Contents
zgelsy - compute the minimum-norm solution to a complex
linear least squares problem
SUBROUTINE ZGELSY(M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
WORK, LWORK, RWORK, INFO)
DOUBLE COMPLEX A(LDA,*), B(LDB,*), WORK(*)
INTEGER M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER JPVT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION RWORK(*)
SUBROUTINE ZGELSY_64(M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
WORK, LWORK, RWORK, INFO)
DOUBLE COMPLEX A(LDA,*), B(LDB,*), WORK(*)
INTEGER*8 M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER*8 JPVT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION RWORK(*)
F95 INTERFACE
SUBROUTINE GELSY([M], [N], [NRHS], A, [LDA], B, [LDB], JPVT, RCOND,
RANK, [WORK], [LWORK], [RWORK], [INFO])
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B
INTEGER :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER, DIMENSION(:) :: JPVT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: RWORK
SUBROUTINE GELSY_64([M], [N], [NRHS], A, [LDA], B, [LDB], JPVT,
RCOND, RANK, [WORK], [LWORK], [RWORK], [INFO])
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, B
INTEGER(8) :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER(8), DIMENSION(:) :: JPVT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: RWORK
C INTERFACE
#include <sunperf.h>
void zgelsy(int m, int n, int nrhs, doublecomplex *a, int
lda, doublecomplex *b, int ldb, int *jpvt, double
rcond, int *rank, int *info);
void zgelsy_64(long m, long n, long nrhs, doublecomplex *a,
long lda, doublecomplex *b, long ldb, long *jpvt,
double rcond, long *rank, long *info);
zgelsy 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 annihi-
lated by unitary transformations 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
more simple.
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) The number of rows of the matrix A. M >= 0.
N (input) The number of columns of the matrix A. N >= 0.
NRHS (input)
The number of right hand sides, i.e., the number
of columns of matrices B and X. NRHS >= 0.
A (input/output)
On entry, the M-by-N matrix A. On exit, A has
been overwritten by details of its complete
orthogonal factorization.
LDA (input)
The leading dimension of the array A. LDA >=
max(1,M).
B (input/output)
On entry, the M-by-NRHS right hand side matrix B.
On exit, the N-by-NRHS solution matrix X.
LDB (input)
The leading dimension of the array B. LDB >=
max(1,M,N).
JPVT (input/output)
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 column of A*P was the k-th column of A.
RCOND (input)
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)
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)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
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)
dimension(2*N)
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knox-
ville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime
I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime
I, Spain