!>
!> This routine is deprecated and has been replaced by routine DGGSVD3.
!>
!> DGGSVD 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)  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) ).
!> 

JOBU

!>          JOBU is CHARACTER*1
!>          = 'U':  Orthogonal matrix U is computed;
!>          = 'N':  U is not computed.
!> 

JOBV

!>          JOBV is CHARACTER*1
!>          = 'V':  Orthogonal matrix V is computed;
!>          = 'N':  V is not computed.
!> 

JOBQ

!>          JOBQ is CHARACTER*1
!>          = 'Q':  Orthogonal matrix Q is computed;
!>          = 'N':  Q is not computed.
!> 

M

!>          M is INTEGER
!>          The number of rows of the matrix A.  M >= 0.
!> 

N

!>          N is INTEGER
!>          The number of columns of the matrices A and B.  N >= 0.
!> 

P

!>          P is INTEGER
!>          The number of rows of the matrix B.  P >= 0.
!> 

K

!>          K is INTEGER
!> 

L

!>          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.
!> 

A

!>          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.
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,M).
!> 

B

!>          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.
!> 

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,P).
!> 

ALPHA

!>          ALPHA is DOUBLE PRECISION array, dimension (N)
!> 

BETA

!>          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
!> 

U

!>          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.
!> 

LDU

!>          LDU is INTEGER
!>          The leading dimension of the array U. LDU >= max(1,M) if
!>          JOBU = 'U'; LDU >= 1 otherwise.
!> 

V

!>          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.
!> 

LDV

!>          LDV is INTEGER
!>          The leading dimension of the array V. LDV >= max(1,P) if
!>          JOBV = 'V'; LDV >= 1 otherwise.
!> 

Q

!>          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.
!> 

LDQ

!>          LDQ is INTEGER
!>          The leading dimension of the array Q. LDQ >= max(1,N) if
!>          JOBQ = 'Q'; LDQ >= 1 otherwise.
!> 

WORK

!>          WORK is DOUBLE PRECISION array,
!>                      dimension (max(3*N,M,P)+N)
!> 

IWORK

!>          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).
!> 

INFO

!>          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.
!> 

!>  TOLA    DOUBLE PRECISION
!>  TOLB    DOUBLE PRECISION
!>          TOLA and TOLB are the thresholds to determine the effective
!>          rank of (A',B')**T. Generally, they are set to
!>                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,
!>                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
!>          The size of TOLA and TOLB may affect the size of backward
!>          errors of the decomposition.
!> 

Univ. of Tennessee

Univ. of California Berkeley

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Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA