= Krylov subspace method
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A <Krylov subspace method> seeks approximations in spaces generated by repeated matrix applications to a starting vector. For a linear system with starting point $x_0$ and residual $r_0=b-Ax_0$, the affine search space is $x_0+K_k(A,r_0)$. Choosing an optimality or orthogonality condition within it gives different algorithms. The <conjugate gradient method> minimizes the <energy norm> of the error for a <symmetric positive-definite matrix>; other methods use residual minimization or different projections. The growth of the <Krylov subspace> can stop early when an invariant subspace is reached.
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