.TH "ggsvd3" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
ggsvd3 \- ggsvd3: SVD, QR iteration
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcggsvd3\fP (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork, info)"
.br
.RI "\fB CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP "
.ti -1c
.RI "subroutine \fBdggsvd3\fP (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)"
.br
.RI "\fB DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP "
.ti -1c
.RI "subroutine \fBsggsvd3\fP (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)"
.br
.RI "\fB SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP "
.ti -1c
.RI "subroutine \fBzggsvd3\fP (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork, info)"
.br
.RI "\fB ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alpha, real, dimension( * ) beta, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)"

.PP
\fB CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CGGSVD3 computes the generalized singular value decomposition (GSVD)
 of an M-by-N complex matrix A and P-by-N complex matrix B:

       U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )

 where U, V and Q are unitary matrices\&.
 Let K+L = the effective numerical rank of the
 matrix (A**H,B**H)**H, 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 unitary
 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**H\&.
 If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x\&.
 In some literature, the GSVD of A and B is presented in the form
                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
 where U and V are orthogonal and X is nonsingular, and 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) )
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP 
.PP
.nf
          JOBU is CHARACTER*1
          = 'U':  Unitary matrix U is computed;
          = 'N':  U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP 
.PP
.nf
          JOBV is CHARACTER*1
          = 'V':  Unitary matrix V is computed;
          = 'N':  V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP 
.PP
.nf
          JOBQ is CHARACTER*1
          = 'Q':  Unitary matrix Q is computed;
          = 'N':  Q is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows of the matrix B\&.  P >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          L is INTEGER

          On exit, K and L specify the dimension of the subblocks
          described in Purpose\&.
          K + L = effective numerical rank of (A**H,B**H)**H\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX 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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX array, dimension (LDB,N)
          On entry, the P-by-N matrix B\&.
          On exit, B contains part of the triangular matrix R if
          M-K-L < 0\&.  See Purpose for details\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is REAL array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is REAL 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
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is COMPLEX array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M unitary matrix U\&.
          If JOBU = 'N', U is not referenced\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          V is COMPLEX array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P unitary matrix V\&.
          If JOBV = 'N', V is not referenced\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is COMPLEX array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q\&.
          If JOBQ = 'N', Q is not referenced\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.

          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\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is REAL array, dimension (2*N)
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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 CTGSJA\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  TOLA    REAL
  TOLB    REAL
          TOLA and TOLB are the thresholds to determine the effective
          rank of (A**H,B**H)**H\&. 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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4
CGGSVD3 replaces the deprecated subroutine CGGSVD\&. 
.RE
.PP

.SS "subroutine dggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)"

.PP
\fB DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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) )\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP 
.PP
.nf
          JOBU is CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP 
.PP
.nf
          JOBV is CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP 
.PP
.nf
          JOBQ is CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows of the matrix B\&.  P >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.

          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\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4
DGGSVD3 replaces the deprecated subroutine DGGSVD\&. 
.RE
.PP

.SS "subroutine sggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alpha, real, dimension( * ) beta, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)"

.PP
\fB SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SGGSVD3 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) )\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP 
.PP
.nf
          JOBU is CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP 
.PP
.nf
          JOBV is CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP 
.PP
.nf
          JOBQ is CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows of the matrix B\&.  P >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL 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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is REAL 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\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is REAL array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is REAL 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
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is REAL array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M orthogonal matrix U\&.
          If JOBU = 'N', U is not referenced\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          V is REAL array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P orthogonal matrix V\&.
          If JOBV = 'N', V is not referenced\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is REAL array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q\&.
          If JOBQ = 'N', Q is not referenced\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.

          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\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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 STGSJA\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  TOLA    REAL
  TOLB    REAL
          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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4
SGGSVD3 replaces the deprecated subroutine SGGSVD\&. 
.RE
.PP

.SS "subroutine zggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)"

.PP
\fB ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZGGSVD3 computes the generalized singular value decomposition (GSVD)
 of an M-by-N complex matrix A and P-by-N complex matrix B:

       U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )

 where U, V and Q are unitary matrices\&.
 Let K+L = the effective numerical rank of the
 matrix (A**H,B**H)**H, 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 unitary
 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**H\&.
 If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x\&.
 In some literature, the GSVD of A and B is presented in the form
                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
 where U and V are orthogonal and X is nonsingular, and 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) )
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU\fP 
.PP
.nf
          JOBU is CHARACTER*1
          = 'U':  Unitary matrix U is computed;
          = 'N':  U is not computed\&.
.fi
.PP
.br
\fIJOBV\fP 
.PP
.nf
          JOBV is CHARACTER*1
          = 'V':  Unitary matrix V is computed;
          = 'N':  V is not computed\&.
.fi
.PP
.br
\fIJOBQ\fP 
.PP
.nf
          JOBQ is CHARACTER*1
          = 'Q':  Unitary matrix Q is computed;
          = 'N':  Q is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A\&.  M >= 0\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The number of columns of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows of the matrix B\&.  P >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          L is INTEGER

          On exit, K and L specify the dimension of the subblocks
          described in Purpose\&.
          K + L = effective numerical rank of (A**H,B**H)**H\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,M)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX*16 array, dimension (LDB,N)
          On entry, the P-by-N matrix B\&.
          On exit, B contains part of the triangular matrix R if
          M-K-L < 0\&.  See Purpose for details\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,P)\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is COMPLEX*16 array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M unitary matrix U\&.
          If JOBU = 'N', U is not referenced\&.
.fi
.PP
.br
\fILDU\fP 
.PP
.nf
          LDU is INTEGER
          The leading dimension of the array U\&. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          V is COMPLEX*16 array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P unitary matrix V\&.
          If JOBV = 'N', V is not referenced\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is COMPLEX*16 array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q\&.
          If JOBQ = 'N', Q is not referenced\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK\&.

          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\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (2*N)
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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 ZTGSJA\&.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters:\fP
.RS 4

.PP
.nf
  TOLA    DOUBLE PRECISION
  TOLB    DOUBLE PRECISION
          TOLA and TOLB are the thresholds to determine the effective
          rank of (A**H,B**H)**H\&. 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\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4
ZGGSVD3 replaces the deprecated subroutine ZGGSVD\&. 
.RE
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.