projections(A, method=None, orth_tol=1e-12, max_refin=3, tol=1e-15)
Uses iterative refinements described in [1] during the computation of Z
in order to cope with the possibility of large roundoff errors.
Matrix A
used in the projection.
Method used for compute the given linear operators. Should be one of:
'NormalEquation': The operators
will be computed using the so-called normal equation approach explained in . In order to do so the Cholesky factorization of
(A A.T)is computed. Exclusive for sparse matrices.
'AugmentedSystem': The operators
will be computed using the so-called augmented system approach explained in . Exclusive for sparse matrices.
'QRFactorization': Compute projections
using QR factorization. Exclusive for dense matrices.
'SVDFactorization': Compute projections
using SVD factorization. Exclusive for dense matrices.
Tolerance for iterative refinements.
Maximum number of iterative refinements.
Tolerance for singular values.
Null-space operator. For a given vector x
, the null space operator is equivalent to apply a projection matrix P = I - A.T inv(A A.T) A
to the vector. It can be shown that this is equivalent to project x
into the null space of A.
Least-squares operator. For a given vector x
, the least-squares operator is equivalent to apply a pseudoinverse matrix pinv(A.T) = inv(A A.T) A
to the vector. It can be shown that this vector pinv(A.T) x
is the least_square solution to A.T y = x
.
Row-space operator. For a given vector x
, the row-space operator is equivalent to apply a projection matrix Q = A.T inv(A A.T)
to the vector. It can be shown that this vector y = Q x
the minimum norm solution of A y = x
.
Return three linear operators related with a given matrix A.
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