!>
!> ZTGSJA computes the generalized singular value decomposition (GSVD)
!> of two complex upper triangular (or trapezoidal) matrices A and B.
!>
!> On entry, it is assumed that matrices A and B have the following
!> forms, which may be obtained by the preprocessing subroutine ZGGSVP
!> from a general M-by-N matrix A and P-by-N matrix B:
!>
!>              N-K-L  K    L
!>    A =    K ( 0    A12  A13 ) if M-K-L >= 0;
!>           L ( 0     0   A23 )
!>       M-K-L ( 0     0    0  )
!>
!>            N-K-L  K    L
!>    A =  K ( 0    A12  A13 ) if M-K-L < 0;
!>       M-K ( 0     0   A23 )
!>
!>            N-K-L  K    L
!>    B =  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.
!>
!> On exit,
!>
!>        U**H *A*Q = D1*( 0 R ),    V**H *B*Q = D2*( 0 R ),
!>
!> where U, V and Q are unitary matrices.
!> R is a nonsingular upper triangular matrix, and D1
!> and D2 are ``diagonal'' matrices, which are of the following
!> structures:
!>
!> 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 ) K
!>             L (  0    0   R22 ) L
!>
!> 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.
!>
!> R = ( 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 computation of the unitary transformation matrices U, V or Q
!> is optional.  These matrices may either be formed explicitly, or they
!> may be postmultiplied into input matrices U1, V1, or Q1.
!> 

JOBU

!>          JOBU is CHARACTER*1
!>          = 'U':  U must contain a unitary matrix U1 on entry, and
!>                  the product U1*U is returned;
!>          = 'I':  U is initialized to the unit matrix, and the
!>                  unitary matrix U is returned;
!>          = 'N':  U is not computed.
!> 

JOBV

!>          JOBV is CHARACTER*1
!>          = 'V':  V must contain a unitary matrix V1 on entry, and
!>                  the product V1*V is returned;
!>          = 'I':  V is initialized to the unit matrix, and the
!>                  unitary matrix V is returned;
!>          = 'N':  V is not computed.
!> 

JOBQ

!>          JOBQ is CHARACTER*1
!>          = 'Q':  Q must contain a unitary matrix Q1 on entry, and
!>                  the product Q1*Q is returned;
!>          = 'I':  Q is initialized to the unit matrix, and the
!>                  unitary matrix Q is returned;
!>          = 'N':  Q is not computed.
!> 

M

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

P

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

N

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

K

!>          K is INTEGER
!> 

L

!>          L is INTEGER
!>
!>          K and L specify the subblocks in the input matrices A and B:
!>          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
!>          of A and B, whose GSVD is going to be computed by ZTGSJA.
!>          See Further Details.
!> 

A

!>          A is COMPLEX*16 array, dimension (LDA,N)
!>          On entry, the M-by-N matrix A.
!>          On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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 COMPLEX*16 array, dimension (LDB,N)
!>          On entry, the P-by-N matrix B.
!>          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
!>          a part of R.  See Purpose for details.
!> 

LDB

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

TOLA

!>          TOLA is DOUBLE PRECISION
!> 

TOLB

!>          TOLB is DOUBLE PRECISION
!>
!>          TOLA and TOLB are the convergence criteria for the Jacobi-
!>          Kogbetliantz iteration procedure. Generally, they are the
!>          same as used in the preprocessing step, say
!>              TOLA = MAX(M,N)*norm(A)*MAZHEPS,
!>              TOLB = MAX(P,N)*norm(B)*MAZHEPS.
!> 

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) = diag(C),
!>            BETA(K+1:K+L)  = diag(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.
!>          Furthermore, if K+L < N,
!>            ALPHA(K+L+1:N) = 0 and
!>            BETA(K+L+1:N)  = 0.
!> 

U

!>          U is COMPLEX*16 array, dimension (LDU,M)
!>          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
!>          the unitary matrix returned by ZGGSVP).
!>          On exit,
!>          if JOBU = 'I', U contains the unitary matrix U;
!>          if JOBU = 'U', U contains the product U1*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 COMPLEX*16 array, dimension (LDV,P)
!>          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
!>          the unitary matrix returned by ZGGSVP).
!>          On exit,
!>          if JOBV = 'I', V contains the unitary matrix V;
!>          if JOBV = 'V', V contains the product V1*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 COMPLEX*16 array, dimension (LDQ,N)
!>          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
!>          the unitary matrix returned by ZGGSVP).
!>          On exit,
!>          if JOBQ = 'I', Q contains the unitary matrix Q;
!>          if JOBQ = 'Q', Q contains the product Q1*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 COMPLEX*16 array, dimension (2*N)
!> 

NCYCLE

!>          NCYCLE is INTEGER
!>          The number of cycles required for convergence.
!> 

INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!>          = 1:  the procedure does not converge after MAXIT cycles.
!> 

!>  MAXIT   INTEGER
!>          MAXIT specifies the total loops that the iterative procedure
!>          may take. If after MAXIT cycles, the routine fails to
!>          converge, we return INFO = 1.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
!>  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
!>  matrix B13 to the form:
!>
!>           U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,
!>
!>  where U1, V1 and Q1 are unitary matrix.
!>  C1 and S1 are diagonal matrices satisfying
!>
!>                C1**2 + S1**2 = I,
!>
!>  and R1 is an L-by-L nonsingular upper triangular matrix.
!>