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4 where A2 L ( H ) is h
;
i H -self-adjoint and invertible and b 2 H . The self-adjointness implies that the spectrum  ( A ) is real. However, we do not assume that A is denite. If an initial guess x 0 2 H is given, we can approximate x by iterates (2.2) x n = x 0 + y n with y n 2K n ( A ;r 0 ) where r 0 = b A x 0 is the initial residual and K n ( A ;r 0 ) = span f r 0 ; A r 0 ;:::; A n 1 r 0 g is the n th Krylov subspace generated with A and r 0 . We concentrate on minimal residual methods here, i.e., methods that construct iterates of the form (2.2) such that the residual r n : = b A x n has minimal k
k H -norm, that is k r n k H = k b A x n

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