Schur complement

The Schur complement, however, has condition number only of the order 1/h.

The following identity holds for a Schur complement of a square matrix:

Emilie Haynsworth was the first to call it the Schur complement.

For performances, the Schur complement method is combined with preconditioning, at least a diagonal preconditioner.

See also Schur complement.

Second, passing to the Schur complement reduces condition number and thus tends to decrease the number of iterations.

After multiplication with the matrix L the Schur complement appears in the upper pxp block.

The equivalence follows from the Schur complement identity applied to the Kirchhoff matrix of the network.

Consequently, we need not store the Schur complement matrix explicitly; it is sufficient to know how to multiply a vector by it.

In that case, the Schur complement of C in V also has a Wishart distribution.

The remaining Schur complement system on the unknowns associated with subdomain interfaces is solved by the conjugate gradient method.

This procedure can be viewed as a Richardson extrapolation for the iterative solution of the equations arising from the Schur complement method.

If M is a positive-definite symmetric matrix, then so is the Schur complement of D in M.

Then the conditional variance of X given Y is the Schur complement of C in V:

In linear algebra and the theory of matrices, the Schur complement of a matrix block (i.e., a submatrix within a larger matrix) is defined as follows.

The Neumann-Neumann method and the Neumann-Dirichlet method are the Schur complement method with particular kinds of preconditioners.

The multiplication of a vector by the Schur complement is a discrete version of the Poincaré-Steklov operator, also called the Dirichlet to Neumann mapping.

In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domain decomposition method, also called iterative substructuring.

This strategy is particularly advantageous if 'A' is diagonal and 'D'−'CA'−1'B' (the Schur complement of 'A') is a small matrix, since they are the only matrices requiring inversion.

The operator of the system solved by BDD is the same as obtained by eliminating the unknowns in the interiors of the subdomain, thus reducing the problem to the Schur complement on the subdomain interface.

The extension to more subdomains is straightforward, but using this method as stated as a preconditioner for the Schur complement system is not scalable with the number of subdomains; hence the need for a global coarse solve.

When the partial differential equation is discretized, for example by finite elements or finite differences, the discretization of the Poincaré-Steklov operator is the Schur complement obtained by eliminating all degrees of freedom inside the domain.