here we square the unsymmetric (or biorthogonal) Lanczos method for computing the eigenvalues of nonsymmetric matrices. Three forms of restarted squared Lanczos methods for solving unsymmetric linear systems of equations were derived. Numerical experiments with unsymmetric (in)definite linear systems of equations comparing these methods to a restarted (orthogonal) Krylov subspace iterative method showed that the new methods are competitive and they require that a fixed small number of direction vectors be stored in the main memory.