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version 1.1, 2010/01/26 15:22:46 version 1.19, 2023/08/07 08:39:21
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   *> \brief \b ZGGGLM
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZGGGLM + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggglm.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggglm.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggglm.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
   *                          INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, LDB, LWORK, M, N, P
   *       ..
   *       .. Array Arguments ..
   *       COMPLEX*16         A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
   *      $                   X( * ), Y( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:
   *>
   *>         minimize || y ||_2   subject to   d = A*x + B*y
   *>             x
   *>
   *> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
   *> given N-vector. It is assumed that M <= N <= M+P, and
   *>
   *>            rank(A) = M    and    rank( A B ) = N.
   *>
   *> Under these assumptions, the constrained equation is always
   *> consistent, and there is a unique solution x and a minimal 2-norm
   *> solution y, which is obtained using a generalized QR factorization
   *> of the matrices (A, B) given by
   *>
   *>    A = Q*(R),   B = Q*T*Z.
   *>          (0)
   *>
   *> In particular, if matrix B is square nonsingular, then the problem
   *> GLM is equivalent to the following weighted linear least squares
   *> problem
   *>
   *>              minimize || inv(B)*(d-A*x) ||_2
   *>                  x
   *>
   *> where inv(B) denotes the inverse of B.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of rows of the matrices A and B.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of columns of the matrix A.  0 <= M <= N.
   *> \endverbatim
   *>
   *> \param[in] P
   *> \verbatim
   *>          P is INTEGER
   *>          The number of columns of the matrix B.  P >= N-M.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,M)
   *>          On entry, the N-by-M matrix A.
   *>          On exit, the upper triangular part of the array A contains
   *>          the M-by-M upper triangular matrix R.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A. LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is COMPLEX*16 array, dimension (LDB,P)
   *>          On entry, the N-by-P matrix B.
   *>          On exit, if N <= P, the upper triangle of the subarray
   *>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
   *>          if N > P, the elements on and above the (N-P)th subdiagonal
   *>          contain the N-by-P upper trapezoidal matrix T.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] D
   *> \verbatim
   *>          D is COMPLEX*16 array, dimension (N)
   *>          On entry, D is the left hand side of the GLM equation.
   *>          On exit, D is destroyed.
   *> \endverbatim
   *>
   *> \param[out] X
   *> \verbatim
   *>          X is COMPLEX*16 array, dimension (M)
   *> \endverbatim
   *>
   *> \param[out] Y
   *> \verbatim
   *>          Y is COMPLEX*16 array, dimension (P)
   *>
   *>          On exit, X and Y are the solutions of the GLM 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,N+M+P).
   *>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
   *>          where NB is an upper bound for the optimal blocksizes for
   *>          ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.
   *>
   *>          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 A in the
   *>                generalized QR factorization of the pair (A, B) is
   *>                singular, so that rank(A) < M; the least squares
   *>                solution could not be computed.
   *>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
   *>                factor T associated with B in the generalized QR
   *>                factorization of the pair (A, B) is singular, so that
   *>                rank( A B ) < N; the least squares solution could not
   *>                be computed.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup complex16OTHEReigen
   *
   *  =====================================================================
       SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,        SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
      $                   INFO )       $                   INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2) --  *  -- LAPACK driver routine --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LDB, LWORK, M, N, P        INTEGER            INFO, LDA, LDB, LWORK, M, N, P
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      $                   X( * ), Y( * )       $                   X( * ), Y( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:  
 *  
 *          minimize || y ||_2   subject to   d = A*x + B*y  
 *              x  
 *  
 *  where A is an N-by-M matrix, B is an N-by-P matrix, and d is a  
 *  given N-vector. It is assumed that M <= N <= M+P, and  
 *  
 *             rank(A) = M    and    rank( A B ) = N.  
 *  
 *  Under these assumptions, the constrained equation is always  
 *  consistent, and there is a unique solution x and a minimal 2-norm  
 *  solution y, which is obtained using a generalized QR factorization  
 *  of the matrices (A, B) given by  
 *  
 *     A = Q*(R),   B = Q*T*Z.  
 *           (0)  
 *  
 *  In particular, if matrix B is square nonsingular, then the problem  
 *  GLM is equivalent to the following weighted linear least squares  
 *  problem  
 *  
 *               minimize || inv(B)*(d-A*x) ||_2  
 *                   x  
 *  
 *  where inv(B) denotes the inverse of B.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N       (input) INTEGER  
 *          The number of rows of the matrices A and B.  N >= 0.  
 *  
 *  M       (input) INTEGER  
 *          The number of columns of the matrix A.  0 <= M <= N.  
 *  
 *  P       (input) INTEGER  
 *          The number of columns of the matrix B.  P >= N-M.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,M)  
 *          On entry, the N-by-M matrix A.  
 *          On exit, the upper triangular part of the array A contains  
 *          the M-by-M upper triangular matrix R.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1,N).  
 *  
 *  B       (input/output) COMPLEX*16 array, dimension (LDB,P)  
 *          On entry, the N-by-P matrix B.  
 *          On exit, if N <= P, the upper triangle of the subarray  
 *          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;  
 *          if N > P, the elements on and above the (N-P)th subdiagonal  
 *          contain the N-by-P upper trapezoidal matrix T.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B. LDB >= max(1,N).  
 *  
 *  D       (input/output) COMPLEX*16 array, dimension (N)  
 *          On entry, D is the left hand side of the GLM equation.  
 *          On exit, D is destroyed.  
 *  
 *  X       (output) COMPLEX*16 array, dimension (M)  
 *  Y       (output) COMPLEX*16 array, dimension (P)  
 *          On exit, X and Y are the solutions of the GLM problem.  
 *  
 *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The dimension of the array WORK. LWORK >= max(1,N+M+P).  
 *          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,  
 *          where NB is an upper bound for the optimal blocksizes for  
 *          ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.  
 *  
 *          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.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit.  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *          = 1:  the upper triangular factor R associated with A in the  
 *                generalized QR factorization of the pair (A, B) is  
 *                singular, so that rank(A) < M; the least squares  
 *                solution could not be computed.  
 *          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal  
 *                factor T associated with B in the generalized QR  
 *                factorization of the pair (A, B) is singular, so that  
 *                rank( A B ) < N; the least squares solution could not  
 *                be computed.  
 *  
 *  ===================================================================  *  ===================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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 *     ..  *     ..
 *     .. External Functions ..  *     .. External Functions ..
       INTEGER            ILAENV        INTEGER            ILAENV
       EXTERNAL           ILAENV         EXTERNAL           ILAENV
 *     ..  *     ..
 *     .. Intrinsic Functions ..  *     .. Intrinsic Functions ..
       INTRINSIC          INT, MAX, MIN        INTRINSIC          INT, MAX, MIN
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 *  *
 *     Quick return if possible  *     Quick return if possible
 *  *
       IF( N.EQ.0 )        IF( N.EQ.0 ) THEN
      $   RETURN           DO I = 1, M
               X(I) = CZERO
            END DO
            DO I = 1, P
               Y(I) = CZERO
            END DO
            RETURN
         END IF
 *  *
 *     Compute the GQR factorization of matrices A and B:  *     Compute the GQR factorization of matrices A and B:
 *  *
 *            Q'*A = ( R11 ) M,    Q'*B*Z' = ( T11   T12 ) M  *          Q**H*A = ( R11 ) M,    Q**H*B*Z**H = ( T11   T12 ) M
 *                   (  0  ) N-M             (  0    T22 ) N-M  *                   (  0  ) N-M                 (  0    T22 ) N-M
 *                      M                     M+P-N  N-M  *                      M                         M+P-N  N-M
 *  *
 *     where R11 and T22 are upper triangular, and Q and Z are  *     where R11 and T22 are upper triangular, and Q and Z are
 *     unitary.  *     unitary.
 *  *
       CALL ZGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),        CALL ZGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ),
      $             WORK( M+NP+1 ), LWORK-M-NP, INFO )       $             WORK( M+NP+1 ), LWORK-M-NP, INFO )
       LOPT = WORK( M+NP+1 )        LOPT = INT( WORK( M+NP+1 ) )
 *  *
 *     Update left-hand-side vector d = Q'*d = ( d1 ) M  *     Update left-hand-side vector d = Q**H*d = ( d1 ) M
 *                                             ( d2 ) N-M  *                                               ( d2 ) N-M
 *  *
       CALL ZUNMQR( 'Left', 'Conjugate transpose', N, 1, M, A, LDA, WORK,        CALL ZUNMQR( 'Left', 'Conjugate transpose', N, 1, M, A, LDA, WORK,
      $             D, MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )       $             D, MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO )
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          CALL ZCOPY( M, D, 1, X, 1 )           CALL ZCOPY( M, D, 1, X, 1 )
       END IF        END IF
 *  *
 *     Backward transformation y = Z'*y  *     Backward transformation y = Z**H *y
 *  *
       CALL ZUNMRQ( 'Left', 'Conjugate transpose', P, 1, NP,        CALL ZUNMRQ( 'Left', 'Conjugate transpose', P, 1, NP,
      $             B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,       $             B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y,
Removed from v.1.1  
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  Added in v.1.19

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