#include "petscmat.h" PetscErrorCode MatSetNullSpace(Mat mat,MatNullSpace nullsp)Logically Collective on Mat and MatNullSpace
mat | - the matrix | |
nullsp | - the null space object |
For inconsistent singular systems (linear systems where the right hand side is not in the range of the operator) you also likely should call MatSetTransposeNullSpace(). This allows the linear system to 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).
Krylov solvers can produce the minimal norm solution to the least squares problem by utilizing MatNullSpaceRemove().
Level:advanced
Location:src/mat/interface/matrix.c
Index of all Mat routines
Table of Contents for all manual pages
Index of all manual pages