naginterfaces.library.lapackeig.zgelsy¶
- naginterfaces.library.lapackeig.zgelsy(a, b, jpvt, rcond)[source]¶
zgelsy
computes the minimum norm solution to a complex linear least squares problemusing a complete orthogonal factorization of . is an matrix which may be rank-deficient. Several right-hand side vectors and solution vectors can be handled in a single call.
For full information please refer to the NAG Library document for f08bn
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08bnf.html
- Parameters
- acomplex, array-like, shape
The matrix .
- bcomplex, array-like, shape
The right-hand side matrix .
- jpvtint, array-like, shape
If , the th column of is permuted to the front of , otherwise column is a free column.
- rcondfloat
Used to determine the effective rank of , which is defined as the order of the largest leading triangular sub-matrix in the factorization of , whose estimated condition number is .
Suggested value: if the condition number of is not known then (where is machine precision, see
machine.precision
) is a good choice. Negative values or values less than machine precision should be avoided since this will cause to have an effective that could be larger than its actual rank, leading to meaningless results.
- Returns
- acomplex, ndarray, shape
has been overwritten by details of its complete orthogonal factorization.
- bcomplex, ndarray, shape
The solution matrix .
- jpvtint, ndarray, shape
If , the th column of was the th column of .
- rankint
The effective rank of , i.e., the order of the sub-matrix . This is the same as the order of the sub-matrix in the complete orthogonal factorization of .
- Raises
- NagValueError
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: .
- Notes
The right-hand side vectors are stored as the columns of the matrix and the solution vectors in the matrix .
zgelsy
first computes a factorization with column pivotingwith defined as the largest leading sub-matrix whose estimated condition number is less than . The order of , , is the effective rank of .
Then, is considered to be negligible, and is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization
The minimum norm solution is then
where consists of the first columns of .
- References
Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug
Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore