CGELSY(l) ) CGELSY(l)
CGELSY - compute the minimum-norm solution to a complex linear least squares problem
SUBROUTINE CGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, RWORK, INFO
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
INTEGER JPVT( * )
REAL RWORK( * )
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
CGELSY computes the minimum-norm solution to a complex linear least squares problem:
minimize || A * X - B ||
using a complete orthogonal factorization of A. A is an M-by-N matrix which may be rank-
Several right hand side vectors b and solution vectors x can be handled in a single call;
they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS
solution matrix X.
The routine first computes a QR factorization with column pivoting:
A * P = Q * [ R11 R12 ]
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated condition number is less
than 1/RCOND. The order of R11, RANK, is the effective rank of A.
Then, R22 is considered to be negligible, and R12 is annihilated by unitary transforma-
tions from the right, arriving at the complete orthogonal factorization:
A * P = Q * [ T11 0 ] * Z
[ 0 0 ]
The minimum-norm solution is then
X = P * Z' [ inv(T11)*Q1'*B ]
[ 0 ]
where Q1 consists of the first RANK columns of Q.
This routine is basically identical to the original xGELSX except three differences:
o The permutation of matrix B (the right hand side) is faster and
o The call to the subroutine xGEQPF has been substituted by the
the call to the subroutine xGEQP3. This subroutine is a Blas-3
version of the QR factorization with column pivoting.
o Matrix B (the right hand side) is updated with Blas-3.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of matrices B and X.
NRHS >= 0.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A has been overwritten by details of its
complete orthogonal factorization.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) COMPLEX array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B. On exit, the N-by-NRHS solution
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).
JPVT (input/output) INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted to the front of AP,
otherwise column i is a free column. On exit, if JPVT(i) = k, then the i-th col-
umn of A*P was the k-th column of A.
RCOND (input) REAL
RCOND is used to determine the effective rank of A, which is defined as the order
of the largest leading triangular submatrix R11 in the QR factorization with piv-
oting of A, whose estimated condition number < 1/RCOND.
RANK (output) INTEGER
The effective rank of A, i.e., the order of the submatrix R11. This is the same
as the order of the submatrix T11 in the complete orthogonal factorization of A.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. The unblocked strategy requires that: LWORK >=
MN + MAX( 2*MN, N+1, MN+NRHS ) where MN = min(M,N). The block algorithm requires
that: LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS ) where NB is an
upper bound on the blocksize returned by ILAENV for the routines CGEQP3, CTZRZF,
CTZRQF, CUNMQR, and CUNMRZ.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
RWORK (workspace) REAL array, dimension (2*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
LAPACK version 3.0 15 June 2000 CGELSY(l)