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# MatSetNullSpace

attaches a null space to a matrix.

## Synopsis

    #include "petscmat.h"
    PetscErrorCode MatSetNullSpace(Mat mat, MatNullSpace nullsp)

Logically Collective

## Input Parameters

- mat - the matrix

- nullsp - the null space object

## Notes

This null space is used by the `KSP` linear solvers to solve singular systems.

Overwrites any previous null space that may have been attached. You can remove the null space from the matrix object by calling this routine with an nullsp of `NULL`

For inconsistent singular systems (linear systems where the right-hand side is not in the range of the operator) the `KSP` residuals will not converge to zero but the linear system will still be solved in a least squares sense.

The fundamental theorem of linear algebra (Gilbert Strang, Introduction to Applied Mathematics, page 72) states that the domain of a matrix A (from \(R^n\) to \(R^m\) (m rows, n columns) \(R^n\) = the direct sum of the null space of A, n(A), + the range of \(A^T\), \(R(A^T)\). Similarly \(R^m\) = direct sum n(\(A^T\)) + R(A). Hence the linear system \(A x = b\) has a solution only if b in R(A) (or correspondingly b is orthogonal to n(\(A^T\))) and if x is a solution then x + alpha n(A) is a solution for any alpha. The minimum norm solution is orthogonal to n(A). For problems without a solution the solution that minimizes the norm of the residual (the least squares solution) can be obtained by solving A x = \hat{b} where \hat{b} is b orthogonalized to the n(\(A^T\)). This \hat{b} can be obtained by calling `MatNullSpaceRemove()` with the null space of the transpose of the matrix.

If the matrix is known to be symmetric because it is an `MATSBAIJ` matrix or one as called `MatSetOption`(mat,`MAT_SYMMETRIC` or possibly `MAT_SYMMETRY_ETERNAL`,`PETSC_TRUE`); this routine also automatically calls `MatSetTransposeNullSpace()`.

The user should call `MatNullSpaceDestroy()`.

## See Also

Matrices, `Mat`, `MatCreate()`, `MatNullSpaceCreate()`, `MatSetNearNullSpace()`, `MatGetNullSpace()`, `MatSetTransposeNullSpace()`, `MatGetTransposeNullSpace()`, `MatNullSpaceRemove()`, `KSPSetPCSide()`

## Level

advanced

## Location

src/mat/interface/matrix.c

## Examples

src/ksp/ksp/tutorials/ex50.c
src/ksp/ksp/tutorials/ex32.c
src/ksp/ksp/tutorials/ex59.c
src/ksp/ksp/tutorials/ex35.cxx
src/ksp/ksp/tutorials/ex36.cxx
src/ksp/ksp/tutorials/ex34.c
src/dm/impls/stag/tutorials/ex4.c
src/dm/impls/stag/tutorials/ex3.c
src/dm/impls/stag/tutorials/ex2.c
src/ksp/ksp/tutorials/ex29.c

Index of all Mat routines
Table of Contents for all manual pages
Index of all manual pages