.TH "ctgsja.f" 3 "Wed Oct 15 2014" "Version 3.4.2" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
ctgsja.f \- 
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBctgsja\fP (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)"
.br
.RI "\fI\fBCTGSJA\fP \fP"
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine ctgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN, integerK, integerL, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, realTOLA, realTOLB, real, dimension( * )ALPHA, real, dimension( * )BETA, complex, dimension( ldu, * )U, integerLDU, complex, dimension( ldv, * )V, integerLDV, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( * )WORK, integerNCYCLE, integerINFO)"

.PP
\fBCTGSJA\fP  
.PP
\fBPurpose: \fP
.RS 4

.PP
.nf
 CTGSJA 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 CGGSVP
 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.
.fi
.PP
 
.RE
.PP
\fBParameters:\fP
.RS 4
\fIJOBU\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fIJOBV\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fIJOBQ\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows of the matrix B.  P >= 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
\fIK\fP 
.PP
.nf
          K is INTEGER
.fi
.PP
.br
\fIL\fP 
.PP
.nf
          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 CTGSJA.
          See Further Details.
.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(N-K+1:N,1:MIN(K+L,M) ) 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, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
          a part of R.  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
\fITOLA\fP 
.PP
.nf
          TOLA is REAL
.fi
.PP
.br
\fITOLB\fP 
.PP
.nf
          TOLB is REAL

          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)*MACHEPS,
              TOLB = MAX(P,N)*norm(B)*MACHEPS.
.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) = 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
            BETA(K+L+1:N)  = 0.
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is COMPLEX array, dimension (LDU,M)
          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
          the unitary matrix returned by CGGSVP).
          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.
.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)
          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
          the unitary matrix returned by CGGSVP).
          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.
.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)
          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
          the unitary matrix returned by CGGSVP).
          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.
.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 (2*N)
.fi
.PP
.br
\fINCYCLE\fP 
.PP
.nf
          NCYCLE is INTEGER
          The number of cycles required for convergence.
.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.
          = 1:  the procedure does not converge after MAXIT cycles.
.fi
.PP
 
.RE
.PP
\fBInternal Parameters: \fP
.RS 4

.PP
.nf
  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.
.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
\fBDate:\fP
.RS 4
November 2011 
.RE
.PP
\fBFurther Details: \fP
.RS 4

.PP
.nf
  CTGSJA 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.
.fi
.PP
 
.RE
.PP

.PP
Definition at line 378 of file ctgsja\&.f\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.