*> \brief <b> ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * ), B( LDB, * ), C( * ), D( * ),
* $ WORK( * ), X( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZGGLSE solves the linear equality-constrained least squares (LSE)
*> problem:
*>
*> minimize || c - A*x ||_2 subject to B*x = d
*>
*> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
*> M-vector, and d is a given P-vector. It is assumed that
*> P <= N <= M+P, and
*>
*> rank(B) = P and rank( (A) ) = N.
*> ( (B) )
*>
*> These conditions ensure that the LSE problem has a unique solution,
*> which is obtained using a generalized RQ factorization of the
*> matrices (B, A) given by
*>
*> B = (0 R)*Q, A = Z*T*Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. 0 <= P <= N <= M+P.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(M,N)-by-N upper trapezoidal matrix T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX*16 array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
*> contains the P-by-P upper triangular matrix R.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is COMPLEX*16 array, dimension (M)
*> On entry, C contains the right hand side vector for the
*> least squares part of the LSE problem.
*> On exit, the residual sum of squares for the solution
*> is given by the sum of squares of elements N-P+1 to M of
*> vector C.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is COMPLEX*16 array, dimension (P)
*> On entry, D contains the right hand side vector for the
*> constrained equation.
*> On exit, D is destroyed.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is COMPLEX*16 array, dimension (N)
*> On exit, X is the solution of the LSE problem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,M+N+P).
*> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
*> where NB is an upper bound for the optimal blocksizes for
*> ZGEQRF, CGERQF, ZUNMQR and CUNMRQ.
*>
*> 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1: the upper triangular factor R associated with B in the
*> generalized RQ factorization of the pair (B, A) is
*> singular, so that rank(B) < P; the least squares
*> solution could not be computed.
*> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor
*> T associated with A in the generalized RQ factorization
*> of the pair (B, A) is singular, so that
*> rank( (A) ) < N; the least squares solution could not
*> ( (B) )
*> be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16OTHERsolve
*
* =====================================================================
SUBROUTINE ZGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( * ), D( * ),
$ WORK( * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3,
$ NB4, NR
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGGRQF, ZTRMV,
$ ZTRTRS, ZUNMQR, ZUNMRQ
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
MN = MIN( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 .OR. P.GT.N .OR. P.LT.N-M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -7
END IF
*
* Calculate workspace
*
IF( INFO.EQ.0) THEN
IF( N.EQ.0 ) THEN
LWKMIN = 1
LWKOPT = 1
ELSE
NB1 = ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'ZGERQF', ' ', M, N, -1, -1 )
NB3 = ILAENV( 1, 'ZUNMQR', ' ', M, N, P, -1 )
NB4 = ILAENV( 1, 'ZUNMRQ', ' ', M, N, P, -1 )
NB = MAX( NB1, NB2, NB3, NB4 )
LWKMIN = M + N + P
LWKOPT = P + MN + MAX( M, N )*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGGLSE', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Compute the GRQ factorization of matrices B and A:
*
* B*Q**H = ( 0 T12 ) P Z**H*A*Q**H = ( R11 R12 ) N-P
* N-P P ( 0 R22 ) M+P-N
* N-P P
*
* where T12 and R11 are upper triangular, and Q and Z are
* unitary.
*
CALL ZGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ),
$ WORK( P+MN+1 ), LWORK-P-MN, INFO )
LOPT = WORK( P+MN+1 )
*
* Update c = Z**H *c = ( c1 ) N-P
* ( c2 ) M+P-N
*
CALL ZUNMQR( 'Left', 'Conjugate Transpose', M, 1, MN, A, LDA,
$ WORK( P+1 ), C, MAX( 1, M ), WORK( P+MN+1 ),
$ LWORK-P-MN, INFO )
LOPT = MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
* Solve T12*x2 = d for x2
*
IF( P.GT.0 ) THEN
CALL ZTRTRS( 'Upper', 'No transpose', 'Non-unit', P, 1,
$ B( 1, N-P+1 ), LDB, D, P, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 1
RETURN
END IF
*
* Put the solution in X
*
CALL ZCOPY( P, D, 1, X( N-P+1 ), 1 )
*
* Update c1
*
CALL ZGEMV( 'No transpose', N-P, P, -CONE, A( 1, N-P+1 ), LDA,
$ D, 1, CONE, C, 1 )
END IF
*
* Solve R11*x1 = c1 for x1
*
IF( N.GT.P ) THEN
CALL ZTRTRS( 'Upper', 'No transpose', 'Non-unit', N-P, 1,
$ A, LDA, C, N-P, INFO )
*
IF( INFO.GT.0 ) THEN
INFO = 2
RETURN
END IF
*
* Put the solutions in X
*
CALL ZCOPY( N-P, C, 1, X, 1 )
END IF
*
* Compute the residual vector:
*
IF( M.LT.N ) THEN
NR = M + P - N
IF( NR.GT.0 )
$ CALL ZGEMV( 'No transpose', NR, N-M, -CONE, A( N-P+1, M+1 ),
$ LDA, D( NR+1 ), 1, CONE, C( N-P+1 ), 1 )
ELSE
NR = P
END IF
IF( NR.GT.0 ) THEN
CALL ZTRMV( 'Upper', 'No transpose', 'Non unit', NR,
$ A( N-P+1, N-P+1 ), LDA, D, 1 )
CALL ZAXPY( NR, -CONE, D, 1, C( N-P+1 ), 1 )
END IF
*
* Backward transformation x = Q**H*x
*
CALL ZUNMRQ( 'Left', 'Conjugate Transpose', N, 1, P, B, LDB,
$ WORK( 1 ), X, N, WORK( P+MN+1 ), LWORK-P-MN, INFO )
WORK( 1 ) = P + MN + MAX( LOPT, INT( WORK( P+MN+1 ) ) )
*
RETURN
*
* End of ZGGLSE
*
END