Krylov Methods

rustebra provides iterative solvers and eigenvalue methods in rustebra::krylov:

• Power iteration for the dominant (largest-magnitude) eigenvalue
• Inverse power iteration for the eigenvalue nearest an arbitrary shift
• Conjugate Gradient (CG) for solving symmetric positive-definite linear systems
• Lanczos iteration, which builds an orthonormal basis of a Krylov subspace and the symmetric tridiagonal matrix a symmetric operator projects onto within it
• Arnoldi iteration, the non-symmetric counterpart to Lanczos, which builds an orthonormal basis of a Krylov subspace and the upper Hessenberg matrix a general operator projects onto within it
• GMRES(m), a restarted iterative solver for general (possibly non-symmetric) linear systems, built on top of Arnoldi iteration

Unlike the direct decompositions in Decompositions, these refine an estimate (or a basis) over many iterations and can fail to converge — or, for Lanczos, to extend the basis further — within a given budget, in addition to the usual dimension and non-finite-value failure modes.