Sinposis

THE MATHEMATICAL THEORY OF KRYLOV SUBSPACE METHODS WITH A FOCUS ON SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS IS GIVEN A DETAILED TREATMENT IN THIS PRINCIPLES-BASED BOOK. STARTING FROM THE IDEA OF PROJECTIONS, KRYLOV SUBSPACE METHODS ARE CHARACTERISED BY THEIR ORTHOGONALITY AND MINIMISATION PROPERTIES. PROJECTIONS ONTO HIGHLY NONLINEAR KRYLOV SUBSPACES CAN BE LINKED WITH THE UNDERLYING PROBLEM OF MOMENTS, AND THEREFORE KRYLOV SUBSPACE METHODS CAN BE VIEWED AS MATCHING MOMENTS MODEL REDUCTION. THIS ALLOWS ENLIGHTENING REFORMULATIONS OF QUESTIONS FROM MATRIX COMPUTATIONS INTO THE LANGUAGE OF ORTHOGONAL POLYNOMIALS, GAUSS-CHRISTOFFEL QUADRATURE, CONTINUED FRACTIONS, AND, MORE GENERALLY, OF VOROBYEV'S METHOD OF MOMENTS. USING THE CONCEPT OF CYCLIC INVARIANT SUBSPACES, CONDITIONS ARE STUDIED THAT ALLOW THE GENERATION OF ORTHOGONAL KRYLOV SUBSPACE BASES VIA SHORT RECURRENCES.