ZGELSX - routine is deprecated and has been replaced by
routine ZGELSY
SYNOPSIS
SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT,
RCOND, RANK, WORK, RWORK, INFO )
INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
INTEGER JPVT( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
PURPOSE
This routine is deprecated and has been replaced by rou
tine ZGELSY. ZGELSX 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-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 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 anni
hilated by unitary transformations from the right, arriv
ing 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.
ARGUMENTS
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
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*16 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*16 array, dimension
(LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B.
On exit, the N-by-NRHS solution matrix X. If m >=
n and RANK = n, the residual sum-of-squares for
the solution in the i-th column is given by the
sum of squares of elements N+1:M in that column.
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 an initial column, otherwise it is a free col
umn. Before the QR factorization of A, all ini
tial columns are permuted to the leading posi
tions; only the remaining free columns are moved
as a result of column pivoting during the factor
ization. On exit, if JPVT(i) = k, then the i-th
column of A*P was the k-th column of A.
RCOND (input) DOUBLE PRECISION
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 factor
ization with pivoting of A, whose estimated condi
tion 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 fac
torization of A.
WORK (workspace) COMPLEX*16 array, dimension
(min(M,N) + max( N, 2*min(M,N)+NRHS )),
(2*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value
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