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Let A be a n n s.p.d. matrix. Given a convergent method based on a matrix M (either l1smoother or the forward Gauss-Seidel) and consider M^T as well (it will be either the l1-smoother, i.e., M^T = M, or the backward Gauss-Seidel, if M is the forward Gauss-Seidel). We also need the matrix P corresponding to a set of aggregates, i.e., the relation vertex aggregate constructed for the sparsity graph of the sparse matrix A for example by the Lubys algorithm. We also need the coarse matrix Ac = P^T AP . The two-level algorithm requires function that solves equations with Ac (use the LDL^T or LU factorization of the s.p.d. matrix Ac).

The symmetric two-level algorithm takes as input a vector r and provides on output r=B-lr, which is defined by the following steps, Solve for y, My=r. Compute coarse r.h.s. r. = PT (r - Ay). Solve the coarse problem for ye, Anye = r. (use LDLT or LU facorization of Ac). . Update y = y + Pye. Solve for correction z, M"z=r - Ay. Update y := y + z. .r=y. The above mapping B-l: r T is referred to as the two-level preconditioner, and B is sometimes denoted Brl. The symmetric two-level algorithm takes as input a vector r and provides on output r=B-lr, which is defined by the following steps, Solve for y, My=r. Compute coarse r.h.s. r. = PT (r - Ay). Solve the coarse problem for ye, Anye = r. (use LDLT or LU facorization of Ac). . Update y = y + Pye. Solve for correction z, M"z=r - Ay. Update y := y + z. .r=y. The above mapping B-l: r T is referred to as the two-level preconditioner, and B is sometimes denoted Brl