
CGELSY(l) ) CGELSY(l)
NAME
CGELSY  compute the minimumnorm solution to a complex linear least squares problem
SYNOPSIS
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
REAL RCOND
INTEGER JPVT( * )
REAL RWORK( * )
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
PURPOSE
CGELSY computes the minimumnorm solution to a complex linear least squares problem:
minimize  A * X  B 
using a complete orthogonal factorization of A. A is an MbyN matrix which may be rank
deficient.
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 MbyNRHS right hand side matrix B and the NbyNRHS
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 minimumnorm 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
more simple.
o The call to the subroutine xGEQPF has been substituted by the
the call to the subroutine xGEQP3. This subroutine is a Blas3
version of the QR factorization with column pivoting.
o Matrix B (the right hand side) is updated with Blas3.
ARGUMENTS
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 MbyN 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 MbyNRHS right hand side matrix B. On exit, the NbyNRHS solution
matrix X.
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 ith 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 ith col
umn of A*P was the kth 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 ith argument had an illegal value
FURTHER DETAILS
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. QuintanaOrti, Depto. de Informatica, Universidad Jaime I, Spain
G. QuintanaOrti, Depto. de Informatica, Universidad Jaime I, Spain
LAPACK version 3.0 15 June 2000 CGELSY(l) 
