dggsvd3.f

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*> \brief <b> DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGSVD3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
*                           LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
*                           LWORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBQ, JOBU, JOBV
*       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P, LWORK
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
*      $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
*      $                   V( LDV, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGSVD3 computes the generalized singular value decomposition (GSVD)
*> of an M-by-N real matrix A and P-by-N real matrix B:
*>
*>       U**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )
*>
*> where U, V and Q are orthogonal matrices.
*> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
*> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
*> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
*> following structures, respectively:
*>
*> If M-K-L >= 0,
*>
*>                     K  L
*>        D1 =     K ( I  0 )
*>                 L ( 0  C )
*>             M-K-L ( 0  0 )
*>
*>                   K  L
*>        D2 =   L ( 0  S )
*>             P-L ( 0  0 )
*>
*>                 N-K-L  K    L
*>   ( 0 R ) = K (  0   R11  R12 )
*>             L (  0    0   R22 )
*>
*> where
*>
*>   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
*>   S = diag( BETA(K+1),  ... , BETA(K+L) ),
*>   C**2 + S**2 = I.
*>
*>   R is stored in A(1:K+L,N-K-L+1:N) on exit.
*>
*> If M-K-L < 0,
*>
*>                   K M-K K+L-M
*>        D1 =   K ( I  0    0   )
*>             M-K ( 0  C    0   )
*>
*>                     K M-K K+L-M
*>        D2 =   M-K ( 0  S    0  )
*>             K+L-M ( 0  0    I  )
*>               P-L ( 0  0    0  )
*>
*>                    N-K-L  K   M-K  K+L-M
*>   ( 0 R ) =     K ( 0    R11  R12  R13  )
*>               M-K ( 0     0   R22  R23  )
*>             K+L-M ( 0     0    0   R33  )
*>
*> where
*>
*>   C = diag( ALPHA(K+1), ... , ALPHA(M) ),
*>   S = diag( BETA(K+1),  ... , BETA(M) ),
*>   C**2 + S**2 = I.
*>
*>   (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
*>   ( 0  R22 R23 )
*>   in B(M-K+1:L,N+M-K-L+1:N) on exit.
*>
*> The routine computes C, S, R, and optionally the orthogonal
*> transformation matrices U, V and Q.
*>
*> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
*> A and B implicitly gives the SVD of A*inv(B):
*>                      A*inv(B) = U*(D1*inv(D2))*V**T.
*> If ( A**T,B**T)**T  has orthonormal columns, then the GSVD of A and B is
*> also equal to the CS decomposition of A and B. Furthermore, the GSVD
*> can be used to derive the solution of the eigenvalue problem:
*>                      A**T*A x = lambda* B**T*B x.
*> In some literature, the GSVD of A and B is presented in the form
*>                  U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
*> where U and V are orthogonal and X is nonsingular, D1 and D2 are
*> ``diagonal''.  The former GSVD form can be converted to the latter
*> form by taking the nonsingular matrix X as
*>
*>                      X = Q*( I   0    )
*>                            ( 0 inv(R) ).
*> \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] 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.  P >= 0.
*> \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.
*>          K + L = effective numerical rank of (A**T,B**T)**T.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit, A contains the triangular matrix R, or part of R.
*>          See Purpose for details.
*> \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 DOUBLE PRECISION array, dimension (LDB,N)
*>          On entry, the P-by-N matrix B.
*>          On exit, B contains the triangular matrix R if M-K-L < 0.
*>          See Purpose for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>
*>          On exit, ALPHA and BETA contain the generalized singular
*>          value pairs of A and B;
*>            ALPHA(1:K) = 1,
*>            BETA(1:K)  = 0,
*>          and if M-K-L >= 0,
*>            ALPHA(K+1:K+L) = C,
*>            BETA(K+1:K+L)  = S,
*>          or if M-K-L < 0,
*>            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
*>            BETA(K+1:M) =S, BETA(M+1:K+L) =1
*>          and
*>            ALPHA(K+L+1:N) = 0
*>            BETA(K+L+1:N)  = 0
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU,M)
*>          If JOBU = 'U', U contains the M-by-M 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 DOUBLE PRECISION array, dimension (LDV,P)
*>          If JOBV = 'V', V contains the P-by-P 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 DOUBLE PRECISION array, dimension (LDQ,N)
*>          If JOBQ = 'Q', Q contains the N-by-N 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] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION 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 >= 1.
*>
*>          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] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*>          On exit, IWORK stores the sorting information. More
*>          precisely, the following loop will sort ALPHA
*>             for I = K+1, min(M,K+L)
*>                 swap ALPHA(I) and ALPHA(IWORK(I))
*>             endfor
*>          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, the Jacobi-type procedure failed to
*>                converge.  For further details, see subroutine DTGSJA.
*> \endverbatim
*
*> \par Internal Parameters:
*  =========================
*>
*> \verbatim
*>  TOLA    DOUBLE PRECISION
*>  TOLB    DOUBLE PRECISION
*>          TOLA and TOLB are the thresholds to determine the effective
*>          rank of (A**T,B**T)**T. 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
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggsvd3
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*
*> \par Further Details:
*  =====================
*>
*>  DGGSVD3 replaces the deprecated subroutine DGGSVD.
*>
*  =====================================================================
      SUBROUTINE dggsvd3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
     $                    LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ,
     $                    WORK, LWORK, IWORK, 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 ..
      CHARACTER          JOBQ, JOBU, JOBV
      INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
     $                   lwork
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   beta( * ), q( ldq, * ), u( ldu, * ),
     $                   v( ldv, * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            WANTQ, WANTU, WANTV, LQUERY
      INTEGER            I, IBND, ISUB, J, NCYCLE, LWKOPT
      DOUBLE PRECISION   ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           lsame, dlamch, dlange
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dggsvp3, dtgsja, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      wantu = lsame( jobu, 'U' )
      wantv = lsame( jobv, 'V' )
      wantq = lsame( jobq, 'Q' )
      lquery = ( lwork.EQ.-1 )
      lwkopt = 1
*
*     Test the input arguments
*
      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( n.LT.0 ) THEN
         info = -5
      ELSE IF( p.LT.0 ) THEN
         info = -6
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -10
      ELSE IF( ldb.LT.max( 1, p ) ) THEN
         info = -12
      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
      ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
         info = -24
      END IF
*
*     Compute workspace
*
      IF( info.EQ.0 ) THEN
         CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb,
     $                 tola,
     $                 tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
     $                 work, -1, info )
         lwkopt = n + int( work( 1 ) )
         lwkopt = max( 2*n, lwkopt )
         lwkopt = max( 1, lwkopt )
         work( 1 ) = dble( lwkopt )
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DGGSVD3', -info )
         RETURN
      END IF
      IF( lquery ) THEN
         RETURN
      ENDIF
*
*     Compute the Frobenius norm of matrices A and B
*
      anorm = dlange( '1', m, n, a, lda, work )
      bnorm = dlange( '1', p, n, b, ldb, work )
*
*     Get machine precision and set up threshold for determining
*     the effective numerical rank of the matrices A and B.
*
      ulp = dlamch( 'Precision' )
      unfl = dlamch( 'Safe Minimum' )
      tola = max( m, n )*max( anorm, unfl )*ulp
      tolb = max( p, n )*max( bnorm, unfl )*ulp
*
*     Preprocessing
*
      CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
     $              tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
     $              work( n+1 ), lwork-n, info )
*
*     Compute the GSVD of two upper "triangular" matrices
*
      CALL dtgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
     $             tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
     $             work, ncycle, info )
*
*     Sort the singular values and store the pivot indices in IWORK
*     Copy ALPHA to WORK, then sort ALPHA in WORK
*
      CALL dcopy( n, alpha, 1, work, 1 )
      ibnd = min( l, m-k )
      DO 20 i = 1, ibnd
*
*        Scan for largest ALPHA(K+I)
*
         isub = i
         smax = work( k+i )
         DO 10 j = i + 1, ibnd
            temp = work( k+j )
            IF( temp.GT.smax ) THEN
               isub = j
               smax = temp
            END IF
   10    CONTINUE
         IF( isub.NE.i ) THEN
            work( k+isub ) = work( k+i )
            work( k+i ) = smax
            iwork( k+i ) = k + isub
         ELSE
            iwork( k+i ) = k + i
         END IF
   20 CONTINUE
*
      work( 1 ) = dble( lwkopt )
      RETURN
*
*     End of DGGSVD3
*
      SUBROUTINE dggsvd3( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, …
      END