Iterative Solution of Linear Systems

• iterative solution of large linear systems is one of the most im-portant numerical tasks in scientific computing (they occur in the discretization of both ODE and PDE) • direct solvers are often not competitive: – too large number of unknowns (cf. PDE in 3D) – sparse matrices (classical elimination destroys sparsity) – anyway only approximations, thus no need for exact solu-tion (this holds esp. in the nonlinear case, where a system of linear equations occurs in each step of some outer itera-tion for the nonlinearity) • objective: “for 3 digits, you need 10 steps " – no matter how big the number n of unknowns is • however typically: speed of convergence deteriorates with in-creasing n