text: *> \brief \b SGGSVP
  *
  *  =========== DOCUMENTATION ===========
  *
  * Online html documentation available at 
  *            http://www.netlib.org/lapack/explore-html/ 
  *
  *> \htmlonly
  *> Download SGGSVP + dependencies 
  *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sggsvp.f"> 
  *> [TGZ]</a> 
  *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sggsvp.f"> 
  *> [ZIP]</a> 
  *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sggsvp.f"> 
  *> [TXT]</a>
  *> \endhtmlonly 
  *
  *  Definition:
  *  ===========
  *
  *       SUBROUTINE SGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
  *                          TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
  *                          IWORK, TAU, WORK, INFO )
  * 
  *       .. Scalar Arguments ..
  *       CHARACTER          JOBQ, JOBU, JOBV
  *       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
  *       REAL               TOLA, TOLB
  *       ..
  *       .. Array Arguments ..
  *       INTEGER            IWORK( * )
  *       REAL               A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
  *      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
  *       ..
  *  
  *
  *> \par Purpose:
  *  =============
  *>
  *> \verbatim
  *>
  *> SGGSVP computes orthogonal matrices U, V and Q such that
  *>
  *>                    N-K-L  K    L
  *>  U**T*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
  *>                 L ( 0     0   A23 )
  *>             M-K-L ( 0     0    0  )
  *>
  *>                  N-K-L  K    L
  *>         =     K ( 0    A12  A13 )  if M-K-L < 0;
  *>             M-K ( 0     0   A23 )
  *>
  *>                  N-K-L  K    L
  *>  V**T*B*Q =   L ( 0     0   B13 )
  *>             P-L ( 0     0    0  )
  *>
  *> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
  *> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
  *> otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
  *> numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T. 
  *>
  *> This decomposition is the preprocessing step for computing the
  *> Generalized Singular Value Decomposition (GSVD), see subroutine
  *> SGGSVD.
  *> \endverbatim
  *
  *  Arguments:
  *  ==========
  *
  *> \param[in] JOBU
  *> \verbatim
  *>          JOBU is CHARACTER*1
  *>          = 'U':  Orthogonal matrix U is computed;
  *>          = 'N':  U is not computed.
  *> \endverbatim
  *>
  *> \param[in] JOBV
  *> \verbatim
  *>          JOBV is CHARACTER*1
  *>          = 'V':  Orthogonal matrix V is computed;
  *>          = 'N':  V is not computed.
  *> \endverbatim
  *>
  *> \param[in] JOBQ
  *> \verbatim
  *>          JOBQ is CHARACTER*1
  *>          = 'Q':  Orthogonal matrix Q is computed;
  *>          = 'N':  Q is not computed.
  *> \endverbatim
  *>
  *> \param[in] M
  *> \verbatim
  *>          M is INTEGER
  *>          The number of rows of the matrix A.  M >= 0.
  *> \endverbatim
  *>
  *> \param[in] P
  *> \verbatim
  *>          P is INTEGER
  *>          The number of rows of the matrix B.  P >= 0.
  *> \endverbatim
  *>
  *> \param[in] N
  *> \verbatim
  *>          N is INTEGER
  *>          The number of columns of the matrices A and B.  N >= 0.
  *> \endverbatim
  *>
  *> \param[in,out] A
  *> \verbatim
  *>          A is REAL array, dimension (LDA,N)
  *>          On entry, the M-by-N matrix A.
  *>          On exit, A contains the triangular (or trapezoidal) matrix
  *>          described in the Purpose section.
  *> \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 REAL array, dimension (LDB,N)
  *>          On entry, the P-by-N matrix B.
  *>          On exit, B contains the triangular matrix described in
  *>          the Purpose section.
  *> \endverbatim
  *>
  *> \param[in] LDB
  *> \verbatim
  *>          LDB is INTEGER
  *>          The leading dimension of the array B. LDB >= max(1,P).
  *> \endverbatim
  *>
  *> \param[in] TOLA
  *> \verbatim
  *>          TOLA is REAL
  *> \endverbatim
  *>
  *> \param[in] TOLB
  *> \verbatim
  *>          TOLB is REAL
  *>
  *>          TOLA and TOLB are the thresholds to determine the effective
  *>          numerical rank of matrix B and a subblock of A. Generally,
  *>          they are set to
  *>             TOLA = MAX(M,N)*norm(A)*MACHEPS,
  *>             TOLB = MAX(P,N)*norm(B)*MACHEPS.
  *>          The size of TOLA and TOLB may affect the size of backward
  *>          errors of the decomposition.
  *> \endverbatim
  *>
  *> \param[out] K
  *> \verbatim
  *>          K is INTEGER
  *> \endverbatim
  *>
  *> \param[out] L
  *> \verbatim
  *>          L is INTEGER
  *>
  *>          On exit, K and L specify the dimension of the subblocks
  *>          described in Purpose section.
  *>          K + L = effective numerical rank of (A**T,B**T)**T.
  *> \endverbatim
  *>
  *> \param[out] U
  *> \verbatim
  *>          U is REAL array, dimension (LDU,M)
  *>          If JOBU = 'U', U contains the orthogonal matrix U.
  *>          If JOBU = 'N', U is not referenced.
  *> \endverbatim
  *>
  *> \param[in] LDU
  *> \verbatim
  *>          LDU is INTEGER
  *>          The leading dimension of the array U. LDU >= max(1,M) if
  *>          JOBU = 'U'; LDU >= 1 otherwise.
  *> \endverbatim
  *>
  *> \param[out] V
  *> \verbatim
  *>          V is REAL array, dimension (LDV,P)
  *>          If JOBV = 'V', V contains the orthogonal matrix V.
  *>          If JOBV = 'N', V is not referenced.
  *> \endverbatim
  *>
  *> \param[in] LDV
  *> \verbatim
  *>          LDV is INTEGER
  *>          The leading dimension of the array V. LDV >= max(1,P) if
  *>          JOBV = 'V'; LDV >= 1 otherwise.
  *> \endverbatim
  *>
  *> \param[out] Q
  *> \verbatim
  *>          Q is REAL array, dimension (LDQ,N)
  *>          If JOBQ = 'Q', Q contains the orthogonal matrix Q.
  *>          If JOBQ = 'N', Q is not referenced.
  *> \endverbatim
  *>
  *> \param[in] LDQ
  *> \verbatim
  *>          LDQ is INTEGER
  *>          The leading dimension of the array Q. LDQ >= max(1,N) if
  *>          JOBQ = 'Q'; LDQ >= 1 otherwise.
  *> \endverbatim
  *>
  *> \param[out] IWORK
  *> \verbatim
  *>          IWORK is INTEGER array, dimension (N)
  *> \endverbatim
  *>
  *> \param[out] TAU
  *> \verbatim
  *>          TAU is REAL array, dimension (N)
  *> \endverbatim
  *>
  *> \param[out] WORK
  *> \verbatim
  *>          WORK is REAL array, dimension (max(3*N,M,P))
  *> \endverbatim
  *>
  *> \param[out] INFO
  *> \verbatim
  *>          INFO is INTEGER
  *>          = 0:  successful exit
  *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
  *> \endverbatim
  *
  *  Authors:
  *  ========
  *
  *> \author Univ. of Tennessee 
  *> \author Univ. of California Berkeley 
  *> \author Univ. of Colorado Denver 
  *> \author NAG Ltd. 
  *
  *> \date November 2011
  *
  *> \ingroup realOTHERcomputational
  *
  *> \par Further Details:
  *  =====================
  *>
  *>  The subroutine uses LAPACK subroutine SGEQPF for the QR factorization
  *>  with column pivoting to detect the effective numerical rank of the
  *>  a matrix. It may be replaced by a better rank determination strategy.
  *>
  *  =====================================================================
        SUBROUTINE SGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
       $                   TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
       $                   IWORK, TAU, WORK, INFO )
  *
  *  -- LAPACK computational routine (version 3.4.0) --
  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  *     November 2011
  *
  *     .. Scalar Arguments ..
        CHARACTER          JOBQ, JOBU, JOBV
        INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
        REAL               TOLA, TOLB
  *     ..
  *     .. Array Arguments ..
        INTEGER            IWORK( * )
        REAL               A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
       $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
  *     ..
  *
  *  =====================================================================
  *
  *     .. Parameters ..
        REAL               ZERO, ONE
        PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
  *     ..
  *     .. Local Scalars ..
        LOGICAL            FORWRD, WANTQ, WANTU, WANTV
        INTEGER            I, J
  *     ..
  *     .. External Functions ..
        LOGICAL            LSAME
        EXTERNAL           LSAME
  *     ..
  *     .. External Subroutines ..
        EXTERNAL           SGEQPF, SGEQR2, SGERQ2, SLACPY, SLAPMT, SLASET,
       $                   SORG2R, SORM2R, SORMR2, XERBLA
  *     ..
  *     .. Intrinsic Functions ..
        INTRINSIC          ABS, MAX, MIN
  *     ..
  *     .. Executable Statements ..
  *
  *     Test the input parameters
  *
        WANTU = LSAME( JOBU, 'U' )
        WANTV = LSAME( JOBV, 'V' )
        WANTQ = LSAME( JOBQ, 'Q' )
        FORWRD = .TRUE.
  *
        INFO = 0
        IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
           INFO = -1
        ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
           INFO = -2
        ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
           INFO = -3
        ELSE IF( M.LT.0 ) THEN
           INFO = -4
        ELSE IF( P.LT.0 ) THEN
           INFO = -5
        ELSE IF( N.LT.0 ) THEN
           INFO = -6
        ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
           INFO = -8
        ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
           INFO = -10
        ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
           INFO = -16
        ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
           INFO = -18
        ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
           INFO = -20
        END IF
        IF( INFO.NE.0 ) THEN
           CALL XERBLA( 'SGGSVP', -INFO )
           RETURN
        END IF
  *
  *     QR with column pivoting of B: B*P = V*( S11 S12 )
  *                                           (  0   0  )
  *
        DO 10 I = 1, N
           IWORK( I ) = 0
     10 CONTINUE
        CALL SGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO )
  *
  *     Update A := A*P
  *
        CALL SLAPMT( FORWRD, M, N, A, LDA, IWORK )
  *
  *     Determine the effective rank of matrix B.
  *
        L = 0
        DO 20 I = 1, MIN( P, N )
           IF( ABS( B( I, I ) ).GT.TOLB )
       $      L = L + 1
     20 CONTINUE
  *
        IF( WANTV ) THEN
  *
  *        Copy the details of V, and form V.
  *
           CALL SLASET( 'Full', P, P, ZERO, ZERO, V, LDV )
           IF( P.GT.1 )
       $      CALL SLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
       $                   LDV )
           CALL SORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
        END IF
  *
  *     Clean up B
  *
        DO 40 J = 1, L - 1
           DO 30 I = J + 1, L
              B( I, J ) = ZERO
     30    CONTINUE
     40 CONTINUE
        IF( P.GT.L )
       $   CALL SLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB )
  *
        IF( WANTQ ) THEN
  *
  *        Set Q = I and Update Q := Q*P
  *
           CALL SLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
           CALL SLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
        END IF
  *
        IF( P.GE.L .AND. N.NE.L ) THEN
  *
  *        RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
  *
           CALL SGERQ2( L, N, B, LDB, TAU, WORK, INFO )
  *
  *        Update A := A*Z**T
  *
           CALL SORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A,
       $                LDA, WORK, INFO )
  *
           IF( WANTQ ) THEN
  *
  *           Update Q := Q*Z**T
  *
              CALL SORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q,
       $                   LDQ, WORK, INFO )
           END IF
  *
  *        Clean up B
  *
           CALL SLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB )
           DO 60 J = N - L + 1, N
              DO 50 I = J - N + L + 1, L
                 B( I, J ) = ZERO
     50       CONTINUE
     60    CONTINUE
  *
        END IF
  *
  *     Let              N-L     L
  *                A = ( A11    A12 ) M,
  *
  *     then the following does the complete QR decomposition of A11:
  *
  *              A11 = U*(  0  T12 )*P1**T
  *                      (  0   0  )
  *
        DO 70 I = 1, N - L
           IWORK( I ) = 0
     70 CONTINUE
        CALL SGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO )
  *
  *     Determine the effective rank of A11
  *
        K = 0
        DO 80 I = 1, MIN( M, N-L )
           IF( ABS( A( I, I ) ).GT.TOLA )
       $      K = K + 1
     80 CONTINUE
  *
  *     Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
  *
        CALL SORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA,
       $             TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
  *
        IF( WANTU ) THEN
  *
  *        Copy the details of U, and form U
  *
           CALL SLASET( 'Full', M, M, ZERO, ZERO, U, LDU )
           IF( M.GT.1 )
       $      CALL SLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
       $                   LDU )
           CALL SORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
        END IF
  *
        IF( WANTQ ) THEN
  *
  *        Update Q( 1:N, 1:N-L )  = Q( 1:N, 1:N-L )*P1
  *
           CALL SLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
        END IF
  *
  *     Clean up A: set the strictly lower triangular part of
  *     A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
  *
        DO 100 J = 1, K - 1
           DO 90 I = J + 1, K
              A( I, J ) = ZERO
     90    CONTINUE
    100 CONTINUE
        IF( M.GT.K )
       $   CALL SLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA )
  *
        IF( N-L.GT.K ) THEN
  *
  *        RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
  *
           CALL SGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
  *
           IF( WANTQ ) THEN
  *
  *           Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
  *
              CALL SORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU,
       $                   Q, LDQ, WORK, INFO )
           END IF
  *
  *        Clean up A
  *
           CALL SLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA )
           DO 120 J = N - L - K + 1, N - L
              DO 110 I = J - N + L + K + 1, K
                 A( I, J ) = ZERO
    110       CONTINUE
    120    CONTINUE
  *
        END IF
  *
        IF( M.GT.K ) THEN
  *
  *        QR factorization of A( K+1:M,N-L+1:N )
  *
           CALL SGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
  *
           IF( WANTU ) THEN
  *
  *           Update U(:,K+1:M) := U(:,K+1:M)*U1
  *
              CALL SORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
       $                   A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
       $                   WORK, INFO )
           END IF
  *
  *        Clean up
  *
           DO 140 J = N - L + 1, N
              DO 130 I = J - N + K + L + 1, M
                 A( I, J ) = ZERO
    130       CONTINUE
    140    CONTINUE
  *
        END IF
  *
        RETURN
  *
  *     End of SGGSVP
  *
        END
  