# MatGetSchurComplement#

Obtain the Schur complement from eliminating part of the matrix in another part.

## Synopsis#

#include "petscksp.h"
PetscErrorCode MatGetSchurComplement(Mat A, IS isrow0, IS iscol0, IS isrow1, IS iscol1, MatReuse mreuse, Mat *S, MatSchurComplementAinvType ainvtype, MatReuse preuse, Mat *Sp)


Collective

## Input Parameters#

• A - matrix in which the complement is to be taken

• isrow0 - rows to eliminate

• iscol0 - columns to eliminate, (isrow0,iscol0) should be square and nonsingular

• isrow1 - rows in which the Schur complement is formed

• iscol1 - columns in which the Schur complement is formed

• mreuse - MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX, use MAT_IGNORE_MATRIX to put nothing in S

• ainvtype - the type of approximation used for the inverse of the (0,0) block used in forming Sp: MAT_SCHUR_COMPLEMENT_AINV_DIAG, MAT_SCHUR_COMPLEMENT_AINV_LUMP, MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG, or MAT_SCHUR_COMPLEMENT_AINV_FULL

• preuse - MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX, use MAT_IGNORE_MATRIX to put nothing in Sp

## Output Parameters#

• S - exact Schur complement, often of type MATSCHURCOMPLEMENT which is difficult to use for preconditioning

• Sp - approximate Schur complement from which a preconditioner can be built A11 - A10 inv(DIAGFORM(A00)) A01

## Notes#

Since the real Schur complement is usually dense, providing a good approximation to Sp usually requires application-specific information.

Sometimes users would like to provide problem-specific data in the Schur complement, usually only for special row and column index sets. In that case, the user should call PetscObjectComposeFunction() on the *S matrix and pass mreuse of MAT_REUSE_MATRIX to set “MatGetSchurComplement_C” to their function. If their function needs to fall back to the default implementation, it should call MatGetSchurComplement_Basic().

MatCreateSchurComplement() takes as arguments the four submatrices and returns the virtual Schur complement (what this function returns in S).

MatSchurComplementGetPmat() takes the virtual Schur complement and returns an explicit approximate Schur complement (what this returns in Sp).

In other words calling MatCreateSchurComplement() followed by MatSchurComplementGetPmat() produces the same output as this function but with slightly different inputs. The actually submatrices of the original block matrix instead of index sets to the submatrices.

## Developer Note#

The API that includes MatGetSchurComplement(), MatCreateSchurComplement(), MatSchurComplementGetPmat() should be refactored to remove redundancy and be clearer and simpler.

KSP: Linear System Solvers, MatCreateSubMatrix(), PCFIELDSPLIT, MatCreateSchurComplement(), MatSchurComplementAinvType