# MatCreateSchurComplement#

Creates a new Mat that behaves like the Schur complement of a matrix

## Synopsis#

#include "petscksp.h"
PetscErrorCode MatCreateSchurComplement(Mat A00, Mat Ap00, Mat A01, Mat A10, Mat A11, Mat *S)


Collective

## Input Parameters#

• A00,A01,A10,A11 - the four parts of the original matrix A = [A00 A01; A10 A11] (A11 is optional)

• Ap00 - preconditioning matrix for use in ksp(A00,Ap00) to approximate the action of A^{-1}

## Output Parameter#

• S - the matrix that behaves as the Schur complement S = A11 - A10 ksp(A00,Ap00) A01

## Notes#

The Schur complement is NOT explicitly formed! Rather, this function returns a virtual Schur complement that can compute the matrix-vector product by using formula S = A11 - A10 A^{-1} A01 for Schur complement S and a KSP solver to approximate the action of A^{-1}.

All four matrices must have the same MPI communicator.

A00 and A11 must be square matrices.

MatGetSchurComplement() takes as arguments the index sets for the submatrices and returns both the virtual Schur complement (what this returns) plus a sparse approximation to the Schur complement (useful for building a preconditioner for the Schur complement) which can be obtained from this matrix with MatSchurComplementGetPmat()

## Developer Note#

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

KSP: Linear System Solvers, MatCreateNormal(), MatMult(), MatCreate(), MatSchurComplementGetKSP(), MatSchurComplementUpdateSubMatrices(), MatCreateTranspose(), MatGetSchurComplement(), MatSchurComplementGetPmat(), MatSchurComplementSetSubMatrices()